Lund E. Finite Element Based Design Sensitivity Analysis and Optimization (1994)(en)(234s)

Institute of Mechanical Engineering

Aalborg University

Special Report no. 23

Finite Element Based

Design Sensitivity Analysis

and Optimization

Ph.D. Dissertation

by

Erik Lund

Copyright c 1994 Erik Lund

Reproduction of material contained in this report is permitted provided the source is given.

Additional copies are available at the cost of printing and mailing from Helle W. Mane,

Institute of Mechanical Engineering, Aalborg University, Pontoppidanstraede 101, DK-

9220 Aalborg East, Denmark. Telephone +45 98 15 42 11 ext. 3506, FAX +45 98 15 14

11. Questions and comments are most welcome and can be directed to the author at the

same adress or by electronic mail: [email protected].

Printed at Aalborg University, April 1994.

ISBN 87-89206-01-0

Preface

This dissertation has been submitted to the Faculty of Technology and Science of Aalborg

University, Aalborg, Denmark, in partial ful�lment of the requirements for the technical

Ph.D. degree.

The project underlying this thesis has been carried out from August 1991 to April 1994

at the Institute of Mechanical Engineering at Aalborg University. The project has been

supervised by Professor, Dr.techn. Niels Olho� to whom I am most indebted for his

vivid engagement, inspiring and supporting guidance, immense competence, and his time

demanding e�orts in providing good research conditions and facilities for our research

group. I am also most indebted to my friend and collegue, Associate Professor, Ph.D.

John Rasmussen for many invaluable suggestions, support, collaborations, and inspiring

discussions.

Furthermore, I would like to thank my colleagues and friends, M.Sc. Lars Krog & M.Sc.

Oluf Krogh for many inspiring discussions and suggestions, Professor Alexander P. Seyra-

nian, Moscow State Lomonosov University, Russia, for our joint work on multiple eigen-

values in structural optimization problems, and the other colleagues at the Institute of

Mechanical Engineering.

Finally, I want to thank my �anc�e Dorte for her support and understanding during periods

with much work and less spare time; her love has been an invaluable support.

The present work has been supported by the Danish Technical Research Council's \Pro-

gramme of Research on Computer Aided Engineering Design".

Erik Lund

Aalborg, April 1994

1

Abstract

The objective of the present Ph.D. project is to develop, implement and integrate methods

for structural analysis, design sensitivity analysis and optimization into a general purpose

computer aided environment for interactive structural design, analysis, design sensitivity

analysis, synthesis, and design optimization. This system is named \ODESSY" (Optimum

DESign SYstem) and is integrated with commercially available CAD systems.

The �nite element library used for the analysis has facilities for solution of linear types

of analysis problems, i.e., static stress analysis, natural frequency analysis, steady state

thermal analysis, thermo-elastic analysis, eigenfrequency analysis with initial stress sti�-

ening e�ects due to mechanical or thermal loads, and linear buckling analysis with the

possibility of including thermo-elastic e�ects. Many di�erent isoparametric �nite elements

are described and implemented in the system.

A reliable tool for design sensitivity analysis is a prerequisite for performing interactive

structural design, synthesis and optimization. General expressions for design sensitivity

analysis of all implemented types of analysis problems are derived with respect to shape

as well as sizing and material design variables. The method of design sensitivity analysis

used is the direct approach, and the semi-analytical method where derivatives of various

�nite element matrices and vectors are approximated by �rst order �nite di�erences is

adopted. However, the traditional semi-analytical method may yield severely erroneous

results for certain types of problems involving shape design variables. Therefore, a new

semi-analytical method based on \exact" numerical di�erentiation of element matrices is

developed and implemented for all types of �nite elements and design variables in ODESSY.

It is demonstrated by several examples that this new approach of semi-analytical design

sensitivity analysis is computationally e�cient and completely eliminates the inaccuracy

problem.

A general and exible method of formulating problems of mathematical programming is

developed. The method enables formulation and solution of problems involving local,

integral, min/max, max/min and possibly non-di�erentiable user de�ned functions in any

conceivable mix. The mathematical formulation is based on the bound formulation and

the implementation involves a parser capable of interpreting and performing symbolic

di�erentiation of the user de�ned functions. This database module can also be used for

graphical visualization of user de�ned mathematical expressions for design criteria.

The special case of solving structural optimization problems involving multiple eigenvalues

is discussed. The main di�culty associated with multiple eigenvalues is the lack of usual

Fr�echet di�erentiability with respect to changes in design, i.e., multiple eigenvalues can only

be expected to be directionally di�erentiable. Necessary optimality conditions are discussed

and iterative numerical algorithms for solution of such design problems are developed and

used for solution of a design problem.

Several examples illustrate how ODESSY can be used very e�ectively for interactive engi-

neering design with focus on design sensitivity analysis, synthesis, and optimization.

3

Abstrakt

Formlet med dette Ph.D. projekt er at udvikle, implementere og integrere metoder for

strukturel analyse, design somhedsanalyse og optimering i et generelt, datamatassisteret

system for interaktiv strukturel design, analyse, design somhedsanalyse, syntese og de-

signoptimering. Dette system er blevet dbt \ODESSY" (Optimum DESign SYstem) og er

integreret med kommercielt tilgngelige CAD-systemer.

Finite element biblioteket, der bruges til analysen, har faciliteter for lsning af linere

typer analyseproblemer, dvs., statisk spndingsanalyse, analyse af frie svingninger, steady

state termisk analyse, termo-elastisk analyse, egenfrekvensanalyse med initiale spndingsaf-

stivende e�ekter pga. mekaniske/termiske belastninger og liner bulingsanalyse med mu-

lighed for at inkludere termo-elastiske e�ekter. Mange forskellige isoparametriske element-

typer er beskrevet og implementeret i systemet.

Et plideligt vrktj til design somhedsanalyse er en forudstning for at kunne udfre inter-

aktiv strukturel design, syntese og optimering. Generelle udtryk for somhedsanalyse

af de implementerede typer af analyseproblemer er udledt for form-, tykkelse- og ma-

terialedesignvariable. Den direkte metode er valgt til somhedsanalysen, og den semi-

analytiske metode, hvor a edede af diverse elementmatricer og -vektorer approksimeres

med frste ordens di�erenser, anvendes. Imidlertid kan den traditionelle semi-analytiske

metode give fuldstndig forkerte resultater for visse typer af problemer, der involverer for-

mdesignvariable. Derfor er en ny semi-analytisk metode, der er baseret p \eksakt" nu-

merisk di�erentiation af elementmatricer, blevet udviklet og implementeret for alle typer

af elementer og designvariable i ODESSY. Denne nye semi-analytiske metode for design -

somhedsanalysen er beregningsmssig e�ektiv og eliminerer unjagtighedsproblemet, hvilket

illustreres ved adskillige eksempler.

En generel og eksible metode til at formulere matematiske programmeringsproblemer er

udviklet. Metoden gr det muligt at formulere og lse problemer, der involverer lokale, inte-

grale, min/max, max/min og mulige ikke-di�erentiable brugerde�nerede funktioner i en-

hver tnkelig kombination. Den matematiske formulering er baseret p bound-formuleringen,

og implementeringen indeholder en fortolker, der kan forst og udfre symbolsk di�erenti-

ation af brugerde�nerede funktioner. Dette databasemodul kan ogs bruges til interaktiv,

gra�sk visualisering af brugerde�nerede matematiske udtryk for designkriterier.

Det specielle til de vedrrende lsning af strukturelle optimeringsproblemer med multiple

egenvrdier behandles. Den strste vanskelighed forbundet med multiple egenvrdier er man-

glen p normal Fr�echet-di�erentiabilitet med hensyn til designndringer, dvs. multiple egen-

vrdier kan kun forventes at vre retningsdi�erentiable. Ndvendige optimalitetskriterier

diskuteres og iterative numeriske algoritmer til lsning af sdanne designproblemer prsen-

teres og anvendes til lsning af et designproblem.

Adskillige eksempler illustrerer, at ODESSY er et srdeles e�ektivt vrktj for interaktiv

strukturel design med fokus p designflsomhedsanalyse, syntese og optimering.

4

Chapter

1

Introduction

Engineering activity has always involved endeavours towards optimization, and

this particularly holds true for the �eld of engineering design. Earlier, engineering

design was conceived as a kind of \art" that demanded great ingenuity and experience

of the designer, and the development of the �eld was characterized by gradual evolution

in terms of continual improvement of existing types of engineering designs. The design

process generally was a sequential \trial and error" process where the designer's skills and

experience were most important prerequisites for successful decisions for the \trial" phase.

However, nowaday's strong technological competition which requires reduction of design

time and costs of products with high quality and functionality, and current emphasis on

saving of energy, saving and re-use of material resources, consideration of environmental

problems, etc., often involve creation of new products for which prior engineering ex-

perience is totally lacking. Development of such products must naturally lend itself on

application of scienti�c methods such as structural analysis, design sensitivity analysis,

and optimization.

The scienti�c research in the �eld of structural optimization has increased rapidly during

recent decades. The increasing interest in this �eld has been strongly boosted by the ad-

vent of reliable general analysis methods like the �nite element method, methods of design

sensitivity analysis, and methods of mathematical programming, along with the exponen-

tially increasing speed and capacity of digital computers. The rapid development of the

�eld of optimum design also re ects a natural shift of emphasis from analysis to synthesis.

The research activities in the �elds of design sensitivity analysis and optimum design can

be found in, e.g., the review papers by Haug (1981), Schmit (1982), Olho� & Taylor (1983),

Haftka & Grandhi (1986), Ding (1986), and Haftka & Adelman (1989) and the proceedings

from various conferences and symposia published by Haug & Cea (1981), Morris (1982),

Eschenauer & Olho� (1983), Atrek, Gallagher, Ragsdell & Zienkiewicz (1984), Bennett &

Botkin (1986), Mota Soares (1987), Rozvany & Karihaloo (1988), Eschenauer & Thierauf

(1989), Eschenauer, Mattheck & Olho� (1991), Bends�e & Mota Soares (1993), Rozvany

(1993), Haug (1993), Pedersen (1993), Herskovits (1993) and Gilmore, Hoeltzel, Azarm &

Eschenauer (1993).

5

6

Rational engineering design and optimization based on the concept of integration of �nite

element analysis, design sensitivity analysis, and optimization by mathematical program-

ming, was early undertaken by Zienkiewicz & Campbell (1973), Kristensen & Madsen

(1976), Pedersen (1981, 1983), Pedersen & Laursen (1983), Esping (1983), Braibant &

Fleury (1984), Eschenauer (1986) and Santos & Choi (1989), and this work has paved the

road for development of large, practice-oriented optimization systems, see, e.g., Braibant

& Fleury (1984), Sobieszanski-Sobieski & Rogers (1984), Bennett & Botkin (1985), Esp-

ing (1986), Stanton (1986), Eschenauer (1986), Botkin, Yang & Bennett (1986), H�ornlein

(1987), Arora (1989), Rasmussen (1989, 1990), Choi & Chang (1991), and Rasmussen,

Lund, Birker & Olho� (1993).

However, until recently, these methods for rational engineering design and optimization

have not, in general, been adopted by designers in industry as stated by Santos & Choi

(1989). Only major companies in �elds like aeronautical, aerospace, mechanical, nuclear,

civil, and o�-shore engineering have adopted these tools for rational design and optimiza-

tion, although their use have proved successful for many di�erent kinds of applications

concerning structural synthesis. The reasons for this lack of use are many.

It may be di�cult, and may even prove impossible, to formulate one or more simple

performance criteria for optimization. The designer must be able to de�ne the objective

function, identify all important constraints, identify the levels of constraint limits, and

enter this information into the structural optimization system. Very often, in applications,

during the design optimization process, additional constraints may become important that

the designer did not foresee in the initial stages of formulation of the design problem. The

structural optimization system must also contain very general and exible facilities for

de�nition of the design problem in order to cover possible de�nitions. If such facilities are

not present, designers in industry will not use the optimization systems as their design

criteria in many cases have been established during many years of practice. If designers in

industry cannot de�ne their usual design criteria, they will typically rather abstain from

the use of structural optimization than being forced into changing their well-tested design

criteria just because of a limitation in the software system.

Furthermore, another possible reason for the lack of adoptivity of structural optimization

in the industry may be that developers of structural optimization systems have been too

focused on selling the systems as automated \black boxes" for obtaining optimum designs

of structures. In this way designers in industry have not become familiar with design

sensitivity analysis which is used to calculate all necessary derivatives of criterion functions,

e.g., derivatives of stresses, displacements, frequencies, user de�ned functions, etc., with

respect to some design parameters. This can also be seen from the fact that only a few

�nite element systems, for example MSC-NASTRAN, have facilities for design sensitivity

analysis although this tool can be just as valuable as structural analysis in the design

process. Of course, it is possible to employ structural synthesis procedures which do

not need derivative information, but apart from the synthesis method, design sensitivity

analysis provides the designer with valuable information about the sensitivity of the design

with respect to various parameters which can be used for a revision of these parameters.

Finally, a structural optimization system must be integrated in a general purpose, interac-

Chapter 1. Introduction 7

tive CAD environment as stated by Fleury & Braibant (1987), see also Braibant & Fleury

(1984), Stanton (1986), Botkin, Yang & Bennett (1986), Santos & Choi (1989), Gu &

Cheng (1990), Rasmussen (1990, 1991), and Botkin, Bajorek & Prasad (1992), if designers

shall use the system. If the structural optimization system is integrated in a CAD environ-

ment that the designer is familiar with, the system will much easier be accepted for daily

use in the design process.

The above-mentioned necessary features for a general purpose computer aided environ-

ment for interactive rational design, structural analysis, design sensitivity analysis, and

optimization involve many di�erent topics and require expertice in quite di�erent �elds.

The development of such a computer aided environment must therefore naturally be carried

out as a collaboration between several people.

1.1 Programme of Research on Computer Aided En-

gineering Design

The research on computer aided design and mechanical engineering was initiated at the

Institute of Mechanical Engineering, Aalborg University, in the mid-eighties and further

intensi�ed by inclusion in a research framework programme under the Danish Technical

Research Council, the \Programme of Research on Computer Aided Design". This research

framework programme was carried out in the period 1989-1993 and extended for the pe-

riod 1993-1997 under the name \Programme of Research on Computer Aided Engineering

Design". The participants in this programme are the Institute of Mechanical Engineering,

Aalborg University, and the Institute of Engineering Design, the Department of Solid Me-

chanics, and the Mathematical Institute, all of the Technical University of Denmark. This

thesis has been carried out within this programme under the project \Engineering Design

Optimization".

The primary objective of this programme is to further the research and development of

new methods, techniques and tools for concurrent (or integrated) computer aided design

of mechanical products, systems and components which are critical in terms of cost, de-

velopment time, functionality and quality.

This research on computer aided design and mechanical engineering at the Institute of Me-

chanical Engineering has resulted in several Ph.D. theses in the �eld of optimum design,

see Rasmussen (1989), Kibsgaard (1991), and Thomsen (1992). In the beginning, work

within the �eld focused on formulating general strategies, gaining experience and imple-

menting prototypes of the proposed concepts. My collegue John Rasmussen developed the

prototype shape optimization system \CAOS" (Computer Aided Optimization System)

which is a tool for interactive shape optimization of planar geometries subjected to static

loads. The problems are de�ned by means of the AutoCAD system, using a design element

concept as described by Fleury (1987). CAOS was further developed by Birker & Lund

(1991) to include three-dimensional structures and thermo-elastic problems, see also Ras-

mussen, Lund, Birker & Olho� (1993). Kibsgaard developed a general shape optimization

system based on the publicly available �nite element program Modulef, see Kibsgaard, Ol-

8 1.1. Programme of Research on Computer Aided Engineering Design

ho� & Rasmussen (1989) and Kibsgaard (1991). However, the computational organization

in Modulef proved less suitable for optimization purposes and this led to the decision of

writing our own analysis code. It might be advantageous to use a commercial �nite element

system for the analysis due to the generality of the analysis facilities but we wanted to

develop a kind of computer engineering design laboratory where experiments concerning,

for example, structural analysis, design sensitivity analysis, mathematical programming,

and links between CAD environments and structural optimization systems can be carried

out. Furthermore, the tools available for software developers have reached a high standard,

making it easier to develop large and complex software systems.

In 1991 the development of a general purpose fully three-dimensional computer aided en-

vironment for interactive structural design, analysis, design sensitivity analysis, synthesis,

and engineering design optimization was initiated. This system is named \ODESSY"

(Optimum DESign SYstem) and is based on the experiences obtained by developing the

above-mentioned systems. ODESSY is being developed by several people at the Institute

of Mechanical Engineering and is programmed in ANSI C, except for the postprocessing

facilities which are written in C++. The ANSI C programming language has many ad-

vantages such as dynamic allocation of memory, facilities for de�nition of complex data

structures, and portability between di�erent computer platforms.

ODESSY

Preprocessor

CAD integration

Finite element modulesfor structural analysis

Modules for structuraldesign sensitivity analysis Postprocessor

Database module forextraction of results

Interactive shape, sizing,or material what-if studyand design optimization

Topology optimization

Figure 1.1: Main features of ODESSY.

The main features of ODESSY can be seen in Fig. 1.1. The features shown in shaded boxes

are all covered in this thesis whereas the other features are only described super�cially.

ODESSY is being integrated with the commercial CAD systems AutoCAD and Pro/Engi-

neer, thereby setting rational design facilities directly at the disposal of the designer. In

Fig. 1.2 it is illustrated how ODESSY can be used as a tool for interactive engineering

design.

The di�erent tasks shown in Fig. 1.2 are not necessarily performed sequentially and several

of them may be omitted, but the �gure illustrates the facilities available for the design en-

gineer. It is the hope of the people involved in the development of ODESSY that designers


Interactiveengineering designusing ODESSY

Define parameterized designmodel in CAD environment.

Convert design model to afinite element analysis modelusing the preprocessor

Perform structural analyses

Monitor analysis results anduser defined design criteriausing database module

Perform design sensitivity analysesw.r.t. specific parameters

Monitor design sensitivitiesof various design criteria

Perform what-if study andobtain improved design

Define optimization problemand perform automaticdesign optimization

Export improved design modelto CAD environment

Figure 1.2: Interactive engineering design using ODESSY.

in industry will adopt these methods for rational design, synthesis, and optimization.

1.2 Objective of Ph.D. Project and Contents of The-

sis

The objective of the present Ph.D. project is to develop, implement and integrate methods

for structural analysis, design sensitivity analysis and optimization into the general purpose

computer aided design system ODESSY.

More precisely, a general �nite element module for structural analysis must be developed

(in cooperation with colleagues). The facilities for structural analysis have been limited to

linear types of analysis problems. A number of reliable �nite elements must be implemented

in the system.

A reliable tool for design sensitivity analysis is the basis for doing interactive structural

design, synthesis, and optimization by means of ODESSY. E�cient and reliable methods

for design sensitivity analysis of all implemented analysis types and �nite element types

must therefore be developed and implemented. Furthermore, it must be possible to evalu-

ate user de�ned design criteria and their derivatives w.r.t. speci�c design parameters, and

a general and exible module for de�nition of structural optimization problems must be

available.

Chapter 2 of this thesis gives a general introduction to the facilities in ODESSY for para-

metric modelling. Basic concepts like design variables, design models, and modi�ers are

introduced and a brief overview of facilities for de�nition of design models is given.

10 1.2. Objective of Ph.D. Project and Contents of Thesis

Chapter 3 provides a description of the analysis capabilities in ODESSY for which struc-

tural synthesis and optimization can be performed. Furthermore, the implemented �nite

elements are described.

Chapter 4 is devoted to derivations of general expressions for design sensitivity analysis.

Di�erent approaches to design sensitivity analysis are discussed and the direct approach is

chosen. This approach is computationally e�cient and based on implicit di�erentiation of

the equations obtained when the continuum equations are discretized. The semi-analytical

method where derivatives of various element matrices are approximated by �rst order �nite

di�erences is used due to its ease of implementation.

Chapter 5 treats the problem of obtaining \exact" numerical derivatives of various �nite

element matrices as inaccuracy of the �nite di�erences involved in the semi-analytical

method may lead to erroneous design sensitivities. A new and e�cient method of \ex-

act" semi-analytical design sensitivity analysis has been developed and implemented in

ODESSY. \Exact" numerical derivatives are derived for element matrices of all imple-

mented �nite elements w.r.t. shape as well as sizing and material design variables. The

e�ciency of the new semi-analytical method is also discussed.

Chapter 6 contains several examples of design sensitivity analysis of structural problems in

order to demonstrate the validity of the methods of design sensitivity analysis derived in

the two preceding chapters. Some of the examples are used to illustrate that the di�erent

methods for design sensitivity analysis have been implemented correctly in ODESSY while

other examples are used to emphasize the importance of using the new semi-analytical

method of design sensitivity analysis that has been developed.

Chapter 7 gives a description of the implementation of a general and exible method of

formulating problems of mathematical programming in structural optimization systems.

The mathematical formulation is based on the so-called bound formulation, and the imple-

mentation involves a parser capable of interpreting and performing symbolic di�erentiation

of the user de�ned functions. This database module can also be used for interactive mon-

itoring of user de�ned design criteria.

Chapter 8 is devoted to the problem of solving structural optimum design problems with

multiple eigenvalues. When several eigenvalues of a structural problem coalesce and attain

the same numerical value, special attention must be made due to lack of usual di�erentia-

bility properties of multiple eigenvalues with respect to changes in design. Necessary opti-

mality conditions for optimum solutions are derived and iterative numerical algorithms for

eigenvalue optimization problems involving multiple eigenvalues are described. Examples

are given to illustrate the e�ciency of a proposed mathematical programming approach.

Chapter 9 presents four examples of interactive engineering design with ODESSY. It is

shown how design sensitivity display and what-if studies can be used to improve the design

of a turbine disk. The use of the general and exible method for de�nition of structural

optimization problems as described in Chapter 7 is illustrated by optimizing the shape

of the turbine disk, taking complex design constraints into account. Furthermore, the

turbine disk is redesigned using a ceramic material where the probability of failure must

be reduced. Finally, the hood of a Mazda 323 automobile is shape optimized with the


objective of maximizing the fundamental frequency of vibrations.

The present work is summarized in the conclusions in Chapter 10 where possible future

extensions also are discussed.

12 1.2. Objective of Ph.D. Project and Contents of Thesis

Chapter

2

Parametric Modelling for Optimum

Design

2.1 Introduction

This chapter is devoted to a general description of the parametric modelling facilities

available in ODESSY. This preprocessing part of ODESSY is mainly being developed

by my collegue John Rasmussen, and the aim is therefore to give an overview of the

geometric facilities available for de�ning structural optimization problems and to introduce

some basic concepts.

The motivation behind developing a system like ODESSY is the current trend toward

uni�cation of engineering design tools which were previously developed and used separately,

namely geometric modelling in the form of CAD systems, and mechanical analysis and

design sensitivity analysis using the �nite element method. The united capabilities of

these techniques allow for a major step forward in mechanical engineering design, i.e.,

from tools aimed at analysis to tools directly aimed at synthesis.

Before the parametric modelling facilities are described, some basic concepts for structural

design optimization are introduced.

2.2 Basic Concepts

The label structural design optimization identi�es the type of design problem where the

set of structural parameters is subdivided into so-called preassigned parameters and design

variables, and the problem consists in determining optimum values of the design variables

such that they maximize or minimize a speci�c function termed the objective (or criterion,

or cost) function, while satisfying a set of geometrical and/or behavioural requirements

which are speci�ed prior to design and are called constraints.

13

14 2.2. Basic Concepts

2.2.1 Design Variables

In order to introduce the possible design variables of the problem it is convenient to de-

�ne the speci�cations necessary to perform an ordinary analysis using the �nite element

method. In general, such an analysis requires information about (see, e.g., Cook, Malkus

& Plesha (1989) and Mouritsen (1992)):

1. Geometry (domain shape of the structure, division into �nite elements, and

kinematic boundary conditions).

2. Actions (loads acting on the structure).

3. Constitution (physical properties of the materials, and properties of the used

�nite elements).

If a design sensitivity analysis and/or optimization is to be performed, a number of design

variables must be de�ned. The design variables will be denoted by

ai; i = 1; : : : ; I (2.1)

and are assembled in the vector a. The design variables can be categorized as follows, see,

e.g., Olho� & Taylor (1983):

1. Geometrical design variables:

(a) Sizing design variables: describe cross-sectional properties of struc-

tural components like dimensions, cross-sectional areas or moments

of inertia of bars, beams, columns, and arches; or thicknesses of mem-

branes, plates, and shells.

(b) Con�gurational design variables: describe the coordinates of the

joints of discrete structures like trusses and frames; or the form of the

center-line or mid-surface of continuous structures like curved beams,

arches, and shells.

(c) Shape design variables: govern the shape of external boundaries and

surfaces, or of interior interfaces of a structure. Examples are the

cross-sectional shape of a torsion rod, column, or beam; the bound-

ary shape of a disk, plate, or shell; the surface shape of a three-

dimensional component; or the shape of interfaces within a structural

component made of di�erent materials.

(d) Topological design variables: describe the type of structure, num-

ber of interior holes, etc., for a continuous structure. For a discrete

structure like a truss or frame, these variables describe the number,

spatial sequence, and mutual connectivity of members and joints.

2. Material design variables: represent constitutive parameters of isotropic mate-

rials, or, e.g., stacking sequence of lamina, and concentration and orientation

of �bers in composite materials.

Chapter 2. Parametric Modelling for Optimum Design 15

3. Support design variables: describe the support (or boundary) conditions, i.e.,

the number, positions, and types of support for the structure.

4. Loading design variables: describe the positioning and distribution of external

loading which in some cases may be at the choice of the designer.

5. Manufacturing design variables: parameters pertaining to the manufacturing

process(es), surface treatment, etc., which in uence the properties and cost of

the structure.

The topological type of design variables is not covered by the presentation in this re-

port, but topology optimization using the homogenization method, see Bends�e & Kikuchi

(1988), Bends�e (1989, 1994), Bends�e & Rodrigues (1990, 1991), and Diaz & Kikuchi

(1992), and theory on optimal orientations of orthotropic materials, see Pedersen (1989,

1990, 1991), is being implemented in ODESSY by my collegue Lars Krog, see Olho�, Krog

& Thomsen (1993). The topology optimization can be extremely e�cient in �nding a

good topology for the structure which then can be further improved by subsequent sizing

or shape optimization, see Olho�, Bends�e & Rasmussen (1991), Bends�e, Rodrigues &

Rasmussen (1991), Bremicker, Chirehdast, Kikuchi & Papalambros (1991), Rasmussen,

Thomsen & Olho� (1993), Olho�, Lund & Rasmussen (1992), and Rasmussen, Lund &

Birker (1992).

Furthermore, the support and loading design variables are currently only available in the

form of position type of variables.

In the following the design variables of the structural design problem is, for simplicity,

generally divided into the three groups

� Generalized shape design variables.

� Sizing design variables.

� Material design variables.

Here, the group of shape design variables include con�gurational variables as well as posi-

tion type of variables for support and loading. Furthermore, the types of manufacturing

design variables covered can be included in the above-mentioned three groups.

2.3 Introduction to the Design Model Concept

Having de�ned the possible design variables of the structural design problem considered,

a convenient way of linking them to the �nite element analysis model must be available.

In case of sizing and material design variables, the design variables may be linked directly

to the analysis model, but this approach is not suitable for shape design variables. Geo-

metrically, shape optimization is much more di�cult to handle than sizing (and material)

optimization, because it involves a successive shape change of the model, and the descrip-

tion in the following is therefore mainly with attention to shape optimization. In the early

16 2.3. Introduction to the Design Model Concept

days of structural shape design sensitivity analysis and optimization, attempts were made

to use the �nite element model directly as design model, i.e., to use node coordinates as

design variables, see e.g., Zienkiewicz & Campbell (1973). It turns out that this method

has at least four serious drawbacks (see, e.g., Ding (1986) and Rodrigues (1988)):

� The number of design variables can become very large.

� It is di�cult to ensure compatibility and slope continuity between boundary

nodes.

� It is di�cult to maintain an adequate �nite element mesh during the domain

shape updating process when a shape optimization is performed.

� The structural shape design sensitivities might not be accurate unless high

order �nite element types are used.

Based on these experiences, most computer aided environments for interactive structural

shape design and optimization are founded on an important distinction between the design

model and the analysis model as demonstrated by, e.g., Braibant & Fleury (1984), Esping

(1984), Rasmussen (1990), and Olho�, Bends�e & Rasmussen (1991).

The design model is a variable description of the domain shape of the structure but sizing

as well as material design variables can also be linked to the design model. It can be closely

connected with a CAD model as described by Rasmussen (1990), and Rasmussen, Lund,

Birker & Olho� (1993), and it is totally distinct from the �nite element model that is used

for the analysis.

The design model may consist of so-called design elements as presented by Braibant &

Fleury (1984) and Bennett & Botkin (1985). The boundaries (or surfaces in case of a

three-dimensional model) of the design elements can be curves of almost any character,

i.e., piecewise straight lines, arcs, b-splines with speci�ed degree of continuity, bezier curves,

Coons patches, etc. It is therefore very simple to generate relatively complicated geometries

with a small number of design elements. The shapes of the boundaries are controlled by a

number of control points, also often termed \master nodes". Then, shape design variables

can be de�ned to control the positions of these master nodes, and possible sizing as well

as material design variables can be linked to each design element.

2.3.1 The Prototype Shape Optimization System CAOS

The design model concept in ODESSY is based on the experience with the prototype shape

optimization system CAOS which were developed by Rasmussen (1989) and further devel-

oped by Birker & Lund (1991), see also Rasmussen (1990), Olho�, Bends�e & Rasmussen

(1991), Rasmussen (1991), Rasmussen, Lund & Birker (1992), Olho�, Lund & Rasmussen

(1992), and Rasmussen, Lund, Birker & Olho� (1933). The design elements in CAOS are

either topologically quadrilateral design elements in case of a two dimensional structure,

or topologically hexahedral design elements in case of a three dimensional structure. This


approach enables simple mapping mesh generation but may require quite many design el-

ements even for simple geometries. The mapping mesh generation technique is illustrated

on Fig. 2.1.

r

s

xy

transformation

transformation

r s

t

xy

z

Figure 2.1: Mapping mesh generation technique in 2D and 3D.

In CAOS, the shape of boundaries or surfaces of design elements are controlled by a

number of master nodes. In the case of de�ning a shape optimization problem, this creates

an evident connection between the generalized shape design variables (the positions of the

master nodes) and the shape of the geometry, and thus provides a simple parametric model

of the structure.

At �rst glance, the design element approach used in CAOS seems to be an adequate solution

to the problem of parameterizing a geometry and coupling it with structural optimization.

However, there are also a few inherent problems which manifests themselves when the

geometries get just slightly complicated. To illustrate this, consider in Fig. 2.2 the CAOS

design model of a, from a geometrical point of view, very simple 3D structure, the so-called

wishbone which was �rst presented by Br�am�a & Rosengren (1990).

In spite of the geometrical simplicity of the model, as many as 30 design elements with

202 boundaries were required to obtain an acceptable �nite element modeling of the struc-

ture. Furthermore, the parametrization required as many as 308 generalized shape design

parameters to control translation directions for master nodes. A subsequent linking of

parameters reduced the number of independent variables to 112, but this is still far more

than what would naturally be assigned to a structure of this complexity. Generating and

checking such a design model manually is, even with the use of an interactive graphics

environment, a time consuming and error prone task. For a detailed description of this

example, refer to Birker & Lund (1991) and Rasmussen, Lund & Birker (1992).

18 2.4. The Design Model Concept in ODESSY

Figure 2.2: CAOS design model of wishbone structure comprising 30 design elements, 202

boundaries, and 308 translation directions.

2.4 The Design Model Concept in ODESSY

Based on the experiences obtained by developing CAOS, a much more general and geo-

metrically versatile design model concept is implemented in ODESSY, re ecting the desire

of integrating the system with three dimensional CAD systems for advanced geometric

modelling, i.e., including Constructive Solid Geometry (CSG) techniques. The problem of

generating parameterized geometric models of even complicated geometries can be solved

by state of the art CAD systems, and the challenge therefore lies in the creation of inter-

faces between the CAD system and the structural optimization system. My collegues John

Rasmussen and Anders Kristensen are currently working on solutions to this problem, and

a geometric formulation with solid modelling capability is being developed based on the

commercial CAD system Pro/Engineer. A more simple interface to the AutoCAD system

has also been implemented by my collegues Oluf Krogh & John Rasmussen.

The design model concept in ODESSY is based on a hierarchical parametric design model

because the desire to be able to parameterize virtually any property of the model has

been the focal point in the development of ODESSY from the very beginning. ODESSY

is based on a totally parametric modelling system in which data are initially divided into

two categories or levels of increasing complexity, Level0 and Level1. Entities in both levels

are always de�ned by user de�ned names, rather than by numbers.

Level0 is the \ground" level. This level contains geometrical entities which are independent

of other entities in the model. This would typically be points, vectors and scalar numbers.

In the terminology of CAOS and other traditional shape optimization systems, master

nodes would be Level0 entities. Other possible Level0 entities could be modules of elasticity

or thicknesses. Level0 is characterized by the property that it is the only level containing

real numbers.


Level1 contains entities which depend only on Level0 information. Boundary curves and

surfaces are typical examples. Their shapes are de�ned solely by the position of the master

nodes. Level1 could contain, for instance, circles, curves, boxes, tetrahedrons, cylinders,

cones, etc. The actual shapes of all entities in Level1 are de�ned by pointers to Level0

information. Level1 thus consists only of integer type information.

Level0 and Level1 of the geometric model of ODESSY cannot handle the problem of �nding

implicitly de�ned curves and surfaces which may result from the union of several solid

components, i.e., when CSG techniques are used. Yet another level of information, Level2,

is devised for this purpose. Level2 is not meant for processing by ODESSY itself but by

an external solid modelling system. In this context, the solid modeler may be thought of

as a geometric compiler which transforms high level geometric assemblies into lower level

curves and surfaces that can be handled directly by the more primitive geometric system

of ODESSY. For further descriptions of this approach of integration with solid modelling

CAD systems, see Rasmussen, Lund & Olho� (1993a, 1993b).

2.5 Design Variables and Modi�ers

The parametric nature of the design model comes automatically from the division of infor-

mation into interdependent levels. We simply control the model by modifying the entities

of Level0. Because of the hierarchical construction of the model, modi�cations of Level0

entities automatically lead to corresponding changes in Level1. In order to improve the

possibility of de�ning the design space, data in Level0 are controlled by the use of so-called

modi�ers. As previously mentioned, traditional systems, like CAOS, usually contain only

one type of modi�er, i.e., translational transformations (also called move directions). Other

possible modi�ers may be point scaling, line scaling, and rotation. This way, Level0 enti-

ties may be subjected to one or several transformations thus controlling the overall design.

The design variables of the problem will then simply be the amount of transformation, i.e.,

the rotation angle or the scale factor.

In this framework, for instance the move directions of the ODESSY system are realized

as translation modi�ers coupled with design variables which specify the magnitude of

the translation. Each master node is Level0 information, and the entire design model

therefore changes with relocation of the master nodes. Similarly, Level0 information may

be subjected to:

� translation modi�ers

� scaling modi�ers

� rotation modi�ers

� combinations of the three above-mentioned modi�ers

In designing the structure of links between design variables, modi�ers and Level0 informa-

tion, the following points shall be observed:

20 2.5. Design Variables and Modi�ers

S

S S

SS

S

S

SPP

1 2

34

1 1

1

2

3

4

1

design variable= 45°

modifier:rotate aroundP

Figure 2.3: Example of rotation modi�er and design variable a�ecting the four points S1,

S2, S3, and S4.

� It shall be possible to have the same design variable a�ect several Level0 entities

simultaneously through the same modi�er as illustrated in Fig. 2.3.

� It shall be possible to have the same design variable a�ect several Level0 entities

simultaneously through di�erent modi�ers.

� It shall be possible to use the same modi�er for several di�erent links between

design variables and Level0 entities. In a problem comprising, say, 50 master

nodes which can move freely in the plane, it would su�ce to de�ne two modi-

�ers, namely an x-translation and a y-translation, rather than having to de�ne

two modi�ers for each individual master node.

� It shall be possible to combine any number of modi�ers into a new compound

modi�er, for instance simultaneous rotation and scale.

These requirements can be ful�lled by creating link data structures of the following form:

LINK: [one design variable] ! [any number of modi�ers] ! [one Level0 entity]

This construction ful�ls any of the requirements above. For instance, given a design

variable \angle" and a modi�er \rotation", the example of Fig. 2.3 is realized by the

links:

LINK: angle ! rotation ! S1




In case of material or sizing design variables, translation modi�ers may be used to a�ect the

current value of the chosen variable, e.g., Young's modulus or plate thickness. Currently,

material and sizing parameters are assumed to be constant within each design element.


2.6 Mesh Generation

The implementation of good algorithms for mesh generation is very important in a struc-

tural shape optimization system. It must be possible to specify desired element sizes

within each design element, and the mesh must remain adequate during the domain shape

updating process when a shape optimization is performed. It may be advantageous to

include adaptive mesh generation in a shape optimization system, see, e.g., Diaz, Kikuchi,

Papalambros & Taylor (1983), Bennett & Botkin (1985), and Kikuchi, Chung, Torigaki &

Taylor (1986), but such facilities are currently not implemented.

Mapping mesh generation techniques are available in both two and three dimensions, but

in order to overcome the problem of having to divide the design model into quadrangular

or hexagonal design elements as necessary for mapping mesh generation, ODESSY also

allows for the use of unstructured free meshing. Currently, a modi�ed version of an algo-

rithm by George (1988) is implemented. This algorithm can subdivide an arbitrary planar

domain into triangles, and in ODESSY, the algorithm is extended to cover curved surfaces.

Furthermore, another unstructured mesh generation algorithm can generate a mixed mesh

predominantly of quadrangular elements, supplemented with a few triangular elements.

Robust free meshing of arbitrary volumes, which is a very challenging area, is currently

being implemented by my collegue Anders Kristensen.

After this basic overview of the facilities for de�nition of design models in ODESSY, the

analysis facilities will be presented.

22 2.6. Mesh Generation

Chapter

3

Analysis Capabilities for Structural

Optimization

3.1 Introduction

In this chapter a general description of the �nite element module in ODESSY is

given with focus on the types of analysis problems for which both analysis and design

sensitivity analysis have been implemented. Furthermore, descriptions of the implemented

�nite elements are given.

The �nite element module in ODESSY has been developed by several people at the Insti-

tute of Mechanical Engineering, but the main code has been developed by my collegue Oluf

Krogh and myself. The implementation of subroutines for solution of the �nite element

discretized equations has been performed by Oluf Krogh. When I started this project, Oluf

Krogh had implemented a pro�le skyline solver for solution of linear static equations based

on algorithms in Dhatt & Touzot (1984). Furthermore, he had implemented the Subspace

iteration method for determination of the lowest eigenvalues of a structural eigenvalue

problem, see Bathe & Ramaswamy (1980), Bathe (1982), and Dhatt & Touzot (1984), and

some simple elements were implemented.

My contribution to the �nite element module has mainly been

� Implementation of a family of isoparametric �nite elements with capabilities

for both analysis and design sensitivity analysis.

� Development of general and e�cient storing schemes for strains and stresses

for any combination of �nite element types.

� Extension of the analysis facilities to include thermal analysis, linear buckling

analysis, and any mixture of the available types of analyses.

� General system development.

23

24 3.2. Implemented Analysis Types for Structural Optimization

3.2 Implemented Analysis Types for Structural Op-

timization

In the following the di�erent types of analysis problems that are covered for both analysis

and design sensitivity analysis in ODESSY are brie y decribed. The intention with this

section is mainly to give an overview, as the di�erent analysis problems are described

more detailed in Chapter 4 where general expressions for design sensitivity analysis are

developed. Furthermore, the notation of di�erent matrices and vectors is introduced. Only

linear analysis types have yet been implemented in ODESSY.

Static Stress Analysis

The most common analysis type with the �nite element method is static stress analysis.

The global equilibrium equation of a �nite element discretized structural problem with

linearly elastic response is given by

KD = F (3.1)

where K is the global sti�ness matrix, D the nodal displacement vector and F is the

consistent nodal force vector. These global matrices and vectors are, as always in the �nite

element method, assembled from element matrices and vectors, i.e.,

K =Xne

k; D =Xne

d; F =Xne

f (3.2)

Here, k is the element sti�ness matrix, d the element nodal displacement vector, f the

consistent element nodal force vector, and ne is the number of �nite elements used to

discretize the structure. In ODESSY, the solution of Eq. 3.1 is carried out by a Crout

decomposition scheme as described in Dhatt & Touzot (1984).

Free Vibration Analysis

The free vibration analysis problem covered is a real, symmetric, structural eigenvalue

problem where the �nite element formulation is

K�j = !2jM�j; j = 1; : : : ; n (3.3)

K is the global sti�ness matrix, M the global mass matrix, !j the eigenfrequency, �j the

corresponding eigenvector, and n denotes the dimension of the problem.

Thermal Analysis

The �nite element equilibrium equation for a steady state heat conduction problem is given

by

Kth T = Q (3.4)

where Kth is the global thermal \sti�ness matrix" involving contributions from element

heat conduction matrices and coe�cients of the temperature vector T arising from convec-

tion boundary conditions, and Q is the thermal load vector involving forcing terms due to

Chapter 3. Analysis Capabilities for Structural Optimization 25

heat addition processes, e.g., heat ux. The solution of Eq. 3.4 is similar to the solution

of a static stress analysis problem.

Thermo-Elastic Analysis

This analysis type consists of a steady state thermal analysis followed by a static analysis.

First the temperature �eld T is found from the solution of Eq. 3.4, resulting in thermal

strains "th. Then Eq. 3.1 is solved by taking the thermal e�ects into account.

Eigenfrequency Analysis with Initial Stress Sti�ening

When calculating eigenfrequencies of a structure, it may be necessary to take into account

initial stress sti�ening e�ects due to mechanical loading. First a static analysis is per-

formed, resulting in element stresses �. These stresses are used to generate element initial

stress sti�ness matrices k� (also termed geometric sti�ness matrices) which represent the

initial stress sti�ening e�ects due to the loads applied to the structure. The conventionel

global sti�ness matrix K is then augmented with the global initial stress sti�ening matrix

K� resulting in a modi�ed form of Eq. 3.3, i.e.,

(K+K�)�j = !2jM�j; j = 1; : : : ; n (3.5)

Eigenfrequency Analysis with Thermal Loading and Initial Stress Sti�ening

The eigenfrequency analysis taking initial stress sti�ening e�ects into account can be ex-

tended taking thermal e�ects into account, i.e., the stress sti�ening e�ects may originate

from a thermo-elastic analysis taking both thermal and mechanical loading into consider-

ation.

Linear Buckling Analysis

It is possible to perform linear buckling analysis which is restricted to the assumptions

that the structure fails suddenly and the structure has constant sti�ness e�ects for all

loads up to the bifurcation point. These assumptions make this analysis type limited in its

applications as results will be most often on the unconservative, unsafe side. In practice,

linear buckling analysis can be used to inexpensively �nd an upper bound of the load

carrying capability of a structure.

If buckling analyses are to be used safely for structural optimization of real-life structures,

it is most often necessary to perform non-linear analyses resulting in a very complicated

design sensitivity analysis, see, e.g., Ringertz (1992), and large displacement buckling

analysis has therefore not been implemented in ODESSY.

The �nite element formulation of a linear buckling analysis can be written as

(K+ �jK�)�j = 0; j = 1; : : : ; n (3.6)

26 3.3. Finite Element Types Implemented for Structural Optimization

where K is the global sti�ness matrix, K� the initial stress sti�ness matrix established

from an initial static stress analysis, n the dimension of the problem, �j the buckling load

factor, and �j is the corresponding eigenvector of displacements.

Linear Buckling Analysis with Thermal Loading

Finally, the linear buckling analysis can be extended by taking thermal loading into ac-

count, i.e., the buckling load factors found correspond to the load vector resulting from

both thermal and mechanical loading.

3.3 Finite Element Types Implemented for Structural

Optimization

Although many di�erent �nite elements have been implemented in ODESSY, the following

description will be restricted to the isoparametric family of �nite elements that I have

implemented and used for solving structural design and optimization problems. Besides

the elements described here, di�erent types of spring, beam, truss, frame, and sandwich

plate and shell �nite elements are currently available in the system.

The isoparametric �nite elements can be divided into the following three groups

1. 3D solid �nite elements.

2. 2D solid �nite elements.

3. Mindlin plate and shell �nite elements.

The element matrices and vectors necessary for performing the di�erent types of analyses

described in Section 3.2 have been implemented for all these isoparametric �nite elements.

The types of elements implemented will be only brie y described in the following while

more detailed descriptions are given in Appendices A, B, and C. Furthermore, all elements

are described in Chapter 5, where \exact" numerical derivatives of element matrices and

vectors are found as a basis for a new method of design sensitivity analysis.

3.3.1 3D Solid Isoparametric Finite Elements

An 8-node and a 20-node isoparametric �nite element as illustrated in Fig. 3.1 have been

implemented.

A detailed description of element matrices and vectors for these two elements is given in

Appendix A. It should be noted that the code for computing the sti�ness matrix and the

consistent nodal load vector for the 3D isoparametric �nite elements has been implemented

by my collegues Niels Kristian Bau-Madsen and Oluf Krogh, respectively, whereas I have

implemented all other matrices and vectors for these elements.

Chapter 3. Analysis Capabilities for Structural Optimization 27

x,u y,v

z,w

Figure 3.1: 8- and 20-node isoparametric �nite elements.

3.3.2 2D Solid Isoparametric Finite Elements

Five di�erent 2D solid isoparametric �nite elements have been implemented in ODESSY

as illustrated in Fig. 3.2. They are all formulated in a uni�ed way for both plane stress,

plane strain, and axisymmetric situations.

(r,u)x,u

y,v(z,w)

axis of rotationalsymmetry( )

Figure 3.2: 3-, 4-, 6-, 8-, and 9-node 2D isoparametric �nite elements. Text in parantheses

refer to standard notations for problems with rotational symmetry.

It should be noted that element matrices for the two 2D solid triangular elements can

be formulated analytically. For the 6-node element it is then necessary that the element

sides are straight and nodes at element sides are midside nodes. Such derivations for

plane stress elements can be found in Pedersen (1973) and for axisymmetric elements in

Ladefoged (1988). Although it is advantageous to have analytic expressions for element

28 3.3. Finite Element Types Implemented for Structural Optimization

matrices, I have nevertheless chosen the standard approach of using numerical integration

because the 2D solid isoparametric �nite elements can be formulated in a uni�ed way

for both plane stress, plane strain, and axisymmetric situations as shown in Appendix B.

Furthermore, using the isoparametric formulation we have the advantage that the 6-node

element can have straight as well as curved edges.

The 2D solid �nite elements are described in detail in Appendix B.

3.3.3 Isoparametric Mindlin Plate and Shell Finite Elements

θ

θyi

zx

y

iw

z u x

y

v

θxi i

i

i

Figure 3.3: 3-, 4-, 6-, 8-, and 9-node isoparametric Mindlin plate �nite elements.

Six di�erent Mindlin plate �nite elements have been implemented in ODESSY. The ele-

ments have 3-, 4-, 6-, 8-, and 9-nodes, respectively, and are shown in Fig. 3.3. The 6-, 8-,

and 9-node elements can have straight as well as curved boundaries in the element plane.

The 9-node element is implemented both as the Lagrange plate element and as the \het-

erosis" plate �nite element as described in Section C.9 in Appendix C, where a detailed

description of the elements is given.

These six plate elements can also be used as at shell elements as described in Section C.10

in Appendix C. This approach of generating at shell elements from plate elements has

worked well for the problems studied, but may cause problems for strongly curved surfaces

where it is necessary to use many elements in order to obtain a proper �nite element model.

This is also discussed in Section C.10 in Appendix C.

After the presentation in this chapter of the analysis problems that can be solved and the

type of �nite elements that can be used, the subsequent chapter will focus on the design

sensitivity analysis.

Chapter

4

General Expressions for Design

Sensitivity Analysis

4.1 Introduction

In a general purpose computer aided environment for interactive structural design

and optimization, design sensitivity analysis is the basic enabling tool, and general ex-

pressions for design sensitivity analysis will be given in this chapter. The types of design

variable of the structural design problem can be shape as well as sizing or material vari-

ables and they are denoted by ai; i = 1; : : : ; I. It is now the aim to establish expressions for

design sensitivities of various criteria with respect to these design variables in an accurate

and e�cient way.

The overall �nite di�erence (OFD) approach to sensitivity analysis and the consequences

of using �nite di�erence schemes are described �rst. This approach to sensitivity analysis

is not computationally e�cient, but can be used as a reliable reference method.

Next, three di�erent methods for design sensitivity analysis are brie y described and it is

chosen to use the so-called discrete version of the direct approach.

Then the direct approach to design sensitivity analysis of static problems is discussed. The

direct approach is based on di�erentiation of the equations obtained when the continuum

equations are discretized by the �nite element method, and this method is very e�cient.

Using this approach it is necessary to determine derivatives of element sti�ness matrices,

and the semi-analytical (S-A) method where these derivatives are approximated by �rst

order �nite di�erences is chosen. Expressions for design sensitivities of displacements,

stresses and compliance are found with respect to single and simultaneous change of design

variables.

Next, the sensitivity analysis of thermo-elastic problems is discussed and the direct ap-

proach is used again. The sensitivities of the thermal problem are taken into account for

thermo-elastic problems.

Finally, eigenvalue design problems are considered. The direct approach can be used

29

30 4.2. Overall Finite Di�erence Approach to Sensitivity Analysis

to simple (distinct) eigenvalues but in the case of multiple (repeated) eigenvalues the

sensitivity analysis is more complicated. Then it is necessary to use perturbation techniques

in order to develop sensitivity expressions for multiple eigenvalues, as the eigenvalues are no

longer di�erentiable functions of the design in the normal (Fr�echet) sense. The sensitivity

analysis again will be presented both for perturbations of a single design variable ai and

for simultaneous change of several design variables.

4.2 Overall Finite Di�erence Approach to Sensitivity

Analysis

The simplest �nite di�erence approximation is the �rst order forward (or backward) dif-

ference approximation. If displacements, stresses, compliance, mass, or any other property

calculated by the analysis module is denoted by a function fj(a) of the design variables

ai, the overall forward �nite di�erence approximation �fj=�ai to the sensitivity @fj=@aiis given by

@fj(a)

@ai' �fj(a1; : : : ; aI)

�ai

=fj(a1; : : : ; ai +�ai; : : : ; aI)� fj(a1; : : : ; ai; : : : ; aI)

�ai(4.1)

If the derivatives of the function fj are sought for n design variables, the overall �nite

di�erence (OFD) method requires n additional analyses. Therefore this method is com-

putationally costly and mainly used as a reference method whose limit with regard to

accuracy is known to be set only by the solution procedure, the discretization, and the

usual accuracy capabilities of the applied �nite element.

Whenever a �nite di�erence scheme is used to approximate derivatives, there are two

sources of error: truncation and condition errors. The truncation error is a result of the

neglected terms in the Taylor series expansion of the perturbed function fj(a1; : : : ; ai +

�ai; : : : ; aI). This source of error can be reduced by using a small perturbation �ai. The

condition error is the di�erence between the numerical evaluation of the function and its

exact value. Contributions to the condition error are, e.g., computational round-o� errors

or errors due to an iterative solution process which is terminated early. The round-o�

errors are normally small for most computers unless the perturbation �ai is very small.

These opposite demands to the magnitude of the perturbation may give rise to the so-called

\step-size dilemma", see, e.g., Haftka & Adelman (1989). If we select the perturbation to

be small, so as to reduce the truncation error, the result may be an excessive condition

error. In some cases there may not even be any perturbation which results in su�ciently

small errors.

As noted by Haftka & Adelman (1989), the \step-size dilemma" may be reduced if a

higher order �nite di�erence approximation is used, e.g., a second order central di�erence

approximation, but this implies additional computational cost.

Chapter 4. General Expressions for Design Sensitivity Analysis 31

4.3 Selection of Method for Design Sensitivity Ana-

lysis

As the overall �nite di�erence (OFD) method is computationally ine�cient for the design

sensitivity analysis, another approach has to be used. The derivatives of structural response

in principle can be calculated at three stages.

We can (I) di�erentiate the continuum equations de�ning the response of the system, using

the material derivative concept of continuum mechanics, then discretize the problem and

solve it by using an adjoint variable technique as described by, e.g., Haug & Rousselet

(1980a, 1980b), Cea (1981), Zolesio (1981), Choi (1985), Haug, Choi & Komkov (1986),

Haug & Choi (1986), Haber (1987), Dems & Mr�oz (1983, 1984, 1993), and Dems & Haftka

(1989). This continuum approach is known as the material derivative (or speed) method.

Using this variational approach, the derivatives can be expressed in terms of boundary

integrals which are computationally inexpensive to evaluate. Unfortunately, there are

considerable numerical di�culties associated with the evaluation of the boundary integrals,

see Yang & Botkin (1986), especially for low-order elements which do not model a curved

boundary well, see Yang & Choi (1985). This can be avoided by using domain instead

of boundary integrals, see Choi & Haug (1983) and Choi & Seong (1986), but then the

numerical e�ciency decreases. The main advantage of the continuum approach, as stated

by Haftka & Grandhi (1986), seems to be the generality of its results as the method is

equally applicable to �nite element, boundary element (see Mota Soares, Rodrigues & Choi

(1984)), or any other numerical or analytical solution technique. Furthermore, sensitivity

calculations can be carried out outside existing �nite element codes, using postprocessing

data only. Thus, the design sensitivity analysis software does not have to be embedded in

an existing �nite element code, see Choi, Santos & Frederick (1985) and Santos & Choi

(1989). This is a great advantage if the analysis module is a commercial �nite element

package like ANSYS, NASTRAN, COSMOS, etc.

We can (II) use the direct approach which is based on di�erentiation of the equations ob-

tained when the continuum equations have been discretized (in this case by the �nite element

method). This approach, which is also known as the implicit di�erentiation approach, thus

has a reversed order of discretization and di�erentiation compared to the continuum ap-

proach described above. For static design sensitivity analysis, we can distinguish between

two types of the direct approach. In the (A) discrete version, the displacement sensitivity

�eld @D=@ai is calculated for the whole structure for each design variable ai; i = 1; : : : ; I,

while the (B) adjoint method calculates the derivative of a function g(D) of displacements

D. The adjoint method requires the solution for each desired function g(D), i.e., if the

number of stress and displacement functions needed for formulating and solving the opti-

mization problem is less than the number of design variables, then the adjoint method is

more e�cient than the discrete version of the direct approach to design sensitivity analysis.

The direct approach is, in both the discrete and adjoint version, very popular due to its

ease of implementation compared to the continuum approach.

Finally, we can (III) di�erentiate directly the computer program used to solve the structural

32 4.4. Design Sensitivity Analysis of Displacements

response, see, e.g., Wexler (1987) and Masmoudi, Broudiscou & Guillaume (1993). This

approach of automatic di�erentiation of a function de�ned by its program (in Fortran, C,

etc.) by using Taylor expansion is very interesting, but has not been considered here as it

seems to require substantial available memory in the computer, see Masmoudi, Broudiscou

& Guillaume (1993).

The issue of which approach of sensitivity analysis is better is a much debated subject

and several authers, e.g., Yang & Botkin (1986), Choi & Twu (1988), and Haftka &

Adelman (1989), have presented comparisons between the variational methods and the

direct approach and their relative merits.

The conclusive argument for selection of method for design sensitivity analysis has been

that the continuum approach takes a lot of analytical work in order to develop the expres-

sions for design sensitivities as can be seen in the references mentioned. I have wanted

to implement many di�erent kinds of �nite element types and at the same time be able

to handle many di�erent kinds of design variables. For this purpose the direct approach

seems much easier to implement and to be just as applicable to solve the problem as the

continuum approach. If we were going to use a commercial �nite element package for the

analysis, the continuum approach might be advantageous as postprocessing techniques can

be employed, but we have decided to write our own analysis code due to less promising

experiences of implementing design sensitivity analysis in an existing �nite element code,

such as, e.g., Modulef, see Kibsgaard (1991).

So, the direct approach to design sensitivity analysis has been chosen due to its ease of

implementation, and it will be shown in the following that this method is very e�cient.

The direct approach will be only used in the discrete version as the adjoint method is not

suited for the way optimization problems are formulated in ODESSY, cf. the description

in Chapter 7.

4.4 Design Sensitivity Analysis of Displacements

In the displacement based �nite element method, the design sensitivity analysis of various

criteria is based on sensitivities of the displacement �eld. Thus, when the displacement

sensitivities are known, e.g., stress and compliance sensitivities are easily computed.

The global equilibrium equation of a �nite element discretized structural design problem

with linearly elastic response is given by

KD = F (4.2)

where K is the global sti�ness matrix, D is the nodal displacement vector and F is the

consistent nodal force vector. The solution of Eq. 4.2 is carried out by Gaussian elim-

ination reformulated in a two phase process that does not require to modify K and F

simultaneously. It is thereby possible to solve Eq. 4.2 for additional load cases, i.e. several

right hand sides, without much additional computational e�ort. The time consuming part

of solving Eq. 4.2 is the factorization of the global sti�ness matrix K, in which this matrix

is basically decomposed into the product LU, where L is a lower triangular matrix (with


elements only on the diagonal and below) and U is a upper triangular matrix. Varieties of

this basic decomposition are, for example, the LDU, Crout, and Cholesky decomposition

schemes, see, e.g., Dhatt & Touzot (1984). In ODESSY, the Crout decomposition scheme

has been implemented.

When the sti�ness matrix K has been decomposed into the product LU, Eq. 4.2 can be

rewritten as

LUD = F (4.3)

Using this decomposed form of the sti�ness matrix, it is only necessary to solve a triangular

set of equations which is quite trivial. First a vector V is found by forward substitution

LV = F (4.4)

and then the displacement vector D can be found by back substitution

UD = V (4.5)

Having determined the displacement vector D, the design sensitivity analysis of displace-

ments now can be considered.

The direct approach to obtain design sensitivities of the displacement �eld is based on

implicit di�erentiation of the global equilibrium equation. If Eq. 4.2 is di�erentiated with

respect to a design variable ai and the terms are rearranged, the following discrete version

of the direct approach for the displacement sensitivities @D=@ai is obtained

K(a)@D

@ai= �@K(a)

@aiD +

@F

@ai; i = 1; : : : ; I (4.6)

Eq. 4.6 is of the same form as Eq. 4.2, so the factorized sti�ness matrix K in the form LU

can be reused, and only the new right hand side which is termed the pseudo load vector

need to be calculated before the sensitivities @D=@ai for each design variable ai can be

found by forward and back substitution. This approach is therefore much more e�cient for

the design sensitivity analysis than the OFD approach described in Section 4.2. It is seen

that the pseudo load vector is the load that must be applied to the structure to produce

the displacement sensitivity �eld due to changes of a design variable ai. The derivatives

@F=@ai of the force vector are easily calculated (are zero for design independent loads),

and then the determination of @D=@ai in Eq. 4.6 only requires calculation of the design

sensitivities @K=@ai of the sti�ness matrix. These derivatives are normally calculated at

the element level, i.e.@K

@ai=Xne

@k

@ai; i = 1; : : : ; I (4.7)

where k is the element sti�ness matrix and ne is the number of �nite elements.

If the design sensitivities @k(a)=@ai are determined analytically before their numerical

evaluation, the approach is called analytical design sensitivity analysis, and if they are

determined by numerical di�erentiation, the method is called semi-analytical (S-A) de-

sign sensitivity analysis, cf. Zienkiewicz & Campbell (1973), Esping (1983), Cheng & Liu

34 4.4. Design Sensitivity Analysis of Displacements

(1987), and Haftka & Adelman (1989). That is, in semi-analytical design sensitivity anal-

ysis the derivatives of the element matrices are approximated by �rst order forward �nite

di�erences (or another �nite di�erence scheme)

@k(a)

@ai' �k(a1; : : : ; aI)

�ai

=k(a1; : : : ; ai +�ai; : : : ; aI)� k(a1; : : : ; ai; : : : ; aI)

�ai(4.8)

The method of analytical design sensitivity analysis is very di�cult to implement in a

general purpose shape design system which contains many di�erent kinds of shape design

variables and �nite element types. Thus, a large amount of analytical work and program-

ming will be required in order to develop analytic expressions for derivatives of various

sti�ness matrices with respect to possible shape design variables. Some examples of deter-

mining analytical derivatives of an element sti�ness matrix with respect to a speci�c kind

of shape design variable can be found in, e.g., Braibant & Fleury (1984), Wang, Sun &

Gallagher (1985), and El-Sayed & Zumwalt (1991).

It is much more attractive to use the method of semi-analytical (S-A) design sensitivity

analysis in this context, as it is easy to implement for many di�erent kinds of shape design

variables and �nite element types, because simple and computationally inexpensive �rst

order �nite di�erences are used. Therefore, the method of S-A sensitivity analysis is very

popular and, in most cases, this method is very e�cient and reliable.

As a �nite di�erence approximation is involved in this method, both truncation and condi-

tions errors may occur. Furthermore, for shape design variables the perturbation �ai must

be selected su�ciently small, so that the elements do not become distorted. A strongly

distorted mesh may result in changing accuracy of the solution and thereby give the deriva-

tives a spurious contribution, see Botkin (1988). This problem is avoided in ODESSY by

selecting the perturbation �ai of a shape design variable ai so that boundary nodes are

perturbed less than 1=1000 of the smallest side length of the elements in the structure.

It should be noted, however, that the approximate �rst order �nite di�erence calculation

of derivatives of element sti�ness matrices used in the S-A sensitivity method may result

in severe inaccuracy problems for shape design variables due to truncation and condition

errors. This problem and solutions to it will be addressed in Chapter 5.

: perturbed elements

Figure 4.1: Perturbation of boundary elements for shape design variables.

The e�ciency of the design sensitivity analysis can be increased by using a so-called \ac-

tive element" strategy. When the domain shape is perturbed in case of a shape design


variable ai, only �nite elements situated at the surface are perturbed (and named active)

as illustrated in Fig. 4.1 for a two-dimensional domain where the perturbation has been

strongly exaggerated. The design boundary in Fig. 4.1 is modelled by a quadratic b-spline

and the master node shown is translated in the direction speci�ed by the vector. The

active (perturbed) �nite elements are hatched in Fig. 4.1 where the unperturbed mesh

also is shown. The active element strategy implies that in the assembly of element matrix

derivatives, only active elements contribute, and the pseudo load vector in Eq. 4.6 can

thereby be calculated at the element level in the following computationally e�cient way

�@K@ai

D+@F

@ai=Xnae

�@k

a

@aida +

@fa

@ai

!; i = 1; : : : ; I (4.9)

where nae is the number of active �nite elements, ka is the element sti�ness matrix, da is

the element displacement vector, and fa is the element load vector for an active element.

The choice of perturbing the entire �nite element mesh or perturbing only boundary nodes

has originated many a dispute. Braibant & Fleury (1984) and Botkin (1988), among oth-

ers, have advocated that it is necessary to move the interior nodes because the direct object

of the optimization is the �nite element model, rather than the real structure. Pedersen

(1988) has argued that only boundary nodes should be perturbed as relocation of interior

nodes may lead to a substantial risk to \maximize the errors of the �nite element model,

rather than minimize the physical stress concentration". Kibsgaard (1991) has made sev-

eral comparisons between these two methods of mesh perturbations, and he obtained best

results perturbing only boundary nodes. The physical argument to support this obser-

vation, as noted by Kibsgaard (1991), is that only the structure in the vicinity of design

boundaries are subject to changes as the design changes. The rest of the structure is un-

changed and therefore can be kept unchanged as well. If the interior nodes are perturbed,

thereby resulting in di�erences in the sti�ness matrix other than for the boundary ele-

ments, these di�erences would not re ect changes in the mechanical properties but rather

inaccuracies of the sensitivity analysis.

Furthermore, it is di�cult to perturb interior nodes using unstructured mesh generators

as a new mesh topology for the perturbed design model may appear. This is not allowed

as it would re ect the accuracy of a di�erent analysis model.

Based on numerical experience and the arguments above, this \design boundary layer"

approach of using one-element-deep sensitivity calculations for shape design variables has

been adopted in ODESSY due to the advantageous increase of numerical e�ciency.

4.5 Design Sensitivity Analysis of Stresses

When the displacement design sensitivities have been calculated it is straight-forward to

calculate stress design sensitivities as will be shown in the following.

The �nite element expression for the element stresses �(x; y; z) = f�x �y �z �xy �yz �xzgTmay be written

�(a) = E "(a) = EB(a) d(a) (4.10)

36 4.6. Design Sensitivity Analysis of Compliance

where the constitutive matrix E is independent of the design, B is the strain-displacement

matrix, and d is the element nodal displacement vector.

One way to calculate the stress design sensitivities is to use a �rst order forward �nite

di�erence approximation, i.e.

@�

@ai' �(a+�ai)� �(a)

�ai(4.11)

where

�(a +�ai) = E "(a +�ai) = EB(a+�ai) d(a +�ai) (4.12)

The perturbed strain-displacement matrix B(a+�ai) is easily calculated and the element

displacement vector for the perturbed design can be approximated by a �rst order Taylor

series expansion

d(a +�ai) ' d(a) +@d(a)

@ai�ai (4.13)

where the element displacement sensitivities @d=@ai are determined by the solution of Eq.

4.6.

4.6 Design Sensitivity Analysis of Compliance

The compliance C can be calculated as

C = DT F (4.14)

By di�erentiating Eq. 4.14 the following expression for the compliance design sensitivity

is obtained@C

@ai=@DT

@aiF+DT @F

@ai(4.15)

All terms on the right hand side are known from the calculation of displacement sensitivi-

ties, see Eq. 4.6, so the compliance design sensitivity is easily evaluated.

4.7 Simultaneous Change of Design Variables

In the following, displacements, stresses, compliance, mass, or any other property (except

for multiple eigenvalues) calculated by the analysis module is denoted by a function fj. If

some or all of the design variables ai are changed simultaneously then the linear increment

of the function fj can be found by using �rst order Taylor series expansions

�fj =rTfj �a (4.16)

where rfj denotes the gradient vector of fj and �a is the vector of changes of the design

variables ai

rfj =

@fj

@a1; : : : ;

@fj

@aI

!; �a = (�a1; : : : ;�aI) (4.17)


Eq. 4.16 is valid due to di�erentiability of the above mentioned vector �elds and functions

fj with respect to the design variables.

These notations are useful for parametric studies of functions fj as well as for formulation

of optimization problems.

4.8 Inclusion of Thermo-Elastic E�ects

In many engineering examples, thermo-elastic e�ects due to a temperature distribution in

the structure have to be considered. The �nite element equilibrium equation for a steady

state heat conduction problem is given by

Kth T = Q (4.18)

where Kth is the global thermal \sti�ness matrix" involving contributions from element

heat conduction matrices and coe�cients of the temperature vector T arising from con-

vection boundary conditions, and Q is the thermal load vector involving forcing terms due

to heat addition processes, e.g., heat ux.

Eq. 4.18 can be solved for temperatures T by standard solution procedures as described in

Section 4.4, and having calculated the temperature distribution, its in uence on the static

�nite element analysis can be included. The temperature T at a given point gives rise to

thermally induced strains "th given by

"th =

n"thx "thy "thz thxy

thyz

thxz

oT=nT T T 0 0 0

oT�; T = T � T0 (4.19)

where � is a matrix containing thermal expansion coe�cients, T is the temperature at the

given point, and T0 is the temperature at which the structure is free of thermally induced

strains (typically 20�C).

It is not possible to add these thermally induced strains directly to the mechanical strains

in order to calculate the induced stress �eld. This is due to the necessity of taking struc-

tural boundary conditions and internal resistance into account, i.e., a structure is free of

thermally induced stresses if its supports do not inhibit thermal expansion or contraction.

Instead the thermally induced strains "th are used to calculate a consistent global nodal

force vector Fth due to the thermally induced strains

Fth =Xne

ZBT E "th d (4.20)

where is the domain of the �nite element and ne is the number of �nite elements. This

nodal force vector due to thermally induced strains is added to the mechanical nodal

force vector in Eq. 4.2 when solving the static equilibrium equations. This results in a

displacement vector D used to calculate the element strains ", see Eq. 4.10, and then the

total element stresses �, due to mechanical and thermal strains, are given by

� = E(B d� "th) (4.21)

38 4.9. Design Sensitivity Analysis of Eigenvalues

where E is the constitutive matrix,B is the strain-displacement matrix and d is the element

displacement vector. In this way boundary conditions are considered when calculating

thermal stresses.

In the design sensitivity analysis the sensitivities of the temperatures T can be found

in the same way as described for displacement sensitivities in Section 4.4. Eq. 4.18 is

di�erentiated with respect to a design variable ai; i = 1; : : : ; I, and rearranging the terms,

the following expression for the temperature sensitivities @T=@ai is obtained

Kth(a)@T

@ai= �@K

th(a)

@aiT +

@Q

@ai(4.22)

The factorized global thermal \sti�ness matrix" can be reused as in the case of sensitivity

analysis of displacements, and the new right hand side which can be termed the thermal

pseudo load vector can be obtained using the S-A approach as described previously for

static design sensitivity analysis, see Eqs. 4.7, 4.8, and 4.9. When the new right hand side

has been determined, the temperature sensitivities @T=@ai can be calculated by forward

and back substitution reusing the factorized global thermal \sti�ness matrix".

Having obtained the temperature sensitivities @T=@ai, the perturbed thermally induced

strains "th(a+�ai) and the perturbed nodal force vector Fth(a+�ai) are easily calculated,

and �nite di�erence approximations of sensitivities of thermally induced strains and the

corresponding nodal force vector can be evaluated. These sensitivities are included in the

static design sensitivity analysis, whereby Eq. 4.6 becomes

K(a)@D

@ai= �K(a)

@aiD +

@F

@ai+

@Fth

@ai(4.23)

In a similar way, the thermally induced strain sensitivities are included in the stress sensi-

tivities in Eq. 4.11 by using Eqs. 4.12 and 4.21, and the thermo-elastic e�ects are thereby

included in the design sensitivity analysis.

4.9 Design Sensitivity Analysis of Eigenvalues

It is a well known fact that the design sensitivity analysis of eigenvalues is problematic in

the case of multiple eigenvalues, i.e., the case where two or more eigenvalues attain exactly

the same value. In this case, the eigenvalues are no longer di�erentiable functions of the

design in the normal Fr�echet sense. In the following it is described how design sensitivities

of both simple (distinct) and multiple (repeated) eigenvalues can be obtained.

The eigenvalue analysis problem considered can be either free vibration frequency analysis,

eigenfrequency analysis including initial stress sti�ening e�ects, or linear buckling analysis

as described in Section 3.2. For such real, symmetric, structural eigenvalue problems, the

�nite element formulation, in general, can be written as

K�j = �jM�j; j = 1; : : : ; n (4.24)

where K and M are symmetric, positive de�nite matrices, �j is the eigenvalue and �j

is the corresponding eigenvector. Depending on the type of analysis problem the global


K and M matrices consist of contributions from either element sti�ness, mass or initial

stress sti�ness matrices. The dimension of the problem is denoted by n, so Eq. 4.24 has n

solutions consisting of eigenvalues �j and corresponding eigenvectors �j. The eigenvalues

are all real and represent eigenfrequencies or linear buckling load factors depending on

the type of analysis problem. In ODESSY, Eq. 4.24 is solved by the Subspace iteration

method, see Bathe (1982), for the lowest eigenvalues �j. The eigenvalues can be ordered

by magnitude as

0 < �1 � �2 � : : : � �j � : : : � �n (4.25)

In the following it is assumed that the eigenvectors have been M-orthonormalized, i.e.,

�Tj M�k = �jk; j; k = 1; : : : ; n (4.26)

where �jk denotes Kronecker's delta.

If Eq. 4.24 is premultiplied by �Tj the following expression is obtained

�Tj K�k = �j�jk; j; k = 1; : : : ; n (4.27)

meaning that the eigenvectors are also K-orthogonal.

So far multiple eigenvalues and corresponding eigenvectors have not been mentioned. In

this case the eigenvectors are not unique. In fact, an in�nite number of linear combinations

of the eigenvectors corresponding to a multiple eigenvalue will satisfy Eqs. 4.24 and 4.26.

However, a set of M-orthonormal eigenvectors which span the subspace that corresponds

to a multiple eigenvalue can always be chosen. In other words, if it is assumed that �jhas multiplicity N (i.e., �j = �j+1 = : : : = �j+N�1), then N eigenvectors �j; : : : ;�j+N�1,

which span the N -dimensional subspace corresponding to the eigenvalues of magnitude �jand satisfy the orthogonality conditions in Eqs. 4.26 and 4.27, can be chosen.

4.9.1 Design Sensitivity Analysis of Simple Eigenvalues

As before it is assumed that the design variables of the structural design problem is denoted

by ai; i = 1; : : : ; I, and the goal is to obtain expressions for eigenvalue sensitivities with

respect to these design variables. It is also assumed that the components of the K and M

matrices are smooth functions of design variables ai.

The direct approach to obtain the eigenvalue sensitivities is to di�erentiate Eq. 4.24 with

respect to a design variable ai assuming that �j is simple

@K

@ai�j + (K� �jM)

@�j

@ai=@�j

@aiM�j + �j

@M

@ai�j; i = 1; : : : ; I (4.28)

By premultiplying Eq. 4.28 by �Tj and making use of Eq. 4.24, the following expression is

obtained for the eigenvalue sensitivity in case of simple eigenvalues �j, see, e.g., Courant

& Hilbert (1953) and Wittrick (1962)

@�j

@ai= �

Tj

@K

@ai� �j

@M

@ai

!�j; i = 1; : : : ; I (4.29)


where the term �Tj M�j = 1 due to theM-orthonormalization, Eq. 4.26, has been omitted.

It appears that the only unknown quantities in Eqs. 4.29 are the derivatives of the K

and M matrices. As in the case of static S-A design sensitivity analysis, these derivatives

are calculated at the element level as in Eqs. 4.7 and 4.8, and then the eigenvalue design

sensitivities can be determined.

If all the design variables ai are changed simultaneously, then, as described in Section 4.7,

due to the di�erentiability of simple eigenvalues with respect to the design variables, the

linear increment of the simple eigenvalue �j can be found in the form

��j =rT�j �a (4.30)

where r�j denotes the gradient vector of �j and �a is the vector of changes of the design

variables ai

r�j =

@�j

@a1; : : : ;

@�j

@aI

!; �a = (�a1; : : : ;�aI) (4.31)

These notations are useful for parametric studies of eigenvalues as well as for formulation

of optimization problems.

4.9.2 Design Sensitivity Analysis of Multiple Eigenvalues

When the solution of the generalized eigenvalue problem in Eq. 4.24 yields a N -fold

multiple eigenvalue~� = �j; j = 1; : : : ; N (4.32)

where, for convenience, the repeated eigenvalues have been numbered from 1 to N , then

the computation of the sensitivities of this eigenvalue is not straight-forward. This is

due to the fact that the eigenvectors �j; j = 1; : : : ; N , of the repeated eigenvalues are not

unique. Thus, any linear combination of the eigenvectors will satisfy the original eigenvalue

problem, Eq. 4.24.

In the following sensitivity analysis we shall use such eigenvectors ~�j which remain con-

tinuous with design changes, see Courant & Hilbert (1953). For this purpose linear com-

binations of eigenvectors �k are introduced

~�j =NXk=1

�jk�k; j = 1; : : : ; N (4.33)

where �jk are unknown coe�cients to be determined.

Works by Courant & Hilbert (1953), Wittrick (1962), and Lancaster (1964) have provided

a basis for calculating the sensitivities of multiple eigenvalues. It is shown that the de-

sign sensitivities of multiple eigenvalues can be found by formulation and solution of a

subeigenvalue problem.

Let us �rst consider a small change "�ai of a single, arbitrarily chosen design parameter

ai where " is a small positive parameter. Due to the perturbation of this design variable

the K and M matrix will be incremented, i.e., the new matrices become

K+ "@K

@ai�ai and M + "

@M

@ai�ai; i = 1; : : : ; I (4.34)


Then multiple eigenvalues and corresponding eigenvectors for the perturbed design can be

written as

�j(ai + "�ai) = ~�+ "�j(ai;�ai) + o("); j = 1; : : : ; N (4.35)

�j(ai + "�ai) = ~�j + "�j(ai;�ai) + o("); j = 1; : : : ; N (4.36)

where �j and �j are unknown eigenvalue and eigenvector sensitivities, respectively, and

o(") represents higher order terms.

Substituting Eqs. 4.34, 4.35, and 4.36 into the main eigenvalue problem in Eq. 4.24, we

obtain in the �rst approximation @K

@ai� ~�

@M

@ai

!~�j + (K� ~�M)�j = �jM~�j (4.37)

Premultiplying this equation by �Ts gives

�Ts

@K

@ai� ~�

@M

@ai

!~�j = �j�

TsM

~�j; s = 1; : : : ; N (4.38)

Here the term �Ts (K� ~�M)�j = �

Tj (K� ~�M)�s drops out because �s is the eigenvector

corresponding to ~�.

Recalling that ~�j is the linear combination in Eq. 4.33 of the original eigenvectors �k,

from Eq. 4.38 the following system of linear algebraic equations of unknown coe�cients

�jk is obtained

NXk=1

�jk

�Ts

@K

@ai� ~�

@M

@ai

!�k � �j�sk

!= 0; s = 1; : : : ; N (4.39)

where the M-orthonormalization, Eq. 4.26, has been used.

A nontrivial solution to these equations only exists if the determinant of the system is

equal to zero

det

��Ts

@K

@ai� ~�

@M

@ai

!�k � ��sk

�� = 0; s; k = 1; : : : ; N; i = 1; : : : ; I (4.40)

This is the main equation for determining the coe�cients �j; j = 1; : : : ; N , of the power

series in Eq. 4.35 which represent the sensitivities of the multiple eigenvalue ~� with respect

to changes �ai of a single design parameter ai. As in the case of simple eigenvalues, the

derivatives of the K and M matrix, respectively, must be calculated �rst, and then the

subeigenvalue problem of Eq. 4.40 is easily formulated and solved.

If the o�-diagonal terms in the quadratic matrix of dimension N in Eq. 4.40 are equal

to zero, the eigenvalues of this matrix, i.e., the directional derivatives of the multiple

eigenvalue ~�, are equal to the traditional Fr�echet derivatives obtained by using Eq. 4.29.

Let us consider the general case when all the design variables ai; i = 1; : : : ; I, are changed

simultaneously. It should be noted that multiple eigenvalues are not di�erentiable in the

common sense, i.e., not Fr�echet-di�erentiable, see, e.g., Haug, Choi & Komkov (1986). This

means that the expression for the eigenvalue increments in Eqs. 4.30 is no longer valid.


Thus, to �nd the sensitivities of multiple eigenvalues it is necessary to use directional

derivatives in the design space.

For this purpose, for the vector of design variables a = (a1; : : : ; aI), a varied form a+ "e is

considered, where e is an arbitrary vector of variation e = (e1; : : : ; eI) with the unit norm

kek =qe21 + : : :+ e2I = 1 and " is a small positive parameter. The vector e represents

a direction in the design space along which the design variables ai are changed, and "

represents the magnitude of the perturbation in this direction.

As a result of perturbation of the vector a the matrices K and M are incremented and

become

K+ "IX

i=1

@K

@aiei; M+ "

IXi=1

@M

@aiei (4.41)

Using expansions for �j and �j in the form

�j = ~�+ "�j + o("); j = 1; : : : ; N (4.42)

�j = ~�j + "�j + o("); j = 1; : : : ; N (4.43)

and performing the same manipulations as earlier, instead of Eq. 4.40 the following N -th

order equation for determining the sensitivities �j of the eigenvalues �j is obtained:

det

��IX

i=1

�Ts

@K

@ai� ~�

@M

@ai

!�kei � ��sk

�� = 0; s; k = 1; : : : ; N (4.44)

If the generalized gradient vectors fsk of dimension I are introduced

fsk =

�Ts

@K

@a1� ~�

@M

@a1

!�k; : : : ;�

Ts

@K

@aI� ~�

@M

@aI

!�k

!(4.45)

then Eq. 4.44 takes the form

det�� fTske� ��sk

�� = 0; s; k = 1; : : : ; N (4.46)

Note that fsk = fks due to the symmetry of the matrices K andM. Also note the notation

used here for the generalized gradient vectors fsk. The subscripts refer to the modes from

which the generalized gradient vector is calculated, i.e., fTske is a scalar product.

Thus, knowing the eigenvectors �k; k = 1; : : : ; N , corresponding to the multiple eigenvalue~�, the generalized gradient vectors fsk can be constructed and the sensitivities �j; j =

1; : : : ; N , for any vector of variation e can be determined, i.e. for any direction in the

space of the design variables. The quantities �j constitute the directional derivatives of

the multiple eigenvalue ~�, cf. Eq. 4.42. In this form Eq. 4.46 was obtained by Bratus &

Seyranian (1983), and Seyranian (1987), see also Haug & Rousselet (1980b), Masur (1984,

1985), and Haug, Choi & Komkov (1986).

In many cases it is expedient to eliminate the unit vector e from Eq. 4.46 and establish

a formula for determining the increments ��j; j = 1; : : : ; N , of the N -fold eigenvalue ~�

subject to a given vector �a = (�a1; : : : ;�aI) of actual increments of the design variables


ai; i = 1; : : : ; I. To this end, we multiply each of the components in Eq. 4.46 by ", note

from the foregoing that "e =�a and "�j = ��j; j = 1; : : : ; N , and obtain

det�� fTsk�a� �sk��

�� = 0; s; k = 1; : : : ; N (4.47)

If we solve this N -th order algebraic equation for ��, we obtain the increments �� =

��j; j = 1; : : : ; N , of the N -fold eigenvalue corresponding to the vector �a of actual

increments of the design variables.

As in the case of semi-analytical (S-A) design sensitivity analysis of static problems, pos-

sible inaccuracies in the approximate numerical di�erentiation of the element sti�ness

matrices can lead to severe inaccuracy problems in the sensitivities of eigenvalues. This

problem will be investigated in Chapter 5.

The technique described above in principle solves the problem of design sensitivity anal-

ysis of eigenvalues. However, in the case of multiple eigenvalues, the problem of non-

di�erentiability continues to be a potential source of di�culty in relation to the numerical

optimization procedure based on common derivative information. Fortunately, in a numer-

ically based optimization system, the case of exactly coalescing eigenvalues is very unlikely

for most common structures, and in most cases this problem does not have a signi�cant

impact on the optimization procedure. However, for plate and shell structures, possibly

reinforced by sti�eners, multiple eigenvalues frequently occur and therefore Chapter 8 is

devoted to the special case of optimization of multiple eigenvalues.

Chapter

5

\Exact" Numerical Di�erentiation of

Element Matrices

5.1 Introduction

This chapter is devoted to the problem of obtaining \exact" numerical derivatives

of various �nite element matrices as the accuracy of the �rst order �nite di�erence

approximations in the semi-analytical (S-A) method, see Eq. 4.8, is strongly dependent on

the chosen size of perturbation �ai of a shape design variable ai. This dependency arises

as the element matrices generally depend non-linearly on shape design variables, and in

some cases, so small perturbations are needed that computational round-o� errors become

the problem. The inaccuracy problem associated with the S-A method in connection with

shape design variables is described in Section 5.2.

In order to avoid dependence on the chosen perturbation �ai, the goal is to construct a

method for \exact" numerical di�erentiation of element matrices based on computationally

inexpensive �rst order �nite di�erences. Here and in the following, derivatives obtained

by numerical di�erentiation will be termed \exact derivatives" if they have no truncation

error due to neglection of higher order terms in their Taylor series expansion and are exact

except for computational round-o� errors.

This goal may seem unattainable, but a closer study of the functions that form the element

matrices reveals that the same mathematical forms are common for large groups of �nite

elements. For instance, the element matrices for all isoparametric elements with trans-

lational degrees of freedom and isoparametric Mindlin plate and shell elements depend

on the same class of functions. Similarly, the element matrices of a large class of �nite

elements comprising Bernoulli-Euler beam and Kirchho� plate and shell elements have a

similar mathematical structure. The members of these classes of matrices in general de-

pend non-linearly on the design variables, but are de�ned within a special mathematical

form. These element functions will be described in Section 5.3 where it is shown that the

mathematical form implies that their approximate numerical derivatives, computed by a

45

46 5.2. Problem of Inaccuracy in the Traditional Semi-Analytical Method

usual �rst order �nite di�erence scheme, can be upgraded to \exact" derivatives by simple

multiplication by appropriate correction factors. The values of these correction factors can

be very easily pre-computed and be used throughout the procedure of design sensitivity

analysis. It follows as a remarkable side-e�ect of the \exactness" that the results become

totally independent of the magnitude of the perturbation.

In Section 5.4 \exact" numerical derivatives of various element matrices of 3D solid isopara-

metric �nite elements will be found by using the results of Section 5.3. Similar derivations

are given in Sections 5.5 and 5.6 for 2D isoparametric solid elements and isoparametric

Mindlin plate �nite elements, respectively. Due to simplicity, only shape design variables

are considered in these sections. Thus, it is assumed that the element matrices of a partic-

ular �nite element only depend on shape design variables, i.e., design variables related to

nodal coordinates of elements. It is assumed that an element matrix depends on a given

sub-set from among the total set of shape design variables ai; i = 1; : : : ; Is, which in this

context is considered to be the global coordinates of the �nite element nodal points. This

sub-set of the design variables is assumed to be renumbered and denoted by aj; j = 1; : : : ; J ,

where J < Is.

Section 5.7 describes how derivatives of various element matrices can be found with re-

spect to generalized shape design variables, e.g., positions of master nodes as described in

Chapter 2, by using the results obtained in Sections 5.4, 5.5, and 5.6.

In Section 5.8 \exact" numerical derivatives of various element matrices are given for all

kinds of design variables, i.e., including sizing as well as material design variables.

Finally, the numerical e�ciency of the new method of S-A design sensitivity analysis and

the actual implementation is described in Section 5.9.

5.2 Problem of Inaccuracy in the Traditional Semi-

Analytical Method

Recent references have demonstrated that the method of S-A sensitivity analysis may

su�er serious accuracy drawbacks in particular types of problems involving shape design

variables. A similar inaccuracy problem is not found if the analytical method or the overall

�nite di�erence technique of sensitivity analysis are employed. The problem must therefore

be attributed to the numerical di�erentiation of the �nite element sti�ness matrix that is

inherent in the S-A method, cf. Eqs. 4.7 and 4.8.

The inaccuracy problem associated with the S-A method was �rst discovered by Barthe-

lemy, Chon & Haftka (1988) and the problem was �rst encountered with a car model made

of 3D beam elements. The authors had previously used the overall �nite di�erence (OFD)

approach to design sensitivity analysis, see Section 4.2, but shifted to the S-A approach

due to its superiority with respect to computational e�ciency. The S-A method worked

excellently for sizing variables but for certain global dimensions of the car, i.e., for a few of

the shape design variables, inaccuracy problems were detected. For these shape variables

the S-A method proved to be very sensitive to the step size used in the �nite di�erence

Chapter 5. \Exact" Numerical Di�erentiation of Element Matrices 47

approximation of the sti�ness matrix derivative, cf. Eqs. 4.7 and 4.8. Because the car

model was complex, Barthelemy, Chon & Haftka studied a cantilever beam subjected to a

moment at the free end. For this model problem where the length of the beam was taken

as a shape design variable they observed the same inaccuracy problem associated with the

S-A method for the sensitivity of the lateral tip displacement. The OFD method was used

as a reference method and the only problem encountered with this method was the usual

\step-size dilemma" as discussed in Section 4.2.

Two types of abnormal errors have been noticed by Barthelemy & Haftka (1988). For

beam- and plate-like structures, (I) the numerical errors of displacement sensitivities with

respect to shape design variables such as length increase quadratically with �nite element

mesh re�nement. This observation would not be expected in advance as a traditionel �nite

element analysis is known to converge to a stationary point with mesh re�nement if a

proper �nite element is used, so this abnormal error type was surprising.

Another type of error problem appears in problems involving linearly elastic bending of

long-span beam-like structures. Here, (II) the errors of displacement sensitivities with

respect to beam length increase rapidly with the length of the beam, i.e., with the beam

aspect ratio. Barthelemy & Haftka (1988) observed that the error problem is not symp-

tomatic of only beam elements but can be expected for beam- and plate-like structures, no

matter what kind of �nite element is used to model them.

Barthelemy & Haftka (1988) and Pedersen, Cheng & Rasmussen (1989) traced the inaccu-

racy problem for the beam model to the basic concept of the pseudo load vector, see Eq.

4.6, which is the load that must be applied to the structure to produce the displacement

sensitivity �eld. In many cases of shape variations, the sensitivity displacement �eld is not

a reasonable displacement �eld for the structure and its boundary conditions. For exam-

ple, for beam- or plate-like structures, the displacement sensitivity to a length dimension

is dominated by shear rather than bending and in order to produce the unlikely shear-

dominated �elds, the pseudo load must include large self-cancelling components. These

components will contain small truncation errors due to the �nite di�erence scheme used

and these errors are ampli�ed into large errors in the displacement sensitivity.

A local error index for the S-A truncation error was developed by Barthelemy, Chon

& Haftka (1988) for beam and plane truss elements based on an examination of error

contribution of each �nite element. Using these error indices they found that elements

which undergo large rigid body motions contribute the most to the total error. This

observation was later analyzed in detail by Cheng & Olho� (1991, 1993) who introduced

a rigid body motion test for detection of errors in the S-A sensitivities.

Pedersen, Cheng & Rasmussen (1989) studied the same model problem of a cantilever beam

with a tip moment and observed that the components of the beam element sti�ness matrix

depend on the design variable considered, i.e., the element length, in three di�erent powers.

Thus, using a �rst order forward �nite di�erence approximation of the sti�ness matrix

derivative, the derivatives of these components are of di�erent accuracy. These uneven

truncation errors result in relative errors of the pseudo forces that are di�erent from those

of the pseudo moments in the pseudo load vector. These di�erent inaccuracies of the pseudo

load components due to the forward (or backward) �nite di�erence scheme are shown by

48 5.2. Problem of Inaccuracy in the Traditional Semi-Analytical Method

Pedersen, Cheng & Rasmussen to be the reason for the two types of error problems for

this model problem, i.e., they showed for this model problem that the sensitivity error is

proportional to the relative length di�erence, but unfortunately it is also proportional to

the square of the number of �nite elements. Furthermore, they showed that a second order

central �nite di�erence scheme cannot remove the inaccuracy problem. However, using

a central �nite di�erence approximation of the sti�ness matrix derivative, the sensitivity

error is proportional to the square of the relative length di�erence and thereby much

reduced. The dependence on the square of the number of elements is not removed.

This important work of error analysis by Pedersen, Cheng & Rasmussen (1989) was ex-

tended by Olho� & Rasmussen (1991a) by deriving the analytical solution to the global set

of �nite element equations for the S-A design sensitivity analysis problem for any degree of

discretization. This enabled the authors to precisely identify and explain the source of the

numerical inaccuracy problem to originate from the forward �nite di�erence approximation

of the sti�ness matrix. The non-uniform distribution of pseudo load errors is such that it

has a very critical in uence on the displacement design sensitivities as the �nite element

mesh is re�ned because the values of the latter, which should be mesh independent, re-

sult from subtractions and additions of an increasing number of increasingly large terms

as the number of �nite elements increases with the mesh re�nement. They also showed

mathematically that the error of the sensitivity of the lateral tip displacement increases

quadratically with the number of �nite elements used to model the beam, see also Fenyes

& Lust (1991).

Di�erent approaches have been suggested for improvement of inaccurate S-A design sensi-

tivities. Haftka & Adelman (1989) have advocated the use of central di�erences instead of

forward di�erences which implies additional computational cost. The use of second order

central �nite di�erences reduces the error problem as described above but it cannot remove

it. Furthermore, central di�erences involve a doubling of the computation time needed to

calculate the sti�ness matrix derivative. Therefore, the approach of using higher order

�nite di�erences has not been chosen. Cheng, Gu & Zhou (1989) proposed an alternate

forward/backward �nite di�erence scheme which preserves the computationally e�ciency

of the S-A method. Cheng, Gu & Wang (1991) introduced a second order correction

method but none of these methods can completely eliminate the inaccuracy problem. Ol-

ho� & Rasmussen (1991b) developed an e�cient method of error elimination by \exact"

numerical di�erentiation which eliminates errors associated with mesh re�nement and de-

sign variable perturbation for a class of problems that comprises the beam model problem.

Their initial approach to remove the inaccuracy problem forms the basis for the method

of \exact" numerical di�erentiation which will be described later in this chapter.

A recent paper by Cheng & Olho� (1991, 1993) is also devoted to the problem of error

elimination, and a geometrical-physical interpretation of the in uence of the error associ-

ated with the �rst order forward di�erence approximation �k=�ai to the element sti�ness

derivative @k=@ai, cf. Eq. 4.8, is presented. The contribution from the matrix derivative

to the element pseudo loads are given by �(@k=@ai)d, see Eq. 4.6 and 4.9, where d is the

vector of nodal displacements of the element. Now, any displacement vector d can be sub-

divided into three vectors: a vector dt for the rigid body translation, a vector dr for the


rigid body rotation, and a vector for the remaining part of d which is associated with the

actual deformation of the �nite element. Along the same lines as a proper �nite element

should possess the rigid body motion capabilities kdt = 0 and kdr = 0 (vanishing of the

element nodal forces associated with dt and dr), it is clear that all the components of the

element pseudo loads associated with a rigid body translation and rotation, respectively,

should vanish such that �(@k=@ai)dt = 0 and �(@k=@ai)dr = 0. Cheng & Olho� show

that the latter conditions are satis�ed for both sizing and shape design variables when

analytical design sensitivities @k=@ai are used.

Now, Cheng & Olho� show that if the design sensitivities are replaced by their forward

di�erence approximations, then the conditions �(�k=�ai)dt = 0 and �(�k=�ai)dr = 0

are also both satis�ed if ai is a sizing design variable. However, if ai is a shape design vari-

able, only the former condition is satis�ed, i.e., only the approximate element pseudo loads

associated with a rigid body translation vanish. Hence the latter condition is generally

not satis�ed if ai is a shape variable which means that the components of the approximate

element pseudo loads that correspond to a rigid body rotation do not vanish in general,

i.e., �(�k=�ai)dr 6= 0. This fact was also shown later by Mlejnek (1992).

Thus, the conclusion from Cheng & Olho� (1991, 1993) is that in S-A design sensitivity

analysis with a shape design variable ai, the use of the �rst order �nite di�erence approx-

imation �k=�ai in Eq. 4.8 generally introduces an error in the form of an extra moment

to the exact element pseudo loads, and that this extra moment is the resultant of the

approximate pseudo loads associated with the rigid body rotation of the element. The ex-

tra (error) pseudo moments from each �nite element become aggregated as a nonuniform

distribution of errors of the system level pseudo loads in the process of assembling the

system level, see Eq. 4.7 (with @ replaced by � throughout), and are responsible for the

errors of the approximate displacement design sensitivities �D=�ai.

It is the conclusion of the above-mentioned papers concerning the accuracy of the S-A

method of design sensitivity method, that the numerical di�erentiation of the �nite ele-

ment sti�ness matrix that is inherent in the S-A method in certain cases may result in

serious inaccuracy problems. The problems may occur for design sensitivities with respect

to structural shape design variables in problems where the displacement �eld is character-

ized by rigid body rotations which are large relative to actual deformations of the �nite

elements, i.e., for example in problems involving linearly elastic bending of long-span,

beam-like structures, and of plate and shell structures. However, it should be noted that

the S-A method works excellently for most problems and the inaccuracy problem is only

encountered for the above-mentioned type of design sensitivity analysis problems.

In order to obtain a robust method for design sensitivity analysis for all possible design

problems, the goal has been to construct a method for \exact" numerical di�erentiation

of the �nite element matrices and in order to maintain the computationally e�ciency the

method must be based on �rst order �nite di�erences, i.e., higher order �nite di�erences

as central di�erences are not used. Such a method has been developed and published by

Olho�, Rasmussen & Lund (1992) and Lund & Olho� (1993a, 1993b), and the remainder

of this chapter is devoted to a presentation of this new method of S-A design sensitivity

analysis.

50 5.3. \Exact" Numerical Di�erentiation of Special Element Functions

5.3 \Exact" Numerical Di�erentiation of Special El-

ement Functions

The foundation for the new S-A method using \exact" numerical di�erentiation of �nite

element matrices is the observation, that element functions g, used in the de�nition of

element matrices for large groups of �nite elements, have the same mathematical form.

This observation and the results described in this section were made by Niels Olho� in

initial studies of the inaccuracy problem.

For instance, the element matrices for all isoparametric elements with translational degrees

of freedom and isoparametric Mindlin plate and shell elements depend on the same class

of element functions g. Similarly, the element matrices of a large class of �nite elements

comprising Bernoulli-Euler beam and Kirchho� plate and shell elements have a common

mathematical structure. The latter kind of �nite elements will not be discussed in this

chapter but a description of \exact" numerical di�erentiation of such elements can be found

in Olho�, Rasmussen & Lund (1992).

It should be noted that in the present context, the design variables are assumed to be shape

design variables in the form of the global coordinates of the �nite element nodal points. The

element matrices of a particular �nite element only depend on a given sub-set from among

the total set of shape design variables ai; i = 1; : : : ; Is. This sub-set of the design variables

is renumbered and denoted by aj; j = 1; : : : ; J , where J < Is. J is, at maximum, equal to

the total number of nodal coordinates of the element.

In the case of isoparametric �nite elements, the element matrices turn out to depend on

element functions g, which, in general, depend non-linearly on the design variables, i.e.,

on the nodal coordinates. The element functions g are incomplete polynomia and de�ned

by the following form:

g(a1; : : : ; aJ) = pj(a1; : : : ; aj�1; aj+1; : : : ; aJ) + qj(a1; : : : ; aj�1; aj+1; : : : ; aJ) � (aj)rj ;rj 2 @ [ f0g; j = 1; : : : ; J (5.1)

Thus, a function g is such that for any j, the term pj and the coe�cient qj 6= 0 are

independent of aj. The design variable aj appears in one and only one power rj which

belongs to the set @ [ 0 of non-negative integers. Although not stated explicitly, pj and

qj, and therefore the element function g, will generally depend on the local coordinates

within the �nite element. Typically, g may represent the determinant or components of the

Jacobian matrix or components of other matrices in the de�nition of an element matrix.

It should be noted that for isoparametric elements, the power rj is always 0 or 1, but for

reasons of generality, the presentation in this section covers any power rj, thereby making

the theory generally applicable to other types of elements, cf. Olho�, Rasmussen & Lund

(1992).

Before numerical derivatives of an element function are considered, let us de�ne the fol-

lowing standard �nite di�erence operators d(k)ajfor numerical di�erentiation of a function

f with respect to a design variable aj:

d(0)ajf = 0


d(1)ajf =

�f

�aj=

1

�aj(f(aj +�aj)� f(aj))

d(2)ajf =

1

2�aj(f(aj +�aj)� f(aj ��aj)) (5.2)

d(3)ajf =

1

6�aj(�f(aj + 2�aj) + 6f(aj +�aj)� 3f(aj)� 2f(aj ��aj))

d(4)ajf =

1

12�aj(�f(aj + 2�aj) + 8f(aj +�aj)� 8f(aj ��aj) + f(aj � 2�aj))

: : :

In Eq. 5.2, the symbol k in d(k)ajdesignates the order of the complete polynomial for which

the standard �nite di�erence operator d(k)ajyields the exact numerical derivative of the

function f .

An element function g as de�ned by Eq. 5.1 is now substituted into the usual �rst or-

der forward di�erence expression for numerical di�erentiation with respect to aj, i.e., the

formula for k = 1 in Eq. 5.2,

�g

�aj=

1

�aj(g(aj +�aj)� g(aj)) (5.3)

=1

�aj(pj + qj � (aj +�aj)

rj � pj � qj � (aj)rj) ; j = 1; : : : ; J

Then, in Eq. 5.2, set k equal to the value of rj associated with aj, i.e.,

@g

@aj= d(k=rj)aj

g; j = 1; : : : ; J (5.4)

and introduce the symbol �j for the relative perturbation of aj, i.e.

�j =�ajaj

> 0; j = 1; : : : ; J (5.5)

Substitute now g from Eq. 5.1 into the right hand side of Eq. 5.4, and consider the

formulas in Eq. 5.2 for f = g and k = rj = 0; 1; 2; : : :. For each value of rj, the following

proportional relationship is easily established between the analytical (and exact) derivative

@g=@aj and its �rst order �nite di�erence approximation �g=�aj:

@g

@aj= crj

�g

�aj; rj 2 @ [ f0g; j = 1; : : : ; J (5.6)

The proportionality factors crj are called correction factors and are found to depend only

on �j (for crj > 1)

c0 = 0

c1 = 1

c2 =�1 +

1

2�j

��1

c3 =�1 + �j +

1

3�2j

��1(5.7)

52 5.3. \Exact" Numerical Di�erentiation of Special Element Functions

c4 =�1 +

3

2�j + �2j +

1

4�3j

��1: : :

ck =

0@1k

k�1Xp=0

k

p

!�(k�1)�pj

1A�1

where the latter expression for ck is given in terms of binomial coe�cients.

∆∆

a ,...,a ,...,a

a

g

a

1 j J

j

j

∆

aj

∆

g= crj

g

aj

( )g

∂∂

Figure 5.1: Illustration of relationsship between the analytical derivative @g=@aj and its

�rst order �nite di�erence approximation �g=�aj.

In Fig. 5.1 this relationship of proportionality between the analytical derivative @g=@ajof an element function g and its �rst order �nite di�erence approximation �g=�aj is

illustrated.

The following important points concerning the above development should be noted:

� Eqs. 5.5 - 5.7 represent an e�cient and simple computational scheme for deter-

mining \exact derivatives" of element functions g by application of correction

factors to computationally inexpensive �rst order di�erences. Truncation errors

are completely avoided.

� The correction factors in Eq. 5.7 are independent of the actual values of the

design variables and can therefore be precomputed for a selected �nite value of

�j > 0. These values of the correction factors are then applicable in all future

sensitivity analyses, provided that the original value of �j is observed when

computing the value of �aj = �jaj to be used in Eq. 5.6.

Hence, using Eq. 5.5 the absolute perturbation �j can be eliminated in Eq. 5.6 which may

be rewritten as

@g

@aj= crj

�g

�aj=

crj�jaj

( g((1 + �j)aj)� g(aj) ) ; rj 2 @ [ 0; j = 1; : : : ; J (5.8)


where the value of �j > 0 has been preselected.

Eqs. 5.7 and 5.8 are the main expressions to be used when calculating derivatives of various

element matrices for isoparametric �nite elements.

5.4 3D Solid Isoparametric Finite Elements

In this section the method of \exact" numerical di�erentiation described in Section 5.3 is

used to determine \exact" numerical derivatives of element matrices and vectors for 3D

isoparametric hexahedral (brick) �nite elements with respect to shape design variables in

the form of nodal coordinates. These elements are described in Appendix A where all

element matrices and vectors are given. Local coordinates, domain, and nodal degrees of

freedom for these isoparametric �nite elements can be seen in Fig. 5.2.

x,u y,v

z,w

ξ

ηζ

Figure 5.2: Domain, node numbering, and nodal degrees of freedom of 3D isoparameric

�nite elements.

5.4.1 Derivative of Element Sti�ness Matrix

The element sti�ness matrix k is given by

k =ZBT EB jJj d (5.9)

Here, is the domain of the �nite element described in curvilinear, non-dimensional �-�-�

coordinates for the element, see Fig. 5.2, and jJj is the determinant of the Jacobian matrixJ which at each point de�nes the transformation of di�erentials d�, d�, and d� into dx,

dy, and dz. Like J, the strain-displacement matrix B depends on coordinates of the nodal

points, whereas the constitutive matrix E depends only on the constitutive parameters

of the assumed linearly elastic material. The expressions for the Jacobian J and for the

strain-displacement matrix B are given in the following.

54 5.4. 3D Solid Isoparametric Finite Elements

Let us recall that within the isoparametric formulation of a �nite element with an arbitrary

number n of nodal points, the same set of shape functions

Ni = Ni(�; �; �); i = 1; : : : ; n (5.10)

is used for interpolation of global x, y, and z coordinates from nodal values xi, yi, and ziand of displacements functions u, v, and w from nodal values ui, vi, and wi, i.e.,

x =nXi=1

Nixi; y =nXi=1

Niyi; z =nXi=1

Nizi (5.11)

u(x; y; z) =nXi=1

Niui; v(x; y; z) =nXi=1

Nivi; w(x; y; z) =nXi=1

Niwi (5.12)

Shape functions Ni; i = 1; : : : ; n, for the two implemented 3D solid isoparametric �nite

elements are given in Appendix A in Tables A.1 and A.2.

In terms of the vector di of nodal degrees of freedom

di = fui vi wigT ; i = 1; : : : ; n (5.13)

the element nodal vector d containing nodal displacements is

d = fdT1 dT2 : : :dTi : : : dTngT (5.14)

and the strain vector function " is

"(x; y; z) = f"x "y "z xy yz xzgT (5.15)

with their mutual relationship de�ned by

" = B d (5.16)

The strain-displacement matrix B is determined by operating on the shape functions Ni,

and it is found that

B = [ b1 b2 : : : bi : : : bn ] (5.17)

where the submatrix bi, which is associated with the nodal point i of the �nite element,

has the form

bi =

266666666664

Ni;x 0 0

0 Ni;y 0

0 0 Ni;z

Ni;y Ni;x 0

0 Ni;z Ni;y

Ni;z 0 Ni;x

377777777775; i = 1; : : : ; n (5.18)

Here, the derivatives of the shape functions Ni with respect to x, y, and z are given by8>><>>:Ni;x

Ni;y

Ni;z

9>>=>>; =

2664�;x �;x �;x

�;y �;y �;y

�;z �;z �;z

37758>><>>:Ni;�

Ni;�

Ni;�

9>>=>>; = �

8>><>>:Ni;�

Ni;�

Ni;�

9>>=>>; ; i = 1; : : : ; n (5.19)


where the matrix � is the inverse

� = J�1 (5.20)

of the Jacobian

J =

2664x;� y;� z;�

x;� y;� z;�

x;� y;� z;�

3775 =

nXi=1

2664Ni;�xi Ni;�yi Ni;�zi

Ni;�xi Ni;�yi Ni;�zi


3775 (5.21)

Note that J is expressed in terms of the derivatives of Ni; i = 1; : : : ; n, with respect to the

curvilinear element coordinates �, �, and � and of the coordinates (xi; yi; zi), i = 1; : : : ; n,

of each of the n nodal points of the �nite element.

Now all terms necessary for calculating the element sti�ness matrix in Eq. 5.9 are de�ned,

and in order to determine its derivative, Eq. 5.9 is di�erentiated with respect to any of

the shape design variables aj; j = 1; : : : ; J , leading to

@k

@aj=Z

"@BT

@ajEB+BT E

@B

@aj

#jJj d +

ZBT EB

@jJj@aj

d; (5.22)

j = 1; : : : ; J

Since the elasticity matrix E is symmetric, each of the two matrix terms in the �rst integral

in Eq. 5.22 is equal to the transpose of the other, and the matrix term in the second integral

is symmetric in itself. Application of this observation can reduce Eq. 5.22 to a more simple

expression. Introducing the notation [ ]S for the operation

[C]S =1

2

�CT +C

�(5.23)

of symmetrization of a quadratic matrix C, Eq. 5.22 can be rewritten as

@k

@aj=Z2

"BT E

@B

@aj

#S

jJj d +

"ZBT EB

@jJj@aj

d

#S

; j = 1; : : : ; J (5.24)

If a matrix B(j) is de�ned as

B(j) =@B

@aj+

B

2jJj@jJj@aj

(5.25)

then Eq. 5.24 can be written in compact form as

@k

@aj= 2

� ZBT E B(j) jJj d

�S

; j = 1; : : : ; J (5.26)

It is seen that Eq. 5.26 is of the same form as Eq. 5.9 so existing subroutines for com-

putation of k can be used to compute the derivative @k=@aj , provided that the strain-

displacement matrix B can be substituted by B(j) determined from Eq. 5.25 and the full

matrix k is calculated so that the symmetrization operation, Eq. 5.23, of the matrix in

Eq. 5.26 can be carried out.

However, in the following the form of Eq. 5.22 will be preferred instead of the compact

form of Eq. 5.26 as it makes it easier to follow the derivations given.

It is seen from Eq. 5.22 that the derivatives of the determinant jJj and of the components

in the strain-displacement matrix B must be determined before the derivatives @k=@aj ,

j = 1; : : : ; J , of the element sti�ness matrix can be calculated.


5.4.2 \Exact" Numerical Di�erentiation of jJj and B

In this section it will be shown that \exact" numerical derivatives of the Jacobian J and

of the strain-displacement matrix B can be obtained on the basis of \exact" numerical

di�erentiation by applying Eqs. 5.1, 5.7, and 5.8 in Section 5.3.

Note �rst that all the shape functions Ni; i = 1; : : : ; I, of the �nite element depend only

on the non-dimensional curvilinear coordinates �, �, and � within the element and thus

are independent of the actual geometry of the element. The shape functions and their

derivatives with respect to local coordinates are therefore independent of the shape design

variables aj; j = 1; : : : ; J .

Consider �rst the Jacobian matrix J in Eq. 5.21. The determinant jJj of this matrix can

be found as jJj = J11cof(J11) + J22cof(J22) + J33cof(J33), where cof denotes cofactor. It

is thereby seen that any coordinate x, y, or z will appear only linearly in the expression

for the determinant jJj. Thus, the scalar jJj will be either independent or a linear functionof any of the shape design variables aj, and the derivative of jJj can therefore be found

by applying Eqs. 5.8 where the correction factor crj = 1, cf. Eq. 5.7. The following

expression for the derivative @jJj=@aj ; j = 1; : : : ; J , is obtained:

@jJj@aj

=�J

�aj=

1

�jaj( jJ((1 + �j)aj)j � jJ(aj)j ) ; j = 1; : : : ; J (5.27)

The computation of derivatives of components of the strain-displacement matrixB requires

di�erentiation of bi with respect to aj and hence of the derivatives of Ni;x, Ni;y, and Ni;z,

see Eqs. 5.17, 5.18, and 5.19. This involves di�erentiation of the matrix �, and since the

components �qp of this matrix are given by �qp = jJj�1cof(Jpq), these components cannotbe di�erentiated exactly on the basis of a simple polynomial approximation. This di�culty

can be circumvented by di�erentiating the identity �J = I, where I is the identity matrix,

which gives@�

@aj= �� @J

@aj�; j = 1; : : : ; J (5.28)

From Eq. 5.21 it is seen that each of the components of the Jacobian matrix J is either

independent or a linear function of any of the shape design variables aj; j = 1; : : : ; J ,

from among the set of coordinates xi, yi, and zi of the element nodal points. Hence, the

derivative of each of the components in the Jacobian matrix J can be determined using

Eq. 5.8.

The derivative @B=@aj can be written as

@B

@aj=

"@b1@aj

@b2@aj

: : :@bi@aj

: : :@bn@aj

#(5.29)


where

@bi@aj

=

266666666666666666666664

@Ni;x

@aj0 0

0@Ni;y

@aj0

0 0@Ni;z

@aj@Ni;y

@aj

@Ni;x

@aj0

0@Ni;z

@aj

@Ni;y

@aj@Ni;z

@aj0

@Ni;x

@aj

377777777777777777777775

; i = 1; : : : ; n (5.30)

The derivatives of Ni;x, Ni;y, and Ni;z, i.e., the components in the strain-displacement

matrix B, can then be found using Eqs. 5.8, 5.19, and 5.28:

@

@aj

8>><>>:Ni;x

Ni;y

Ni;z

9>>=>>; =

@�

@aj

8>><>>:Ni;�

Ni;�

Ni;�

9>>=>>;

= �� @J@aj

�

8>><>>:Ni;�

Ni;�

Ni;�

9>>=>>; (5.31)

= �J�1 @J@aj

8>><>>:Ni;x

Ni;y

Ni;z

9>>=>>;

= �J�1 1

�jaj[ J((1 + �j)aj)� J(aj)]

8>><>>:Ni;x

Ni;y

Ni;z

9>>=>>; ;

i = 1; : : : ; n; j = 1; : : : ; J

Now all terms needed for the \exact" numerical derivative of the sti�ness matrix, cf. Eq.

5.22, are found using �rst order �nite di�erences, i.e., using \exact" semi-analytical design

sensitivity analysis.

In fact, only a comparatively small amount of additional work is required to derive the

corresponding expressions for analytical design sensitivity analysis. Such derivations can

be found in Wang, Sun & Gallagher (1985), El-Sayed & Zumwalt (1991), and Olho�,

Rasmussen & Lund (1992). These analytical expressions have not been implemented in

ODESSY because the method of \exact" numerical di�erentiation is just as accurate within

computational round-o� errors. Furthermore, as many di�erent kinds of generalized shape

design variables are available in ODESSY, it would take a lot of work to obtain analyti-

cal sensitivities of the relations between derivatives of generalized shape design variables

Am; m = 1; : : : ;M (governing positions of master nodes controlling the shape of, e.g.,

b-splines), and derivatives of shape design variables aj; j = 1; : : : ; J , which represent ele-

ment nodal coordinates. If analytical sensitivity analysis is not carried out in all evaluation

steps, then there is no reason to introduce it here, so we will stay within the semi-analytical


approach to design sensitivity analysis. In Section 5.7 it is described how derivatives with

respect to generalized shape design variables Am can be related to derivatives with respect

to shape design variables aj.

5.4.3 Stress Sensitivities

The stress components for a particular �nite element are assembled in the stress vector

function �(x; y; z) = f�x �y �z �xy �yz �xzgT which is given by

� = E�"� "

th�= E

�Bd� "

th�

(5.32)

in terms of the elasticity matrix E, strain-displacement matrix B, element nodal displace-

ment vector d, and possible initial thermal strains "th. These thermally induced strains

are given by

"th =


thyz

thxz

oT=nT T T 0 0 0

oT�; T = T � T0 (5.33)



strains.

The stress design sensitivities @�=@aj with respect to any shape design variable aj; j =

1; : : : ; J , becomes

@�

@aj= E

@B

@ajd+B

@d

@aj� @"th

@aj

!; j = 1; : : : ; J (5.34)

where@"th

@aj=

(@T

@aj

@T

@aj

@T

@aj0 0 0

)T

�; j = 1; : : : ; J (5.35)

The sensitivities @B=@aj are given by Eqs. 5.29, 5.30, and 5.31 and the element nodal

displacement sensitivities @d=@aj and the temperature sensitivities @T=@aj for the element

are known from the solutions of Eqs. 4.6 and 4.22, respectively.

The stress sensitivities can be calculated in a similar way for 2D solid isoparametric ele-

ments and isoparametric Mindlin plate and shell elements, so Eqs. 5.34 and 5.35 can be

adopted immediately for these elements.

5.4.4 Derivative of Element Mass Matrix

The consistent element mass matrix m is given by

m =Z%NT N jJj d (5.36)

Here, is the domain of the �nite element in its local coordinate system, see Fig. 5.2, % the

mass density, N contains shape functions Ni, and jJj is the determinant of the Jacobianmatrix J.


The derivative of the mass matrix can be found by di�erentiating Eq. 5.36 with respect

to any of the shape design variables aj; j = 1; : : : ; J , i.e.,

@m

@aj=Z%NT N

@jJj@aj

d; j = 1; : : : ; J (5.37)

The derivative of the determinant jJj of the Jacobian matrix is given by Eq. 5.27 so

all terms needed for calculating the \exact" numerical derivative of the mass matrix are

known.

5.4.5 Derivative of Element Initial Stress Sti�ness Matrix

Next derivatives of the element initial stress sti�ness matrix, also called element geometric

sti�ness matrix, are considered. In the derivation of element initial stress sti�ness matrices

it is convenient to reorder nodal degrees of freedom by introducing the element displace-

ment vector d�, where translational d.o.f. are reordered so that �rst all x-direction d.o.f.

are given, then y, and then z as follows

d� = fu1 u2 : : : ui : : : un v1 v2 : : : vi : : : vn w1 w2 : : : wi : : : wngT (5.38)

Relating d.o.f. to the reordered element vector d� the element initial stress sti�ness matrix

k� for the 3D isoparametric �nite elements is given by

k� =ZGT SG jJj d (5.39)


coordinates for the element, see Fig. 5.2, G a matrix obtained by appropriate di�erentia-

tion of shape functions Ni, S a matrix of initial stresses, and jJj is the determinant of theJacobian matrix J.

The matrix G is given by

G =

2664g 0 0

0 g 0

0 0 g

3775 (5.40)

where each submatrix g is given by

g =

2664Ni;x

Ni;y

Ni;z

3775 ; i = 1; : : : ; n (5.41)

The stress matrix S is given by

S =

2664s 0 0

0 s 0

0 0 s

3775 (5.42)

and each submatrix s is de�ned as

s =

2664�x �xy �xz

�xy �y �yz

�xz �yz �z

3775 (5.43)


Here �x, �xy, etc., are stresses found by an initial static stress analysis.

If Eq. 5.39 is di�erentiated with respect to any of the design variables aj; j = 1; : : : ; J ,

the following expression for the \exact" numerical derivative of the initial stress sti�ness

matrix is obtained

@k�@aj

=Z

"@GT

@ajSG +GT @S

@ajG +GT S

@G

@aj

#jJj d

+ZGT SG

@jJj@aj

d; j = 1; : : : ; J (5.44)

Thus, it is necessary to �nd the derivatives of the components in the stress matrix S, of

the determinant jJj of the Jacobian, and of the components of the matrix G.

The derivatives of the components in the stress matrix S, i.e., the stress sensitivities,

are given by Eq. 5.34, and the derivative of the determinant jJj is given by Eq. 5.27.

The matrix G contains the components Ni;x, Ni;y, and Ni;z, i.e., the same components as

involved in the de�nition of strain-displacement matrix B, see Eqs. 5.17, 5.18, 5.40, and

5.41. The derivatives of these components are given by Eq. 5.31. Thus, all terms necessary

for evaluating the \exact" numerical derivative of the element initial stress sti�ness matrix,

cf. Eq. 5.44, are now found.

5.4.6 Derivative of Thermal Element \Sti�ness Matrix"

The thermal element \sti�ness matrix" consists of contributions from the heat conduction

matrix kth given by

kth =ZBthT

�Bth jJj d (5.45)

Here, is the domain of the �nite element in its local coordinate system, see Fig. 5.2,

Bth a matrix obtained by appropriate di�erentiation of shape functions Ni, � the thermal

conductivity matrix, and jJj is the determinant of the Jacobian matrix J. If the material

is isotropic, � can simply be replaced by the scalar �, the conductivity coe�cient.

The matrix Bth is given by

Bth =hbth1 bth2 : : : bthi : : : bthn

i(5.46)

where the submatrix bthi , which is associated with the nodal point i of the �nite element,

has the form

bthi =

2664Ni;x

Ni;y

Ni;z

3775 ; i = 1; : : : ; n (5.47)

In case of boundary conditions in terms of convection heat transfer, the thermal \sti�ness

matrix" receives additional contributions given by the element matrix h

h =Z!2

NT hN jJj d! (5.48)

Here, !2 is the surface of the �nite element described in curvilinear, non-dimensional ��,� � �, or � � � coordinates for the element, for which the convection boundary condition


is applied. N contains shape functions Ni, h is the convection coe�cient speci�ed, and jJjis the determinant of the Jacobian matrix J for the surface !2.

If Eqs. 5.45 and 5.48 are di�erentiated with respect to any shape design variable aj; j =

1; : : : ; J , the following expressions for the derivatives of the thermal \sti�ness matrices"

are obtained

@kth

@aj=Z

24@BthT

@aj�Bth +BthT

�@Bth

@aj

35 jJj d +

ZBthT

�Bth @jJj@aj

d (5.49)

and@h

@aj=Z!2

NT hN@jJj@aj

d!; j = 1; : : : ; J (5.50)

The derivative of the determinant jJj is given by Eq. 5.27, and derivatives of the compo-

nents Ni;x, Ni;y, and Ni;z in the matrix Bth, see Eqs. 5.46 and 5.47, are given by Eq. 5.31.

All terms required for evaluating the \exact" numerical derivatives in Eqs. 5.49 and 5.50

are therefore found.

5.4.7 Derivative of Consistent Load Vector

In this section \exact" numerical derivatives will be found for the consistent element load

vector f which is given by

f =ZNT FB jJj d +

Z!

NT FS jJj d! (5.51)

where is the domain of the �nite element in its local coordinate system, FB represents

body forces, ! the surface described in curvilinear, non-dimensional � � �, � � �, or � � �

coordinates for the element at which surface forces FS are applied, and N contains shape

functions Ni. In the surface integral, N and jJj are evaluated on !.

If initial thermally induced strains have to be taken into account, the consistent nodal

force vector f th due to thermally induced strains is calculated as

f th =ZBT E "th jJj d (5.52)

where "th is an element vector containing thermally induced strains as given by 5.33.

Taking the same approach as previously, the load vectors in Eqs. 5.51 and 5.52 is di�eren-

tiated with respect to any of the design variables aj; j = 1; : : : ; J , leading to the following

expressions for the \exact" derivatives

@f

@aj=ZNT FB

@jJj@aj

d +Z!

NT FS

@jJj@aj

d!; j = 1; : : : ; J (5.53)

and

@f th

@aj=Z

"@BT

@ajE "th +BT E

@"th

@aj

#jJj d +

ZBT E "th

@jJj@aj

d; (5.54)

j = 1; : : : ; J

Here, the derivative @"th=@aj of the element thermally induced strains is given by Eq. 5.35,

@jJj=@aj is given by Eq. 5.27, and \exact" numerical derivatives of the strain-displacementmatrix B are given by Eqs. 5.29, 5.30, and 5.31.

62 5.5. 2D Isoparametric Finite Elements

5.4.8 Derivative of Consistent Thermal Flux Vector

Finally, \exact" numerical derivatives of the thermal ux vector will be determined for the

3D isoparametric �nite elements.

The consistent thermal nodal ux vector q is given by

q =Z!1

NT qS jJj d! +Z!2

NT h Te jJj d! (5.55)

where the �rst term derives from speci�ed ux at the surface !1 and the latter term from

a speci�ed convection boundary condition at surface !2. The surfaces !1, !2 are described

in curvilinear, non-dimensional � � �, � � �, or � � � coordinates for the element. The

scalar qS is prescribed ux normal to the surface !1, N contains shape functions Ni that

are evaluated on the surface !, jJj the determinant of the Jacobian matrix for the surface

!, h the convection coe�cient speci�ed, and Te is the environmental temperature speci�ed

for the convection boundary condition.

The derivative @q=@aj then can be determined by

@q

@aj=Z!1

NT qS@jJj@aj

d! +Z!2

NT h Te@jJj@aj

d! (5.56)

where @jJj=@aj is given by Eq. 5.27.

Now \exact" numerical derivatives have been determined for all implemented element

matrices and vectors of the 3D isoparametric �nite elements.

5.5 2D Isoparametric Finite Elements

Next, \exact" numerical derivatives of element matrices and vectors of the implemented 3-,

4-, 6-, 8-, and 9-node 2D isoparametric �nite are considered. These elements are formulated

for both plane stress, plane strain, and axisymmetric situations and they are described in

Appendix A where all element matrices and vectors are given. Local coordinates, domain,

and nodal degrees of freedom for these 2D isoparametric �nite elements can be seen in Fig.

5.3.

Shape functions for the implemented 3-, 4-, 6-, 8-, and 9-node 2D elements are given in

Apppendix B in Tables B.4, B.1, B.5, B.2, and B.3, respectively.

In case of triangular isoparametric elements, the interpolation functions are de�ned con-

veniently in terms of non-dimensional area coordinates �1, �2, and �3 within the element

as shown in Fig. 5.3. Only two of the three dimensionless area coordinates are mutually

independent, due to the area constraint relation

�1 + �2 + �3 = 1 (5.57)

If �1 and �2 are selected as independent coordinates the following relations are obtained

�1 = �; �2 = �; �3 = 1� � � � (5.58)


y,v

(z,w)

axis of rotationalsymmetry

x,u

ξ

η

ξξ

ξ

( )

3

1

2

(r,u)

Figure 5.3: Domain, node numbering, and nodal degrees of freedom of 2D solid iso-

parameric �nite elements. Local, non-dimensional coordinates �, � and area

coordinates �1, �2, �3 are shown for the quadrilateral and triangular elements,

respectively. Text in parantheses refer to standard notations for problems with

rotational symmetry.

Derivatives with respect to � and �, with the constraint in Eq. 5.57 taken into account,

can be found as@

@�=

@

@�1� @

@�3;

@

@�=

@

@�2� @

@�3(5.59)

The element matrices for the triangular elements can then be formulated similarly to the

quadrilateral elements by using Eqs. 5.57 - 5.59.



k =Z!

BT EB jJj d! (5.60)

Here, ! is the domain of the �nite element in its local coordinate system, see Fig. 5.3, and

jJj is the determinant of the Jacobian matrix J which at each point de�nes the transfor-

mation of di�erentials d� and d� into dx and dy. Like J, the strain-displacement matrix

B depends on coordinates of the nodal points, whereas the constitutive matrix E depends

only on the constitutive parameters of the assumed linearly elastic material.

For the plane stress and strain situations, d! = t d� d�, where t is the thickness of the

�nite element, and for axisymmetric structures we have d! = 2� r d� d�, where r is the

radius at the integration point. The axis of rotational symmetry is assumed to be parallel

with the y-axis as shown in Fig. 5.3.

The expressions for the Jacobian J and for the strain-displacement matrix B can be found

in a uniform way for plane stress, plane strain and axisymmetric �nite elements, if the

strain vector " is de�ned in the following way

"(x; y) = f"x "y xy "zgT (5.61)


In order to have the relations between standard notations for axisymmetric problems and

the notations used here, it should be noted that r = x, z = y, w = v, "r = "x, "z = "y,

rz = xy, and "� = "z as indicated on Fig. 5.3.

The element nodal vector d containing nodal displacements is given by

d = fdT1 dT2 : : :dTi : : : dTngT (5.62)

where the vector di of nodal degrees of freedom is

di = fui vigT ; i = 1; : : : ; n (5.63)

The relation between the strain vector " and the displacement vector d is

" = B d (5.64)

The strain-displacement matrix B can be found to have the following form

B = [ b1 b2 : : : bi : : : bn ] (5.65)


has the form

bi =

26666664

Ni;x 0

0 Ni;y

Ni;y Ni;x

Ni

r0

37777775 ; i = 1; : : : ; n (5.66)

The fourth row in bi is used only in case of an axisymmetric problem. The derivatives of

the shape functions with respect to x and y are given by(Ni;x

Ni;y

)=

"�;x �;x

�;y �;y

#(Ni;�

Ni;�

)= �

(Ni;�

Ni;�

); i = 1; : : : ; n (5.67)


� = J�1 (5.68)

of the Jacobian

J =

"x;� y;�

x;� y;�

#=

nXi=1

"Ni;�xi Ni;�yi

Ni;�xi Ni;�yi

#(5.69)

Now all terms necessary for calculating the element sti�ness matrix in Eq. 5.60 are de�ned.

The \exact" numerical derivative of the element sti�ness matrix is found, as before, by

di�erentiating Eq. 5.60 with respect to any of the shape design variables aj; j = 1; : : : ; J ,

and because the derivative for the axisymmetric case is di�erent from the plane stress or

strain situations, expressions are given for both situations. For the plane stress or strain

case we have:

@k

@aj=

Z Z "@BT

@ajEB+BT E

@B

@aj

#jJj t d� d� (5.70)

+Z Z

BT EB@jJj@aj

t d� d�; j = 1; : : : ; J (5.71)


and for the axisymmetric case:

@k

@aj=

Z Z "@BT

@ajEB+BT E

@B

@aj

#jJj 2� r d� d�

+Z Z

BT EB

"@jJj@aj

r + jJj @r@aj

#2� d� d�; j = 1; : : : ; J (5.72)

Hence, it is necessary to determine the derivative @B=@aj of the strain-displacement ma-

trix, the derivative @jJj=@aj of the determinant of the Jacobian matrix, and the derivative

@r=@aj of the radius r.

The derivative of the determinant of the Jacobian matrix J is given by Eq. 5.27 and the

derivatives of Ni;x and Ni;y needed for the evaluation of the derivative @B=@aj are given

by 5.31. However, for completeness of the derivations given here, Eqs. 5.29, 5.30, and 5.31

are rewritten to the 2D case.

The derivative @B=@aj can be written as

@B

@aj=

"@b1@aj

@b2@aj

: : :@bi@aj

: : :@bn@aj

#(5.73)

where

@bi@aj

=

266666666666664

@Ni;x

@aj0

0@Ni;y

@aj@Ni;y

@aj

@Ni;x

@aj@�Ni

r

�@aj

0

377777777777775; i = 1; : : : ; n (5.74)

The derivatives @Ni;x=@aj and @Ni;y=@aj are given by Eq. 5.31, i.e.,

@

@aj

(Ni;x

Ni;y

)= �J�1 1

�jaj[ J((1 + �j)aj)� J(aj)]

(Ni;x

Ni;y

); (5.75)

i = 1; : : : ; n; j = 1; : : : ; J

and the derivative of the fourth row in the submatrix bi can be found by applying Eqs.

5.1 and 5.8, i.e.,

@�Ni

r

�@aj

= Ni

@�1r

�@aj

=

8><>:�Ni

r2@r

@ajif aj is a x-coordinate

0 if aj is a y-coordinate(5.76)

Here it is assumed that the axis of rotational symmetry is parallel with the y-axis as shown

in Fig. 5.3.


Finally, the derivative @r=@aj , see Eq. 5.76 can be calculated as

@r

@aj=

�r

�aj=

1

�jaj(r((1 + �j)aj)� r(aj)) ; j = 1; : : : ; J (5.77)

Using Eqs. 5.27, 5.73, 5.74, 5.75, 5.76, and 5.77, the derivative @k=@aj of the element

sti�ness matrix k, cf. Eqs. 5.71 and 5.72, can now be determined.


Next \exact" numerical derivatives of the consistent element mass matrix m are derived.

The mass matrix m is given by

m =Z!

%NT N jJj d! (5.78)

Here, ! is the domain of the �nite element described in curvilinear, non-dimensional �� coordinates for the element, see Fig. 5.3, % the mass density, N contains shape functions

Ni, and jJj is the determinant of the Jacobian matrix J.

Taking the same approach as before, Eq. 5.78 is di�erentiated with respect to a design

variable aj leading to the following result for the plane stress or strain case:

@m

@aj=Z Z

%NT N@jJj@aj

t d� d� (5.79)


@m

@aj=Z Z

%NT N

"@jJj@aj

r + jJj @r@aj

#2� d� d�; j = 1; : : : ; J (5.80)

The derivative @jJj=@aj is given by Eq. 5.27 and the derivative @r=@aj is given by Eq.

5.77.


As in the case of 3D isoparametric �nite elements, it is convenient to de�ne the element

initial stress sti�ness matrix k� in terms of the reordered displacement vector d�

d� = fu1 u2 : : : ui : : : un v1 v2 : : : vi : : : vngT (5.81)

Relating d.o.f. to the reordered element vector d�, the element initial stress sti�ness matrix


k� =Z!

GT S G jJj d! (5.82)

Here, ! is the domain of the �nite element in its local coordinate system, see Fig. 5.3, G a

matrix obtained by appropriate di�erentiation of shape functions Ni, S a matrix of initial

stresses, and jJj is the determinant of the Jacobian matrix J.



G =

"g 0

0 g

#(5.83)


g =

"Ni;x

Ni;y

#; i = 1; : : : ; n (5.84)


S =

"s 0

0 s

#(5.85)

where each submatrix s is de�ned as

s =

"�x �xy

�xy �y

#(5.86)

Here �x, �xy, etc., are stresses determined by an initial static stress analysis.

Di�erentiating Eq. 5.82 with respect to any of the shape design variables aj; j = 1; : : : ; J ,

leads to the following expression for the \exact" numerical derivative in the plane stress

or strain case:

@k�@aj

=Z Z "

@GT

@ajS G +GT @S

@ajG +GT S

@G

@aj

#jJj t d� d�

+Z Z

GT S G@jJj@aj

t d� d�; j = 1; : : : ; J (5.87)


@k�@aj

=Z Z "

@GT

@ajS G +GT @S

@ajG +GT S

@G

@aj


+Z Z

GT SG

"@jJj@aj

r + jJj @r@aj

#2� d� d�; j = 1; : : : ; J (5.88)

The derivative @jJj=@aj is given by Eq. 5.27, the derivatives of the components in the

matrix G, see Eqs. 5.83 and 5.84, are given by Eq. 5.75, the sensitivities @S=@aj are given

by Eq. 5.34, and the derivative @r=@aj is given by Eq. 5.77.



matrix kth given by

kth =Z!

BthT�Bth jJj d! (5.89)

Here, ! is the domain of the �nite element described in curvilinear, non-dimensional �� coordinates for the element, see Fig. 5.3, Bth a matrix obtained by appropriate di�erenti-

ation of shape functions Ni, � the thermal conductivity matrix, and jJj is the determinant


of the Jacobian matrix J. If the material is isotropic, � can simply be replaced by the

scalar �, the conductivity coe�cient.



i(5.90)


has the form

bthi =

"Ni;x

Ni;y

#; i = 1; : : : ; n (5.91)



h =Z�2

NT hN jJj d� (5.92)

Here, �2 is the boundary of the �nite element described in curvilinear, non-dimensional �

or � coordinates for the element, for which the convection boundary condition is applied.

N contains shape functions Ni, h is the convection coe�cient speci�ed, and jJj is thedeterminant of the Jacobian matrix J for the boundary �2. For the plane stress and strain

situations, d� = t d�, where t is the thickness, and for axisymmetric structures we have

d� = 2� r d�, where r is the radius.

Di�erentiating Eqs. 5.89 and 5.92 with respect to a shape design variable aj; j = 1; : : : ; J ,

leads to the following expressions for the \exact" numerical derivatives for the plane stress

or strain case:

@kth

@aj=Z Z 2

4@BthT

@aj�Bth +BthT

�@Bth

@aj

35 jJj t d� d�+Z Z

BthT�Bth @jJj

@ajt d� d� (5.93)

and@h

@aj=Z�2

NT hN@jJj@aj

t d�; j = 1; : : : ; J (5.94)

For the axisymmetric case we obtain:

@kth

@aj=

Z Z 24@BthT

@aj�Bth +BthT

�@Bth

@aj

35 jJj 2� r d� d�

+Z Z

BthT�Bth

"@jJj@aj

r + jJj @r@aj

#2� d� d�; (5.95)

and

@h

@aj=

Z�2

NT hN

"@jJj@aj

r + jJj @r@aj

#2� d�; j = 1; : : : ; J (5.96)

Here, the derivative @jJj=@aj is given by Eq. 5.27, the derivative of the components in the

matrix Bth, see Eqs. 5.90 and 5.91, are given by Eq. 5.75, and the derivative @r=@aj is

given by Eq. 5.77.



Next, \exact" numerical derivatives will be derived for the consistent element load vector

f which is given by

f =Z!

NT FB jJj d! +Z�

NT FS jJj d� (5.97)

where ! is the domain of the �nite element in its local coordinate system, FB represents

body forces, � the boundary described in curvilinear, non-dimensional � or � coordinates

for the element at which boundary forces FS are applied, and N contains shape functions

Ni. In the boundary integral, N and jJj are evaluated on �.



f th =Z!

BT E "th jJj d! (5.98)

where "th is an element vector containing thermally induced strains, i.e.,

"th =

n"thx "thy thxy "

thz

oT=nT T 0 T

oT�; T = T � T0 (5.99)



strains.

Taking the same approach as previously, the load vectors in Eqs. 5.97 and 5.98 are di�eren-

tiated with respect to any of the design variables aj; j = 1; : : : ; J , leading to the following

expressions for the \exact" derivatives for the plane stress or strain case:

@f

@aj=Z Z

NT FB

@jJj@aj

t d� d� +ZNT FS

@jJj@aj

t d�; j = 1; : : : ; J (5.100)

and

@f th

@aj=

Z Z "@BT

@ajE "th +BT E

@"th

@aj

#jJj t d� d�

+Z Z

BT E "th@jJj@aj

t d� d�; j = 1; : : : ; J (5.101)

For the axisymmetric case we have:

@f

@aj=

Z ZNT FB

"@jJj@aj

r + jJj @r@aj

#2� d� d�

+ZNT FS

"@jJj@aj

r + jJj @r@aj

#2� d�; j = 1; : : : ; J (5.102)

and

@f th

@aj=

Z Z "@BT

@ajE "th +BT E

@"th

@aj


+Z Z

BT E "th"@jJj@aj

r + jJj @r@aj

#2� d� d�; j = 1; : : : ; J (5.103)

70 5.6. Isoparametric Mindlin Plate and Shell Finite Elements

Here, the derivative @"th=@aj of the element thermally induced strains is found by combin-

ing Eqs. 5.35 and 5.99, @jJj=@aj is given by Eq. 5.27, \exact" numerical derivatives of the

strain-displacement matrix B are given by Eqs. 5.73, 5.74, and 5.75, and the derivative

@r=@aj is given by Eq. 5.77.


Now \exact" numerical derivatives of the thermal ux vector q will be determined for the

2D isoparametric �nite elements.


q =Z�1

NT qS jJj d�+Z�2

NT h Te jJj d� (5.104)

where the �rst term derives from speci�ed ux at the boundary �1 and the latter term

from a speci�ed convection boundary condition at boundary �2. The boundaries �1, �2are described in curvilinear, non-dimensional � or � coordinates for the element. The

scalar qS is prescribed ux normal to the boundary �1, N contains shape functions Ni that

are evaluated on the boundary �, jJj the determinant of the Jacobian matrix J for the

boundary �, h the convection coe�cient speci�ed, and Te is the environmental temperature

speci�ed for the convection boundary condition.

The \exact" numerical derivative for the plane stress or strain case is given by:

@q

@aj=Z�1

NT qS@jJj@aj

t d�+Z�2

NT h Te@jJj@aj

t d�; j = 1; : : : ; J (5.105)


@q

@aj=

Z�1

NT qS

"@jJj@aj

r + jJj @r@aj

#2� d�

+Z�2

NT h Te

"@jJj@aj

r + jJj @r@aj

#2� d�; j = 1; : : : ; J (5.106)

Here, the derivative @jJj=@aj is given by Eq. 5.27 and the derivative @r=@aj is given by

Eq. 5.77.

5.6 Isoparametric Mindlin Plate and Shell Finite El-

ements

In this section the method of \exact" numerical di�erentiation is used to determine \exact"

numerical derivatives of element matrices and vectors for isoparametric Mindlin plate and

shell �nite elements. These elements are described in Appendix C where all element

matrices and vectors are given. The four main assumptions of the Mindlin plate theory,

which is a \thick" plate theory where transverse shear strains are accounted for, are also

given in Appendix C.


These Mindlin plate elements have in-plane membrane capability, where the elements are

formed by combining a plane membrane element, i.e., a 2D solid isoparametric element,

with a standard Mindlin plate bending element. These plate elements can be transformed

into at shell elements as described in Section C.10 in Appendix C. Using this approach of

generating at shell elements, all element matrices for the shell elements are established in

the local coordinate system for the Mindlin plate element, see Fig. 5.4, and all derivations

in this section are therefore only given for the Mindlin plate elements.

Local coordinates, domain, and nodal degrees of freedom for these isoparametric �nite

plate elements can be seen in Fig. 5.4.

z x

y

zw xu

v

y

ξ

η

ξξξ

ii

i1

32

i

θxi

θyi

Figure 5.4: Domain, node numbering, and nodal degrees of freedom of isoparametric

Mindlin plate �nite elements. Local, non-dimensional coordinates �, � and

area coordinates �1, �2, �3 are shown for the quadrilateral and triangular ele-

ments, respectively.

It is important to notice that all element matrices for the Mindlin elements are given in

terms of standard right-hand-rule rotations �x, �y as illustrated in Figs. 5.4. When the

Mindlin plate theory is derived it is normally done using the rotations �1, �2 which are

de�ned as�1 = ��y�2 = �x

(5.107)

The use of rotations �1, �2 greatly simplify the algebra when developing the Mindlin plate

theory but as the use of standard right-hand-rule rotations is most common in �nite element

programs, standard right-hand-rule rotations �x, �y are used here.

Shape functions for the implemented 3-, 4-, 6-, 8-, and 9-node Mindlin plate �nite elements

are similar to those used for the 2D solid elements, i.e., the shape functions given in

Apppendix B in Tables B.4, B.1, B.5, B.2, and B.3, respectively.

In case of triangular isoparametric elements, Eqs. 5.57, 5.58, and 5.59 in Section 5.5 are

used as relations between the non-dimensional coordinates �, � and area coordinates �1,

�2, �3.



The element sti�ness matrix k for a Mindlin plate �nite element is given by

k =Z!

BT DM B jJj d! (5.108)

where DM is the elasticity matrix, B the generalized strain-displacement matrix, jJj thedeterminant of the Jacobian matrix J, and ! is the domain of the �nite element described

in curvilinear, non-dimensional �� coordinates for the element as shown in Fig. 5.4. The

matrices J and B depend on the coordinates of the nodal points, whereas DM depends

only on the thickness t of the plate and on the constitutive parameters of the assumed

linearly elastic material. The elasticity matrix DM is given in Appendix C in Eqs. C.7,

C.8, and C.9.

In terms of the vector di of nodal degrees of freedom, cf. Fig. 5.4

di = fui vi wi �xi �yigT ; i = 1; : : : ; n (5.109)

the relationship between the vector d of generalized displacements of element nodal points

d = fdT1 dT2 : : :dTi : : : dTngT (5.110)

and the generalized strain vector " takes the standard form " = B d where B is given by

B = [ b1 b2 : : : bi : : : bn ] (5.111)

If the generalized strain vector " is de�ned as

"(x; y) =

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

u;x

v;y

u;y +v;x�1;x

�2;y

�1;y + �2;x

�2 � w;y

�1 � w;x

9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>;

=

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

u;x

v;y

u;y +v;x��y;x�x;y

��y;y + �x;x

�x � w;y

��y � w;x

9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>;

(5.112)

then the submatrix bi, which is associated with the nodal point i of the �nite element, has

the form

bi =

26666666666666664

Ni;x 0 0 0 0

0 Ni;y 0 0 0

Ni;y Ni;x 0 0 0

0 0 0 0 �Ni;x

0 0 0 Ni;y 0

0 0 0 Ni;x �Ni;y

0 0 �Ni;y Ni 0

0 0 �Ni;x 0 �Ni

37777777777777775

; i = 1; : : : ; n (5.113)


It is seen that the �rst three rows in bi originate from the membrane part, the next three

from the bending terms and the last two rows from the shear part. The derivatives of the

shape functions Ni with respect to x and y are given by Eq. 5.67 in Subsection 5.5.1.

Having de�ned the matrices involved in the element sti�ness matrix k, the \exact" numer-

ical derivative is, as before, found by di�erentiating Eq. 5.108 with respect to any shape

design variable aj; j = 1; : : : ; J , which leads to

@k

@aj=Z!

"@BT

@ajDM B+BT DM

@B

@aj

#jJj d! +

Z!

BT DM B@jJj@aj

d!; (5.114)

j = 1; : : : ; J

Here, the derivative @jJj=@aj is given by Eq. 5.27 and the derivative @B=@aj of the

generalized strain-displacement matrix B can be found using the results in Subsection

5.5.1. The derivatives @Ni;x=@aj and @Ni;y=@aj are given by Eq. 5.75, and as the shape

functions Ni only depend on the local, non-dimensional coordinates � and �, we have

@Ni

@aj= 0; j = 1; : : : ; J (5.115)

Using these equations the \exact" numerical derivative @k=@aj in Eq. 5.114 can now be

determined.



m =Z!

%NT N jJj d! (5.116)

Here, ! is the domain of the �nite element in its local coordinate system, % the mass

density, N contains shape functions Ni, and jJj is the determinant of the Jacobian matrix

J.

Di�erentiating Eq. 5.116 with respect to any of the shape design variables aj leads to

@m

@aj=Z!

%NT N@jJj@aj

d!; j = 1; : : : ; J (5.117)

where the derivative @jJj=@aj is given by Eq. 5.27.


In the derivation of element initial stress sti�ness matrices it is convenient to reorder

and omit some nodal degrees of freedom by introducing the reordered, condensed element

displacement vector d� that only contains translational degrees of freedom. These trans-

lational d.o.f. are reordered so that �rst all x-direction d.o.f. are given, then y, and then

z as follows

d� = fu1 u2 : : : ui : : : un v1 v2 : : : vi : : : vn w1 w2 : : : wi : : : wngT (5.118)


Relating d.o.f. to the reordered, condensed element vector d�, the element initial stress

sti�ness matrix k� for an isoparametric Mindlin plate �nite element is given by

k� =Z!

GT S G jJj d! (5.119)

where ! is the domain of the �nite element described in curvilinear, non-dimensional ��coordinates for the element, G a matrix obtained by appropriate di�erentiation of shape

functions Ni, S a matrix of initial membrane stresses, and jJj is the determinant of theJacobian matrix J.

The matrix G for the isoparametric Mindlin plate �nite element is given by

G =

2664g 0 0

0 g 0

0 0 g

3775 (5.120)

where each submatrix g is de�ned as

g =

"Ni;x

Ni;y

#; i = 1; : : : ; n (5.121)

The stress matrix S has the following form

S =

2664s 0 0

0 s 0

0 0 s

3775 (5.122)


s =

"�x �xy

�xy �y

#(5.123)

Here �x, �xy, etc., are membrane stresses in the plate found by an initial static stress

analysis.

Now, Eq. 5.119 is di�erentiated with respect to any of the design variables aj; j = 1; : : : ; J :

@k�@aj

=Z!

"@GT

@ajSG +GT @S

@ajG +GT S

@G

@aj

#jJj d!

+ZGT S G

@jJj@aj

d!; j = 1; : : : ; J (5.124)

Here, the derivatives of the components in the stress matrix S, i.e., the stress sensitivities,

are given by Eq. 5.34, and the derivative of the determinant jJj is given by Eq. 5.27. The

matrix G contains the components Ni;x and Ni;y, and the derivatives of these components

are given by Eq. 5.75. Using these equations, all terms necessary for evaluating the

\exact" numerical derivative of the element stress sti�ness matrix for an isoparametric

Mindlin plate �nite element, cf. Eq. 5.124, are now found.



Next, the \exact" numerical derivative of the thermal element \sti�ness matrix" is derived

for isoparametric Mindlin plate �nite elements. The thermal element sti�ness matrix

consists of contributions from the heat conduction matrix kth given by

kth =Z!

BthT�Bth jJj d! (5.125)

Here, ! is the domain of the �nite element in its local coordinate system, Bth a matrix

obtained by appropriate di�erentiation of shape functions Ni, � the thermal conductivity

matrix, and jJj is the determinant of the Jacobian matrix J. If the material is isotropic,

� can be simply replaced by the scalar �, the conductivity coe�cient. The matrix Bth is

given by Eqs. 5.90 and 5.91.



h =Z!2

NT hN jJj d! +Z�2

NT hN jJj t d� (5.126)

The convection boundary condition is applied to either the surface !2 or the boundary

�2 of the �nite element, both described in local coordinates. N contains shape functions

Ni, t is the thickness of the element, h the convection coe�cient speci�ed, and jJj is thedeterminant of the Jacobian matrix J for the surface !2 or the boundary �2.

Di�erentiating Eqs. 5.125 and 5.126 with respect to any shape design variable aj; j =

1; : : : ; J , leads to

@kth

@aj=

Z!

24@BthT

@aj�Bth +BthT

�@Bth

@aj

35 jJj d!

+Z!

BthT�Bth @jJj

@ajd! (5.127)

and@h

@aj=Z!2

NT hN@jJj@aj

d! +Z�2

NT hN@jJj@aj

t d�; j = 1; : : : ; J (5.128)

Here, the derivative of the determinant jJj is given by Eq. 5.27, and derivatives of the

components Ni;x and Ni;y in the matrix Bth, see Eqs. 5.90 and 5.91, are given by Eq. 5.75.

All terms required for evaluating the \exact" numerical derivatives in Eqs. 5.127 and 5.128

are therefore found.


The consistent element load vector f is given by

f =Z!

NT FB jJj d! +Z!

NT FS jJj d! +Z�

NT FS jJj t d� (5.129)



body forces, � is the boundary described in curvilinear, non-dimensional � or � coordi-

nates for the element, t the thickness, FS represents surface forces, and N contains shape

functions Ni. In the surface and boundary integrals, N and jJj are evaluated on ! and �,

respectively.



f th =Z!

BT E "th jJj d! (5.130)

where "th is an element vector containing thermally induced strains and it is given by Eq.

5.33, where "thz is omitted.

Di�erentiating Eqs. 5.129 and 5.130 with respect to any shape design variable aj; j =

1; : : : ; J , leads to

@f

@aj=Z!

NT FB

@jJj@aj

d! +Z!

NT FS

@jJj@aj

d! +Z�

NT FS

@jJj@aj

t d� (5.131)

and

@f th

@aj=Z!

"@BT

@ajE "th + "

th E@BT

@aj

#jJj d! +

Z!

BT E "th@jJj@aj

d!; (5.132)

j = 1; : : : ; J

Here, the derivative @"th=@aj of the thermally induced element strains can be found using

Eqs. 5.35, @jJj=@aj is given by Eq. 5.27, and \exact" numerical derivatives of the strain-

displacement matrix B are found by combining by Eqs. 5.111, 5.113, 5.75, and 5.115.


Finally, \exact" numerical derivatives of the thermal ux vector will be determined for the

isoparametric Mindlin plate �nite elements.


q =Z!1

NT qS jJj d! +Z�1

NT qS jJj t d�

+Z!2

NT h Te jJj d! +Z�2

NT h Te jJj t d� (5.133)

where the �rst two terms derive from speci�ed ux at either surface !1 or boundary �1and the two latter terms from speci�ed convection boundary conditions at surface !2 or

boundary �2. The surfaces !1, !2 and boundaries �1, �2 are described in curvilinear,

non-dimensional coordinates for the element, and t is the thickness of the element. The

scalar qS is prescribed ux normal to the surface !1 or the boundary �1, N contains shape

functions Ni that are evaluated on the surface ! or the boundary �, jJj the determinantof the Jacobian matrix J for the surface ! or the boundary �, h the convection coe�cient


speci�ed, and Te is the environmental temperature speci�ed for the convection boundary

condition.

Not surprisingly, we di�erentiate Eq. 5.133 with respect to any of the shape design variables

aj; j = 1; : : : ; J , leading to

@q

@aj=

Z!1

NT qS@jJj@aj

d! +Z�1

NT qS@jJj@aj

t d�

+Z!2

NT h Te@jJj@aj

d! +Z�2

NT h Te@jJj@aj

t d�; j = 1; : : : ; J (5.134)

Here, the derivative @jJj=@aj is given by Eq. 5.27.

Now \exact" numerical derivatives with respect to any shape design variable aj; j =

1; : : : ; J , have been found for element matrices and vectors of all the implemented isopara-

metric �nite elements.

5.7 Element Derivatives w.r.t. Generalized Shape De-

sign Variables

In this section it will be shown how the element derivatives obtained in Sections 5.4, 5.5,

and 5.6 with respect to shape design variables aj; j = 1; : : : ; J , which represent nodal

coordinates of the individual �nite element, can be associated with generalized shape

design variables Am; m = 1; : : : ;M , which may represent, e.g., positions of master nodes

controlling the shape of a Coons patch surface, a b-spline, etc. Di�erent modi�ers like

translation, scaling, and rotation types as described in Chapter 2 will be covered, and it

will be shown how to treat combinations of these modi�ers when they are linked to a single

shape design variable.

As described in Chapter 2, the coordinates of �nite element nodes are never used as

individual shape design variables as this may lead to a very large number of design variables

and other major drawbacks as poor convergence properties and di�culties in ensuring

compatibility and slope continuity between boundary nodes. Therefore, it is advantageous

to introduce a comparatively small number M of shape design variables Am; m = 1; : : : ;M ,

which, through a suitable set of shape functions for the design boundary, control the large

number of design variables ai; i = 1; : : : ; Is, which are all element nodal coordinates that

are selected as design variables. In the description given here it is assumed that the \design

boundary layer technique", where only �nite element nodes at the design boundary are

subjected to perturbation, see Fig. 4.1 in Section 4.4, is adopted. This implies that the

computation of element matrices at perturbed values of the design variables is as limited

as possible.

Introducing the set of generalized shape design variables Am; m = 1; : : : ;M , it is only

necessary to performM sensitivity analyses at a given step of redesign. For a static design

sensitivity analysis, for example, Eq. 4.6 can be rewritten in terms of any of the generalized

78 5.7. Element Derivatives w.r.t. Generalized Shape Design Variables

shape design variables Am; m = 1; : : : ;M :

K@D

@Am

= � @K

@Am

D +@F

@Am

; m = 1; : : : ;M (5.135)

which is restricted to the pseudo load vector � @K

@AmD+ @F

@Am.

Thus, instead of solving Eq. 4.6 for each pseudo load associated with a design variable

ai; i = 1; : : : ; Is, in the form of a nodal coordinate, the single pseudo loads are superposed

�a priori, resulting in a pseudo load vector that corresponds to the generalized shape design

variable Am. Mechanically, this may be conceived as the superposition principle, in terms

of pseudo loads and corresponding displacement sensitivities.

Although a particular Am may not be related to all ai; i = 1; : : : ; Is, for the sake of

generality it is assumed that Am = Am(a1; : : : ; aIs). Application of the chain rule then

yields@K

@Am

=IsXi=1

@K

@ai

@ai

@Am

; m = 1; : : : ;M (5.136)

with summation over all element nodal coordinates that are selected as design variables

ai; i = 1; : : : ; Is. The derivative @F=@Am is established in a similar way.

Consider now an element sti�ness matrix k in global coordinates of any of the �nite

elements of the discretized structure. The possible nodal coordinates of a particular element

that play the role of shape design variables are denoted by aj; j = 1; : : : ; J , and generally

constitute a small subset of the total set of shape design variables ai; i = 1; : : : ; Is, in terms

of nodal coordinates. Thus, the derivative @K=@ai in Eq. 5.136 can be calculated as

@K

@ai=Xnae

@k

@aj

��aj=ai

(5.137)

where nae is the number of active (perturbed) elements in the design boundary layer as

described in Section 4.4. Each of these elements may contain one or more of the parameters

ai; i = 1; : : : ; Is, as a nodal point coordinate design variable aj.

Eqs. 5.136 and 5.137 are applicable to all derivatives of element matrices and vectors

de�ned in the preceding sections, and the formulas for \exact" numerical di�erentiation

of element matrices and vectors are now applicable to generalized shape design variables

when the mesh sensitivities @ai=@Am have been established.

5.7.1 Boundary Shape Representation

In order to determine the derivatives @ai=@Am, a general description of the boundary shape

representation is given. This is done for the general case of a design boundary surface of

a solid structural domain. It is straight-forward to specialize from the general case to the

more commonly treated cases of boundary curves for planar structural domains.

Fig. 5.5 illustrates the design boundary surface, which may be conceived to be a part,

or the full surface, of a solid continuum structure. The shape of the boundary surface is

assumed to be controlled by M master nodes Sm; m = 1; : : : ;M , and the �nite element


x

y

z

S

S

Sx

x

x

y y

y

z

zz

Q Q

QT

T

T

T

11

1

1

tt t

t

m

1

M

Figure 5.5: Design boundary surface of solid body.

nodal points on the boundary surface are denoted by Qt; t = 1; : : : ; T . In the general

case, the surface may have been generated by some kind of mesh generator which do not

contain analytical descriptions of the surface representation. In ODESSY, unstructured

mesh generators may be used to generate the �nite elements on the surface, and this

mesh generation may include a smoothing process of the boundary surface nodes. It is

thereby not possible to have an explicit relation between the geometric design element

description and the �nite element nodes at the surface. Nevertheless, it will be assumed

that the design boundary can be described by some smooth shape interpolation functions

Rm, m = 1; : : : ;M , that are functions of some non-dimensional curvilinear coordinates �kthat are embedded in the surface, but explicit expressions for these shape interpolation

functions Rm are not available in the general case.

If mapping techniques are used to generate the design surface, it is possible to calculate

analytical derivatives @Rm(�k)=@Am. This has been done by, e.g., Yao & Choi (1989)

for Bezier surfaces, by Choi & Chang (1991) for geometric surfaces and other surface

representations, and for B-spline curves by Braibant & Fleury (1984), but it is not possible

in the general case to determine the derivatives of these shape interpolation functions

Rm(�k). This has also been realized by Botkin (1992) and Botkin, Bajorek & Prasad

(1992) in their approach of using feature-based structural design and fully automatic mesh

generation techniques in shape optimization.

Each of the master nodes Sm; m = 1; : : : ;M , which controls the shape of the design

boundary surface, can be characterized by the vector

som = fxom yom zomgT (5.138)

which contains the global coordinates of the initial position of the m-th master node Sm.

The coordinate vector c of any �nite element nodal point Qt; t = 1; : : : ; T , on the design


boundary surface can now be written symbolically as

c =MX

m=1

Rm(�k) [som + G(Am)] (5.139)

Here, the vector

c = fx y zgT (5.140)

contains the set of global coordinates for any �nite element point on the design boundary

surface and the vector G(Am) represents some transformation of the m-th master node

Sm as a function of the generalized shape design variable Am.

The master node transformation G(Am) may, in ODESSY, be controlled by

� translation modi�ers

� scaling modi�ers

� rotation modi�ers

� combinations of the three above-mentioned modi�ers

For simplicity, it is assumed here that only one shape design variable Am controls the posi-

tion of a master node Sm, but in reality one design variable may control many master nodes

or the position of a speci�c master node may depend on several shape design variables.

Now, let the set of shape design variables ai; i = 1; : : : ; Is, be associated with the design

boundary surface under study and assume, for reasons of generality, that the set ai; i =

1; : : : ; Is, comprises all xt, yt, and zt, t = 1; : : : ; T , coordinates of the �nite element nodal

points Qt; t = 1; : : : ; T , that belong to the design boundary as shown in Fig. 5.5.

The problem is now to establish the derivative of the coordinate vector, i.e., the mesh

sensitivities @ai=@Am for the implemented master node transformations G(Am).

Translation Modi�ers

If the master node transformationG(Am) represents a translation modi�er, the coordinate

vector c of any �nite element nodal point Qt; t = 1; : : : ; T , on the design boundary surface

can be written as

c =MX

m=1

Rm(�k) [som + Amnm] (5.141)

Here, nm is a unit vector in global coordinates that represents the translation direction,

i.e., the move direction for the m-th master node. The generalized shape design variable

Am represents the movement of the m-th master node in the direction nm relative to its

initial location given by som. This is illustrated in Fig. 5.6.

Now, from Eq. 5.141 it is seen that the coordinate vector c, and thereby the shape design

variables ai; i = 1; : : : ; I, I = 3T , depend linearly on the parameters Am; m = 1; : : : ;M .


x

y

z

S

S

Sx

x

x

y y

y

z

zz

Q Q

QT

T

T

T

11

1

1

tt t

t

m

1

M

n

nA1 1

1

A n

n

M M

M

A

n

nm m

m

Figure 5.6: Translation modi�ers for design boundary surface.

Hence, the mesh sensitivities @ai=@Am in case of a translation modi�er can be computed

by means of \exact" numerical di�erentiation using Eq. 5.8, with crj = 1; i.e.,

@ai

@Am

=�ai�Am

=1

�mAm

(ai((1 + �m)Am)� ai(Am)) ; (5.142)

i = 1; : : : ; I; I = 3T; m = 1; : : : ;M

Having determined the mesh sensitivities @ai=@Am, it is now possible to determine \exact"

numerical derivatives of element matrices and vectors with respect to generalized shape

design variables Am; m = 1; : : : ;M , cf. Eqs. 5.136 and 5.137, ifAm is linked to a translation

modi�er.

Scaling Modi�ers

Next, master node transformations G(Am) in terms of scaling modi�ers are considered. In

this case, the coordinate vector c of any �nite element nodal point Qt; t = 1; : : : ; T , on the

design boundary surface can be written as

c =MX

m=1

Rm(�k) [pm + (som � pm)(1 + Am)] (5.143)

Here, the generalized shape design variable Am represents the scaling of the distance be-

tween the m-th master node and the point Pm, which is characterized by the coordinate

vector pm. The value of Am is assumed to be equal to zero in the initial geometry. Scaling

modi�ers are illustrated in Fig. 5.7.


x

y

z

S

S

Sx

x

x

y y

y

z

zz

Q Q

QT

T

T

T

11

1

1

tt t

t

m

1

M

(1+ )1

smo

mp- )(

s1o

1p- )(

(1+ )m

sMo

Mp- )( (1+ )MP P

P

1 m

M

A

A

A

Figure 5.7: Scaling modi�ers for design boundary surface.

As in case of translation modi�ers, the coordinate vector c, and thereby the shape design

variables ai; i = 1; : : : ; I, I = 3T , depend linearly on the parameters Am; m = 1; : : : ;M ,

and Eq. 5.142 can be used to compute the mesh sensitivities @ai=@Am.

Rotation Modi�ers

In case of rotation modi�ers, here illustrated for the two-dimensional case but implemented

for the general three-dimensional case in ODESSY, the coordinate vector c of any �nite

element nodal point Qt; t = 1; : : : ; T , on the design boundary surface can be written as

c =MX

m=1

Rm(�k)hpm + (som � pm)

TT(Am)i

(5.144)

Here, the generalized shape design variable Am represents the rotation of the m-th master

node around the point Pm, which is characterized by the coordinate vector pm. Am denotes

the angle of rotation in the x � y plane, see Fig. 5.8, and T(Am) is the transformation

matrix given by

T(Am) =

2664

cos(Am) sin(Am) 0

�sin(Am) cos(Am) 0

0 0 1

3775 (5.145)

It is seen from Eqs. 5.144 and 5.145 that the coordinate vector c does not depend linearly

on the parameters Am; m = 1; : : : ;M , and furthermore, the method of \exact" numerical

di�erentiation cannot yield \exact" numerical derivatives of the transformation matrix

T. Alternatively, one could di�erentiate Eq. 5.144 and use analytical expressions for

the derivative of the transformation matrix T. However, such an evalution of derivatives

@c=@Am is very di�cult to implement in general in the mesh generation routines, and it


x

y

AS

m

m

boundary curvePm

s pm mo -( ) T ( )T

m

s pm mo -

z

A

Figure 5.8: Rotation modi�er for design boundary.

is therefore decided to use �rst order �nite di�erence approximations, i.e.,

@ai

@Am

� �ai�Am

; i = 1; : : : ; I; I = 3T; m = 1; : : : ;M (5.146)

Combinations of Modi�ers

If several modi�ers are linked to the same generalized shape design variable Am, and

thereby evaluated sequentially, it is chosen to use �rst order �nite di�erences to approxi-

mate the mesh sensitivities @ai=@Am, i.e.,

@ai

@Am

' �ai�Am

; i = 1; : : : ; I; I = 3T; m = 1; : : : ;M (5.147)

Eq. 5.147 may yield the \exact" numerical derivative, depending on the types of modi�ers

combined, cf. the preceding subsections.

In general, the derivatives @ai=@Am; m = 1; : : : ;M , are evaluated by the preprocessor using

the following steps:

84 5.8. Element Derivatives for Arbitrary Design Variables

1. Perturb the generalized design variable Am in question.

2. Call design-model-building routines which update the design model.

3. Call boundary surface remeshing routines to update �nite element mesh.

4. Subtract the new boundary surface mesh from the original mesh to obtain mesh

sensitivities @ai=@Am, cf. Eqs. 5.147.

Now the relations @ai=@Am, m = 1; : : : ;M , have been determined for the possible types

of generalized shape design variables Am implemented in ODESSY, and the new semi-

analytical method of design sensitivity analysis using \exact" numerical di�erentiation is

thereby applicable for all types of shape variables implemented.

5.8 Element Derivatives for Arbitrary Design Vari-

ables

The aim of this section is to describe how the new S-A method for design sensitivity

analysis can be applied for arbitrary types of design variables, i.e., shape as well as sizing

and material design variables. The description is given for 2D solid isoparametric �nite

elements but is easily applied to 3D solid and Mindlin plate and shell �nite elements.

Each of the design variables ai; i = 1; : : : ; I, can be divided into the three following groups:

1. Generalized shape design variables Am; m = 1; : : : ;M , which are transformed

to shape design variables in the form of nodal coordinates ai; i = 1; : : : ; Is.

2. Sizing design variables ai; i = 1; : : : ; It.

3. Material design variables ai; i = 1; : : : ; Im.

Element derivatives w.r.t. the �rst group containing shape design variables have been

covered in the preceding sections and the new S-A method for design sensitivity analysis

using \exact" numerical di�erentiation of element matrices for the two other types of design

variables will be described in the following.

5.8.1 Element Derivatives w.r.t. Sizing Design Variables

For a 2D solid isoparametric �nite element as described in Section 5.5, the thickness t

may be chosen as a sizing design variable ai; i = 1; : : : ; It, in case of plane stress or strain

situations.

As an example, take the sti�ness matrix k given by Eq. 5.60, where d! = t d� d�. The

derivative @k=@ai is easily computed as

@k

@ai=@k

@t=Z!

BT EB jJj d� d�; i = 1; : : : ; It (5.148)

The derivatives of other element matrices and vectors for 2D solid and Mindlin plate and

shell �nite elements w.r.t. sizing design variables ai; i = 1; : : : ; Is, in the form of element

thicknesses are easily established in a similar way.


5.8.2 Element Derivatives w.r.t. Material Design Variables

All types of material parameters like Young's modulus, mass density, conductivity co-

e�cients, shear modulus, and components of the constitutive matrix and orientation of

anisotropy for a composite material can be chosen as material design variables ai; i =

1; : : : ; Im. Using the same example as for sizing variables, the sti�ness matrix k given

by Eq. 5.60 for a 2D solid element depends on material parameters in the constitutive

matrix E, and choosing one of these material parameters as a material design variable, the

derivative @k=@ai is found as

@k

@ai=Z!

BT @E

@aiB jJj d!; i = 1; : : : ; Im (5.149)

where the derivative @E=@ai is easily computed analytically.

Derivatives of all other element matrices and vectors w.r.t. material design variables are

established using the same approach as above.

Now derivatives of all implemented element matrices and vectors have been calculated

in an accurate way for a variety of design variables. The approach of \exact" numerical

di�erentiation of element matrices is used for shape design variables whereas analytical

derivatives can be used for sizing and material design variables. The actual implementation

of the new S-A method is described in the following section.

5.9 Implementation and Numerical E�ciency of New

S-A Method

In this section the numerical e�ciency and the actual implementation of the new S-A

method are discussed. The description is exempli�ed by means of 3D solid �nite elements

and is similar for all other types of implemented elements. First derivatives w.r.t. shape

design variables are discussed as these are most complicated and then sizing as well as

material design variables are covered. The actual implementation of the new S-A method

is a comprimise between accuracy, numerical e�ciency, and ease of implementation so the

traditional S-A method using �rst order �nite di�erences is applied for some sensitivities if

it leads to improved e�ciency or easier implementation without signi�cant loss of accuracy.

In Chapter 6 several examples will demonstrate the accuracy of the new S-A method.

5.9.1 Derivatives w.r.t. Shape Design Variables

Derivative of Sti�ness Matrix

When the derivative of the global sti�ness matrixK is computed, it is done at the element

level as described by Eqs. 4.7 and 4.9, restricting the summation of element matrix deriva-

tives to the active (perturbed) �nite elements. The \design boundary layer" approach of

using one-element-deep sensitivity calculations for shape design variables is adopted.

86 5.9. Implementation and Numerical E�ciency of New S-A Method

For 3D solid elements, the derivative of the element sti�ness matrix k w.r.t. element shape

design variables aj; j = 1; : : : ; J , in the form of nodal coordinates is given by Eqs. 5.22

and 5.26, i.e.,@k

@aj= 2

� ZBT E B(j) jJj d

�S

; j = 1; : : : ; J (5.150)

where the operation [ ]S of symmetrization of a quadratic matrix, cf. Eq. 5.23, has been

employed and the matrix B(j) is de�ned by Eq. 5.25, i.e.,

B(j) =@B

@aj+

B

2jJj@jJj@aj

(5.151)

The element derivatives @k=@aj yield the \exact" numerical derivatives and they only

need to be calculated once for each possible perturbed nodal coordinate ai; i = 1; : : : ; Is,

of which aj; j = 1; : : : ; J , is a small subset. This leads to the following procedural steps for

establishing the derivatives @K=@Am in the design sensitivity analysis w.r.t. generalized

shape design variables Am; m = 1; : : : ;M :

1. If design sensitivity analysis w.r.t. �rst generalized shape design variable A1:

(a) The preprocessor evaluates all generalized shape design variables

Am; m = 1; : : : ;M , in order to determine the set ai; i = 1; : : : ; Is,

of possible perturbed nodal coordinates. Then it perturbs the vari-

able A1 and updates the surface mesh. Mesh sensitivities @ai=@A1

are then available, see Eqs. 5.147.

(b) The �nite element module evaluates all element derivatives @k=@ai,

i = 1; : : : ; Is, cf. Eqs. 5.137 and 5.150, and store them on disk.

(c) The global sti�ness matrix derivative @K=@A1 is evaluated using Eqs.

5.136, 5.137, and 5.147.

2. Else design sensitivity analysis w.r.t. Am; m = 2; : : : ;M :

(a) The preprocessor perturbs shape design variable Am leading to mesh

sensitivities @ai=@Am.

(b) The �nite element module evaluates the derivative @K=@Am based on

the element derivatives @k=@ai stored on disk and mesh sensitivities

@ai=@Am.

The perturbation �Am of a generalized shape design variable Am is set to 1=1000 of the

smallest side length of the �nite elements in the structure as described in Section 4.4 in

order to obtain accurate mesh sensitivities @ai=@Am. It should be noted that the mesh

sensitivities are \exact" numerical derivatives in case of translation or scaling modi�er

links to the generalized shape design variable Am, cf. Eq. 5.142.

The e�ciency of the new S-A approach to obtain the derivative @K=@Am of the global

sti�ness matrixK compared with the traditional method is very much problem dependent.


If the number M of generalized shape design variables Am is small, then the traditional

S-A method, where the element sti�ness matrix derivatives are approximated by �rst order

�nite di�erences �k=�Am for all nae perturbed elements, is more e�cient as the number

Is of element derivatives @k=@ai, see Eq. 5.150, involved in the new S-A method will be

substantially larger than nae .

However, increasing the number M of generalized design variables, the new S-A method

becomes more e�cient because the element derivatives @k=@ai only needs to be calcu-

lated once and the derivative @K=@Am is quickly evaluated from these element derivatives

whereas the traditional S-A method needs to evaluate �k=�Am for nae perturbed elements

for all design variables Am; m = 1; : : : ;M .

Furthermore, the ratio of e�ciency between the traditional and the new S-A method is

very much dependent on the computer used. The new S-A method needs to store more

element derivatives on disk than the traditional method, so the disk access speed is also

an important factor.

In general, for a few generalized shape design variables Am the new S-A method may be

twice as slow to determine the global sti�ness matrix derivative @K=@Am (or even slower

in special cases), but having a larger number of design variables, the two methods are

comparable in e�ciency and for a large number of generalized shape design variables, say

50, the new S-A method may be considerably faster.

For most problems, the traditional S-A method yields su�ciently accurate derivatives of

sti�ness matrices, but for many beam- and plate-like structures, it is absolute necessary

to use the new S-A method in order to obtain satisfactory results as will be demonstrated

by numerical examples in Chapter 6.

Derivatives of Mass Matrix and Thermal \Sti�ness Matrix"

The implementation of \exact" numerical derivatives for the global mass matrixM and the

global thermal \sti�ness matrix"Kth is very similar to the implementation for the sti�ness

matrix as described above. The description of the e�ciency of the new S-A method given

above can also be applied directly to these element derivatives but it should be noted that

the traditional S-A method based on �rst order �nite di�erences has worked well for these

matrix derivatives for all the design sensitivity analysis problems that I have studied.

Stress Sensitivities

The element stress sensitivities are calculated using \exact" numerical di�erentiation as

described in Subsection 5.4.3, see Eqs. 5.34 and 5.35, which are easily rewritten to the

situation of design sensitivity analysis w.r.t. a generalized shape design variable Am; m =

1; : : : ;M :@�

@Am

= E

@B

@Am

d+B@d

@Am

� @"th

@Am

!; m = 1; : : : ;M (5.152)


where@"th

@Am

=

(@T

@Am

@T

@Am

@T

@Am

0 0 0

)T

�; m = 1; : : : ;M (5.153)

The element nodal displacement vector d, the displacement sensitivities @d=@Am and the

temperature sensitivities @T=@Am are known from the solutions of Eqs. 4.2, 5.135, and

4.22, respectively, and the strain-displacement matrix B is given by Eqs. 5.17, 5.18, and

5.19. Finally, the derivative @B=@Am can be computed by application of the chain rule,

i.e.,@B

@Am

=IsXi=1

@B

@ai

@ai

@Am

; m = 1; : : : ;M (5.154)

The derivative @B=@Am will be non-zero only for elements in the perturbed \boundary

layer", i.e., elements that have nodal coordinates at the design boundary surface. Moreover,

sinceB is independent of all design variables other than those associated with the particular

�nite element, i.e., aj; j = 1; : : : ; J , then

@B

@Am

=

8>><>>:

JXj=1

@B

@aj

@aj

@Am

for elements in the \boundary layer"

0 for all other elements, m = 1; : : : ;M

(5.155)

where the mesh sensitivities @aj=@Am are available from Eq. 5.147. It should be noted that

the summation in Eq. 5.155 is only carried out over those shape design variables aj; j =

1; : : : ; J , that are associated with nodal points of the element on the design boundary

surface.

In this way \exact" numerical derivatives of stresses are obtained. This approach of ob-

taining stress sensitivities is slightly slower than the traditional S-A method because the

derivative @B=@Am must be established for elements in the \design boundary surface

layer", but the time it takes to perform these additional computations is not perceptible

in a design sensitivity analysis.

Derivatives of Load Vectors

Many numerical experiments have shown that usual �nite di�erence approximations of

the derivatives of the nodal load vector f , see Eqs. 5.51 and 5.52, and the thermal ux

vector q, see Eqs. 5.55, are su�ciently accurate, and therefore the expressions for the

derivatives of these element vectors using \exact" numerical di�erentiation are currently

not implemented. Furthermore, it has been easier reusing the traditional S-A method for

these computations when implementing the new S-A method in ODESSY.

Derivative of Initial Stress Sti�ness Matrix

There is a minor di�erence between calculating the derivative of the initial stress sti�ness

matrix and of the sti�ness matrix because part of the initial stress sti�ness matrix derivative

is dependent on the stress sensitivities calculated.


The \exact" numerical derivative of the element initial stress sti�ness matrix is given by

Eqs. 5.44, and using the operation [ ]S of symmetrization of a quadratic matrix, cf. Eq.

5.23, the \exact" numerical derivative @k�=@aj can be written as

@k�@aj

= 2� Z

GT S G(j) jJj d

�S

+ZGT @S

@ajG jJj d; j = 1; : : : ; J (5.156)

where the matrix G(j) is de�ned as

G(j) =@G

@aj+

G

2jJj@jJj@aj

(5.157)

It is seen that the �rst part of Eq. 5.156 only needs to be calculated once whereas the

second part needs to be evaluated for the each sensitivity analysis w.r.t. a generalized shape

design variable Am. The new S-A method is therefore substantially slower for calculating

the initial stress sti�ness matrix derivative than the traditional S-A method, but Eq. 5.156

always leads to accurate sensitivities. This will be exempli�ed by numerical examples in

Chapter 6.

5.9.2 Derivatives w.r.t. Sizing or Material Design Variables

The implementation of the new S-A method for sizing and material design variables is very

similar to the implementation for shape design variables as described above, and therefore

only the di�erences will be pointed out.

Analytical derivatives similar to Eqs. 5.148 and 5.149 in Section 5.8 are implemented for

derivatives of sti�ness, mass, thermal \sti�ness", and for initial stress sti�ness matrices

and stress sensitivities, whereas the traditional S-A method is applied for derivatives of

load vectors.

The analytical derivatives of element matrices are computed and stored on disk during the

design sensitivity analysis w.r.t. the �rst design variable ai as described in the preceding

subsection for calculation of \exact" numerical derivatives of element sti�ness matrices

w.r.t. shape design variables. These derivatives are calculated only once and then reused

for all other design variables, when necessary.

This approach to design sensitivity analysis is in most cases more e�cient than the tradi-

tional S-A method, because the element derivatives only need to be calculated once, and

they are faster to evaluate than the �rst order �nite di�erences used in the traditional S-A

method.

The actual implementation and the e�ciency of the new S-A method has now been dis-

cussed, and the accuracy of this approach to design sensitivity analysis will be demon-

strated by several examples in the following chapter.

Chapter

6

Examples of Design Sensitivity

Analysis

6.1 Introduction

In this chapter several examples of design sensitivity analysis are presented. The

examples are used to illustrate the implementation of the methods for design sensitivity

analysis that have been described in preceding Chapters 4 and 5. Some of the examples are

used to illustrate that the di�erent methods for design sensitivity analysis have been imple-

mented correctly in ODESSY while other examples are used to emphasize the importance

of using the new semi-analytical (S-A) method of design sensitivity analysis described in

Chapter 5 for certain types of problems.

In Section 6.2 the initial design for the classical problem of shape optimization of a planar

�llet is subjected to static design sensitivity analysis w.r.t. a generalized shape design

variable. Both the OFD method, the traditional S-A method, and the new S-A method

are shown to give accurate stress sensitivities for this problem.

Next the cantilever beam example introduced by Barthelemy & Haftka (1988) for study of

the inaccuracy problem associated with the traditional S-A method, as described in Section

5.2, is presented in Section 6.3. The results presented for this model problem illustrate the

accuracy of the new S-A method and demonstrate the shortcoming of the traditional S-A

method.

In Section 6.3 a two-material cantilever beam subjected to thermal loading is presented.

This example again witnesses inaccuracy problems for the traditional S-A method when

the displacement �eld is characterized by large rigid body rotations relative to actual

deformations of the �nite elements. This is also illustrated by a plate example in Section

6.5 where a thin plate clamped at all edges is subjected to a point load at the midpoint.

The next example in Section 6.6 concerns design sensitivity analysis of a plate reinforced

by sti�neners. The objective here is to illustrate that the sensitivity analysis of both simple

and multiple eigenfrequencies can be carried out in an accurate way based on the results

91

92 6.2. Static Design Sensitivity Analysis of Classical Fillet Problem

of Sections 4.9.1 and 4.9.2. In Section 6.7 the thickness of the reinforced plate is reduced,

resulting in a dense spectrum of eigenfrequencies. By this example it is illustrated how

di�cult it can be to decide from numerical results the multiplicity of a speci�c eigenvalue,

with the risk of computing erroneous sensitivities in case of a wrong estimation of the

multiplicity.

Finally, two plate examples presented in Sections 6.8 and 6.9 illustrate that the inaccuracy

problem associated with the traditional S-A method for static design sensitivity analysis

also manifests itself in sensitivity analysis of eigenvalues. The sensitivity analysis of free

transverse vibration frequencies of a thin clamped square plate is shown to give erroneous

results for the traditional S-A method, and the inaccuracy problem is even worse for

sensitivities of eigenfrequencies of a vibrating plate with in-plane loads, i.e., a problem

where initial stress sti�ening e�ects are taken into account.

Altogether, the examples in this chapter demonstrate the accuracy of the implemented

methods for design sensitivity analysis of static, thermo-elastic, and dynamic problems.

The examples also emphasize the superiority of the new S-A method to the traditional S-A

method for sensitivity analysis of beam- and plate-like structures.

6.2 Static Design Sensitivity Analysis of Classical Fil-

let Problem

The �rst example is used to illustate that both the OFD, the traditional S-A, and the

new S-A method have been implemented properly in ODESSY for static design sensitivity

analysis w.r.t. generalized shape design variables. The initial design for the classical

problem of shape optimization of a planar �llet in plane stress is used as a test example.

Fig. 6.1 shows schematically the loading and boundary conditions for the problem after use

has been made of symmetry conditions. The oblique �llet design boundary is modelled by

a quadratic B-spline controlled by �ve master nodes, each of which is assigned a translation

modi�er as shown in Fig. 6.1. A free mesh generation algorithm is used for discretization

of the structural domain and has divided it into 173 �nite elements. The mesh is a mixture

of 6- and 8-node isoparametric �nite elements for plane stress and the side length ` of the

smallest element is speci�ed to be 1 mm. The �llet has unit thickness and is modelled by

an isotropic material.

The sensitivity of the von Mises reference stress at the �nite element nodal point Q1 is

studied here with respect to perturbation of the generalized shape design variable A5 which

represents translation of the master node S5 in the direction n5, see Fig. 6.1.

With the discretization shown in Fig. 6.1, the von Mises reference stress �vM at the point

Q1 is 241.0 MPa, and the sensitivity ��vM=�A5 of this stress component is calculated by

the OFD method, by the traditional S-A method using �rst order �nite di�erence approxi-

mations for derivatives of element matrices and vectors, and by the new S-A method using

\exact" numerical di�erentiation of element matrices and vectors. The computationally

expensive OFD technique, cf. Eq. 4.1, is chosen as a reference method whose limit with

regard to design sensitivity accuracy is known to be set only by the solution procedure, the

Chapter 6. Examples of Design Sensitivity Analysis 93

S

S

Q

1

5

1

30

20

20100

45

100 MPa

this design boundary is modelled by a quadraticB-spline controlled by five master nodes

translation modifier is linked to master nodeS5n5

Figure 6.1: Classical �llet example.

discretization, and the usual analysis accuracy capabilities of the applied �nite element.

The perturbation �A5 is varied from 10�1 to 10�9 of the smallest side length ` of the �nite

elements in the structure and the results obtained are given in Table 6.1.

The stress sensitivities obtained by the three di�erent methods are seen to agree within

at least four digits when the relative perturbation �`=` of the boundary elements is less

than 10�2, whereas the sensitivities obtained by the OFD and the traditional S-A method

for larger perturbations di�er slightly due to the large perturbations used. The relative

perturbation is normally, as default in ODESSSY, set to 1=1000 of the smallest side length

` of the �nite elements in the structure as described in Section 4.4, i.e., for this example

�A5 would be set to 10�3 as default. All three methods are seen to give accurate results

for this perturbation, but it is worth noticing that the new S-A method is completely

independent of the perturbation used.

The present example indicates correct implementation of the three di�erent methods for

design sensitivity analysis of static problems with respect to generalized shape design

variables, but does not endow the new S-A method with priority over the traditional S-A

approach. This is because a problem has been considered in which the displacement �eld

is characterized by very small rigid body rotations relative to actual deformations of the

�nite elements, cf. the discussion in Section 5.2. It is very important to notice that under

such conditions, the traditional S-A method is completely reliable.

6.3 Static Design Sensitivity Analysis of Long Can-

tilever Beam

Consider next an example in which, still within the usual linear theory of elasticity, the

displacement �eld entails dominance of rigid-body rotation relative to actual deformation

94 6.3. Static Design Sensitivity Analysis of Long Cantilever Beam

Table 6.1: Computed von Mises reference stress sensitivity��vM=�A5 at the �nite element

nodal point Q1.

Boundary

element

length

perturbation Perturbation

OFD

method

Traditional

S-A method

New

S-A method

�`` �A5

��vM�A5

��vM�A5

��vM�A5

10�1

10�2

10�3

10�4

10�5

10�6

10�7

10�8

10�9

10�1

10�2

10�3

10�4

10�5

10�6

10�7

10�8

10�9

23:47

23:61

23:62

23:62

23:62

23:62

23:62

23:62

23:61

29:56

23:61

23:62

23:62

23:62

23:62

23:62

23:62

23:61

23:62

23:62

23:62

23:62

23:62

23:62

23:62

23:62

23:62

of the �nite elements in a subdomain of the structure. The example is the cantilever beam

problem used by Barthelemy & Haftka (1988) to study the inaccuracy problem associated

with the traditional S-A method as described in Section 5.2. The problem pertains to a

slender cantilever beam of given length L and aspect ratio 50 as shown in Fig. 6.2. The

beam is subjected to a given tip load P at the free end and is modelled by a regular pattern

of 8-node isoparametric serendipity �nite elements. The length of the element sides are

equal and denoted by `. The design sensitivity �vL=�L of the tip displacement vL with

respect to the length L of the beam is studied, and the \design boundary layer" approach

of using one-element-deep sensitivity information is adopted.

It has been demonstrated by numerical experiments in Barthelemy & Haftka (1988) that

the displacement design derivative �vL=�L determined by the traditional S-A method for

this beam problem is subject to severe inaccuracy problems. This fact is illustrated by

the results presented in Table 6.2 for di�erent values of the relative perturbation �`=` of

the lengths ` of the �nite elements in the \design boundary layer". The results are based

on a discretization of the beam into 200 x 4 eight-node isoparametric �nite elements as

indicated in Fig. 6.2, the beam has length L = 100 and unit thickness, the load P has

unit value, Young's modulus is set to 2.1�105 MPa, and Poisson's ratio = 0.3. Since the

beam is long, sensitivity results in Table 6.2 may be compared with analytical results for

@vL=@L stated in the caption of Table 6.2 for a corresponding Bernoulli-Euler beam.It is seen from the results in Table 6.2 that for values of the relative perturbation �`=` as

small as 10�5, even the sign is wrong for the sensitivities obtained by the traditional S-A


y,v

x,uv

P

L

L

Figure 6.2: Finite element model of long cantilever beam with aspect ratio 50.

Table 6.2: Computed displacement design sensitivities �vL=�L for cantilever beam mod-

elled by 200 x 4 eight-node isoparametric serendipity �nite elements for plane

stress.

(Computed displacement vL = 2.3807. Bernoulli-Euler comparison beam has vL = 2.3809

and @vL=@L = 0.1429.)

Boundary

element

length

perturbation

Beam length

perturbation

OFD

method

Traditional

S-A method

New

S-A method

�``

�LL

�vL�L

�vL�L

�vL�L

10�1

10�2

10�3

10�4

10�5

10�6

2:5 � 10�22:5 � 10�32:5 � 10�42:5 � 10�52:5 � 10�62:5 � 10�7

0:1429

0:1428

0:1428

0:1436

0:1523

0:2006

�3235:�324:8�32:25�3:096�0:17710:1344

0:1428

0:1428

0:1428

0:1428

0:1428

0:1428

method. Furthermore, for perturbations less than 10�3 the OFD method starts to diverge

due to truncation errors, showing that it is di�cult to obtain accurate sensitivities for this

beam problem.

As expected, sensitivities obtained by use of the new S-A method are independent of the

perturbation �`=` and attain values that agree within 0:02% with the sensitivity value of

the corresponding Bernoulli-Euler comparison beam. This illustrates the \exactness" of

the new S-A method, and furthermore, it demonstrates that the \design boundary layer"

approach of using one-element-deep sensitivity information gives accurate results.

96 6.4. Thermo-Elastic Sensitivity Analysis of Two-Material Beam

It should be noted that the traditional S-A method would yield more accurate sensitivities

if a second order central di�erence approximation of the sti�ness matrix derivative is used,

as the sensitivity error then is proportional to the square of the length perturbation. When

�rst order �nite di�erences are used the sensitivity error is linearly dependent on the length

perturbation. This can also be seen from Table 6.2. However, in case of both �rst and

second order �nite di�erence approximations, the sensitivity error for the traditional S-A

method for this problem will be proportional to the square of the number of elements used

to model the beam in the length direction. This has been shown by Pedersen, Cheng &

Rasmussen (1989) and Olho� & Rasmussen (1991a).

6.4 Thermo-Elastic Sensitivity Analysis of Two-Mate-

rial Beam

Next, a two-material cantilever beam subjected to thermal loading is considered. The

previous cantilever beam example illustrated the inaccuracy problem associated with the

traditional S-A method when the displacement �eld contains large rigid body rotations

relative to usual strains, and because such displacement �elds occur often in thermo-elastic

problems, inaccuracy problems can be expected for the traditional S-A method.

aluminium

steelinterface

y,v

x,u

v

L

L

= 373 K

= 573 K

T

T

Figure 6.3: Two-material cantilever beam with aspect ratio 25.

The cantilever beam consists of two material domains of equal heights and di�erent tem-

peratures are speci�ed at the upper and lower boundary of the beam as shown in Fig. 6.3.

The beam aspect ratio is 25 whereas the previous beam example in Section 6.3 had an

aspect ratio of 50. The length of the beam is set to L = 50, i.e., the height of the beam

is 2. As in the previous beam example, the derivative of the tip displacement, �vL=�L,

is studied for di�erent values of relative perturbations �`=` of the lengths ` of the �nite

elements in the \design boundary layer".

The two materials have the following properties:The �nite element model consists of 400 eight-node elements, giving the tip displacement

vL = 0:6714 due to the thermal loading. Displacement sensitivities �vL=�L are presented

in Table 6.3.

The results in Table 6.3 emphasize the possible need for using the new S-A method for

thermo-elastic design sensitivity analysis. The S-A method needs so small relative pertur-


Steel Aluminium

Young's modulus : 210000 MPa 70000 MPa

Poisson's ratio : 0.3 0.3

Conductivity coe�cient : 27 � 10�6 W/(mm2�K) 202 � 10�6 W/(mm2�K)Expansion coe�cient : 1:2 � 10�5 K�1 2:5 � 10�5 K�1

Table 6.3: Computed displacement design sensitivities�vL=�L for two-material cantilever

beam modelled by 100 x 4 eight-node isoparametric serendipity �nite elements

for plane stress. The beam aspect ratio is 25.

Boundary

element

length

perturbation

Beam length

perturbation

OFD

method

Traditional

S-A method

New

S-A method

�``

�LL

�vL�L

�vL�L

�vL�L

10�1

10�2

10�3

10�4

10�5

10�6

5:0 � 10�25:0 � 10�35:0 � 10�45:0 � 10�55:0 � 10�65:0 � 10�7

0:02686

0:02684

0:02684

0:02683

0:02658

0:02534

�310:3�31:05�3:081�0:2840�0:0043140:02292

0:02684

0:02684

0:02684

0:02684

0:02684

0:02684

bations as 10�6 just in order to obtain sensitivities of correct sign, and if a more slender

beam is studied, this error problem is increased. The OFD method produces accurate

results unless very small perturbations are used, and the new S-A method yields very ac-

curate results for all perturbations. It should be noted that in the current implementation

of the new S-A method in ODESSY, a �nite di�erence approximation is involved in the

calculation of the derivatives of the load vectors as described in Subsection 5.9.1, but this

approximation is seen not to a�ect the accuracy of the sensitivities obtained.

It should be noted that no inaccuracy problem has been detected in the thermal sensitivity

analysis computing temperature sensitivities and thereby thermally induced strain sensi-

tivities using the traditional S-A approach. The inaccuracy problem encountered here is

entirely associated with the displacement sensitivities due to a thermal load.

98 6.5. Static Design Sensitivity Analysis of Thin Clamped Square Plate

6.5 Static Design Sensitivity Analysis of Thin Clamp-

ed Square Plate

Next let us consider another example where the traditional method of S-A static design

sensitivity analysis may result in severe errors, cf. Barthelemy & Haftka (1988). The

example concerns a square plate clamped at all edges and subjected to a concentrated

load at the midpoint. As indicated in Fig. 6.4, one quarter of the plate is modelled by 100

4-node isoparametric Mindlin plate �nite elements.

P

L L

Figure 6.4: Finite element model of one quarter of square plate.

The length/thickness ratio is 100, and the sensitivity of the lateral displacement at the point

of load application with respect to the side length L is computed for di�erent perturbations.

The perturbations are con�ned to the elements adjacent to the lines of symmetry.

The results are presented in Table 6.4 and con�rm that the traditional S-A method is

strongly dependent on the size of the relative perturbation �L=L, whereas the new S-A

method gives accurate sensitivities. The inaccuracies of the OFD method for the pertur-

bations 10�1 and 10�2 are due to these large values.

Other numerical tests of this plate example with di�erent length/thickness ratios have

shown that, as expected, the errors of the traditional S-A method increase rapidly with

increasing length/thickness ratios as the displacement �eld becomes more and more charac-

terized by large rigid body rotations relative to actual deformations of the �nite elements.

6.6 Design Sensitivity Analysis of Reinforced Vibra-

ting Plate

Next a square plate reinforced by ribs as shown in Fig. 6.5 is considered. The aim of

this example is to illustrate how the results of Sections 4.9.1 and 4.9.2 can be used in

eigenfrequency design sensitivity analysis, that is, to show that sensitivities of both simple

and multiple eigenvalues can be computed accurately. The new S-A method constitutes

the basis for the design sensitivity analysis but the traditional S-A method may be applied

as well.


Table 6.4: Computed derivative of lateral displacement wmid at the point of load applica-

tion with respect to side length L. Length/thickness ratio is 100.

Plate length

perturbation

OFD

method

Traditional

S-A method

New

S-A method

�LL

�wmid

�L�wmid

�L�wmid

�L

10�1

10�2

10�3

10�4

10�5

10�6

10�7

10�8

0:6815

0:7997

0:8162

0:8179

0:8181

0:8181

0:8181

0:8181

�254:8�46:14�4:3040:3011

0:7663

0:8129

0:8176

0:8180

0:8181

0:8181

0:8181

0:8181

0:8181

0:8181

0:8181

0:8181

a a ab b

bb

aa

a verticalribs

horizontalribs

plate

Figure 6.5: Square plate reinforced by ribs.

The plate is clamped at all edges and reinforced by two horizontal and two vertical ribs.

The length a=0.5 m, b=0.05 m, the thickness of the ribs is 0.05 m, and the thickness of the

plate is 0.005 m. Both the plate and the ribs are made of steel with the following material

properties

Young's modulus = 210000 MPa

Poisson's ratio = 0.3

Mass density = 7800 Kg/m3

The structure is symmetric and therefore multiple eigenfrequencies are expected. The goal

100 6.6. Design Sensitivity Analysis of Reinforced Vibrating Plate

is to �nd sensitivities of both the simple and the multiple eigenfrequencies with respect to

6 di�erent design variables.

The �nite element model consists of 1156 4-node isoparametric Mindlin plate elements and

the model has 5445 d.o.f. All eigenvectors are M-orthonormalized, see Eq. 4.26, and the

4 lowest eigenfrequencies will be considered. From the analysis it is found that the lowest

eigenfrequency is simple and has the value 92.20 Hz, but the second and third eigenfre-

quency are identical and have the value 161.71 Hz. In this example, the eigenfrequencies

are considered to be identical when the relative di�erence between the values is � 10�4.

The fourth eigenfrequency is simple and equal to 175.03 Hz. The eigenmodes correspond-

ing to the 4 eigenfrequencies can be seen in Figs. 6.6 - 6.9, and the in uence of the ribs is

very clear.

As the second eigenfrequency is multiple, an in�nite number of linear combinations of the

eigenvectors shown in Figs. 6.7 and 6.8 corresponding to the multiple eigenfrequency will

satisfy the general eigenvalue problem in Eq. 4.24 and theM-orthonormalization condition

in Eq. 4.26.

The design sensitivities with respect to changes of single design variables are computed

by two di�erent methods, namely the overall �nite di�erence (OFD) method and the new

semi-analytical (S-A) method.

The OFD method which is used as a reference method implies that the design is perturbed,

a new eigenfrequency analysis is performed, and then the eigenfrequency sensitivities �jare found from

�j ' �fj(a1; : : : ; aI)

�ai=fj(a1; : : : ; ai +�ai; : : : ; aI)� fj(a1; : : : ; ai; : : : ; aI)

�ai(6.1)

The new semi-analytical (S-A) method is based on Eq. 4.29 for the simple eigenfrequencies

and Eq. 4.40 for the multiple eigenfrequencies. Eqs. 4.7, 5.114, 5.117, 5.136, 5.137, and

5.142 yield the \exact" numerical derivatives of sti�ness and mass matrices.

Furthermore, based on the S-A method we shall calculate the sensitivities of all four eigen-

frequencies regarding them as simple, i.e., using only Eq. 4.29 in order to see the conse-

quences of this erroneous assumption.


Figure 6.6: 1. eigenmode. f1 = 92.20 Hz.





The �rst design sensitivity analysis is with respect to the thickness of the plate as shown

in Fig. 6.10.

design variable no. 1:plate thickness

Figure 6.10: Design variable no. 1.

The sensitivities are shown in Table 6.5 and it is seen that the same results are obtained

by the OFD method and the S-A method using Eqs. 4.29 and 4.40.

Increasing the plate thickness is a symmetric design change and therefore we may expect

that the multiple eigenfrequency remains multiple. Note that while the sensitivities of

the other eigenfrequencies are positive then the sensitivity of the lowest eigenfrequency is

negative, i.e., the �rst eigenfrequency will decrease if the thickness of the plate is increased.

These results require some explanation.

The eigenmode corresponding to the lowest eigenfrequency can be characterized as a global

mode while all higher order eigenmodes can be regarded as nearly-local modes for each

of the nine subdomains of the plate, see Figs. 6.6 - 6.9. Thus, increasing the plate

thickness will have little e�ect on the overall sti�ness of the structure as it is mainly

governed by the ribs whereas the thickness of the plate has a large in uence on the total

mass, i.e., the inertia forces. Therefore, increasing the plate thickness will decrease the

lowest eigenfrequency which mainly depends on the overall sti�ness and total mass of the

structure.

The higher order eigenmodes, on the other hand, mainly depend on the local sti�ness and

mass of each of the nine subdomains, and increasing the plate thickness has a larger e�ect

on the local sti�ness than on the local mass of each subdomain. This is the reason why

the higher eigenfrequencies will increase with increasing plate thickness.

For this design change, the erroneous assumption of using Eq. 4.29 to the double eigenfre-


Table 6.5: Eigenfrequency sensitivities with respect to design variable no. 1: plate thick-

ness.

Frequency no. OFD method Eq. 4.29 & 4.40 Eq. 4.29

j�fj�a1

�j@fj@a1

1

2

3

4

�1552:14093:

14093:

31407:

�1552:14094:

14094:

31407:

�1552:14094:

14094:

31407:

quency, see last column in Table 6.5, in fact gives the same sensitivities as obtained by Eq.

4.40. This shows that the in uence of the o�-diagonal terms fT12e = fT21e in the sensitivity

matrix in Eq. 4.40 is very weak for this symmetric design change.

Next, the eigenfrequency sensitivities are found with respect to the thickness of the ribs

as shown in Fig. 6.11. The results are shown in Table 6.6.

design variable no. 2:rib thickness


Table 6.6: Eigenfrequency sensitivities with respect to design variable no. 2: rib thickness.


j�fj�a2

�j@fj@a2

1

2

3

4

1879:

1714:

1714:

304:7

1879:

1714:

1714:

305:1

1879:

1714:

1714:

305:1


It is seen in Table 6.6 that again the same results are obtained by the OFD method and

the S-A method using Eqs. 4.29 and 4.40. All the sensitivities are positive, and again the

multiple eigenfrequency remains multiple with this design change.

Next we want to determine sensitivities of the eigenfrequencies when the position of the

horizontal ribs is changed. The design variable is the distance between the horizontal ribs,

see Fig. 6.12.

design variable no. 3:position of horizontal ribs


The results are shown in Table 6.7.

Table 6.7: Eigenfrequency sensitivities with respect to design variable no. 3: position of

horizontal ribs.


j�fj�a3

�j@fj@a3

1

2

3

4

�40:86�380:6186:9

�169:5

�40:86�380:6186:9

�168:6

�40:86�287:5

93:75

�168:6

It is seen from Table 6.7 that the multiple eigenfrequency ~f = f2= f3 will split when the

distance between the horizontal ribs is increased. Now we can also see di�erences between

the two last columns in Table 6.7 showing the in uence of o�-diagonal terms in Eq. 4.40

which implies that calculation of sensitivities of the double eigenfrequency by means of the

single-modal formula in Eq. 4.29 gives erroneous results.

Next, the distance between the vertical ribs is used as a design variable as shown in Fig.

6.13, and the results are presented in Table 6.8.

The sensitivities for this design variable should be the same as obtained for design variable

no. 3, the distance between the horizontal ribs, except for that the sensitivities for the

multiple eigenfrequency f2 = f3 should be interchanged when using Eq. 4.40. It is seen

that the OFD method does not display this situation as the eigenfrequencies are ordered


design variable no. 4:position of vertical ribs



vertical ribs.


j�fj�a4

�j@fj@a4

1

2

3

4

�40:86�380:6186:9

�166:4

�40:86186:9

�380:6�167:0

�40:8693:75

�287:5�167:0

by magnitude in each analysis when using the OFD method. More importantly, we again

note that application of the single-modal Eq. 4.29 yields erroneous sensitivities of the

multiple eigenfrequency f2=f3.

The next design variable is the width of the horizontal ribs, see Fig. 6.14, and the results

are shown in Table 6.9.

design variable no. 5:width of horizontal ribs


Again, the results obtained by the OFD method and the S-A method using Eqs. 4.29

and 4.40 are very similar, and it is seen that the multiple eigenfrequency splits with this

design change. Furthermore, the sensitivities of the double eigenfrequency obtained by

using Eq. 4.29 even have a wrong sign, so using the erroneous assumption of regarding the


Table 6.9: Eigenfrequency sensitivities with respect to design variable no. 5: width of

horizontal ribs.


j�fj�a5

�j@fj@a5

1

2

3

4

�273:4�866:2

26:38

�401:4

�273:4�866:2

26:38

�400:9

�273:4�719:7�120:1�400:9

eigenfrequencies as simple leads to completely wrong results for this design change.

The last design variable is the width of the vertical ribs as shown in Fig. 6.15.

design variable no. 6:width of vertical ribs


Table 6.10: Eigenfrequency sensitivities with respect to design variable no. 6: width of

vertical ribs.


j�fj�a6

�j@fj@a6

1

2

3

4

�273:4�866:2

26:38

�399:5

�273:426:38

�866:2�399:8

�273:4�120:1�719:7�399:8

As before, the sensitivities with respect to this design variable should be the same as

obtained for design variable no. 5, the width of the horizontal ribs, except for that the


sensitivities for the multiple eigenfrequency f2=f3 should be interchanged when using the

S-A method and Eqs. 4.29 and 4.40.

Again it is seen that very similar results are obtained by the OFD method and the S-

A method when Eqs. 4.29 and 4.40 are used properly, whereas the last column again

witnesses shortcoming of Eq. 4.29 when applied to the double eigenfrequency f2=f3.

Up to now the eigenfrequency sensitivities have been found with respect to single design

changes of each of the 6 design variables thickness of plate, thickness of ribs, position of

horizontal ribs, position of vertical ribs, width of horizontal ribs, and width of vertical ribs.

Let us �nally show that it is possible to determine sensitivities for any direction in the

space of the 6 design variables when some of them are changed simultaneously. The

position of both the horizontal and vertical ribs, see Figs. 6.12 and 6.13, will be changed

simultaneously and again the OFD method is used as a reference. The two design variables

will be given unit increments.

All the generalized gradient vectors fsk in Eq. 4.46 have been calculated and Eq. 4.47 is

used for determining the increments �f =�f2 and �f=�f3 of the multiple eigenfrequency

f2=f3. Eq. 4.30 is used for determining increments of the simple eigenfrequencies f1 and

f4. The sensitivities for this simultaneous design change are shown in Table 6.11.

Table 6.11: Eigenfrequency sensitivities for unit increments of design variable no. 3: posi-

tion of horizontal ribs and design variable no. 4: position of vertical ribs.

Frequency no. OFD method Eq. 4.30 & 4.47

j �fj �fj

1

2

3

4

�81:72�193:8�193:8�335:5

�81:72�193:8�193:8�335:5

It is seen that very accurate results are obtained by using Eqs. 4.30 and 4.47 for determining

sensitivities of single and bimodal eigenfrequencies, respectively, in any direction in the

space of design parameters.

It has now been demonstrated that design sensitivities of simple as well as multiple eigen-

values with respect to single or simultaneous change of design variables can be calculated

very accurately using the S-A approach.

108 6.7. Ribbed Plate with Cluster of Eigenfrequencies

6.7 Ribbed Plate with Cluster of Eigenfrequencies

Next, we shall illustrate how important it is for the design sensitivity analysis to decide cor-

rectly from the numerical results whether some eigenvalues coalesce and become multiple,

or remain distinct.

We consider the same example as before, i.e., the square plate with ribs shown in Fig. 6.5,

but now the thickness of the plate is 212times less, i.e., 0.0020 m. This causes the 9 lowest

eigenfrequencies to become very close as their corresponding eigenmodes can be regarded

as local modes for each of the nine subdomains of the plate, see Figs. 6.16 - 6.25.


Figure 6.16: 1. mode. f1 = 70.40 Hz.

Figure 6.18: 3. mode. f3 = 72.17 Hz.

Figure 6.20: 5. mode. f5 = 72.28 Hz.

Figure 6.22: 7. mode. f7 = 72.32 Hz.

Figure 6.24: 9. mode. f9 = 72.32 Hz.

Figure 6.17: 2. mode. f2 = 72.17 Hz.

Figure 6.19: 4. mode. f4 = 72.25 Hz.

Figure 6.21: 6. mode. f6 = 72.31 Hz.

110 6.7. Ribbed Plate with Cluster of Eigenfrequencies

Figure 6.23: 8. mode. f8 = 72.32 Hz.

Figure 6.25: 10. mode. f10 = 110.0 Hz.


The �rst eigenfrequency is 70.404 Hz and the second through the ninth eigenfrequency are

close to 72.2 Hz. These 9 eigenfrequencies are so close that it is di�cult to decide which

of them are multiple.

If we use as a criterion for identical eigenfrequencies that the relative di�erence between

the frequencies must be � 10�3 then the 8 eigenfrequencies from the second to the ninth

should be considered as multiple, but if we use a tighter criterion as 10�4 then the second

and third should be considered as a double eigenfrequency and the sixth, seventh, eighth

and ninth should be considered as a 4-fold multiple eigenfrequency. If the criterion 10�5 is

used the second and third should be considered as a double eigenfrequency, and similarly

with the seventh and eighth.

We will determine sensitivities of the eigenfrequencies when the position of the horizontal

ribs is changed as shown in Fig 6.12. The sensitivities are calculated (a) by the OFD

method which is used as reference, (b) using Eq. 4.29 and 4.40 and assuming two double

eigenfrequencies, (c) using Eq. 4.29 and 4.40 and assuming one double and one 4-fold

multiple eigenfrequency, (d) using Eq. 4.29 and 4.40 and assuming one 8-fold multiple

eigenfrequency, and (e) only using Eq. 4.29 which is just valid in cases of simple eigenvalues.

The results are shown in Table 6.12.


horizontal ribs.

Freq. OFD Eq. 4.29 & 4.40, Eq. 4.29 & 4.40, Eq. 4.29 & 4.40 and Eq. 4.29

no. method f2=f3 and f2=f3 and f2=f3=f4=f5f7=f8 f6=f7=f8=f9 =f6=f7=f8=f9

j�fj�a5

�j �j �j@fj@ai

1

2

3

4

5

6

7

8

9

10

�203:2�257:1143:5

�71:65�116:8143:8

114:4

143:8

123:5

�13:60

�203:2�257:1143:5

�71:65�116:8143:7

114:4

143:8

123:5

�13:60

�203:2�257:1143:5

�71:65�116:8143:7

114:4

143:8

123:5

�13:60

�203:2�286:5143:5

78:36

�287:1143:7

143:7

143:8

143:8

�13:60

�203:2�256:8143:1

�71:65�116:8143:7

127:8

130:3

123:5

�13:60

It is seen that we obtain the same results by the S-A method assuming either two double

eigenfrequencies f2 = f3 and f7 = f8, or one double eigenfrequency f2= f3 and one 4-fold

multiple f6 = f7 = f8 = f9. These two columns are in excellent agreement with the OFD

112 6.8. Vibration Frequencies of Thin Clamped Square Plate

method but it is not a general situation that di�erent choices of multiplicity lead to the

same results. This is also illustrated by the next column in Table 6.12 because if we

assume having an 8-fold multiple eigenfrequency f2=f3=f4=f5=f6=f7=f8=f9, wrong

sensitivities are obtained for several of the eigenfrequencies, i.e., for f2, f4, f5, f7, and f9.

At last, if we consider all eigenfrequencies as simple, i.e., use Eq. 4.29, then the results

for the multiple eigenfrequencies f2 = f3 and f7 = f8 only di�er slightly from the results

obtained by the OFD method while the other sensitivities are correct. This illustrates that

the in uence of the o�-diagonal terms in the sensitivity matrix in Eq. 4.40 is quite small

for this design change. This is also the reason why the same sensitivities are obtained for

f6, f7, f8, and f9 using Eq. 4.40 independently of whether the assumption f7= f8 or the

assumption f6=f7=f8=f9 is used. In both cases, the o�-diagonal terms in the sensitivity

matrix in Eq. 4.40 are small compared to the diagonal terms for this design change, and

therefore the same sensitivities are obtained for both assumptions.

These results show the importance for the design sensitivity analysis of deciding correctly

whether an eigenfrequency is multiple or simple. Thus, wrong assumptions concerning the

multiplicity of an eigenfrequency may lead to erroneous results.

The importance of deciding correctly the multiplicity of a multiple eigenvalue can be seen

from the sensitivity matrix in Eqs. 4.40 and 4.46. If a wrong multiplicity N is assumed, a

subeigenvalue problem of wrong dimension is posed, and if the o�-diagonal terms fTske; s 6=k, in the sensitivity matrix are non-zero, the eigenvalues of the sensitivity matrix, i.e.,

the directional derivatives, in general will be incorrect. A structural design with a dense

spectrum of eigenfrequencies often occur for plate and shell structures, and special care

must be taken in the sensitivity analysis as observed in this example.

6.8 Vibration Frequencies of Thin Clamped Square

Plate

This example deals with the subject of semi-analytical design sensitivity analysis of vibra-

tion frequencies of a thin clamped square plate, and it will be shown that application of

the traditional S-A method can result in severe errors. The example concerns a square

plate which is clamped at all edges and modelled by 100 9-node \heterosis" isoparametric

Mindlin plate elements, see Fig. 6.26.

L L

Figure 6.26: Finite element model of square plate.


The plate is made of steel with the following material properties

Young's modulus : 210000 MPa

Poisson's ratio : 0.3

Mass density : 7800 Kg/m3

114 6.8. Vibration Frequencies of Thin Clamped Square Plate

The side length L of the square plate is 1:0 m and the thickness 1:0 � 10�3 m, i.e. the

length/thickness ratio is 1000. The lowest eigenfrequency f1 = 8.994 Hz is simple, while

f2=f3=18:36 Hz is a double.

The shape design variable is the side length L of the plate, and the sensitivities of the three

lowest eigenfrequencies will be computed for di�erent perturbations. The overall �nite

di�erence (OFD) method, see Eq. 6.1, is used as a reference method, and furthermore, the

eigenfrequency sensitivities will be calculated by the traditional S-A method, cf. Eqs. 4.7

and 4.8, and by the new semi-analytical method using \exact" numerical di�erentiation of

element matrices.

Table 6.13: Computed derivative of lowest eigenfrequency f1 with respect to side length

L.

Plate length

perturbation

OFD

method

Traditional

S-A method

New

S-A method

�LL

�f1�L

�f1�L

�f1�L

10�2

10�3

10�4

10�5

10�6

10�7

10�8

10�9

10�10

�17:71�17:96�17:99�17:99�17:99�17:99�18:00�17:97�18:13

12188:

1308:

115:7

�4:605�16:65�17:86�17:98�17:99�17:97

�17:99�17:99�17:99�17:99�17:99�17:99�17:99�17:99�17:99

The data presented in Table 6.13 are the sensitivities of the lowest eigenfrequency while

Table 6.14 presents sensitivities of the multiple eigenfrequency. As f2 and f3 remain

multiple with the design change considered, the same values are obtained for sensitivities

�2 and �3, and therefore only results for �2 are listed in Table 6.14.

The results in Table 6.13 and 6.14 con�rm that the traditional S-A method is strongly

dependent on the size of the relative perturbation, and that even the sign of the sensitivity

is wrong unless a very small perturbation is used. The new S-A method, however, yields

accurate sensitivities. The OFD method is used as a reference, and it should be noted

that the inaccuracies of the OFD method for the perturbations 10�1 and 10�2 are due

to these large values. Furthermore, the OFD method becomes inaccurate for very small

perturbations due to numerical round-o� errors.

The traditional S-A method for static design sensitivity analysis is prone to large errors for

problems involving large rigid body rotations relative to actual deformations of the �nite


Table 6.14: Computed sensitivities �2(=�3) of double frequency f2 = f3 with respect to

side length L.

Plate length

perturbation

OFD

method

Traditional

S-A method

New

S-A method

�LL �2 �2 �2

10�2

10�3

10�4

10�5

10�6

10�7

10�8

10�9

10�10

�36:10�36:63�36:71�36:71�36:71�36:72�36:72�36:80�36:19

19415:

2074:

176:2

�15:40�34:58�36:50�36:69�36:71�36:70

�36:71�36:71�36:71�36:71�36:71�36:71�36:71�36:71�36:71

elements as the components of the approximate element pseudo loads that correspond to

to a rigid body rotation dr do not vanish in general, i.e., �(�k=�ai)dr 6= 0 as described

in Section 5.2. As the design sensitivity expressions for eigenvalues involve multiplication

of sti�ness matrix derivatives by the eigenvector, see Eqs. 4.29 and 4.40, the same type of

inaccuracy problem will arise when the S-A method is applied to design sensitivity analysis

of eigenvalues for such structures.

In the current example, the small length/thickness ratio will result in eigenvector dis-

placement �elds where the rigid body rotations are comparatively large relative to the

actual deformations of the �nite elements, and therefore the traditional S-A method yields

completely wrong results unless very small perturbations are used.

6.9 Eigenfrequencies of Vibrating Square Plate with

In-plane Loads

The next example concerns design sensitivity analysis of a square plate which is clamped

at one edge and simply supported at two opposite edges. The plate is subjected to a

uniformly distributed in-plane load of magnitude 200 N/m at the fourth edge. The �nite

element model consists of 100 9-node \heterosis" isoparametric Mindlin plate elements, see

Fig. 6.27.

The plate has the same dimensions and material properties as the plate in the previous

example in Section 6.8.

116 6.9. Eigenfrequencies of Vibrating Square Plate with In-plane Loads

LL

200 N/m

clamped

simply supported

simply supported

Figure 6.27: Finite element model of square plate with in-plane load.

Table 6.15: Computed derivative of lowest eigenfrequency f1 with respect to side length L

for plate without in-plane load. Plate length/thickness ratio is 1000.

Plate length

perturbation

OFD

method

Traditional

S-A method

New

S-A method

�LL

�f1�L

�f1�L

�f1�L

10�2

10�3

10�4

10�5

10�6

10�7

10�8

10�9

10�10

�6:231�6:315�6:324�6:325�6:325�6:320�6:271�4:948�8:880

73253:

7951:

796:3

7:401

1:709

�5:521�6:184�5:804�6:837

�6:325�6:325�6:325�6:325�6:325�6:325�6:325�6:325�6:325

For comparison purposes it is �rst assumed that the in-plane load is absent and then the

lowest frequency is obtained as f1 = 3.167 Hz. The shape design variable is again taken

to be the side length L of the plate, and the sensitivities of the lowest eigenfrequency are

given in Table 6.15.

It is seen from Table 6.15 that there is a good agreement between the results obtained

by the OFD method and the new S-A method for the perturbations 10�4 - 10�6, but for

smaller values of the perturbation the OFD method starts to diverge due to numerical

round-o� errors. The traditional S-A method starts to diverge for perturbations smaller

than 10�8 and does not reach the same sensitivity value as the two other methods.

The errors in the sensitivities obtained by the traditional S-A method are seen to be larger

for this example than for the previous one. This is to be expected as the rigid body


rotations relative to actual deformations of the �nite elements are larger for this example

due to the boundary conditions considered.

Now, consider the case where the plate is subjected to the in-plane load. This implies a

lowering of the �rst eigenfrequency from 3.167 Hz to 2.690 Hz. Again design sensitivities

are calculated, and the results are given in Table 6.16.


for plate with in-plane load = 200 N/m. Plate length/thickness ratio is 1000.

Plate length

perturbation

OFD

method

Traditional

S-A method

New

S-A method

�LL

�f1�L

�f1�L

�f1�L

10�2

10�3

10�4

10�5

10�6

10�7

10�8

10�9

10�10

�6:511�6:583�6:597�6:598�6:598�6:570�6:322�5:477�0:384

103053:

11180:

1122:

106:4

4:714

�5:451�6:383�5:847�7:325

�6:587�6:587�6:587�6:587�6:587�6:587�6:587�6:587�6:587

This example emphasizes the importance of using our modi�ed version of the S-A method.

The new S-A method is independent of the perturbation used, whereas a comparison of

the results obtained by the OFD method and the traditional S-A method for this example

re ects di�culties in obtaining reliable sensitivities. This is due to both comparatively

larger rigid body rotations relative to actual deformations of the �nite elements, and the

necessity of performing two design sensitivity analyses. First the stress sensitivities are

determined in a static sensitivity analysis and then the eigenfrequency sensitivities are

calculated including the initial stress sti�ening e�ects. Inaccuracies in the static sensitivity

analysis hereby become accumulated in the dynamic sensitivity analysis and this is the

reason why there are small di�erences in the results obtained by the OFD method and

the new S-A method. The OFD method is based on �rst order forward �nite di�erences,

and as the problem is strongly non-linear this gives small inaccuracies although small

perturbations are used.

Finally, the same example is considered but now the thickness is increased to 20:0 � 10�3m, i.e. the length/thickness ratio is 50, and the in-plane load is set to 1:5 � 106 N/m.Without the in-plane load the lowest eigenfrequency becomes f1 = 62.97 Hz, and with the

in-plane load included, the lowest eigenfrequency f1 is reduced to 53.93 Hz. The shape

118 6.10. Concluding Remarks on Examples

design variable is still taken to be the side length L of the plate and the design sensitivities

obtained are given in Table 6.17.


for plate with in-plane load = 1:5 � 106 N/m. Plate length/thickness ratio is

50.

Plate length

perturbation

OFD

method

Traditional

S-A method

New

S-A method

�LL

�f1�L

�f1�L

�f1�L

10�2

10�3

10�4

10�5

10�6

10�7

10�8

10�9

10�10

�127:9�129:5�129:6�129:7�129:7�129:7�129:7�129:7�131:1

5024:

430:0

�72:93�123:7�128:8�129:3�129:4�129:4�129:5

�129:5�129:5�129:5�129:5�129:5�129:5�129:5�129:5�129:5

For this example, there is seen to be good agreement between the OFD method and the

new S-A method. The traditional S-A method is seen to converge to the same value as

obtained by the other two methods but it is still dependent of the chosen value of the

perturbation. The sensitivities obtained by the traditional S-A method are much better

for this example than for the previous one, because the length/thickness ratio has been

decreased. This results in smaller rigid body rotations relative to actual deformations of the

�nite elements, and therefore the errors associated with the fact that �(�k=�aj)dr 6= 0

are reduced.

6.10 Concluding Remarks on Examples

Several numerical examples of design sensitivity analysis have been presented in this chap-

ter. The traditional semi-analytical method of design sensitivity analysis has been shown

to give very accurate results in cases where the displacement �eld is characterized by small

rigid body rotations relative to actual deformations of the �nite elements, cf. Section 6.2

where a classical �llet example is studied.

However, in cases where the displacement �eld entails dominance of rigid body rotations rel-

ative to actual deformations of the �nite elements, the traditional semi-analytical method


has been shown to be prone to yield erroneous sensitivities. This is demonstrated for static

design sensitivity analysis of a long cantilever beam in Section 6.3 and of a thin clamped

square plate in Section 6.5. Furthermore, the inaccuracy problem has been demonstrated

to manifest itself in thermo-elastic design sensitivity analysis of a two-material cantilever

beam in Section 6.4. Thus, the inaccuracy problem can also be expected in many thermo-

elastic design problems as displacement �elds with dominance of rigid body rotations often

occur in such problems.

Design sensitivity analysis of simple as well as multiple eigenfrequencies of reinforced vi-

brating plates has been shown to give accurate results in Sections 6.6 and 6.7. One of the

major di�culties in computing sensitivities of multiple eigenvalues is to decide correctly

from numerical results the multiplicity of a given eigenvalue in case of a dense spectrum

of eigenvalues. A wrong estimation of the multiplicity of a repeated eigenvalue is shown

to give erroneous sensitivities in Section 6.7 and this di�culty associated with multiple

eigenvalues, in addition to the lack of usual Fr�echet di�erentiability, has to be taken into

account when solving optimum design problems involving multiple eigenvalues. These

questions will be further discussed in Chapter 8.

Finally, the inaccuracy problem associated with the traditional semi-analytical method is

shown to occur also in dynamic design sensitivity analyses. This is illustrated by com-

putation of design sensitivies of simple as well as multiple free vibration frequencies of a

thin clamped square plate in Section 6.8. Furthermore, if initial stress sti�ening e�ects

are taken into account when computing eigenfrequencies of vibrating plates, in Section 6.9

the error problem is shown to be even worse because errors in stress sensitivities from the

static design sensitivity analysis become accumulated in the dynamic design sensitivity

analysis.

The new approach to semi-analytical design sensitivity analysis based on \exact" numerical

di�erentiation of element matrices has been shown to yield accurate sensitivities for all

studies made and must be considered a very reliable tool in a general purpose computer

aided engineering design system.

120 6.10. Concluding Remarks on Examples

Chapter

7

A General and Flexible Method of

Problem De�nition

7.1 Introduction

This chapter describes the implementation of a general and exible method of for-

mulating problems of mathematical programming in structural optimization systems.

The method enables the formulation and solution of problems involving local, integral,

min/max, max/min and possibly non-di�erentiable user de�ned functions in any conceiv-

able mix. The mathematical formulation is based on the so-called bound formulation, and

the implementation speci�c details involve a parser capable of interpreting and performing

symbolic di�erentiation of the user de�ned functions. The basic data involved in the user

de�ned functions are available through a class of database operations and many di�erent

mathematical operators and functions are implemented.

The problem of mathematical programming consists in determining optimum values of

the design variables such that they maximize or minimize a speci�c function termed the

objective function, while satisfying a set of geometrical and/or behavioural requirements

which are called constraint functions.

In Section 7.2 the traditional way of formulating problems of mathematical programming in

general structural optimization systems is described. Usually, only standard formulations,

such as to minimize the maximum stress with a volume constraint, can be chosen, but in

practical design cases much more complicated problem de�nitions may be necessary.

The mathematical programming problems to be solved can involve criteria of various na-

ture, e.g., integral type functions, max/min or min/max type functions, or local type

functions like stress at a given point. A very general and elegant way of formulating and

solving such mathematical programming problems is to make use of the bound formulation

which is described in Section 7.3.

Having formulated the problem of mathematical programming in a standard form using

the bound formulation where the objective and constraint functions are given as vector

121

122 7.2. Background

functions, a description of how the user can de�ne these vector functions is given. We have

realized that the only way to obtain a exible system for de�nition of structural optimiza-

tion problems is to allow the user to de�ne objective and constraint functions in a home-

made language consisting of a set of operators and functions. Thus, a parser interprets

the de�nition of the user de�ned objective and constraint functions as described in Section

7.4. This results in some database operations as described in Section 7.5 where basic data

are read. The relations between these basic data must be de�ned by some mathematical

operators and functions described in Section 7.6. Having obtained the necessary data and

their relations, the vector functions can be evaluated as described in Section 7.7 and sym-

bolic di�erentiation is easily implemented in each evaluation step. In this way analytical

expressions for derivatives of vector functions are available. When all vector functions have

been evaluated, the mathematical programming problem can be established and solved,

resulting in improved values of the design variables as described in Section 7.8.

The approach described in this chapter is valid for all types of optimization problems

that can be solved by ODESSY, except the case of having multiple eigenvalues in the

optimization problem. This special case is covered in Chapter 8.

Documentation of this approach of de�ning problems of mathematical programming in

structural optimization systems can also be found in Lund & Rasmussen (1994).

7.2 Background

In this section the traditional way of formulating problems of mathematical programming

in general structural optimization systems is described. Some of the �rst available general

structural optimization systems, for instance OASIS, see Esping (1986), SAMCEF, see

Fleury (1987) and Braibant & Morelle (1990), and CAOS, see Rasmussen (1989, 1990)

and Rasmussen, Lund, Birker & Olho� (1993) provide the user with a choice of a number

of prede�ned functions from which the optimization problem can be built, for instance,

weight, de ection, stress, eigenfrequency, and compliance.

Functions like these are made available by equipping the system with a module which can

�nd this information in the database of analysis results. Given the options above, the user

may choose to formulate the problem as, for instance,

Minimize maximum stress

Subject to

weight � some upper limit

lowest eigenfrequency � some lower limit

or alternatively

or any other combination of the functions. We see that the possible problem formulations

of such a system are of very di�erent types. Weigth and compliance are integral type

functions, de ection or stress at a given point are local type functions, and maximum stress

Chapter 7. A General and Flexible Method of Problem De�nition 123

Minimize weight

Subject to

maximum stress � some upper limit

de ection at a given point � some upper limit

compliance � some upper limit

or minimum eigenfrequency are min/max or max/min type criteria which are inherently

non-di�erentiable.

In addition to the mathematical di�culties created by the di�erent natures of these criteria,

it is a programming di�culty to make the system cope with any possible combination that

the user may de�ne. A very general and elegant solution is to make use of the so-called

bound formulation which will be brie y described in Section 7.3.

A system that provides a wide selection of basic functions, possibly more than the above-

mentioned, and enables the user to freely combine them into optimization problems is

a very general tool for structural optimization. However, by developing and using such

systems extensively for some years to solve industrial problems, we have learned that it is

impossible for a system designer to foresee the necessary facilities for all types of problems

that may arise in practical design cases. The universe of relevant design problems is too

large to be covered by any practical �nite set of criterion functions.

Some straight-forward examples of unusual problem de�nitions have been

� cases where some mathematical combination of di�erent stress types at dif-

ferent locations in the structure has been established as the design criterion

through a long time of numerical and experimental investigations and perhaps

practical use. Industries will typically rather abstain from the use of structural

optimization than being forced into changing their well-tested design criteria

just because of a limitation in the software system.

� cases of multicriterion optimization in the form of using a weighted sum of

di�erent functions as one of the criteria of the problem.

� cases of unusual material failure criteria in the form of expressions involving,

for instance, di�erent stress or strain types.

� cases where manufacturing constraints must be included in the de�nition of the

optimization problem.

System developers have the possibility to taylor the system by continuously reprogramming

it to �t the problem at hand but this is not acceptable in the long run. Therefore we

initiated a more fundamental solution of the problem of generality.

124 7.3. Bound Formulation

7.3 Bound Formulation

In this section the bound formulation which is used as a basis for formulating and solving

problems of mathematical programming in ODESSY is described.

In order to keep an overview of a large system like ODESSY, it is absolutely necessary

to maintain a rigorous modularity in the organization of the code and have the modules

perform the necessary exchange of information through well-de�ned interfaces. In relation

to optimizers, this problem is well-known and has been addressed by Vanderplaats (1987)

in connection with his ADS subroutine package.

The optimization concept of ODESSY is based on the formation and solution of a sequence

of explicit subproblems as opposed to systems that work with line-search oriented opti-

mizers. From a theoretical point of view, the subproblem approach is known to be less

stable than line-search algorithms and it is often di�cult or impossible to prove conver-

gence properties for the algorithm. However, from a system point of view, the subproblem

approach is much easier to program and control because it is realized via a �xed sequence

of calculations independently of the iteration history. The subproblem approach works re-

liably in most cases and has gained a signi�cant popularity among developers of structural

optimization systems.

In the formulation of problems of mathematical programming only �rst order approxima-

tions are used, i.e., we do not require our analysis module to provide more than �rst order

sensitivities because higher order derivatives are computationally very costly to compute.

This does not prevent the use of, for instance, a quasi-Newton approach in which second

order approximations are gradually formed based on �rst order information in several it-

erations. However, currently ODESSY only contains two optimizers, namely a sequential

linear programming algorithm (SIMPLEX) and an implementation of the MMA method

by Svanberg (1987), see also Fleury & Braibant (1986).

The usual mathematical programming formulation of a structural optimization problem is

as follows:

Minimize g0(a)

a

Subject to (7.1)

gj(a) � Gj; j = 1; : : : ; J

ai � ai � ai; i = 1; : : : ; I

where I is the number of so-called design variables, ai, and J is the number of constraints

other than side constraints. Maximization problems are easily included in this form by

simply minimizing �g0(a). We notice that non-di�erentiable functions occur very often

in connection with practical structural optimization problems as the result of min/max or

max/min criteria.

The objective and constraint functions gj; j = 0; : : : ; J , are speci�ed by the user as a part

of the problem formulation. These vector functions can be picked and combined freely


among all the analysis results that the system is able to evaluate as will be described in

the next section.

In the interest of simplicity, generality and, above all, ease of programming, a formulation

that enables the system to handle the optimization problem in a uniform way regardless of

the blend of local-, integral- and min/max-criteria is very much desired and can be obtained

by use of the so-called bound formulation, see Bends�e, Olho� & Taylor (1983), Taylor &

Bends�e (1984), and Olho� (1989). This technique very elegantly solves the problem of

generality, and, assuming an adequate sensitivity analysis scheme, it also provides a simple

solution to the non-di�erentiability problem in connection with min/max and max/min

problems.

It is assumed that any function gj in the problem is actually a max over a set of functions

(the formulation for max/min problems is completely equivalent), such that

gj(a) = max gjk(a); j = 0; : : : ; Jk=1;:::;pj (7.2)

The counters pj; j = 0; : : : ; J , designate the number of functions among which gj is given

as the max. In the case of, for instance, a max stress criterion, pj would be the number

of nodal stresses among which the maximum is to be found. Ordinary scalar functions are

just a special case signi�ed by the corresponding pj = 1. This gives us the following form

of 7.1:

Minimize max g0k(a);a k=1;:::;p0

Subject to (7.3)

max gjk(a) � Gj; j = 1; : : : ; Jk=1;:::;pj

ai � ai � ai; i = 1; : : : ; I

The constraint functions gj = max(gjk), j = 1; : : : ; J , do not cause di�erentiability prob-

lems. They may simply be treated separately:

Minimize max g0k(a);a k=1;:::;p0

Subject to (7.4)

gjk(a) � Gj; k = 1; : : : ; pj; j = 1; : : : ; J

ai � ai � ai; i = 1; : : : ; I

This leads to the idea of using a similar approach for the objective function g0 = max(g0k).

In order to make this possible, a new, arti�cial design variable � is introduced and a new,

arti�cial objective function f(�). It is now possible to formulate an equivalent problem in

which the previous non-di�erentiable objective function plays the role of constraints:

Minimize f(�)

126 7.3. Bound Formulation

a; �

Subject to (7.5)

g0k(a) � f(�); k = 1; : : : ; p0;

gjk(a) � Gj; k = 1; : : : ; pj; j = 1; : : : ; J

ai � ai � ai; i = 1; : : : ; I

In this formulation, the only way of reducing the value of the arti�cial objective function

f(�) while ful�lling the constraints, is to simultaneously reduce all the functions comprised

in the original objective, i.e., minimizing the maximum. The arti�cial objective function

f(�) acts as a bound which suppresses the value of the original objective and gives this

technique the name of \bound formulation".

The speci�c form of the arti�cial objective function f(�) has yet to be selected. If a linear

method of mathematical programming is used, there is no reason not to select the simplest

imaginable function of �:

f(�) = � (7.6)

Next 7.6 is inserted in 7.5, and, in order to better cope with functions of varying nature

and scale, we perform a normalization of the problem:

Minimize �

a; �

Subject to (7.7)

g0k(a)

�� 1; k = 1; : : : ; p0;

gjk(a)

Gj

� 1; k = 1; : : : ; pj; j = 1; : : : ; J

ai � ai � ai; i = 1; : : : ; I

As previously mentioned, the counters pj; j = 0; : : : ; J , designate the number of functions

over which the max for criterion j is to be found. In the case of, for instance, minimization

of maximum nodal stress, pj may often be many thousands, and the number of functions in

the optimization becomes very large. In order to prevent this, an \active set strategy" can

be used, which only includes the nodal stresses exceeding a certain fraction of the current

maximum in the problem.

Tableau 7.7 is valid regardless of the blend of vector functions gj; j = 0; : : : ; J , and the

numerical operations performed are therefore identical for any problem that the user could

possibly de�ne. Due to this standardization, the bound formulation has greatly simpli�ed

the programming of the optimization module.

Having obtained a standard format for the mathematical program, a method for the user to

de�ne the problem and a database module for extracting vector functions gj; j = 0; : : : ; J ,

are required.


7.4 Specifying the Problem

Solving many real-life structural design optimization problems has indicated that the only

way to cover all possible combinations of problem speci�cations is to allow the user to

specify the functions of the problem as mathematical relations between a set of basic data

available in the system. Furthermore, the system must be designed such that it is relatively

simple to expand the set of basic information as new analysis facilities are added to the

system. The same is valid for the available mathematical expressions.

The basic data of the system are available through a class of database operations that

extract the required information from the ODESSY analysis result �les. These functions

are very similar to those used in conventional systems to extract the values and sensitivities

of the objective and constraint functions.

Examples of calls of the database functions could be:

nstress(svm, mat=steel, ldc=1)

which returns the nodal von Mises stresses in material \steel" in load case 1, or

weight( )

which returns the overall weight of the structure.

The mathematical operations are available as the usual mathematical operators, for in-

stance addition, subtraction, multiplication and division. In addition to these, a set of

prede�ned functions are available, for instance logarithm, trigonometric functions, maxi-

mum and minimum.

We can imagine a criterion function speci�ed as

maxval [ 1.5�[ nstress(svm, mat=steel) ] + 1.2�[ nstress(st, mat=aluminium) ] ]

which would specify the maximum value of a vector containing a weighted sum of von Mises

stresses in material \steel" and Tresca stresses in material \aluminium". Please notice the

use of brackets as parantheses except in the case of arguments to database funtions. We

shall return to this syntactic peculiarity later.

7.5 Database Operations

The database operations are speci�ed as function calls. Each function extracts a speci�c

type of information from the database, for instance nodal stresses, element strains, com-

pliance or volume. The obtained vector functions can be divided into the three categories:

nodal vector functions, element vector functions, and global vector functions.

The database functions available in the ODESSY system are listed in the following and

possible input speci�cations to each function are also given.

128 7.5. Database Operations

Table 7.1: Database operations for nodal vector functions with arguments.

Description Name Speci�c Speci�c Load Absolute Speci�c Speci�c Additional

type material case value layer point options

Nodal stresses nstress yes yes yes yes yes yes

Nodal strains nstrain yes yes yes yes yes yes

Absolute

Displacements disp yes yes yes yes - yes displace-

ment

Nodal forces force yes yes yes yes - yes Absolute

force

Nodal temp - yes yes yes - yes

temperatures

Nodal thermal tempforce yes yes yes yes - yes

forces

Nodal x,y,z - - - yes - yes

coordinates

Nodal radii rad - - - yes - yes

Boundary

radius of curvature - - - yes - no

curvature

Table 7.2: Database operations for element vector functions with arguments.

Description Name Speci�c Speci�c Load Absolute Speci�c Speci�c Additional

type material case value layer point options

Element estress yes yes yes yes yes yes

stresses

Element estrain yes yes yes yes yes yes

strains

Element elthk - yes - - - -

thicknesses

The extraction of raw data by database operations is always the �rst step in evaluating

a user de�ned criterion function gj. In order to be able to handle all combinations in a

uni�ed way, the system has been designed such that all database operations return the

same data type, a vector of nodal values, VNV.

The components of VNV's are nodal values (may also contain element or global values)


Table 7.3: Database operations for global vector functions with arguments.

Description Name Speci�c Load Additional options

material case

Volume vol yes -

Weight weight yes -

Compliance compliance - yes

Mass moment massinertia yes -

of inertia

Area moment areainertia yes -

of inertia

Eigen- eigenfreq - yes Speci�c number, include all eigenfrequencies

frequencies above or below speci�ed number

Eigen-

frequencies eigenfreqsti� - yes Speci�c number, include all eigenfrequencies

incl. stress above or below speci�ed number

sti�ening

Buckling load buckload - yes Speci�c number, include all buckling load factors

factors above or below speci�ed number

Probability pof yes yes

of failure

characterized by the following data structure (ODESSY is coded in ANSI C):

struct ods nv flong nr; /* Node or element number */

long mat; /* Material number */

long ldc; /* Load case number */

double val; /* Value */

double vali; /* Sensitivity value */

short sign; /* Sign of value */

g

The material number is part of this data structure because a given node may be on the

interface between several materials and have a di�erent value for each material.

VNV's are simply arrays of the structure described above:

struct ods vnv flong n nv; /* Number of components in the vector */

short type; /* Type of vector (nodal, element, or global) */

struct ods nv nv[ ]; /* The actual vector of the type struct ods nv */

g

Some of the database functions, for instance the volume calculation, return only a scalar

130 7.6. Operators and Functions

value. In this case there is no evident need for the data structures described above, but

the subsequent operations are much simpler if data are represented in a uniform way.

7.6 Operators and Functions

The usual way of writing mathematical expressions is by a mix of operators and functions,

for instance

log(3:2 � x+ y)

where the two operators, * and +, are used together with a logarithm function. The usual

mathematical operators and functions are available in ODESSY for operations between

random combinations of VNV's and scalar numbers. Furthermore, some additional conve-

nient operators, like concatenation of two VNV's, have been implemented. The currently

available operators and functions are listed in tables 7.4 and 7.5.

Table 7.4: Implemented operators.

Operator: Description:

+ Addition operators.

- Subtraction operator.

/ Division operator.

* Multiplication operator.

^ Power function.

j Operator which concatenates two vectors.

max Operator which speci�es a maximization of the objective function.

min Operator which speci�es a minimization of the objective function.

maxval Maximum value operator which �nds the maximum value of a vector.

minval Minimum value operator which �nds the minimum value of a vector.

maxvec Operator which compares two vectors and returns a vector consisting of maximum

values for each node (or element).

minvec Operator which compares two vectors and returns a vector consisting of minimum

values for each node (or element).

�\value" \Top" operator which subtracts all values of a vector exceeding \value" multiplied by

the maximum value of the vector. The scalar \value" must be between 0.0 and 1.0.

�\value" \Bottom" operator which subtracts all values of a vector less than \value" multiplied

by the minimum value of the vector. The scalar \value" must be larger than 1.0.

Each operator or function operates on the VNV's de�ned in section 7.5 and is thus capable

of evaluating either initial values or sensitivity values. For each operator and function, a

set of evaluation rules is implemented, e.g., it is not allowed to multiply a VNV of nodal

values with a VNV of element values.


Table 7.5: Implemented functions.

Function: Description:

cos Cosine.

sin Sine.

log Logarithm base e.

7.7 Evaluation Sequence

Each de�nition of an objective or constraint function in the mathematical program is

interpreted by a so-called parser. A parser is well-known as an important part of any

computer programming language compiler. It breaks a mathematical expression into a

sequence of simple operations and keeps track of input and output of each operation. The

user can control the evaluation sequence by subdividing the expression using brackets.

Brackets are used for this purpose to distinguish subdivisions from function arguments.

In the absence of brackets, the parser breaks the expression down based on the usual

matematical rules of precedence, such that a+ b � c is evaluated as a+ [b � c] rather than[a + b] � c.The available operators are internally converted into functions, such that the expression

results in a sequence of function calls each returning one VNV to be used in subsequent

calls. Each function call involves either:

1. A database function de�ning a VNV.

2. Call of an operation on one VNV.

3. Call of an operation between two VNV's.

This decomposition of the evaluation sequence makes it easy to implement analytical ex-

pressions for derivatives of each VNV. Because a maximum of two VNV's and only one

basic operation or function is involved in each function call, symbolic di�erentiation is

easily implemented in each step.

To illustrate the evaluation sequence for di�erent types of VNV's, we consider the example

in Section 9.3 where the shape of a turbine disk is optimized. For this turbine disk the

design optimization problem is de�ned as

where �vM;i is von Mises reference stress at node i and Ti is the nodal temperature. The

non-linear stress constraint is illustrated on Fig. 9.12 in Section 9.3. In ODESSY, this

optimization problem can be de�ned as

min [ massinertia() ]

maxval [ nstress(svm)/ 1.0E6 / [ minvec [ 550; [ 1484.5 - [ 1.5�temp()] ] ] ] ] < 1.0

minval [ curvature() � 5.0 ] > 0.0050

The stress constraint where the von Mises reference stress at every nodal point i is nor-

malized with a temperature dependent non-linear function de�ning the allowable stress

132 7.7. Evaluation Sequence

Minimize the mass moment of inertia

Subject to

maximum normalized stress

8>>>><>>>>:

�vM;i [MPa]550 if Ti � 623K

�vM;i [MPa]1484:5� 1:5Ti if Ti > 623K

� 1

minimum boundary radius of curvature � 5 mm

illustrates the generality of this approach of de�ning problems of mathematical program-

ming.

For these de�nitions of objective and constraint functions, we get the following evaluation

sequences:

Objective function:

VNV1 = massinertia() Database function de�ning VNV1 as the mass moment.

VNV2 = min(VNV1) Call of \min" operator with input VNV1 gives �nal

result.

Constraint function no. 1:

VNV1 = 1.5 VNV1 is de�ned as the digit 1.5.

VNV2 = temp() Database function de�nes VNV2 as a vector containing

nodal temperatures.

VNV3 = mult(VNV1,VNV2) Call of multiplication operator between VNV1 and

VNV2.


VNV5 = minus(VNV4,VNV3) Call of subtraction operator between VNV4 and VNV3.

VNV6 = 550 VNV6 is de�ned as the digit 550.

VNV7 = nstress(svm) Database function de�nes VNV7 as a vector containing

von Mises stresses for all nodes.

VNV8 = 1.0E6 VNV8 is de�ned as the digit 1:0 � 106.VNV9 = minvec(VNV6,VNV5) Call of \minvec" operator between VNV6 and VNV5

returns a vector containing the smallest value of either

VNV6 and VNV5 for each node, see Fig. 9.12.

VNV10 = div(VNV7,VNV8) Call of division operator between VNV7 and VNV8

gives VNV10 containing von Mises stresses in [MPa].


VNV11 = div(VNV10,VNV9) Call of division operator between VNV10 and VNV9

gives VNV11 containing normalized von Mises stresses.

VNV12 = maxval(VNV11) Call of \maxval" operator with input VNV11 gives �-

nal result for constraint function no. 1 in VNV12 and

de�nes the maximum value of VNV12 to be less than

the right hand side (= 1.0). VNV12 is just a copy of

VNV11.

Constraint function no. 2:


VNV2 = curvature() Database function de�nes VNV2 as a vector containing

boundary radius of curvatures for all boundary nodes.

VNV3 = bottom(VNV2,VNV1) Call of \bottom" operator between VNV2 and VNV1

gives VNV3 containing the boundary radii of curvature

which are less than the smallest value multiplied by 5.0.

VNV4 = minval(VNV3) Call of \minval" operator copies VNV3 to �nal result

for constraint function no. 2 in VNV4 and de�nes the

minimum value of VNV4 to be less than the right hand

side (= 0.005).

Each VNV is evaluated for either initial values or sensitivity values, and we see that

symbolic di�erentiation is very easily incorporated in each function call. For instance, by

usual di�erentiation rules, the derivative of VNV10 for constraint function no. 1 is given

as

VNV0

10 =VNV

0

7 VNV8 � VNV7 VNV0

8

(VNV8)2

which is easily evaluated due to the structure of each VNV.

7.8 Using the Database Module

This database module for evaluation of objective and constraint functions gj can be used in

di�erent ways. It is mainly used as a basic part of the iterative solution procedure for mul-

ticriterion optimum design which is realized through systematic sequences of redesign and

reanalysis. This is illustrated on Fig. 7.1 where a ow diagram for structural optimization

with ODESSY is shown.

The user can improve the design by using this procedure for structural optimization. In

each step of redesign, the available analysis modules in ODESSY compute current values

of variables fm of the types listed in Tables 7.1, 7.2, and 7.3. These basic data are used to

evaluate current values of the speci�ed objective and constraint functions gj. The design

sensitivity analysis is carried out for each design variable ai, resulting in derivatives rgj

134 7.8. Using the Database Module

Start optimization

boundary conditions and loads

Structural analysis

Design sensitivity analysis

Optimizer calculates improvedvalues of design variables

Converged?

Stop

yes

no

f

i

m

f

∆

Preprocessor updates geometry,

Database module evaluates objectiveand constraint functions

g

m

Preprocessor perturbs design variableand updates analysis model

a

j

Database module evaluates derivativesof objective and constraint functions

gj

∆

iafor each design

variable

ia

for specified numberof iterations

ia

gj

∆gj

, ,

Figure 7.1: Flow diagram for structural optimization using ODESSY.

of objective and constraint functions. The bound formulation, see Eq. 7.7, is then used to

set up a problem of mathematical programming which is solved by one of the implemented

optimizers, resulting in improved values of the design variables. This iteration process

continues until convergence is achieved or the speci�ed number of iterations is reached.

The database module can also be used in another way as it has been integrated in the

module for postprocessing of the results. The user can, while using the postprocessor,

make interactive function calls to the database module and have any VNV (of nodal or

element values) plotted at the current geometry. Such a facility can be very helpful in the

design phase in order to evaluate some well-tested design criteria or to establish new design

criteria. Furthermore, it is much easier to check the validity of de�nitions of objective and

constraint functions by using this interactive graphical facility.


7.9 Conclusions

In this chapter it has been demonstrated that it is not only possible but also relatively

simple to create a structural optimization system that handles problem formulations com-

prising any mix of local, integral, min/max and max/min functions in a completely uniform

way. The mathematical basis of this is the bound formulation. This, in combination with

a parser capable of interpreting user de�ned expressions, makes it possible to do analysis

and sensitivity analysis of user de�ned mathematical expressions and even performing the

sensitivity analysis by exact and e�cient symbolic di�erentiation of the expressions. The

approach described makes the system very easy to expand with new available mathematical

expressions or with new basic information if new analysis facilities are implemented. The

integration of the database module in the module for postprocessing has made it very easy

to check the validity of de�nitions of objective and constraint functions, and, furthermore,

has equipped the system with a very useful tool for interactive, graphical visualization of

user de�ned mathematical expressions for design criteria.

This development can be seen as one out of many necessary steps that will make structural

optimization facilities applicable and accessible to real-life engineering designers. Problems

involving mixed criteria like, for instance, thermo-elastic stresses are di�cult to handle by

the traditional \trial and error" methods because the consequences of a contemplated

design change may be extremely di�cult to foresee, and they call very much upon rational

design techniques based on optimization.

136 7.9. Conclusions

Chapter

8

Multiple Eigenvalues in Structural

Design Problems

8.1 Introduction

This chapter is devoted to the di�culties of solving structural optimum design prob-

lems with multiple eigenvalues. In Section 4.9 expressions for design sensitivity anal-

ysis of both simple and multiple eigenvalues were derived and it was shown that multiple

eigenvalues can only be expected to be directionally di�erentiable. In Sections 6.6 and 6.7

these expressions for design sensitivity analysis were used successfully to compute changes

of simple and multiple natural transverse vibration frequencies subject to changes of di�er-

ent design parameters of sti�ener reinforced thin elastic plates. The question now is how

to solve optimum design problems with multiple eigenvalues as traditional gradient based

optimization algorithms cannot be applied due to the lack of usual Fr�echet di�erentiability

of multiple eigenvalues.

The di�culties of having multiple eigenvalues in structural design problems have been an

active research area during the last 15 years and references to some important papers in

this area are given in Section 8.2.

The problem of solving optimization problems with multiple eigenvalues is then illustrated

by a simple example involving 2x2 matrices depending on two design variables.

Next the problem of optimization is formulated as maximization of the smallest (simple or

multiple) eigenvalue subject to a constraint of given volume of material of the structure in

Section 8.4. For such an optimization problem necessary optimality conditions are derived

for arbitrary multiplicity of the smallest eigenvalue. The necessary optimality conditions

express (I) linear dependence of a set of generalized gradient vectors of the multiple eigen-

value and the gradient vector of the constraint, and (II) positive semi-de�niteness of a

matrix of the coe�cients of the linear combination. The main advantage of these neces-

sary optimality conditions is, when compared to those obtained by other researchers, that

they do not contain variations of design variables.

137

138 8.2. Background

In Section 8.5 iterative numerical algorithms for eigenvalue optimization problems involving

multiple eigenvalues are presented. The basic idea of the approaches described is to add

constraints on the allowable design changes in case of multiple eigenvalues, whereby design

sensitivity expressions for simple as well as multiple eigenvalues become identical. It is

shown how the derived necessary optimality conditions can be applied for development

of an iterative numerical method for optimization of structural eigenvalues of arbitrary

multiplicity. Furthermore, an e�ective mathematical programming approach for solving

optimization problems is described. The mathematical programming approach is used to

solve the problem of maximizing the lowest eigenvalue of a sti�ener reinforced thin elastic

plate in Section 8.6.

Most of the results presented in this chapter originate from a joint work on multiple

eigenvalues in structural optimization problems between Alexander P. Seyranian, Moscow

State Lomonosov University, Niels Olho� and myself. This joined work can be found in

Seyranian, Lund & Olho� (1994).

8.2 Background

Multiple eigenvalues in the form of buckling loads and natural frequencies of vibration

very often occur in complex structures that depend on many design parameters and have

many degrees of freedom. For example, sti�ener reinforced thin-walled plate and shell

structures have a dense spectrum of eigenvalues, and multiple eigenvalues are found very

often as illustrated in Sections 6.6 and 6.7. Also, symmetry of structural systems may lead

to apperance of several linearly independent buckling modes and vibration modes with

multiple eigenvalues.

In 1977, Olho� & Rasmussen discovered that the optimum buckling load of a clamped-

clamped column of given volume is bimodal. This optimization problem was �rst consid-

ered by Lagrange, and its interesting history is presented in a recent paper by Cox (1992).

Olho� & Rasmussen (1977) showed that the bimodality of the optimum eigenvalue must

be taken into account in the mathematical formulation of the problem in order to ob-

tain the correct optimum solution. They �rst demonstrated that an analytical solution

obtained earlier by Tadjbakhsh & Keller (1962) under the tacit assumption of a simple

buckling load is not optimal, then presented a bimodal formulation of the problem, solved

it numerically, and obtained the correct optimum design. The optimum bimodal buckling

load obtained was later con�rmed to be correct to within a slight deviation of the sixth

digit by analytical solutions obtained independently by Seyranian (1983, 1984) and Masur

(1984). The discovery in 1977 of multiple optimum eigenvalues in structural optimization

problems, and the necessity of applying a bi- or multimodal formulation in such cases,

opened a new �eld for theoretical investigations and development of methods of numerical

analysis and solution.

Prager & Prager (1979), Choi & Haug (1981), and Haug & Choi (1982) presented uni-

modal and bimodal optimum solutions for systems with few degrees of freedom con�rming

apperance of multiple eigenvalues in optimization problems. A wealth of references on mul-

Chapter 8. Multiple Eigenvalues in Structural Design Problems 139

timodal optimization problems and speci�c results for columns, arches, plates and shells

can be found in comprehensive surveys by Olho� & Taylor (1983), Gajewski & Zyczkowski

(1988), Zyczkowski (1989) and Gajewski (1990). A survey of other problems of optimum

design with respect to structural eigenvalues was earlier published by Olho� (1980), see

also Olho� (1981a, 1981b).

One of the main problems related to multiple eigenvalues is their non-di�erentiability in

the common (Fr�echet) sense. This was revealed by Masur & Mr�oz (1979, 1980) and Haug

& Rousselet (1980b). The non-di�erentiability creates di�culties in �nding sensitivities of

multiple eigenvalues with respect to design changes and derivation of necessary optimal-

ity conditions in optimization problems. Choi & Haug (1981) used a Lagrange multiplier

method for bimodal problems and showed that this method which is very useful for di�er-

entiable criteria and constraints may yield incorrect results.

Haug & Rousselet (1980b) proved existence of directional derivatives of multiple eigenvalues

and obtained explicit formulas for derivatives. Bratus & Seyranian (1983) and Seyranian

(1987) presented sensitivity analysis of multiple eigenvalues based on a perturbation tech-

nique and derived necessary optimality conditions. The main advantage of these necessary

optimality conditions is, when compared with those obtained by previous researchers, that

they do not contain variations of design variables. Similar developments were presented by

Masur (1984, 1985). It was with the use of these necessary optimality conditions Seyra-

nian (1983, 1984) and Masur (1984) independently of each other obtained the analytical

solution to the bimodal optimum clamped-clamped column problem mentioned above.

Overton (1988) considered minimization of the maximum eigenvalue of a symmetric matrix.

This problem is similar to the problem of maximizing the minimum eigenvalue. Derivation

of necessary optimality conditions using the bound formulation of such problems had earlier

been presented by Bends�e, Olho� & Taylor (1983) and Taylor & Bends�e (1984). In a

recent paper Cox & Overton (1992) presented new mathematical results for optimization

problems of columns against buckling. They derived necessary optimality conditions using

advanced nonsmooth optimization methods. Their results for optimum columns are in

good agreement with the results obtained earlier by Olho� & Rasmussen (1977), Seyranian

(1983, 1984), and Masur (1984).

Numerical algorithms for solution of structural optimization problems with multiple eigen-

values have been suggested and discussed by, among others, Olho� & Rasmussen (1977),

Choi, Haug & Lam (1982), Choi, Haug & Seong (1983), Olho� & Plaut (1983), Bends�e,

Olho� & Taylor (1983), Myslinski & Sokolowski (1985), Zhong & Cheng (1986), Plaut,

Johnson & Olho� (1986), Gajewski & Zyczkowski (1988), Overton (1988), and Cox &

Overton (1992).

8.3 Illustrative Optimization Problem for Eigenval-

ues

In order to illustrate the di�culties in solving eigenvalue optimization problems with mul-

tiple eigenvalues, let us consider a simple illustrative example. The example involves 2x2

140 8.3. Illustrative Optimization Problem for Eigenvalues

matrices depending on two design variables x and y, and design sensitivities of multiple

eigenvalues are calculated using the expressions given in Section 4.9. The example is taken

from Seyranian, Lund & Olho� (1994) where other simple illustrative examples can be

seen.

The example is described by the following K and M matrices

K =

"1 + x y

y 1� x

#; M =

"1 0

0 1

#(8.1)

The characteristic equation for this system is

�2 � 2�+ 1� x2 � y2 = 0 (8.2)

and

�1;2 = 1�qx2 + y2 (8.3)

The level curves � = c, where c is a constant, for this function are described by the equation

c = 1�qx2 + y2 (8.4)

and

x2 + y2 = (c� 1)2 (8.5)

This surface is a circular cone, see Fig. 8.1.

Figure 8.1: Circular cone surface for eigenvalue �.

Bimodality occurs at x = 0; y = 0, for which we have �1 = �2 = 1.

It can be seen immediately from Eq. 8.3 and Fig. 8.1 that the eigenvalues � are not

di�erentiable at the bimodal point in the usual (Fr�echet) sense. Indeed

@�1;2

@x= � xp

x2 + y2;

@�1;2

@y= � yp

x2 + y2(8.6)


As x ! 0 and y ! 0 the two right-hand side expressions become unde�ned and have

no limit. Furthermore, L'Hopital's Rule cannot help either because the derivatives of the

denominators also tend to zero.

Now we proceed to sensitivity analysis of the double eigenvalue ~� = 1 at x = 0; y = 0,

using the results of Section 4.9.2 in Chapter 4. Taking the direction "e we have

x = "e1; y = "e2;pe1 + e2 = 1 (8.7)

For the sake of simplicity we can introduce the angle � and write the directional vector e

in the form

e1 = cos�; e2 = sin� (8.8)

Let us determine the directional derivatives �1, �2 for the double eigenvalue ~� = 1. The

orthonormalized eigenvectors corresponding to ~� are

�1 =

1

0

!; �2 =

0

1

!(8.9)

Using the expressions in Eqs. 8.1 and 8.9 we obtain the vectors fsk according to Eq. 4.45

fT11 =

�1 0

� 1 0

0 �1

! 1

0

!;�1 0

� 0 1

1 0

! 1

0

!!= (1; 0)

fT12 =

�1 0

� 1 0

0 �1

! 0

1

!;�1 0

� 0 1

1 0

! 0

1

!!= (0; 1) (8.10)

fT22 =

�0 1

� 1 0

0 �1

! 0

1

!;�0 1

� 0 1

1 0

! 0

1

!!= (�1; 0)

Thus, Eq. 4.46 takes the form

det

�� cos�� sin�

sin� � cos��

�� = 0 (8.11)

or

�21;2 = sin2 � + cos2 � = 1 (8.12)

So, �1 = 1 and �2 = �1 for any direction e = (cos�; sin�). Hence, the double eigenvalue~� splits into �1;2 = 1 � " for any direction "e. This means that the bimodal solution

x = 0; y = 0 is the optimum solution to the problem

Maximize min �j; j = 1; 2

x; y(8.13)

This result, of course, can be seen immediately from Fig. 8.1.

8.4 Necessary Optimality Conditions for Eigenvalue

Problems

In this section necessary optimality conditions for eigenvalue optimization problems are

described. The optimization problem considered concerns maximization of the lowest of the

142 8.4. Necessary Optimality Conditions for Eigenvalue Problems

eigenvalues �j; j = 1; : : : ; n, subject to a constant volume constraint and can be formulated

as follows

Maximize min �j; j = 1; : : : ; n (8.14)

a1; : : : ; aI

Subject to

F (a1; : : : ; aI) = 0 (8.15)

This formulation of the structural eigenvalue optimization problem is chosen as it is the

most commonly used.

In the following di�erent situations of optimum fundamental eigenvalues are considered.

8.4.1 Simple Optimum Fundamental Eigenvalue

First the situation where optimum is achieved at the simple lowest eigenvalue �1 with

�1 < �2 � �3 � : : : is studied. In this well-known case, due to Fr�echet di�erentiability

of simple eigenvalues, the necessary optimality condition implies linear dependence of the

gradient vectors of �1 and F

r�1 � 0f0 = 0 (8.16)

where

r�1 =

�T1

@K

@a1� �1

@M

@a1

!�1; : : : ;�

T1

@K

@aI� �1

@M

@aI

!�1

!

(8.17)

f0 =rF =

@F

@a1; : : : ;

@F

@aI

!

and 0 is a positive (Lagrangian) multiplier to be determined from Eqs. 8.15 and 8.16.

8.4.2 N-fold Optimum Fundamental Eigenvalue

Let us consider the general case when in the optimization problem, Eqs. 8.14 and 8.15,

the maximum is attained at an N -fold multiple lowest eigenvalue �1 = �2 = : : : = �N <

�N+1 � : : :. This is a non-di�erentiable case, and we have to use directional derivatives as

described in Section 4.9.

Taking the vector of varied design variables in the form a + "e, kek = 1, we obtain the

directional derivatives � = �j; j = 1; : : : ; N , from Eq. 4.46, i.e.,


�� = 0; s; k = 1; : : : ; N (8.18)

where the generalized gradient vectors fsk are given by Eq. 4.45. Please note that the sub-

scripts of the generalized gradient vector fsk refer to the modes from which it is calculated.


The direction e must ful�l the volume constraint in Eq. 8.15, i.e.,

fT0 e = 0 (8.19)

The necessary optimality conditions are derived for arbitrary multiplicity N of the lowest

multiple eigenvalue in Appendix D and they give much insight in the di�culties in max-

imizing a lowest multiple eigenvalue. In the sequel, a short summary of these conditions

are given.

In general, the optimum point is characterized by the fact that there must be no admissible

direction e for which all � = �j; j = 1; : : : ; N , are of the same sign, otherwise an improving

direction e exists.

In Appendix D it is proved that if the vectors f0; fsk, s; k = 1; : : : ; N , k � s (the total

number of these vectors is equal to (N + 1)N=2 + 1) are linearly independent, then there

exists an improving direction e for which �j > 0; j = 1; : : : ; N . It should be noted that the

linear independence of the vectors is only possible if I � (N + 1)N=2 + 1, where I is the

dimension of the vector a of design variables.

In this way the existence of an improving direction e can easily be checked.

The necessary optimality conditions in case of an N -fold optimum fundamental eigenvalue

can be stated as:

If the vector of design variables a renders a lowest N-fold eigenvalue �1 = �2 = : : : = �N

a maximum, it is necessary that the vectors f0; fsk, s; k = 1; : : : ; N , k � s, are linearly

dependentNX

s;k=1

skfsk � 0f0 = 0 (8.20)

with the coe�cients 0; sk satisfying conditions of positive semi-de�niteness of the sym-

metric matrix sk, s; k = 1; : : : ; N .

The main advantage of these necessary optimality conditions is, when compared to those

obtained by other researchers, that they do not contain variations of design variables.

These necessary optimality conditions are proved in Appendix D both in case of a bimodal

optimum eigenvalue and for arbitrary multiplicity N of the fundamental eigenvalue. The

applicability of the necessary optimality conditions as basis for numerical iterative algo-

rithms for solution of eigenvalue problems involving multiple eigenvalues is described in

the following section.

8.5 Algorithms for Eigenvalue Problems with Multi-

ple Eigenvalues

This section is devoted to descriptions of algorithms that can be used to solve eigenvalue

problems involving multiple eigenvalues. It will be illustrated how the necessary optimality

conditions derived in Section 8.4 and Appendix D can be used as a basis for an iterative

numerical method for solution of eigenvalue optimization problems, and furthermore, an

e�ective mathematical programming approach is described.

144 8.5. Algorithms for Eigenvalue Problems with Multiple Eigenvalues

The basic idea in our algorithms developed for solving the structural eigenvalue optimiza-

tion problem de�ned by Eqs. 8.14 and 8.15 have been that we would like to avoid the

necessity of using directional derivatives. If directional derivatives should be implemented

in the optimization algorithm, a line search method using directional derivatives may pos-

sibly be developed but such an algorithm will be computationally very costly.

For summary, let us rewrite Eq. 4.47 which is used for computing the increments �� =

��j; j = 1; : : : ; N , of the N -fold eigenvalue corresponding to the vector �a of actual

increments of the design variables:

det�� fTsk�a� �sk��

�� = 0; s; k = 1; : : : ; N (8.21)

where the generalized gradient vectors fsk are de�ned by Eq. 4.45.

If the o�-diagonal terms in the sensitivity matrix in Eqs. 8.21 are zero, then the traditional

equations for determining increments of simple eigenvalues appear, i.e., traditional Fr�echet

derivatives are valid. Such a situation can be obtained by adding the constraints

fTsk�a = 0; s; k = 1; : : : ; N; s 6= k (8.22)

For an N -fold multiple eigenvalue, Eqs. 8.22 results in N(N � 1)=2 constraints for the

allowable direction �a, and in case of only a few design variables ai; i = 1; : : : ; I, the

optimization problem may become too constrained. However, adding these constraints

makes it possible to use traditional gradient based algorithms for solving the optimization

problem where the increments ��j of an N -fold multiple eigenvalue can be computed as

��j = fTjj�a; j = 1; : : : ; N (8.23)

Furthermore, as demonstrated by the reinforced plate example in Section 6.7, it is necessary

to decide correctly from the numerical results the multiplicity N of a multiple eigenvalue,

otherwise the directional derivatives computed might be erroneous due to the solution of an

incorrect subeigenvalue problem for determination of directional sensitivities, see Eq. 8.21

and the discussion in Section 6.7. This problem of determining correctly the multiplicity

of an eigenvalue is now avoided by adding the constraints in Eq. 8.22 because Eq. 8.23 is

valid for both simple and multiple eigenvalues.


8.5.1 Optimality Criteria Based Algorithm

In the sequel it will be shown how the necessary optimality conditions derived in Section

8.4 can be used as basis for an optimization algorithm. Such an approach is also described

in Seyranian, Lund & Olho� (1994).

At a given iteration stage, the design is associated with eigenvalues �j; j = 1; : : : ; n, and

it is necessary to decide the multiplicity of the lowest eigenvalue. If the relative di�erence

between adjacent eigenvalues is less than �, which is a small parameter assumed to be

speci�ed, these eigenvalues are considered to be multiple, and the multiplicity of the lowest

eigenvalue determines the actual formulation for calculating the increments �a of the

design variables a.

Simple Eigenvalue

If the lowest eigenvalue �1 is simple, the eigenvalue increment and the volume constraint

can be expressed as

��1 =rT�1 �a = fT11 �a (8.24)

fT0 �a = 0 (8.25)

where f11 and f0 are de�ned by Eqs. 4.45 and 8.17, respectively, and the volume constraint

is assumed to be linear in the design variables. Guided by the results in Subsection 8.4.1,

the vector of increments of the design variables is taken in the single modal form

�a = k(f11 � 0f0) (8.26)

where k is a positive move limit type of scaling factor. At the optimum point, according

to Eqs. 8.16 and 8.17, the vector �a will tend towards the null vector.

The �a priori unknown constant 0 is determined by substituting Eq. 8.26 into Eq. 8.25

which gives

0 =fT0 f11fT0 f0

(8.27)

and substitution of Eq. 8.27 into 8.24 yields the Cauchy-Bunyakowski inequality

��1 = k

fT11f11 �

(fT0 f11)(fT0 f11)

fT0 f0

!� 0 (8.28)

for the increment �1 of the eigenvalue. Thus, at each step of redesign the eigenvalue �1increases while satisfying the volume constraint, Eq. 8.25, and this continues until f11 and

f0 becomes linearly dependent, cf. Eqs. 8.16 and 8.17.

Bimodal Eigenvalue

If the lowest eigenvalue is associated with a bimodal eigenvalue ~� = �1 = �2 or the two

lowest eigenvalues �1 < �2 are very close such that �2 � �1 < �, a bimodal formulation

must be used to determine the increments �a of the vector of design variables a.

146 8.5. Algorithms for Eigenvalue Problems with Multiple Eigenvalues

Guided by the results stated in Lemma 1 and Theorem 1 in Section D.1 in Appendix D,

see also Eq. 8.20, the vector of increments �a is taken as

�a = k ( 11f11 + 2 12f12 + 22f22 � 0f0) (8.29)

where k again is a positive scaling factor.

Obviously, the increments �a for the iterative computational procedure can be chosen in

many ways, but using the approach of adding the constraints given by Eq. 8.22 leads us to

choose the following four simultaneous conditions as a basis for determining the coe�cients

11, 2 12, 22, and 0 in the expression for �a in Eq. 8.29 (where we disregard the scaling

factor k):

��1 = fT11�a = 1 (8.30)

��2 = fT22�a = (1 + �1 � �2) (8.31)

fT12�a = 0 (8.32)

fT0 �a = 0 (8.33)

It is seen from Eqs. 8.30 and 8.31 that we specify the increments ��1 and ��2 with a

view to diminish the (possible) di�erence between �1 and �2 while going in a direction

that increases the bimodal eigenvalue. Now, by substituting Eq. 8.29 into Eqs. 8.30-8.33,

we obtain the following system of equations for determining the unknown coe�cients 11,

2 12, 22, and 0:266664fT11f11 fT11f22 fT11f12 fT11f0

fT22f22 fT22f12 fT22f0fT12f12 fT12f0

symm fT0 f0

377775

8>>>><>>>>:

11

22

2 12� 0

9>>>>=>>>>;=

8>>>><>>>>:

1

1 + �1 � �2

0

0

9>>>>=>>>>;

(8.34)

Having solved Eq. 8.34 for the coe�cients 11, 2 12, 22, and 0, substitution of these

coe�cients into Eq. 8.29 results in a new vector �a of increments of design variables. It

may be necessary to normalize these coe�cients in order to avoid numerical problems as

described in Seyranian, Lund & Olho� (1994).

It should be noted that the determinant of the coe�cient matrix in Eq. 8.34 is non-

negative, and that it only vanishes if the vectors f11, f12, f22, and f0 become linearly

dependent, which is the necessary optimality condition for an optimum bimodal solution.

This approach for a bimodal lowest eigenvalue is easily extended to higher multiplicity

of the lowest eigenvalue, and during the optimization process, the optimization procedure

switches between the di�erent formulations, depending on the multiplicity of the lowest

eigenvalue.

The optimization procedure described in this section has been used successfully in Seyra-

nian, Lund & Olho� (1994) for solving optimization problems for maximum buckling load

of stepped columns on an elastic foundation and my collegue Lars Krog has implemented

this approach in ODESSY for solving topology design problems with eigenvalues. The

major advantage of this approach is that the number of design variables, in most cases,

is reduced. The number of design variables is I but using this approach, only four design


parameters 11, 2 12, 22, and 0 need to be determined in case of a bimodal eigenvalue.

This approach may be very attractive in topology optimization where the number I of

design variables can be very large.

8.5.2 Mathematical Programming Approach

In this section it will be demonstrated that a mathematical programming approach for

solving optimization problems can be used e�ectively. The approach is based on the same

ideas as described in Subsection 8.5.1, i.e., the constraints given by Eq. 8.22 are included

such that the need for using directional derivatives is avoided, but in this mathematical

programming formulation a set of the lowest eigenvalues �j; j = 1; : : : ; m, is included

in each design step, ensuring better convergence properties than the algorithm based on

necessary optimality conditions described in the preceding section where only the lowest

eigenvalue is considered in each design step. The implementation of this mathematical

programming approach has been made in cooperation with my collegue Lars Krog.

The basic idea is to replace the original objective function in Eq. 8.14 with a linearized

prediction of the objective function in the next iteration, i.e., the optimization problem,

cf. Eqs. 8.14 and 8.15, is reformulated as

Maximize min �j +��j; j = 1; : : : ; m (8.35)

�a1; : : : ;�aI

Subject to

F (a1; : : : ; aI) = 0 (8.36)

fTsk�a = 0; s 6= k (8.37)

�ai � �ai � �ai; i = 1; : : : ; I (8.38)

The additional N(N � 1)=2 constraints in Eq. 8.37 for each N -fold multiple eigenvalue

make it possible to compute the linear increment of both simple and multiple eigenvalues

�j in the form

��j = fjj �a (8.39)

where the generalized gradient vector fjj is de�ned by Eq. 4.45, i.e., the linear increment of

eigenvalue �j can be related directly to eigenvector �j, although the eigenvalue is multiple.

The optimization problem in Eqs. 8.35-8.38 is rewritten using the bound formulation

described in Section 7.3, i.e.,

Maximize � (8.40)

�ai; �

148 8.6. Example: Maximization of Lowest Eigenvalue of Ribbed Plate

Subject to

�j +��j � � ; j = 1; : : : ; m (8.41)

F (a1; : : : ; aI) = 0 (8.42)

fTsk�a = 0; s 6= k (8.43)

�ai � �ai � �ai; i = 1; : : : ; I (8.44)

This mathematical programming problem is solved in each design iteration, and as the

process converges to the optimum point, the vector �a tends to the null vector. The

necessary optimality conditions derived in Section 8.4 then can be applied to check that

the optimum solution is obtained.

The mathematical programming approach described in this section has proven to be very

e�ective as will be demonstrated in the next section where this iterative numerical method

will be used for solution of an eigenvalue optimization problem involving multiple eigen-

values.

8.6 Example: Maximization of Lowest Eigenvalue of

Ribbed Plate

In order to illustrate the e�ciency of the mathematical programming approach for solving

eigenvalue optimization problems with multiple eigenvalues as described in the preceding

section, a reinforced plate example is studied.

The aim of the optimization is to maximize the lowest eigenfrequency of the ribbed plate

introduced in Section 6.6, taking a volume constraint into account. The plate is clamped

at all edges and all dimensions can be seen in Fig. 8.2.

The plate is made of steel with the following material properties

Young's modulus = 210000 MPa

Poisson's ratio = 0.3

Mass density = 7800 Kg/m3

The �nite element model consists of 1156 4-node isoparametric Mindlin plate �nite ele-

ments, the lowest 6 eigenfrequencies are calculated, and eigenfrequencies are considered to

be identical if the relative di�erence between the values is � 10�4. The eigenmodes for the

initial design are shown in Fig. 8.3.

In order to improve the design of the plate, �ve design variables are de�ned. For simplicity,

only 5 symmetric design variables are de�ned, that is, the plate thickness, two di�erent

rib thicknesses, the distances a between the ribs, and the width b of the ribs as shown in

Fig. 8.2.


ab b

bb

a

ribs ribs

plate

L

L

: plate thickness

: rib thickness 1

: rib thickness 2

= 0.005 m

= 0.05 m

= 0.05 m

a

b

= 0.5 m

= 0.05 m

L = 1.60 m

Figure 8.2: Initial design and design variables of square plate reinforced by ribs.

f1 = 92.1974 Hz. f2 = 161.7083 Hz.

f3 = 161.7083 Hz. f4 = 175.0327 Hz.

f5 = 176.0789 Hz. f6 = 177.0941 Hz.

Figure 8.3: Eigenmodes for initial design of reinforced plate.

The SIMPLEX algorithm is used as optimizer and 100 design iterations are performed

using a move limit factor of 2%, i.e., the maximum change of a design variable in each

design iteration is 2%. The optimized design is shown in Fig. 8.4.

The side constraint for minimum rib width b becomes active in the optimized design as

could be expected, see, e.g., the discussions about the \solid plate paradox" in Rozvany,

Olho�, Cheng & Taylor (1982), see also Olho� (1974, 1975) and Cheng & Olho� (1981).

The optimization history is shown in Fig. 8.5 which demonstrates a stable convergence


ab b

bb

a

: plate thickness

: rib thickness 1

: rib thickness 2

= 0.006558 m

= 0.1637 m

= 0.1621 m

a

b

= 0.4945 m

= 0.01 m

Figure 8.4: Optimized design.

and comparisons between eigenfrequencies of the initial and the �nal design are given in

Table 8.1. The lowest eigenfrequency is increased by more than 106%.

Iteration number

Eigenfrequency [Hz]

0 10 20 30 40 50 60 70 80 90 10080

100

120

140

160

180

200

220

3 1

2

3

4

5

6

ωωωωωω

Figure 8.5: Optimization history.

It is seen that the lowest eigenfrequency of the optimized design is distinct although the

di�erence between the �rst and second eigenfrequency is less than 0.08%. During the

optimization process the two lowest eigenfrequencies coalesce in several design iterations.

The third and fourth eigenfrequency remain multiple in all design iterations, so the number

of additional constraints given by Eq. 8.22 is one or two in each design iteration, depending

on the multiplicity of the lowest eigenfrequency. The eigenmodes for the optimized design

are shown in Fig. 8.6.

This example illustrates that the mathematical programming approach described in Section

8.5 can be used e�ectively for design problems involving multiple eigenvalues. However,

let us slightly modify the example in order to obtain a multiple lowest eigenfrequency.

Next four supporting springs with sti�nesss k = 109 N/m are added to the structure as


Table 8.1: Result of design optimization for reinforced plate.

Frequency: Initial design Final design

f1 92.1974 190.3866

f2 161.7083 190.5347

f3 161.7083 191.9949

f4 175.0327 191.9949

f5 176.0789 193.2443

f6 177.0941 208.1040

f1 = 190.3866 Hz. f2 = 190.5347 Hz.

f3 = 191.9949 Hz. f4 = 191.9949 Hz.

f5 = 193.2443 Hz. f6 = 208.1040 Hz.

Figure 8.6: Eigenmodes for optimized design of reinforced plate.

illustrated in Fig. 8.7. Obviously, this increases the lowest eigenfrequency and results in

a dense spectrum of eigenfrequencies as can be seen in Fig. 8.8 where the eigenmodes for

the initial design are shown.

Again, 100 design iterations are performed, the lowest eigenfrequency is increased by 29%

and becomes double in the optimized design. None of the side constraints are active in the

design iterations and the optimized design is shown in Fig. 8.9.

The iteration history is illustrated in Fig. 8.10 and comparisons between initial and �nal

values of eigenfrequencies are given in Table 8.2. It is seen that the �rst and second eigen-

frequency and the fourth and �fth eigenfrequency coalesce, and the third eigenfrequency

is nearly equal to the lowest double eigenfrequency as the relative di�erence is less than

2 � 10�4. This dense spectrum of lowest eigenfrequencies in the �nal design results in two


ab b

bb

a

L

L

: plate thickness

: rib thickness 1

: rib thickness 2

= 0.005 m

= 0.05 m

= 0.05 m

a

b

= 0.5 m

= 0.05 m

kk

kk

L = 1.60 m

Figure 8.7: Initial design and design variables of square plate reinforced by ribs and with

four supporting springs.

f1 = 170.2444 Hz. f2 = 175.0327 Hz.

f3 = 176.5525 Hz. f4 = 176.5525 Hz.

f5 = 177.6583 Hz. f6 = 178.9780 Hz.

Figure 8.8: Eigenmodes for initial design of reinforced plate with four springs.

additional constraints given by Eq. 8.22 as eigenfrequencies are considered to be identi-

cal by the optimization algorithm if the relative di�erence between the values is � 10�4,

that is, the lowest and the fourth eigenfrequency is considered double by the optimization

algorithm.


ab b

bb

a

: plate thickness

: rib thickness 1

: rib thickness 2

= 0.007888 m

= 0.04766 m

= 0.07906 m

a

b

= 0.5069 m

= 0.02082 m

Figure 8.9: Optimized design.

The iteration history demonstrates a stable convergence process.

Iteration number

Eigenfrequency [Hz]

0 10 20 30 40 50 60 70 80 90 100170

180

190

200

210

220

230

240

3 1

2

3

4

5

6

ωωωωωω

Figure 8.10: Optimization history.

The eigenmodes for the optimized design can be seen in Fig. 8.11.

This example illustrates that the mathematical programming approach described in Section

8.5 is very e�cient in case of multiple eigenvalues. Correct sensitivity information is used

in each design step due to the additional constraints given by Eq. 8.22, which is the reason

for the stable convergence.


Table 8.2: Result of design optimization for reinforced plate with four springs.

Frequency: Initial design Final design

f1 170.2444 219.6628

f2 175.0327 219.6628

f3 176.5525 219.6980

f4 176.5525 219.9389

f5 177.6583 219.9389

f6 178.9780 231.4708

f1 = 219.6628 Hz. f2 = 219.6628 Hz.

f3 = 219.6980 Hz. f4 = 219.9389 Hz.

f5 = 219.9389 Hz. f6 = 231.4708 Hz.

Figure 8.11: Eigenmodes for optimized design of reinforced plate with four springs.

Chapter

9

Examples of Interactive Engineering

Design with ODESSY

9.1 Introduction

In this chapter it will be illustrated by means of examples how some of the facilities

of ODESSY can be used for interactive engineering design. For reasons of brevity, only

a few representative examples will be presented here; a demonstration of all the facilities

of the system would require a large number of additional examples.

In Section 9.2 it will be illustrated how design sensitivity display and what-if studies can

be used to improve the design of a turbine disk. The use of design sensitivity display can

greatly improve the design synthesis process as colour design sensitivity contours provide

the designer information about which design variables are critical to some performance

measure or the performance measures that are most e�ected by changing a particular

design variable.

The use of design optimization based on mathematical programming is illustrated in Sec-

tion 9.3 for improvement of the design of the turbine disk introduced in the preceding

section. The objective is to reduce the mass moment of inertia in order to increase the ac-

celeration capabilities of the disk, and at the same time, a quite complicated temperature

dependent stress constraint and a manufacturing constraint on the minimum boundary

radius of curvature are taken into account.

The next section illustrates how the turbine disk can be designed using a ceramic material.

For such design cases the probability of failure must be evaluated. A reliability evaluation

based on a two-parameter Weibull distribution has been generally accepted in design of

ceramics and has therefore been implemented in ODESSY. The low mass density of the

ceramic material reduces the mass moment of inertia signi�cantly, compared to the design

made of steel, and the objective then is to reduce the probability of failure of the disk.

Finally, the shape optimization of a shell structure in the form of the hood of a Mazda 323

automobile with the objective of maximizing the fundamental frequency of free vibrations

155

156 9.2. Design Sensitivity Display and What-If Studies of Turbine Disk

is shown.

9.2 Design Sensitivity Display and What-If Studies of

Turbine Disk

With expressions for design sensitivities of displacements, stresses, compliance, eigenvalues,

etc., at hand, this section will brie y illustrate how design sensitivity analysis can be used to

improve engineering designs. A turbine disk will be used as an example, and the relations

between the parameterized design model and the �nite element analysis model are also

illustrated.

Previously, when using the traditional design process, the designer was required to use

intuition and trial and error procedures to �nd ways of improving the design. Nowadays,

by using a structural analysis program which has capabilities for design sensitivity analysis,

the e�ciency of the design process can be highly improved. Through the use of, e.g., colour

stress contour plots together with colour stress design sensitivity contour plots, the engineer

easily identi�es critical regions in which design improvements can be made.

To illustrate this, let us consider the problem of the rotating turbine disk of Figs. 9.1 and

9.2.

Figure 9.1: Initial design of turbine disk.

The disk has blades attached to its circumference and is driven by hot gas. Only the

design of the cross section of the turbine disk is considered so an axisymmetric model can

be used as illustrated in Fig. 9.2. The blades give rise to a �xed centrifugal force which

is modelled by a uniformly distributed load of value 310 MPa at the circumference of the

Chapter 9. Examples of Interactive Engineering Design with ODESSY 157

20

573 K

723

K

100

5010

16°

20

axis ofrevolution

convection

convection

master node

fixed boundarydesign boundary

= 2094 rad/s

R10

ω

310

MP

a

Figure 9.2: Design model of cross section of turbine disk.

disk, and the gas exposes the circumference to a relatively high temperature of 723 K. The

centre of the disk is attached to a relatively cold shaft of temperature 573 K, and forced

convection of heat to the surroundings takes place in the region between the blades and

the shaft. At these boundaries the temperature of the environment is speci�ed to 723 K

and the convection coe�cient is 0.0012 W/(mm2�K). At maximum speed, the disk rotates

at 2094 rad/s (= 20000 rev/min). The disk is made of steel with the mass density 7.75

Kg/mm3, Young's modulus 180000 MPa, Poisson's ratio 0.3, thermal expansion coe�cient

1.2 � 10�5 K�1, and thermal conductivity coe�cient 0.027 W/(K�mm).The design model consists of two design elements. There are two design boundaries, i.e.,

boundaries whose shapes are allowed to change. Each of these shapes are de�ned by the

positions of a number of master nodes, and this creates an evident connection between the

design variables (the movements of the master nodes) and the shape of the geometry.

In this example, the direction of the movement of each master node is constrained to follow

some prede�ned translation directions speci�ed by the designer as shown in Fig. 9.3. Thus,

in this example, the design variables are simply taken to be the sizes of the movements of

the master nodes along the associated translation directions.

In addition, the distribution of �nite element nodes on the boundaries and the desired

�nite element type for each design element must be de�ned. All necessary speci�cations

including loading conditions are assigned to the design model, and the preprocessor with

facilities for automatic mesh generation automatically converts the design model into a

�nite element analysis model as shown in Fig. 9.4. The preprocessor has meshed the

design elements with a mixture of 6 and 9 node isoparametric 2-D axisymmetric �nite

elements. All load speci�cations are automatically converted into consistent nodal loads.

The design model now has been converted into an analysis model, and by changing the


axis ofrevolution

master node

fixed boundary

design boundary

vector defining allowabletranslation direction formaster node

Figure 9.3: Variable design model.

prescribed temperature = 573 K

convection boundary conditions

consistentnodal loads

prescribedtemperature= 723 K

convection boundary conditions

axis ofrevolution

= 2094 rad/sω

Figure 9.4: Finite element analysis model.

values of the design variables, the geometry can be changed parametrically into other

shapes.

Now, as a starting point we would like to:

Decrease the maximum von Mises reference stress

The convection boundary condition as well as the ow of heat from the hot circumference

to the cold shaft clearly change with the design and give rise to a varying stress �eld, and


so do the centrifugal force of the disk. This problem is therefore quite complex, because

the stresses depend on the design, the temperatures, and the forces, which again depend

on the design.

The temperature �eld and the von Mises stress �eld in the initial disk are displayed in

Figs. 9.5 and 9.6, respectively. In Fig. 9.7 the von Mises stresses are shown in the region

near the lower boundary where the maximum value is found.

ODESSY Postprocessor

Name: wheel

Date: Jul 19 1993 19:07

TEMPERATURES

723.00

698.00

673.00

648.00

623.00

598.00

573.00

Figure 9.5: Temperature �eld in turbine disk.


Name: wheel

Date: Jul 19 1993 19:07

STRESS LEVELS

von Mises

5.711E+008

4.761E+008

3.811E+008

2.861E+008

1.911E+008

9.610E+007

1.095E+006

Figure 9.6: von Mises stress �eld in turbine disk.



Name: wheel

Date: Jun 7 1994 16:39

STRESS LEVELS

von Mises

5.711E+008

4.761E+008

3.811E+008

2.861E+008

1.911E+008

9.610E+007

1.095E+006

Figure 9.7: Zoom of region with largest von Mises stress.

As was already indicated in Fig. 9.3, the disk has been assigned 20 shape design variables

which control the shape of the structural domain, and the geometry may be perturbed for

each of these variables in order to display stress design sensitivities with colour contour

plots.

axis ofrevolution

master node

fixed boundary

design boundary

vector defining allowabletranslation direction formaster node

Figure 9.8: The two selected design variables for which sensitivities will be shown.

For the two particular design variables indicated in Fig. 9.8, corresponding stress design

sensitivity �elds are shown in Figs. 9.9 and 9.10. Each stress design sensitivity �eld is

associated with translation of the master node in the direction indicated by the arrow.



Name: wheel

Date: Jun 7 1994 16:41

STRESS LEVELS

von Mises

7.464E+010

5.000E+010

3.750E+010

2.500E+010

1.250E+010

0.000E+000

-9.270E+010

Figure 9.9: von Mises stress design sensitivity plot no. 1.


Name: wheel

Date: Jul 20 1993 19:13

STRESS LEVELS

von Mises

3.938E+010

5.000E+009

2.500E+009

0.000E+000

-2.500E+009

-5.000E+009

-6.428E+009

Figure 9.10: von Mises stress design sensitivity plot no. 2.

Fig. 9.9 shows that most of the large von Mises stresses in this region will be increased

if the master node is moved in the direction of the arrow. In other words, in order to

decrease the large stresses in this region, the master node must be moved in the opposite

direction of the arrow in Fig. 9.9.

Fig. 9.10 shows that the indicated inward movement of the master node results in negative

stress sensitivities in the highly stressed region at the lower boundary, i.e., a design change

of this kind has a desirable decreasing e�ect on the maximum von Mises stress. No doubt,

162 9.3. Shape Optimization of Turbine Disk

this is because such a design change will imply a decrease of the centrifugal forces in the

rotating disk.

When all 20 design perturbations have been performed and the corresponding design sen-

sitivity �elds determined, the designer can do a what-if study, where he decides on suitable

changes of the values of the design variables according to the stress sensitivity information.

The �rst result of a what-if study of the disk example is shown in Fig. 9.11, where changes

of all design variables have been guessed on the basis of the stress sensitivity information

obtained, and without considering, e.g., a constraint on the volume of the disk. This new

design implies a reduction of the maximum von Mises stress from 571 to 453 MPa.


Name: wheel

Date: Jul 20 1993 20:58

STRESS LEVELS

von Mises

4.525E+008

3.772E+008

3.018E+008

2.265E+008

1.511E+008

7.580E+007

4.524E+005

Figure 9.11: von Mises stresses in updated geometry obtained by what-if study.

Based on this improved design, a new design sensitivity study can be performed, and the

design of the disk presumeably can be further improved, if necessary. This leads us to

another use of design sensitivity analysis, namely in engineering design optimization.

9.3 Shape Optimization of Turbine Disk

The previous example illustrated that the designer can improve designs based on design

sensitivity information by doing what-if studies. However, if the designer is confronted with

a problem that involves more than just a few design variables and criteria to be taken into

account, it becomes very di�cult to survey and quantify the design sensitivity information

so as to make decisions regarding values of the design variables that would satisfy design

constraints and be any better than other alternative values. In such situations, facilities

for engineering design optimization based on an iterative solution procedure as described

in Section 7.8 can be used as will be illustrated on the turbine disk example introduced in

the preceding section.


When designing turbine disks or other thermo-elastic problems involving high tempera-

tures, it is very often necessary to consider the fact that the strength, in terms of yield

stress, of most materials is strongly dependent upon the temperature. Metals are often

assumed to have a constant strength up to a certain temperature and then a linear decay

as shown in Fig. 9.12 for the material used for the turbine disk. It is noted that this stress

constraint has a non-di�erentiable behaviour.

Allowable von Mises stress [Mpa]600

500

400

300

200

100

0273 323 373 423 473 523573 623673 723773 823

Temperature [K]

σvM

Figure 9.12: Allowable von Mises stress.

Furthermore, the manufacturing process used for the turbine disk requires the minimum

boundary radius of curvature to be larger than 5 mm.

The objective of the shape optimization is to increase the acceleration capabilities of the

turbine disk, i.e., to minimize the mass moment of inertia, so the de�nition of the structural

optimization problem is


Subject to

maximum normalized von Mises stress � 1


This temperature dependent non-linear stress constraint is realized by normalizing the von

Mises stress at the i'th node with the function shown in Fig. 9.12. It should be noted that

the e�ect of this non-linear stress constraint at a given material point changes during the

optimization process as the temperature �eld changes with design. The de�nition of the

optimization problem can be rewritten as

This de�nition of the structural optimization problem can be realized using the database

module as described in detail in Section 7.7.

The temperature dependent normalized stress constraint is shown for the initial structure

in Fig. 9.13. The maximum value is 1:34 and corresponds to a violation of 34%.

Performing a shape optimization of the turbine disk example using a SIMPLEX algorithm

as optimizer leads to a reduction of the mass moment of inertia from 3:12 � 104 to 2:33 � 104Kg�m2. The stress constraint is now ful�lled as shown in Fig. 9.14. Here, the very even

164 9.4. Design Optimization of Ceramic Components


Subject to

maximum normalized stress

8>>>><>>>>:

�vM;i [MPa]550 if Ti � 623K

�vM;i [MPa]1484:5� 1:5Ti if Ti > 623K

� 1



Name: wheel

Date: Jun 7 1994 16:50

VNV LEVELS

1.340E+000

1.117E+000

8.938E-001

6.709E-001

4.480E-001

2.251E-001

2.182E-003

Vnv: maxval[ nstress(svm)/1.0E6 / [minvec [550; [1484.5 - [1.5*temp()] ] ] ] ] < 1.0

Figure 9.13: Temperature dependent stress constraint for initial design.

distribution of the normalized stress criterion over the domain should be noted. The �nal

design can also be seen in Fig. 9.15.

The �nal minimum boundary radius of curvature is 13:5 mm, i.e. this constraint is not

active in the �nal design. During the optimization process, this manufacturing constraint

became active in some iterations.

This example illustrates the generality and exibility of the database module described in

Chapter 7 for evaluating objective and constraint functions.

9.4 Design Optimization of Ceramic Components

This section is devoted to problems concerning design optimization of ceramic components.

Such design optimization problems are rarely discussed but design with ceramic materials

calls for use of structural shape optimization as design with new materials very often

cannot be based on design rules or engineering tradition. This will be demonstrated for



Name: wheel

Date: Jun 7 1994 16:51

VNV LEVELS

1.000E+000

8.335E-001

6.669E-001

5.004E-001

3.338E-001

1.673E-001

7.709E-004

Vnv: maxval[ nstress(svm)/1.0E6 / [minvec [550; [1484.5 - [1.5*temp()] ] ] ] ] < 1.0

Figure 9.14: Temperature dependent stress constraint for �nal design.

Figure 9.15: Optimized design of turbine disk.

the turbine disk introduced in the preceding sections. Discussions about multicriteria

design optimization of ceramic components can be found in a very interesting paper by

Koski & Silvennoinen (1990) contained in Eschenauer, Koski and Osyczka (1990) where

many other interesting applications of multicriteria design optimization are discussed.

In the last decades there has been an increasing use of ceramic materials in mechanical en-


ginering applications where good wear resistance properties, high hardness, su�cient high-

temperature capability, high sti�ness, and good corrosion resistance are needed. However,

design with ceramic components is di�erent from design with traditional ductile materials

due to the brittle behaviour of ceramics.

The use of ceramic materials for load carrying components involves two basic features

that must be taken into account in the design phase. First, even at high temperature,

the material has very low strain tolerance and practically exhibits no yielding. Thus, the

material behaviour is linearly elastic up to the fracture point where an unstable crack

growth suddenly takes place. Second, there is frequently large scatter in the strength

data so probabilistic methods must be used. A reliability evaluation based on a two-

parameter Weibull distribution has been generally accepted in design of ceramics, see,

e.g., McLean & Hartsock (1989). Weibull developed a probabilistic failure criterion based

only on tensile stresses in the component. Compressive failure is not considered in this

criterion because brittle materials usually fail from tensile stresses due to their very high

compressive strength.

The probability of failure is computed for a ceramic component from its stress �eld by

using the weakest link theory based on the Weibull distribution, i.e., it is assumed that

the weakest crack-stress combination will cause the overall damage, but the location of

this critical point is not known. The mean fracture stress �c is extremely sensitive to

small defects in the material which may be caused by the manufacturing process or by

environmental e�ects during the everyday use, and as the ceramic material properties are

variable due to the random distribution of aws in ceramics, there is a variation of strengths

for various parts of a component. So, the probability of failure of a ceramic component is

a function of the volume of material subjected to tensile stresses.

Accordingly, the mean fracture stress �c is associated with a certain reference volume Vcwhich usually encompasses that part of the test specimen where tensile stresses occur. The

most popular method of generating material data is to use test specimens subjected to four

point bending. The specimens are loaded to fracture and the mean fracture stress �c is

calculated for a corresponding reference volume Vc. Furthermore, the Weibull modulus m

is determined. The Weibull modulus m can be interpreted as a measure of the narrowness

of the strength distribution, i.e., the larger the value m, the smaller is the variation of the

fracture stress of the material.

For uniaxial stress, Weibull established the following function that describes the cumulative

probability of failure Pf of a ceramic component:

Pf = 1� exp��1

m!�m � 1

�c

�m � 1

Vc

� ZV

�mdV

�(9.1)

where � is the tensile stress at a given point. The term

�1

m!�= �

�1

m+ 1

�=Z 1

0t1

m exp [�t] dt (9.2)

is the value of the gamma function � at 1m+ 1 which is easily evaluated.

In order to expand Eq. 9.1 for three-dimensional stress states, the concept of integrating


the normal stress �n around the portion of the unit radius sphere where the normal stress

is positive is generally used, see, e.g., McLean & Hartsock (1989) and Fig. 9.16.

dA = cos d d

σ

σ

σ

ψ

ϕ

ϕ ϕ ψσ

n

1

3

2

Figure 9.16: Geometric variables describing location on the unit sphere.

Thus, the general equation for the probability of failure is

Pf = 1� exp��1

m!�m � 1

�c

�m � 1

Vc

� ZV

2m + 1

2�

(Z 2�

0

Z �=2

0

hcos2 �

��1 cos

2 + �2 sin2 �+ �3 sin

2 �im

cos� d� d

)dV

# (9.3)

The integration over the unit sphere can be carried out by numerical integration for any

value of the Weibull modulus m, or analytically for discrete values of m. In the imple-

mentation in ODESSY, I have chosen the latter approach. When evaluating the volume

integral in Eq. 9.3, the order of Gauss quadrature used for the numerical integration in

general depends on m, but in the current implementation in ODESSY, the Gauss points

used for evaluating the stresses are also used for computing the volume integral. This

approximation might result in inaccurate results in case of large stress gradients.

In the case of a three-dimensional stress state it might be computationally advantageous

to neglect the mutual interrelationsship between the tensile principal stresses whereby the

following expression for the probability of failure is obtained:

Pf = 1� exp��1

m!�m � 1

�c

�m � 1

Vc

� ZV

(�m1 + �m2 + �m3 ) dV�

(9.4)

This approximation might be acceptable if the second and third principal stresses are

small percentages of the maximum principal stress. In case of other stress states, this

approximation may lead to unacceptable errors. The principal stresses are only included

in the volume integral if they are positive, i.e., �i < 0 ) �mi = 0. Both Eqs. 9.3 and 9.4

have been implemented in ODESSY.

To illustrate the use of this theory for determining the probability of failure of ceramic

components, let us redesign the turbine disk introduced in the preceding sections using a

ceramic material. Silicon nitride is chosen as material due to its strength properties at high

temperatures and its low mass density compared to other ceramics, e.g., zirconia. The ma-

terial properties are standard values given by a manufacturer (Kyocera Corporation). The


chosen sintered silicon nitride has mass density 3.20 Kg/mm3, Young's modulus 304000

MPa, Poisson's ratio 0.27, thermal expansion coe�cient 3.0 � 10�6 K�1, and thermal con-

ductivity coe�cient 0.029 W/(K�mm). The data used for computing the probability of

failure are the mean fracture stress �c = 785 MPa (at 723 K), the corresponding reference

volume Vc = 13.5 mm3, and the Weibull modulus m = 13.

It is assumed that the turbine blades have been redesigned in silicon nitride, resulting in

smaller centrifugal forces. The uniformly distributed load of value 310 MPa corresponding

to these centrifugal forces, see Fig. 9.2, is assumed to be reduced to 130 MPa due to the

lower mass density of silicon nitride. All other boundary conditions and load speci�cations

are unchanged but one of the major reasons for using ceramic materials in gas turbines

and turbo chargers is to allow higher working temperatures, whereby the e�ciency can be

improved.

The initial design is shown in Fig. 9.1 and the largest principal stresses are shown in Fig.

9.17 because the probability of failure highly depends on these stresses.


Name: wheelc

Date: Apr 24 1994 18:23

STRESS LEVELS

SIG1

2.397E+008

1.994E+008

1.590E+008

1.187E+008

7.833E+007

3.797E+007

-2.380E+006

Figure 9.17: Largest principal stresses for initial design of ceramic turbine disk.

The probability of failure is computed by using Eq. 9.3 because the �rst and second

principal stresses are of the same magnitude. The probability of failure of the initial design

is computed to 1.47�10�3, i.e., 1 out of 680 turbine disks will fail. The value obtained usingthe simpli�ed Eq. 9.4 di�ers from the value obtained by Eq. 9.3 by a factor of 3 due to

the stress state in the disk.

Next, automatic shape optimization is performed. The mass moment of inertia for the

initial design is 1.29�104 Kg�m2 which is 55% of the mass moment of inertia of the optimized

design in Section 9.3. However, the objective of the optimization process is to decrease

the probability of failure, so the mass moment of inertia is allowed to be increased. The

constraint on the minimum boundary radius of curvature is maintained, so the de�nition


of the structural optimization problem is


Minimize probability of failure

Subject to

the mass moment of inertia � 1.40�104 Kg�m2


Performing a shape optimization, the probability of failure is reduced from 1.47�10�3 to3.22�10�6, i.e., 1 out of 310000 disks will fail. This probability of failure seems to be

su�ciently reduced, but it is always very di�cult to choose an acceptable reliability. The

failure of a turbine disk may lead to a life-threatening situation so it is necessary to be

absolutely certain about material data, loading conditions, and boundary conditions.

The �nal design can be seen in Fig. 9.18, and the largest principal stresses for the optimized

design are shown in Fig. 9.19.

Figure 9.18: Optimized design of ceramic turbine disk.

The constraints on the mass moment of inertia and the minimum boundary radius of

curvature are both active for the �nal design, i.e., the mass moment of inertia of the

�nal design is 40% smaller than that of the optimized design obtained in Section 9.3.

The turbine disk designed and made of silicon nitride therefore has signi�cantly improved

acceleration capabilities.



Name: wheelc

Date: Apr 24 1994 18:19

STRESS LEVELS

SIG1

1.343E+008

1.119E+008

8.945E+007

6.703E+007

4.461E+007

2.218E+007

-2.379E+005

Figure 9.19: Largest principal stresses for �nal design of ceramic turbine disk.

9.5 Shape Optimization of an Automobile Hood

This example involves shape optimization of a shell structure in the form of the hood of

a Mazda 323 automobile with the objective of maximizing the fundamental frequency of

free vibrations. In practical shape design optimization it is usually advantageous to start

out with a relatively simple model in terms of geometry and �nite element representation.

The model is then re�ned iteratively as the process converges towards a good solution and

the designer acquires more knowledge about the nature of the problem. The �nal result

often will be the last in a long sequence of models. However, for reasons of brevity and

the purpose of illustration, we shall start out this example with a fairly accurate geometric

modeling of the original real-life structure and not attempt to generate improved models

along the way.

It is assumed that there is a constraint on the total amount of material such that the weight

of the optimized structure does not exceed the original one. The hood is made from steel

plates with thickness 1.0 mm, Young's modulus = 210 GPa, Poisson's ratio = 0.3 and the

mass density = 7800 kg/m3. The overall dimensions of the hood are approximately 1.25 x

0.85 m and must remain unchanged.

9.5.1 Model

The original geometry, which is visualized in Figs. 9.20 and 9.21, consists of two doubly

curved shells joined by welding. One shell is the external surface of the hood, and the other

shell, which is welded to the inner side of the hood, is a multi-connected set of sti�eners.

A total of 496 boundaries and 238 curved surface patches form the model. The shapes

of the surfaces are controlled by their boundaries which in turn depend on a number of

172 9.5. Shape Optimization of an Automobile Hood

Engine hood

Stiffening shell

Joined area

Figure 9.20: The underside of the original geometry. The black area is the joined region

(joined by welding), the dark gray is the sti�ening shell, and the light gray is

the upper side of the hood.

Engine hood

Stiffening shell

Joined area

B BC

A A

Figure 9.21: The original geometry with boundary conditions.

master nodes as described in Chapter 2. Modi�ers and design variables are de�ned such

that positions, heights and widths of the sti�eners can be changed continuously along

each sti�ener. The outer surface of the engine hood is not changed during the design


optimization as we do not want to a�ect the aerodynamical properties. Symmetry with

respect to the vertical midplane in Fig. 9.21 is imposed on possible design changes, and

the model then has a total of 32 independent design variables.

The �nite element model (see Figs. 9.21 and 9.22) is established using a mix of mapping

and free meshing of curved surfaces. Fig. 9.22 shows a section through the two shells

illustrating their interrelation.

Engine hood

Stiffening shell

Joined area

Figure 9.22: Section through the two shells.

As seen in Fig. 9.21, the structure is supported by two hinges attached to the edge of

the hood, and three support points on the opposite side. The hinges (A) �x all degrees of

freedom except for one rotation. The supports (B) �x the out-of-plane translational degree

of freedom while the support (C) �xes the out-of-plane and vertical translational degrees

of freedom.

Isoparametric shell elements with 6 and 8 nodes are used for the analysis, and the model

comprises a total of more than 20000 degrees of freedom. This modeling gives a rather

crude description of the geometry, but a test of the convergence shows that it is adequate

for the representation of the eigenmodes at hand.

9.5.2 Analysis

The analysis of the original geometry using the Subspace iteration method yields the

following three lowest eigenfrequencies:

f1 = 132 Hz

f2 = 177 Hz

f3 = 198 Hz

We see that none of these eigenfrequencies are multiple. The lowest eigenfrequency corre-

sponds to the eigenmode shown in Fig. 9.23.


Frequency: 132 Hz

Figure 9.23: First eigenmode for initial geometry.

9.5.3 Result

The �nal geometry is shown in Figs. 9.24 and 9.25, and is obtained after 12 iterations using

the SIMPLEX algorithm of ODESSY. The three lowest eigenfrequencies of the optimized

design are found to be:

f1 = 163 Hz

f2 = 182 Hz

f3 = 216 Hz

The minimum eigenfrequency of 163 Hz implies an increase 24% relative to that of the

original structure, and the �nal volume corresponds exactly to the original one.

It is obvious that rather large geometric changes have taken place, and many of the design

variables reach speci�ed maximum (or minimum) allowable values which indicates that the

topology of the initial geometry is not optimum for the case of maximizing the smallest

eigenfrequency. Thus, the �nal geometry may only be considered optimum within this

topology and the limitations we have set for variations of the design variables in the initial

mathematical formulation of the problem.


Engine hood

Stiffening shell

Joined area

Figure 9.24: Shape optimized geometry.

Engine hood

Stiffening shell

Joined area

Figure 9.25: Section through the two shells in optimized geometry.


Frequency: 163 Hz

Figure 9.26: First eigenmode for optimized geometry.

Chapter

10

Conclusions

Many different topics in the �eld of structural analysis, design sensitivity analysis

and optimization are covered in this Ph.D. project. These topics are related to the

development of a variety of capabilities which constitute part of the backbone of the general

purpose computer aided engineering design system ODESSY, and a short summary and

conclusions of the work presented are given in the following.

Analysis Capabilities for Structural Optimization

The �nite element method is used as the analysis tool and we have chosen to write our

own code. This makes it possible to design the system exactly to desired purposes. 19

di�erent isoparametric 2-D solids, 3-D solids, plate and shell �nite elements have been

implemented as described in Chapter 3. The �nite element module has facilities for static

stress analysis, natural frequency analysis, steady state thermal analysis, thermo-elastic

analysis, eigenfrequency analysis with initial stress sti�ening e�ects due to mechanical or

thermal loads, and linear buckling analysis with the possibility of including thermo-elastic

e�ects.

The �nite element library has reached a quite high level and has been programmed in a

very structured way such that it is easy to implement other �nite element types.

Design Sensitivity Analysis

Facilities for design sensitivity analysis have been implemented for all available analysis

modules, �nite element types, and design variable types as described in Chapters 4 and

5. The design variables of the structural design problem can be either geometrical design

variables like sizing or shape variables, material design variables like constitutive parame-

ters of materials, support design variables like the position of supports for the structure,

or loading design variables like the position of external loads applied to the structure.

The method of design sensitivity analysis used is the direct approach which is compu-

tationally e�cient and based on di�erentiation of the state equations. Using the direct

177

178

approach to design sensitivity analysis, derivatives of various �nite element matrices need

to be determined. It is not possible, in ODESSY, to establish analytical relations between

the derivatives of �nite element matrices and the available types of generalized shape

design variables due to the very general mesh generation and parameterization facilities

implemented. As described in Section 5.7, the unstructured mesh generators that may

be used to generate three-dimensional surface meshes include smoothing processes, and

analytical relations between the position of surface �nite element nodes and generalized

shape design variables are therefore not available. The semi-analytical approach where

derivatives of various �nite element matrices and vectors are approximated by computa-

tionally inexpensive �rst order �nite di�erences has therefore been chosen instead of the

analytical approach.

However, as discovered for static problems by Barthelemy & Haftka (1988), the semi-

analytical method of design sensitivity analysis is prone to large errors for certain types

of problems involving shape design variables. The inaccuracy problems may occur for

design sensitivities with respect to structural shape design variables in problems where

the displacement �eld is characterized by rigid body rotations which are large relative to

actual deformations of the �nite elements, i.e., for example in problems involving linearly

elastic bending of long-span, beam-like structures, and of plate and shell structures. This

error problem is entirely due to the �nite di�erence approximation involved in determining

various element matrix derivatives.

There has been developed a modi�ed semi-analytical method that is based on \exact"

numerical di�erentiation of element matrices (\exact" up to round-o� errors) by means of

computationally inexpensive �rst order �nite di�erences. The method of \exact" numerical

di�erentiation has been implemented for all analysis modules, �nite element types, and

design variable types as described in detail in Chapter 5. This method is computationally

e�cient, especially in problems involving many design variables as described in Section

5.9, because \exact" numerical derivatives of element matrices only need to be calculated

once. Furthermore, the method can even be implemented in connection with existing �nite

element codes where di�erent subroutines for computation of element sti�ness matrices are

only available as black-box routines.

The new method of semi-analytical design sensitivity analysis completely eliminates the

inaccuracy problems as has been demonstrated by several numerical examples of design

sensitivity analysis in Chapter 6. The traditional semi-analytical method of design sensi-

tivity analysis has been shown to give very accurate results for a classical �llet example,

i.e., in a case where the displacement �eld is characterized by small rigid body rotations rel-

ative to actual deformations of the �nite elements. For such design problems the traditional

semi-analytical method is completely reliable.

However, in cases where the displacement �eld entails dominance of rigid body rotations rel-

ative to actual deformations of the �nite elements, the traditional semi-analytical method

has been shown to be prone to yield erroneous sensitivities. This has been demonstrated

for static design sensitivity analysis of a long cantilever beam in Section 6.3 and of a thin

clamped square plate in Section 6.5. Furthermore, the inaccuracy problem has also been

demonstrated for thermo-elastic design sensitivity analysis of a two-material cantilever

Chapter 10. Conclusions 179

beam in Section 6.4. The inaccuracy problem can be expected in many thermo-elastic

design problems as displacement �elds with dominance of rigid body rotations often occur

in such design problems.

The design sensitivity analysis of simple as well as multiple eigenvalues of vibrating plates

reinforced by ribs has been shown to yield accurate results in Sections 6.6 and 6.7. One of

the major di�culties in computing sensitivities of multiple eigenvalues is to decide correctly

from numerical results the multiplicity of a given eigenvalue in cases where the spectrum

of eigenvalues is dense. An incorrect estimation of the multiplicity of a repeated eigenvalue

is shown to give erroneous sensitivities in Section 6.7 and this di�culty associated with

multiple eigenvalues, in addition to the lack of usual Fr�echet di�erentiability, has to be

taken into account when solving optimum design problems involving multiple eigenvalues.

Finally, the inaccuracy problem associated with the traditional semi-analytical method is

shown to occur also in dynamic design sensitivity analysis. This is illustrated by com-

putation of design sensitivies of simple as well as multiple free vibration frequencies of a

thin clamped square plate in Section 6.8. Furthermore, as shown in Section 6.9, if initial

stress sti�ening e�ects are taken into account when computing eigenfrequencies of vibrat-

ing plates, the error problem is even worse because errors in stress sensitivities from the

static design sensitivity analysis become accumulated in the dynamic design sensitivity

analysis.

The new approach to semi-analytical design sensitivity analysis based on \exact" numerical

di�erentiation of element matrices has been shown to yield accurate sensitivities for all

studies made and must be regarded as a very reliable and useful tool in a general purpose

computer aided engineering design system.

A General and Flexible Method of Problem De�nition

In Chapter 7 it has been demonstrated that it is not only possible but also relatively simple

to develop a general database module that handles problem formulations comprising any

mix of local, integral, min/max and max/min functions in a completely uniform way. The

mathematical basis for this is the bound formulation. This, in combination with a parser

capable of interpreting user de�ned expressions, makes it possible to perform analyses and

sensitivity analyses of user de�ned mathematical expressions and even to carry out the

sensitivity analysis by exact and e�cient symbolic di�erentiation of the expressions. The

approach described makes it very easy to expand the system by new available mathematical

expressions or by new basic information if new analysis facilities are to be implemented.

The integration of the database module in the module for postprocessing has made it

very easy to check the validity of de�nitions of objective and constraint functions, and,

furthermore, has equipped the system with a very useful tool for interactive, graphical

visualization of user de�ned mathematical expressions for design criteria.

180

Multiple Eigenvalues in Structural Design Problems

In Chapter 8 the di�culties of solving structural optimum design problems with multiple

eigenvalues are described. The basic di�culty is the lack of usual Fr�echet di�erentiability,

suct that traditional gradient based algorithms cannot be used in general, and directional

derivatives must be employed instead.

A very simple example has been used to illustrate the case of a bimodal eigenvalue. The

necessary optimality conditions for an optimum solution to the problem of maximizing the

lowest eigenvalue of a structure with a volume constraint are given, and they provide basic

insight in the di�culties connected with this type of optimization problems.

Based on the experience gained by design sensitivity analysis studies and on the basis of

deriving the necessary optimality conditions, di�erent approaches to the development of

iterative numerical algorithms for eigenvalue optimization problems are described. The

basic idea of the approaches is to add constraints on the allowable design changes in cases

of multiple eigenvalues, whereby design sensitivity expressions for simple as well as mul-

tiple eigenvalues become identical. The major advantage of using this approach is that

the problem of determining the multiplicity of an eigenvalue correctly is no longer crucial

when these additional constraints are considered. It has been shown that the necessary

optimality conditions can be applied for development of an iterative numerical method for

optimization of structural eigenvalues of arbitrary multiplicity. Thus, an e�ective and ele-

gant algorithm based on a mathematical programming approach has been developed. This

mathematical programming approach has been used to solve the problem of maximizing

the lowest eigenfrequency of a plate reinforced by ribs. If four supporting springs are added

to the structure, the optimum solution has a bimodal lowest eigenvalue. The mathematical

programming approach has been shown to work very well for problems involving multiple

eigenvalues.

Interactive Engineering Design with ODESSY

The di�erent topics covered in this thesis have all been investigated with the aim of devel-

oping a general, exible, and reliable computer aided environement for interactive rational

design, and several examples have been presented with a view to illustrate that ODESSY is

a very e�ective tool for interactive engineering design. The use of design sensitivity display

and what-if studies for improving engineering designs has been illustrated via a turbine

disk example. The use of design sensitivity display can greatly improve the design synthesis

process as colour design sensitivity plots provide the designer with information concerning

the in uence of design variables on performance measures. The use of automated design

optimization has been illustrated on various design improvements of the turbine disk and

on shape optimization of a hood of an automobile. Complicated design criteria can be

taken into account, thereby making it easier for designers in industry to adopt and use

ODESSY for design synthesis and optimization.

The optimum design techniques implemented in ODESSY have proved to be invaluable

tools in the development of new products. Not only do they speed up the development

process, but they often enable signi�cant improvements of existing or even classical me-

chanical components. Optimization techniques are also most important in connection with

design with new materials, e.g., ceramics or composites, where design rules and engineering

tradition are often sparse or unavailable.

Further Work

Although a very general design system has been developed, there are still many possibil-

ities for extensions and improvement. The e�orts pertaining to integration of ODESSY

into commercial CAD systems, to develop facilities for automatic three-dimensional mesh

generation, and other topics related to the pre- and postprocessing facilities will continue.

Furthermore, the system should be expanded with respect to the classes of problems that

can be handled in terms of implemented types of �nite elements, analysis capabilities, and

hence new types of structures and design objectives. Currently, only at shell elements

generated from plate �nite elements are available, so improved shell elements should be

implemented. Furthermore, several other three-dimensional solid �nite elements must be

implemented when unstructured three-dimensional mesh facilities become available. Facil-

ities for adaptive mesh generation can improve the reliability of the �nite element model

used during the redesign process and can therefore become a valuable tool for shape op-

timization. The extension into transient and non-linear types of analysis problems is a

large task but such capabilities are very often necessary in order to solve practical real-life

problems. Furthermore, expansions into other �elds of analysis, for instance uid dynam-

ics, acoustics, and magnetic �eld theory are other possibilities. Inclusion of design criteria

concerning fabrication cost can also be very useful. Expansions of analysis and design

facilities into these areas will greatly improve the general applicability of the system.

181

References

The references are listed alphabetically according to the last name of the author.

Arora, J.S. (1989): Interactive Design Optimization of Structural Elements. In: Dis-

cretization Methods and Structural Optimization - Procedures and Applications (Eds. H.A.

Eschenauer & G. Thierauf), pp. 10-16, Springer-Verlag, Berlin.

Arora, J.S.; Haug, E.J. (1979): Methods of Design Sensitivity Analysis in Structural

Optimization. AIAA Journal, Vol. 17, No. 9, pp. 970-974.

Atrek, E. R.; Gallagher, H.; Ragsdell, K.M.; Zienkiewicz, O.C. (1984) (Eds.):

New Directions in Optimum Structural Design, John Wiley & Sons, New York.

Barthelemy, B.; Chon, C.T.; Haftka, R.T. (1988): Sensitivity Approximation of

Static Structural Response. Finite Elements in Analysis and Design, Vol. 4, pp. 249-265.

Barthelemy, B.; Haftka, R.T. (1988): Accuracy Analysis of the Semi-Analytical

Method for Shape Sensitivity Analysis. AIAA Paper 88-2284, Proc. AIAA/ASME/-

ASCE/ASC 29th Structures, Structural Dynamics and Materials Conference, Part 1, pp.

562-581. Also Mechanics of Structures and Machines, Vol. 18, pp. 407-432 (1990).

Bathe, K.-J. (1982): Finite Element Procedures in Engineering Analysis, Prentice-Hall,

New Jersey.

Bathe, K.-J.; Ramaswamy, S. (1980): An Accelerated Subspace Iteration Method.

Computer Methods in Applied Mechanics and Engineering, Vol. 23, pp. 313-331.

Bends�e, M.P. (1989): Optimal Shape as a Material Distribution Problem. Structural

Optimization, Vol. 1, No. 4, pp. 193-202.

Bends�e, M.P. (1994): Methods for the Optimization of Structural Topology, Shape and

Material. Mathematical Institute, Technical University of Denmark, Lyngby, Denmark, 299

pp.

Bends�e, M.P.; Kikuchi, N. (1988): Generating Optimal Topologies in Structural

Design Using a Homogenization Method. Computer Methods in Applied Mechanics and

Engineering, Vol. 117, pp. 197-224.

Bends�e, M.P.; Mota Soares, C.A. (1993) (Eds.): Topology Design of Structures,

Kluwer Academic Publishers, Dordrecht, The Netherlands.

Bends�e, M.P.; Olho�, N.; Taylor, J.E. (1983): A Variational Formulation for

Multicriteria Structural Optimization. Journal of Structural Mechanics, Vol. 11, pp. 523-

544.

183

Bends�e, M.P.; Rodrigues, H.C. (1990): On Topology and Boundary Variations in

Shape Optimization. Control and Cybernetics, Vol. 19, No. 3-4, pp. 9-26.

Bends�e, M.P.; Rodrigues, H.C. (1991): Integrated Topology and Boundary Shape

Optimization of 2-D Solids. Computer Methods in Applied Mechanics and Engineering,

Vol. 87, pp. 15-34.

Bends�e, M.P.; Rodrigues, H.C.; Rasmussen, J. (1991): Topology and Boundary

Shape Optimization As an Integrated Tool for Computer Aided Design. In: Engineering

Optimization in Design Processes (Eds. H.A. Eschenauer, C. Mattheck & N. Olho�), pp.

27-34, Springer-Verlag, Berlin.

Bennett, J.A.; Botkin, M.E. (1985): Structural Shape Optimization with Geometric

Description and Adaptive Mesh Re�nement. AIAA Journal, Vol. 23, No. 3, pp. 458-464.

Bennett, J.A.; Botkin, M.E. (1986) (Eds.): The Optimum Shape. Automated Struc-

tural Design, Plenum Press, New York.

Birker, T.; Lund, E. (1991): Development of a 3D Computer Aided Optimal Design

System, Master Thesis. Institute of Mechanical Engineering, Aalborg University, Denmark.

Botkin, M.E. (1988): Shape Optimization with Multiple Loading Conditions and Mesh

Re�nement. AIAA Paper 88-2299, Proc. AIAA/ASME/ASCE/AHS/ASC 29th Struc-

tures, Structural Dynamics and Materials Conf. (held in Williamsburg, Va., April 18-20),

Vol. 2, pp. 689-695.

Botkin, M.E. (1992): Three-Dimensional Shape Optimization Using Fully Automatic

Mesh Generation. AIAA Journal, Vol. 30, No. 7, pp. 1932-1934.

Botkin, M.E.; Bajorek, D.; Prasad, M. (1992): Feature-Based Structural Design

of Solid Automotive Components. In: DE-Vol.42, Design Theory and Methodology -DTM

'92 (Eds. D.L. Taylor & L.A. Stau�er), pp. 57-64.

Botkin, M.E.; Yang, R.J.; Bennett, J.A. (1986): Shape Optimization of Three-

dimensional Stamped and Solid Automotive Components. In: The Optimum Shape. Au-

tomated Structural Design (Eds. J.A. Bennett & M.E. Botkin), pp. 235-262, Plenum Press,

New York.

Braibant, V. (1986): Shape Sensitivity by Finite Elements. Journal of Structural Me-

chanics, Vol. 14, pp. 209-228.

Braibant, V.; Fleury, C. (1984): Shape Optimal Design Using B-splines. Computer

Methods in Applied Mechanics and Engineering, Vol. 44, pp. 247-267.

Braibant, V.; Morelle, P. (1990): Shape Optimal Design and Free Mesh Generation.

Structural Optimization, Vol. 2, pp. 223-231.

Bratus, A.S.; Seyranian, A.P. (1983): Bimodal Solutions in Eigenvalue Optimiza-

tion Problems. Prikl.Matem.Mekhan., Vol. 47, No. 4, pp. 546-554; Also in: Applied

Mathematics in Mechanics, Vol. 47, pp. 451-457.

Bremicker, M.; Chirehdast, M.; Kikuchi, N.; Papalambros, P. (1991): Integrated

Topology and Shape Optimization in Structural Design. Mechanics of Structures and

Machines, Vol. 19, pp. 551-587.

184

Cea, J. (1981): Problems of Shape Optimal Design. In: Optimization of Distributed

Parameter Structures (Eds. E.J. Haug & J. Cea), Vol. 2, pp. 1005-1048, Sijtho� &

Nordho�, The Netherlands.

Cheng, G.; Gu, Y.; Wang, X (1991): Improvement of Semi-Analytic Sensitivity

Analysis and MCADS. In: Engineering Optimization in Design Processes (Eds. H.A.

Eschenauer, C. Mattheck & N. Olho�), pp. 221-223, Springer-Verlag, Berlin.

Cheng, G.; Gu, Y.; Zhou, Y. (1989): Accuracy of Semi-Analytical Sensitivity Anal-

ysis. Finite Elements in Analysis and Design, Vol. 6, pp. 113-128.

Cheng, G.; Liu, Y. (1987): A New Computational Scheme for Sensitivity Analysis.

Engineering Optimization, Vol. 12, pp. 219-235.

Cheng, G.; Olho�, N. (1981): An Investigation Concerning Optimal Design of Solid

Elastic Plates. International Journal of Solids and Structures, Vol. 17, pp. 305-323.

Cheng, G.; Olho�, N. (1991): New Method of Error Analysis and Detection in Semi-

Analytical Sensitivity Analysis. Report No. 36, Institute of Mechanical Engineering, Aal-

borg University, Denmark, 34 pp.

Cheng, G.; Olho�, N. (1993): Rigid Body Motion Test Against Error in Semi-

Analytical Sensitivity Analysis. Computers & Structures, Vol. 46, No. 3, pp. 515-527.

Choi, K.K. (1985): Shape Design Sensitivity Analysis of Displacement and Stress Con-

straints. Journal of Structural Mechanics, Vol. 13, pp. 27-41.

Choi, K.K.; Chang, K.-H. (1991): Shape Design Sensitivity Analysis and What-

If Workstation For Elastic Solids. Technical Report R-105, Center for Simulation and

Design Optimization and Department of Mechanical Engineering, The University of Iowa,

33 pp.

Choi, K.K.; Haug, E.J. (1981a): Optimization of Structures with Repeated Eigenval-

ues. In: Optimization of Distributed Parameter Structures (Eds. E.J. Haug & J. Cea),

Vol. 1, pp. 219-277, Sijtho� & Nordho�, The Netherlands.

Choi, K.K.; Haug, E.J. (1981b): Repeated Eigenvalues in Mechanical Optimization

Problems. Contemporary Mathematics, Vol. 4, pp. 61-86.

Choi, K.K.; Haug, E.J. (1983): Shape Design Sensitivity Analysis of Elastic Structures.

Journal of Structural Mechanics, Vol. 11, No. 2, pp. 231-269.

Choi, K.K.; Haug, E.J.; Lam, H.L. (1982): A Numerical Method For Distributed Pa-

rameter Structural Optimization Problems with Repeated Eigenvalues. Journal of Struc-

tural Mechanics, Vol. 10, No. 2, pp. 191-207.

Choi, K.K.; Haug, E.J.; Seong, H.G. (1983): An Iterative Method For Finite Di-

mensional Structural Optimization Problems with Repeated Eigenvalues. International

Journal for Numerical Methods in Engineering, Vol. 19, pp. 93-112.

Choi, K.K.; Santos, J.L.T.; Frederick, M.C. (1985): Implementation of Design

Sensitivity Analysis with Existing Finite Element Codes. Center for Computer Aided

Design and Department of Mechanical Engineering, The University of Iowa, 9 pp.

Choi, K.K.; Seong, H.W. (1986): A Domain Method for Shape Design of Built-up

Structures. Computer Methods in Applied Mechanics and Engineering, Vol. 57, pp. 1-15.

185

Choi, K.K.; Twu, S.-L. (1988): On Equivalence of Continuum and Discrete Methods

of Shape Sensitivity Analysis. AIAA Journal, Vol. 27, No. 10, pp. 1418-1424

Clarke, F. (1990): Optimization and Nonsmooth Analysis, Classics in Applied Mathe-

matics 5. Society for Industrial and Applied Mathematics, Philadelphia.

Cook, R.D.; Malkus, D.S.; Plesha, M.E. (1989): Concepts and Applications of

Finite Element Analysis, 3rd Edition, Wiley & Sons, New York.

Courant, R.; Hilbert, D. (1953): Methods of Mathematical Physics, Vol. 1, Interscience

Publishers, New York.

Cox, S.J. (1992): The Shape of the Ideal Column. The Mathematical Intelligencer, Vol.

14, pp. 6-24.

Cox, S.J.; Overton, M.L. (1992): On the Optimal Design of Columns against Buckling.

SIAM Journal on Mathematical Analysis, Vol. 23, No. 2, pp. 287-325.

Dems, K.; Haftka, R.T. (1989): Two Approaches to Sensitivity Analysis for Shape

Variation of Structures. Mechanics of Structures and Machines, Vol. 16, No. 4, pp.

501-522.

Dems, K.; Mr�oz, Z. (1983): Variational Approach by Means of Adjoint Systems to

Structural Optimization and Sensitivity Analysis - I Variation of Material Parameters

within Fixed Domain. International Journal of Solids and Structures, Vol. 19, pp. 677-

692.

Dems, K.; Mr�oz, Z. (1984): Variational Approach by Means of Adjoint Systems to

Structural Optimization and Sensitivity Analysis - II Structure Shape Variation. Interna-

tional Journal of Solids and Structures, Vol. 20, pp. 527-552.

Dems, K.; Mr�oz, Z. (1993): On Shape Sensitivity Approaches in the Numerical Anal-

ysis of Structures. Structural Optimization, Vol. 6, pp. 86-93.

Dhatt, G.; Touzot, G. (1984): The Finite Element Method Displayed, Wiley & Sons,

New York.

Diaz, A.R.; Kikuchi, N. (1992): Solution to Shape and Topology Eigenvalue Opti-

mization Problems using a Homogenization Method. International Journal for Numerical

Methods in Engineering, Vol. 35, pp. 1487-1502.

Diaz, A.R.; Kikuchi, N.; Papalambros, P.; Taylor, J.E. (1983): Design of an

Optimal Grid for Finite Element Methods. Journal of Structural Mechanics, Vol. 11, pp.

215-230.

Ding, Y. (1986): Shape Optimization of Structures: a Literature Survey. Computers &

Structures, Vol. 24, No. 6, pp. 985-1004.

El-Sayed, M.E.M.; Zumwalt, K.W. (1991): E�cient Design Sensitivity Derivatives

for Multi-Load Case Structures as an Integrated Part of Finite Element Analysis. Com-

puters & Structures, Vol. 40, No. 6, pp. 1461-1467.

Eschenauer, H.A. (1986): Numerical and Experimental Investigations of Structural

Optimization of Engineering Designs, Siegen: Bonn + Fries, Druckerei und Verlag.

Eschenauer, H.A.; Koski, J.; Osyczka, A. (1990) (Eds.): Multicriteria Design

Optimization, Springer-Verlag, Berlin.

186

Eschenauer, H.A.; Mattheck, C.; Olho�, N. (1991) (Eds.): Engineering Optimiza-

tion in Design Processes, Springer-Verlag, Berlin.

Eschenauer, H.A.; Olho�, N. (1983) (Eds.): Optimization Methods in Structural

Design, B.I.-Wissenschaftsverlag, Mannheim.

Eschenauer, H.A.; Thierauf, G. (1989) (Eds.): Discretization Methods and Structural

Optimization - Procedures and Applications, Springer-Verlag, Berlin.

Esping, B.J.D. (1983): MinimumWeight Design of Membrane Structures, Ph.D. Thesis,

Dept. Aeronautical Structures and Materials, The Royal Institute of Technology, Stock-

holm, Report 83-1.

Esping, B.J.D. (1984): Minimum Weight Design of Membrane Structures Using Eight

Node Isoparametric Elements and Numerical Derivatives. Computers & Structures, Vol.

19, No. 4, pp. 591-604.

Esping, B.J.D. (1986): The OASIS Structural Optimization System. Computers &

Structures, Vol. 23, No. 3, pp. 365-377.

Fenyes, P.A.; Lust, R.V. (1991): Error Analysis of Semi-Analytical Displacement

Derivatives for Shape and Sizing Variables. AIAA Journal, Vol. 29, pp. 271-279.

Fleury, C. (1987): Computer Aided Optimal Design of Elastic Structures. In: Computer

Aided Optimal Design: Structural and Mechanical Systems (Ed. C.A. Mota Soares), pp.


Fleury, C. (1989): CONLIN: An E�cient Dual Optimizer Based On Convex Approxi-

mation Concepts. Structural Optimization, Vol. 1, pp. 81-89.

Fleury, C.; Braibant, V. (1986): Structural Optimization: A New Dual Method Using

Mixed Variables. International Journal for Numerical Methods in Engineering, Vol. 23,

409-428.

Fleury, C.; Braibant, V. (1987): Structural Synthesis: Towards a Reliable CAD-tool.

In: Computer Aided Optimal Design: Structural and Mechanical Systems (Ed. C.A. Mota

Soares), pp. 137-149, Springer-Verlag, Berlin.

Gajewski, A. (1990): Multimodal Optimal Design of Structural Elements (A Survey).

Mechanika Teoretyczna i Stosowana, Vol. 28, pp. 75-92.

Gajewski, A.; Zyczkowski, M. (1988): Optimal Structural Design Under Stability

Constraints. Kluwer Academic Publishers, Dordrecht-Boston-London.

George, P.L. (1988): Modulef: G�en�eration Automatique de Maillages (in French). Col-

lection Didactique, edit�e par Institut National de Recherche en Informatique et en Au-

tomatique, pp. 75-80. ISBN 2-7261-0522-X, ISSN 0299-0733.

Gilmore, B.J.; Hoeltzel, D.A.; Azarm, S.; Eschenauer, H.A. (1993) (Eds.): Ad-

vances in Design Automation. Proc. 1993 ASME Design Technical Conferences - 19th

Design Automation Conference, Albuquerque, New Mexico, The American Society of Me-

chanical Engineers, New York.

Gu, Y.; Cheng, G. (1990): Structural Shape Optimization Integrated with CAD En-

vironment. Structural Optimization, Vol. 2, pp. 23-28.

187

Haber, R.B. (1987): A New Variational Approach to Structural Shape Sensitivity Anal-

ysis. In: Computer Aided Optimal Design: Structural and Mechanical Systems (Ed. C.A.

Mota Soares), pp. 573-587, Springer-Verlag, Berlin.

Haftka, R.T.; Adelman, H.M. (1989): Recent Developments in Structural Sensitivity

Analysis. Structural Optimization, Vol. 1, pp. 137 - 151.

Haftka, R.T.; Grandhi, R.V. (1986): Structural Shape Optimization - A Survey.

Computer Methods in Applied Mechanics and Engineering, Vol. 57, pp. 91-106.

Haug, E.J. (1981): A Review of Distributed Parameter Structural Optimization Liter-

ature. In: Optimization of Distributed Parameter Structures (Eds. E.J. Haug & J. Cea),


Haug, E.J. (1993) (Ed.): Concurrent Engineering: Tools and Technologies for Mechan-

ical System Design, Springer-Verlag, Berlin.

Haug, E.J.; Cea, J. (1981) (Eds.): Optimization of Distributed Parameter Structures,

Vol. 1: pp. 1-842, Vol. 2: pp. 843-1609, Sijtho� & Nordho�, The Netherlands.

Haug, E.J.; Choi, K.K. (1982): Systematic Occurence of Repeated Eigenvalues in

Structural Optimization. Journal of Optimization Theory and Applications, Vol. 38, No.

2, pp. 251-274.

Haug, E.J.; Choi, K.K. (1986): Material Derivative Methods for Shape Design Sensi-

tivity Analysis. In: The Optimum Shape. Automated Structural Design (Eds. J.A. Bennett

& M.E. Botkin), pp. 29-60, Plenum Press, New York.

Haug, E.J.; Choi, K.K.; Komkov, V. (1986): Design Sensitivity Analysis of Struc-

tural Systems, Academic Press, New York.

Haug, E.J.; Rousselet, B. (1980a): Design Sensitivity Analysis in Structural Mechan-

ics. I: Static Response Variation. Journal of Structural Mechanics, Vol. 8, No. 1, pp.

17-41.

Haug, E.J.; Rousselet, B. (1980b): Design Sensitivity Analysis in Structural Me-

chanics. II: Eigenvalue Variations. Journal of Structural Mechanics, Vol. 8, No. 2, pp.

161-186.

Herskovits, J. (1993): Proc. Structural Optimization '93 - The World Congress on

Optimal Design of Structural Systems, Vol. 1 & 2, Federal University of Rio de Janeiro,

Brazil.

Hughes, T.J.R. (1987): The Finite Element Method, Prentice-Hall, New Jersey.

Hughes, T.J.R.; Cohen, M. (1978): The \Heterosis" Finite Element for Plate Bending.

Computers & Structures, Vol. 9, pp. 445-450.

Hughes, T.J.R.; Cohen, M.; Haroun, M. (1978): Reduced and Selective Integration

Techniques in the Finite Element Analysis of Plates. Nuclear Engineering and Design,

Vol. 46, pp. 203-222.

H�ornlein, H.R.E.M. (1987): Take-o� in Optimum Structural Design. In: Computer

Aided Optimal Design. Structural and Mechanical Systems (Ed. C.A. Mota Soares), NATO

ASI Series F: Computer and System Sciences, Vol. 27, pp. 901-919, Springer-Verlag,

Berlin.

188

Kibsgaard, S. (1991): Analysis and Optimization of FE-discretized Structures, Ph.D.

Thesis. Special Report No. 6, Institute of Mechanical Engineering, Aalborg University,

Denmark.

Kibsgaard, S.; Olho�, N.; Rasmussen, J. (1989): Concept of an Optimization Sys-

tem. In: Proc. OPTI89 - First Int. Conf. on Computer Aided Optimum Design of

Structures: Applications (Eds. C.A. Brebbia & S. Hernandez), pp. 79-88, Computational

Mechanics Publications, Springer-Verlag, Berlin.

Kikuchi, N.; Chung, K.Y.; Torigaki, T.; Taylor, J.E. (1986): Adaptive Finite

Element Methods for Shape Optimization of Linearly Elastic Structures. In: The Optimum

Shape. Automated Structural Design (Eds. J.A. Bennett & M.E. Botkin), pp. 139-170,

Plenum Press, New York.

Koski, J.; Silvennoinen; R. (1990): Multicriteria Design of Ceramic Components. In:

Multicriteria Design Optimization (Eds. H.A. Eschenauer, J. Koski & A. Osyczka), pp.


Kristensen, E.S.; Madsen, N.F. (1976): On the Optimum Shape of Fillets in Plates

Subjected to Multiple In-plane Loading Cases. International Journal for Numerical Meth-

ods in Engineering, Vol. 10, pp. 1007-1019.

Ladefoged, T. (1988): Triangular Ring Element with Analytic Expressions for Sti�ness

and Mass Matrix. Computer Methods in Applied Mechanics and Engineering, Vol. 67, pp.

171-187.

Lancaster, P. (1964): On Eigenvalues of Matrices Dependent on a Parameter. Nu-

merische Mathematik, Vol. 6, pp. 377-387.

Lund, E. (1993): Design Sensitivity Analysis of Finite Element Discretized Structures.

Report No. 55, Institute of Mechanical Engineering, Aalborg University, Denmark, 31

pp. An earlier version of the paper can be found in: Shape Design Sensitivity Analysis

of Finite Element Discretized Structures. In: Lecture Notes for COMETT course on

Computer Aided Optimum Design of Structures, August 1993, Aalborg.

Lund, E.; Birker, T.; Rasmussen, J. (1991): CAOS Computer Aided Optimization

System Ver. 2.1. User's Manual and Reference. Special Report No. 9, Institute of

Mechanical Engineering, Aalborg University, Denmark, 161 pp.

Lund, E.; Olho�, N. (1993a): Reliable and E�cient Finite Element Based Design

Sensitivity Analysis of Eigenvalues. In: Proc. Structural Optimization '93 - The World

Congress on Optimal Design of Structural Systems (Ed. J. Herskovits), Vol. 2, pp. 197-

204, Federal University of Rio de Janeiro, Brazil.

Lund, E.; Olho�, N. (1993b): Shape Design Sensitivity Analysis of Eigenvalues Using

\Exact" Numerical Di�erentiation of Finite Element Matrices. Report No. 54, Institute

of Mechanical Engineering, Aalborg University, Denmark, 19 pp. To appear in Structural

Optimization, Vol. 8 (1994).

Lund, E.; Rasmussen, J. (1994): A General and Flexible Method of Problem De�nition

in Structural Optimization Systems. Report No. 58, Institute of Mechanical Engineering,

Aalborg University, Denmark, 18 pp. To appear in Structural Optimization, Vol. 8 (1994)).

189

Masmoudi, M.; Broudiscou, C.; Guillaume, P. (1993): Automatic Di�erentiation

and Shape Optimization. In: Proc. Structural Optimization '93 - The World Congress on

Optimal Design of Structural Systems (Ed. J. Herskovits), Vol. 2, pp. 189-196, Federal

University of Rio de Janeiro, Brazil.

Masur, E.F. (1984): Optimal Structural Design under Multiple Eigenvalue Constraints.

International Journal of Solids and Structures, Vol. 20, pp. 211-231.

Masur, E.F. (1985): Some Additional Comments on Optimal Structural Design under

Multiple Eigenvalue Constraints. International Journal of Solids and Structures, Vol. 21,

pp. 117-120.

Masur, E.F.; Mr�oz, Z. (1979): Non-stationary Optimality Conditions in Structural

Design. International Journal of Solids and Structures, Vol. 15, pp. 503-512.

Masur, E.F.; Mr�oz, Z. (1980): Singular Solutions in Structural Optimization Problems.

In: Variational Methods in the Mechanics of Solids (Ed. S. Nemat-Nasser), pp. 337-343,

Evanston, Illinois, Pergamon Press, New York.

McLean, A.F.; Hartsock, D.L. (1989): Design with Structural Ceramics. In: Struc-

tural Mechanics (Ed. J.B. Wachtman), pp. 27-97, Academic Press, London.

Mills-Curran, W.C. (1988): Calculation of Eigenvector Derivatives for Structures with

Repeated Eigenvalues. AIAA Journal, Vol. 26, No. 7, pp. 867-871.

Mlejnek, H.P. (1992): Accuracy of Semi-Analytical Sensitivities and Its Improvement

by the \Natural Method". Structural Optimization, Vol. 4, pp. 128-131.

Morelle, P.; Braibant, V. (1991): Multi-Model Optimization of Large Scale Structures.

Samtech, Li�ege, Belgium.

Morris, A.J. (1982) (Ed.): Foundations of Structural Optimization: A Uni�ed Ap-

proach, Chichester, England.

Mota Soares, C.A. (1987) (Ed.): Computer Aided Optimal Design: Structural and

Mechanical Systems, Springer-Verlag, Berlin.

Mota Soares, C.A.; Rodrigues, H.C.; Choi, K.K. (1984): Shape Optimal Struc-

tural Design Using Boundary Elements and Minimum Compliance Techniques. Journal of

Mechanisms, Transmissions and Automation in Design, Vol. 106, pp. 518-523.

Mouritsen, O.�. (1992): Introduktion til Elementmetoden (In Danish), Aalborg Uni-

versity, Denmark, 63 pp.

Myslinski, A.; Sokolowski, J. (1985): Nondi�erentiable Optimization Problems For

Elliptic Systems. SIAM Journal on Control and Optimization, Vol. 23, No. 4, pp. 632-648.

Olho�, N. (1974): Optimal Design of Vibrating Rectangular Plates. International Jour-

nal of Solids and Structures, Vol. 10, pp. 93-109.

Olho�, N. (1975): On Singularities, Local Optima, and Formation of Sti�eners in Op-

timal Design of Plates. In: IUTAM Symposium, Warsaw/Poland 1973 (Eds. A. Sawczuk

& Z. Mr�oz), pp. 82-103, Springer-Verlag, Berlin.

Olho�, N. (1980): Optimal Design with Respect to Structural Eigenvalues. In: Proc.

XVth International IUTAM Congress on Theoretical and Applied Mechanics, Toronto 1980

(Eds. F.P.J. Rimrott & B. Tabarrok), pp. 133-149, Nord-Holland, The Netherlands.

190

Olho�, N. (1981a): Optimization of Columns against Buckling. In: Optimization of

Distributed Parameter Structures (Eds. E.J. Haug & J. Cea), Vol. 1, pp. 152-176, Sijtho�

& Nordho�, The Netherlands.

Olho�, N. (1981b): Optimization of Transversely Vibrating Beams and Rotating Shafts.

In: Optimization of Distributed Parameter Structures (Eds. E.J. Haug & J. Cea), Vol. 1,

pp. 177-199, Sijtho� & Nordho�, The Netherlands.

Olho�, N. (1989): Multicriterion Structural Optimization Via Bound Formulation and

Mathematical Programming. Structural Optimization, Vol. 1, pp. 11-17.

Olho�, N.; Bends�e, M.P.; Rasmussen, J. (1991): On CAD-integrated Structural

Topology and Design Optimization. Computer Methods in Applied Mechanics and Engi-

neering, Vol. 89, pp. 259-279.

Olho�, N., Krog, L.; Thomsen, J. (1993): Bi-Material Topology Optimization. In:

Proc. Structural Optimization '93 - The World Congress on Optimal Design of Structural

Systems (Ed. J. Herskovits), Vol. 1, pp. 327-334, Federal University of Rio de Janeiro,

Brazil.

Olho�, N., Lund, E. (1994): Finite Element Based Engineering Design Sensitivity

Analysis and Optimization. In: Advances in Structural Optimization (Ed. J. Herskovits),

Kluwer (to appear).

Olho�, N.; Lund, E.; Rasmussen, J. (1992): Concurrent Engineering Design Opti-

mization In a CAD Environment. Special Report No. 16, Institute of Mechanical Engi-

neering, Aalborg University, Denmark, 64 pp; Also in: Concurrent Engineering: Tools and

Technologies for Mechanical System Design (Ed. E.J. Haug), pp. 523-586, Springer-Verlag,

Berlin (1993).

Olho�, N.; Plaut, R.H. (1983): Bimodal Optimization of Vibrating Shallow Arches.

International Journal of Solids and Structures, Vol. 19, No. 6, pp. 553-570.

Olho�, N.; Rasmussen, J. (1991a): Study of Inaccuracy in Semi-Analytical Sensitivity

Analysis - A Model Problem. Structural Optimization, Vol. 3, pp. 203-213.

Olho�, N.; Rasmussen, J. (1991b): Method of Error Elimination for a Class of Semi-

Analytical Sensitivity Analysis Problems. In: Engineering Optimization in Design Pro-

cesses (Eds. H.A. Eschenauer, C. Mattheck & N. Olho�), pp. 193-200, Springer-Verlag,

Berlin.

Olho�, N.; Rasmussen, J.; Lund, E. (1992): A Method of \Exact" Numerical Di�er-

entiation for Error Elimination in Finite Element Based Semi-Analytical Shape Sensitivity

Analysis. Special Report No. 10, Institute of Mechanical Engineering, Aalborg University,

Denmark, 79 pp. Also Mechanics of Structures and Machines, Vol. 21, pp. 1-66 (1993).

Olho�, N.; Rasmussen, S.H. (1977): On Single and Bimodal Optimum Buckling

Loads of Clamped Columns. International Journal of Solids and Structures, Vol. 13, pp.

605-614.

Olho�, N.; Taylor, J.E. (1983): On Structural Optimization. Journal of Applied

Mechanics, Vol. 50, pp. 1139-1151.

191

Overton, M.L. (1988): On Minimizing the Maximum Eigenvalue of a Symmetric Matrix.

SIAM Journal on Matrix Analysis and Applications, Vol. 9, No. 2, pp. 256-268.

Pedersen, P. (1973): Some Properties of Linear Strain Triangles and Optimal Finite

Element Models. International Journal of Numerical Methods in Engineering, Vol. 7, pp.

415-429.

Pedersen, P. (1981): Design with Several Eigenvalue Constraints by Finite Elements

and Linear Programming. Journal of Structural Mechanics, Vol. 10, No. 3, pp. 243-271.

Pedersen, P. (1983): A Uni�ed Approach to Optimal Design. In: Optimization Methods

in Structural Design (Eds. H.A. Eschenauer & N. Olho�), pp. 182-187, B.I.-Wissenschafts-

verlag, Mannheim.

Pedersen, P. (1988): Design for Minimum Stress Concentration - Some Practical As-

pects. In: Structural Optimization (Eds. G.I.N. Rozvany & B.L. Karihaloo), pp. 225-232,

Kluwer, Dordrecht, The Netherlands.

Pedersen, P. (1989): On Optimal Orientation of Orthotropic Materials. Structural

Optimization, Vol. 1, pp. 101-106.

Pedersen, P. (1990): Bounds on Elastic Energy in Solids of Orthotropic Materials.

Structural Optimization, Vol. 2, pp. 55-63.

Pedersen, P. (1991): On Thickness and Orientational Design with Orthotropic Materi-

als. Structural Optimization, Vol. 3, pp. 69-78.

Pedersen, P. (1993) (Ed.): Optimal Design with Advanced Materials, Elsevier, The

Netherlands.

Pedersen, P.; Cheng, G.; Rasmussen, J. (1989): On Accuracy Problems for Semi-

Analytical Sensitivity Analysis. Mechanics of Structures and Machines, Vol. 17, No. 3,

pp. 373-384.

Pedersen, P.; Laursen, C.L. (1983): Design for Minimum Stress Concentration by

Finite Elements and Linear Programming. Journal of Structural Mechanics, Vol. 10, No.

4, pp. 375-391.

Plaut, R.H.; Johnson, L.W.; Olho�, N. (1986): Bimodal Optimization of Com-

pressed Columns on Elastic Foundations. Journal of Applied Mechanics, Vol. 53, pp.

130-134.

Prager, S.; Prager, W. (1979): A Note on Optimal Design of Columns. Int. J. Mech.

Sci., Vol. 21, pp. 249-251.

Rasmussen, J. (1989): Development of the Interactive Structural Shape Optimization

System CAOS (in Danish), Ph.D. Thesis, Institute of Mechanical Engineering, Aalborg

University, Denmark, Special Report No. 1a.

Rasmussen, J. (1990): The Structural Optimization System CAOS. Structural Opti-

mization, Vol. 2, pp. 109-115.

Rasmussen, J. (1991): Shape Optimization and CAD. International Journal of Systems

Automation: Research and Applications (SARA), Vol. 89, pp. 259-279.

Rasmussen, J.; Lund, E.; Birker, T. (1992): Collection of Examples, CAOS Opti-

mization System, 3rd edition. Special Report No. 1c, Institute of Mechanical Engineering,

192

Aalborg University, Denmark, 119 pp.

Rasmussen, J.; Lund, E.; Birker, T.; Olho�, N. (1993): The CAOS System.

International Series of Numerical Mathematics, Vol. 110, pp. 75-96, Birkhuser Verlag,

Basel.

Rasmussen, J.; Lund, E.; Olho�, N. (1992): Structural Optimization by the Finite

Element Method. In: Lecture Notes for European School CIMI/COMETT on Computer

Aided Design of Structures, Journees Universitaires, 13-17 July 1992, Nice, France.

Rasmussen, J.; Lund, E.; Olho�, N. (1993a): Parametric Modeling and Analysis

for Optimum Design. In: Proc. Structural Optimization '93 - The World Congress on

Optimal Design of Structural Systems (Ed. J. Herskovits), Vol. 2, pp. 407-414, Federal

University of Rio de Janeiro, Brazil.

Rasmussen, J.; Lund, E.; Olho�, N. (1993b): Integration of Parametric Modeling

and Structural Analysis for Optimum Design. In: Advances in Design Automation. Proc.

1993 ASME Design Technical Conferences - 19th Design Automation Conference, Albu-

querque, New Mexico (Eds. B.J. Gilmore, D.A. Hoeltzel, S. Azarm & H.A. Eschenauer),

Vol. 1, pp. 637-648, The American Society of Mechanical Engineers, New York.

Rasmussen, J.; Lund, E.; Olho�, N.; Birker, T. (1993): Structural Topology and

Shape Optimization with the CAOS System. In: Lecture Notes for COMETT course on

Computer Aided Optimal Design, February 1993, Eindhoven.

Rasmussen, J.; Olho�, N.; Lund, E. (1992): Foundations for Optimum Design

System Development. In: Lecture Notes for Advanced TEMPUS course on Numerical

Methods in Computer Optimal Design, 11-15 maj 1992, Zakopane, Poland.

Rasmussen, J.; Thomsen, J.; Olho�, N. (1993): Topology and Boundary Variation

Design Methods in a CAD System. In: Topology Design of Structures (Eds. M.P. Bends�e

& C.A. Mota Soares), Kluwer Academic Publishers, Dordrecht, The Netherlands.

Ringertz, U. (1992): Optimal Design and Imperfection Sensitivity of Nonlinear Shell

Structures. FFA TN 1992-30, The Aeronautical Research Institute of Sweden, 33 pp.

Rodrigues, H.C. (1988): Shape Optimal Design of Elastic Bodies Using a Mixed Vari-

ational Formulation. Computer Methods in Applied Mechanics and Engineering, Vol. 69,

pp. 29-44.

Rosengren, R. (1989): Shape Optimization of a Saab 9000 Wishbone. SAAB-SCANIA,

Sweden.

Rozvany, G.I.N. (1989): Structural Design via Optimality Criteria - The Prager Ap-

proach to Structural Optimization, Kluwer, Dordrecht, The Netherlands.

Rozvany, G.I.N. (1993) (Ed.): Optimization of Large Structural Systems, Vol. 1: pp.

1-622, Vol. 2: 623-1198, Kluwer, Dordrecht, The Netherlands.

Rozvany, G.I.N.; Olho�, N.; Cheng, G.; Taylor, J.E. (1982): On the Solid Plate

Paradox in Structural Optimization. Journal of Structural Mechanics, Vol. 10, No. 1, pp.

1-32.

Rozvany, G.I.N.; Karihaloo, B.L. (1988) (Eds.): Structural Optimization, Kluwer,

Dordrecht, The Netherlands.

193

Santos, J.L.T.; Choi, K.K. (1989): Integrated Computational Considerations for Large

Scale Structural Design Sensitivity Analysis and Optimization. In: Discretization Methods

and Structural Optimization - Procedures and Applications (Eds. H.A. Eschenauer & G.

Thierauf), pp. 299-307, Springer-Verlag, Berlin.

Seyranian, A.P. (1983): A Solution of a Problem of Lagrange. Dokl. Akad. Nauk

SSSR, Vol. 271, pp. 337-340; Also Sov. Phys. Dokl., Vol. 28, pp. 550-551.

Seyranian, A.P. (1984): On a Problem of Lagrange. Mekhanika Tverdogo Tela, No. 2,

pp. 101-111; Also Mechanics of Solids, Vol. 19, No. 2, pp. 100-111.

Seyranian, A.P. (1987): Multiple Eigenvalues in Optimization Problems. Prikl. Mat.

Mekh., Vol. 51, pp. 349-352; Also Applied Mathematics in Mechanics, Vol. 51, pp. 272-

275.

Seyranian, A.P.; Lund, E.; Olho�, N. (1994): Multiple Eigenvalues in Structural

Optimization Problems. Special Report No. 21, Institute of Mechanical Engineering,

Aalborg University, Denmark, 51 pp. (To appear in Structural Optimization).

Schmit, L.A. (1982): Structural Synthesis - Its Genesis and Development. AIAA Jour-

nal, Vol. 20, pp. 992-1000.

Sobieszanski-Sobieski, J.; Rogers, J.L. (1984): A Programming System for Research

and Applications in Structural Optimization. In: New directions in optimum structural

design (Eds. E. Atrek, R.H. Gallagher, K.M. Ragsdell & O.C. Zienkiewicz), pp. 563-585,

John Wiley & Sons, New York.

Stanton, E.L. (1986): Geometric Modeling for Structural and Material Shape Optimiza-

tion. In: The Optimum Shape. Automated Structural Design (Eds. J.A. Bennett & M.E.

Botkin), pp. 365-383, Plenum Press, New York.

Svanberg, K. (1987): The Method of Moving Asymptotes - A New Method For Struc-

tural Optimization. International Journal for Numerical Methods in Engineering, Vol. 24,

pp. 359-373.

Tadjbakhsh, I.; Keller, J.B. (1962): Strongest Columns and Isoperimetric Inequalities

For Eigenvalues. Journal of Applied Mechanics, Vol. 29, pp. 159-164.

Taylor, J.E.; Bends�e, M.P. (1984): An Interpretation of Min-Max Structural Design

Problems Including a Method for Relaxing Constraints. International Journal of Solids

and Structures, Vol. 20, pp. 301-314.

Thomsen, J. (1992): Optimering af anisotrope materialers egenskaber og konstruktion-

ers topologi (in Danish), Ph.D. Thesis. Special Report No. 14. Institute of Mechanical

Engineering, Aalborg University, Denmark.

Vanderplaats, G.N. (1987): ADS - A FORTRAN program for Automated Design

Synthesis - version 2.01. Engineering Design Optimization, INC.

Wang, S.-Y.; Sun, Y.; Gallagher, R.H. (1985): Sensitivity Analysis in Shape Opti-

mization of Continuum Structures. Computers & Structures, Vol. 20, No. 5, pp. 855-867.

Wexler, A.S. (1987): Automatic Evaluation of Derivatives. Appl. Math. Comp., Vol.

24, pp. 19-46.

194

Wittrick, W.H. (1962): Rates of Change of Eigenvalues, With Reference to Buckling

and Vibration Problems. Journal of the Royal Aeronautical Society, Vol. 66, pp. 590-591.

Yang, R.J.; Botkin, M.E. (1986): The Relationship Between the Variational Approach

and the Implicit Di�erentiation Approach to Shape Design Sensitivities. In: The Optimum

Shape. Automated Structural Design (Eds. J.A. Bennett & M.E. Botkin), pp. 61-77,

Plenum Press, New York.

Yang, R.-J.; Choi, K.K. (1985): Accuracy of Finite Element Based Shape Design

Sensitivity Analysis. Journal of Structural Mechanics, Vol. 13, No. 2, pp. 223-239.

Yao, T.-M.; Choi, K.K. (1989): 3-D Shape Optimal Design and Automatic Finite

Element Regridding. International Journal for Numerical Methods in Engineering, Vol.

28, pp. 369-384.

Zhong, W.; Cheng, G. (1986): Second-order Sensitivity Analysis of Multimodal Eigen-

values and Related Optimization Techniques. Journal of Structural Mechanics, Vol. 14,

No. 4, pp. 421-436.

Zienkiewicz, O.C.; Campbell, J.S. (1973): Shape Optimization and Sequential Linear

Programming. In: Optimum Structural Design, Theory and Applications (Eds. R.H.

Gallagher & O.C.Zienkiewicz), Wiley and Sons, London, pp. 109-126.

Zienkiewicz, O.C.; Taylor, R.L. (1989): The Finite Element Method. Volume 1 & 2,

4th Edition, McGraw-Hill, London.

Zolesio, J.-P. (1981): The Material Derivative (or Speed) Method for Shape Optimiza-

tion. In: Optimization of Distributed Parameter Structures (Eds. E.J. Haug & J. Cea),


Zyczkowski, M. (1989) (ed.): Structural Optimization Under Stability and Vibration

Constraints. Springer, Wien - New York.

195

APPENDIX

197

Appendix

A

3D Solid Isoparametric Finite Elements

Two 3D isoparametric �nite elements have been implemented in ODESSY, that is,

an 8-node and a 20-node element. The 20-node element can have curved element sides

as illustrated in Fig. A.1.

x,u y,v

z,w

ξ

ηζ

ζ

ξ

η

Figure A.1: Domain, node numbering, and nodal degrees of freedom of 8- and 20-node

isoparametric �nite elements.

A.1 Shape Functions for 3D Solid Isoparametric Fi-

nite Elements.

Within the isoparametric formulation of a �nite element with an arbitrary number n of

nodal points, the same set of shape functions

Ni = Ni(�; �; �); i = 1; : : : ; n (A.1)

199

200 A.2. Element Sti�ness Matrix

is used for interpolation of global x, y, and z coordinates from nodal values xi, yi, and ziand of displacements functions u, v, and w from nodal values ui, vi, and wi, i.e.,

x =nXi=1

Nixi; y =nXi=1

Niyi; z =nXi=1

Nizi (A.2)

u(x; y; z) =nXi=1

Niui; v(x; y; z) =nXi=1

Nivi; w(x; y; z) =nXi=1

Niwi (A.3)

Shape functions for the 8-node element are given in Table A.1.

Table A.1: Shape functions for 8-node 3D solid isoparametric element.

N1 =18(1� �)(1� �)(1� �) N2 =

18(1 + �)(1� �)(1� �)

N3 =18(1 + �)(1 + �)(1� �) N4 =

18(1� �)(1 + �)(1� �)

N5 =18(1� �)(1� �)(1 + �) N6 =

18(1 + �)(1� �)(1 + �)

N7 =18(1 + �)(1 + �)(1 + �) N8 =

18(1� �)(1 + �)(1 + �)

and shape functions for the 20-node element are given in Table A.2.

A.2 Element Sti�ness Matrix


k =ZBT EB jJj d (A.4)


coordinates for the element, see Fig. A.1, and jJj is the determinant of the Jacobian matrixJ which at each point de�nes the transformation of di�erentials d�, d�, and d� into dx,

dy, and dz. Like J, the strain-displacement matrix B depends on coordinates of the nodal

points, whereas the constitutive matrix E depends only on the constitutive parameters

of the assumed linearly elastic material which, in the implementation in ODESSY, can

be either isotropic or anisotropic. The expressions for the Jacobian J and for the strain-

displacement matrix B are given in the following.

In terms of the vector di of nodal degrees of freedom

di = fui vi wigT ; i = 1; : : : ; n (A.5)

the element nodal vector d containing nodal displacements is

d = fdT1 dT2 : : :dTi : : : dTngT (A.6)

and the strain vector function " is

"(x; y; z) = f"x "y "z xy yz xzgT (A.7)

Appendix A. 3D Solid Isoparametric Finite Elements 201

Table A.2: Shape functions for 20-node 3D solid isoparametric element.

N9 =14(1� �2)(1� �)(1� �) N10 =

14(1 + �)(1� �2)(1� �)

N11 =14(1� �2)(1 + �)(1� �) N12 =

14(1� �)(1� �2)(1� �)

N13 =14(1� �2)(1� �)(1 + �) N14 =

14(1 + �)(1� �2)(1 + �)

N15 =14(1� �2)(1 + �)(1 + �) N16 =

14(1� �)(1� �2)(1 + �)

N17 =14(1� �)(1� �)(1� �2) N18 =

14(1 + �)(1� �)(1� �2)

N19 =14(1 + �)(1 + �)(1� �2) N20 =

14(1� �)(1 + �)(1� �2)

N1 =18(1� �)(1� �)(1� �)� 1

2(N17 +N12 +N9)

N2 =18(1 + �)(1� �)(1� �)� 1

2(N18 +N10 +N9)

N3 =18(1 + �)(1 + �)(1� �)� 1

2(N19 +N11 +N10)

N4 =18(1� �)(1 + �)(1� �)� 1

2(N20 +N12 +N11)

N5 =18(1� �)(1� �)(1 + �)� 1

2(N17 +N16 +N13)

N6 =18(1 + �)(1� �)(1 + �)� 1

2(N18 +N14 +N13)

N7 =18(1 + �)(1 + �)(1 + �)� 1

2(N19 +N15 +N14)

N8 =18(1� �)(1 + �)(1 + �)� 1

2(N20 +N16 +N15)

with their mutual relationship de�ned by

" = B d (A.8)

The strain-displacement matrix B is determined by operating on the shape functions Ni,

and it is found that

B = [ b1 b2 : : : bi : : : bn ] (A.9)


has the form

bi =

266666666664

Ni;x 0 0

0 Ni;y 0

0 0 Ni;z

Ni;y Ni;x 0

0 Ni;z Ni;y

Ni;z 0 Ni;x

377777777775; i = 1; : : : ; n (A.10)

Here, the derivatives of the shape functions Ni with respect to x, y, and z are given by8>><>>:Ni;x

Ni;y

Ni;z

9>>=>>; =

2664�;x �;x �;x

�;y �;y �;y

�;z �;z �;z

37758>><>>:Ni;�

Ni;�

Ni;�

9>>=>>; = �

8>><>>:Ni;�

Ni;�

Ni;�

9>>=>>; ; i = 1; : : : ; n (A.11)

202 A.3. Consistent Nodal Load Vector


� = J�1 (A.12)

of the Jacobian

J =

2664x;� y;� z;�

x;� y;� z;�

x;� y;� z;�

3775 =

nXi=1

2664Ni;�xi Ni;�yi Ni;�zi



3775 (A.13)

Note that J is expressed in terms of the derivatives of Ni; i = 1; : : : ; n, with respect to the

curvilinear element coordinates �, �, and �, and of the coordinates (xi; yi; zi), i = 1; : : : ; n,

of each of the n nodal points of the �nite element.

A.3 Consistent Nodal Load Vector


f =ZNT FB jJj d +

Z!

NT FS jJj d! (A.14)

where is the domain of the �nite element in its local coordinate system, FB represents

body forces, ! the surface described in curvilinear, non-dimensional � � �, � � �, or � � �

coordinates for the element at which surface forces FS are applied, and N contains shape

functions Ni. In the surface integral, N and jJj are evaluated on !.



f th =ZBT E "th jJj d (A.15)


"th =


thyz

thxz

oT=nT T T 0 0 0

oT�; T = T � T0 (A.16)



strains (typically 20�C). These thermally induced strains are based on Gauss points values

for obtaining the highest accuracy.

A.4 Consistent and Lumped Mass Matrices


m =Z%NT N jJj d (A.17)

Here, is the domain of the �nite element in its local coordinate system, see Fig. A.1, %

the mass density, N contains shape functions Ni, and jJj is the determinant of the Jacobianmatrix J.


Lumped mass matrices are also available in ODESSY. These are generated using the

popular HRZ lumping scheme, see Cook, Malkus and Plesha (1989). The idea of this

method is to use only the diagonal terms of the consistent mass matrix, but to scale them

in such a way that the total mass of the element is preserved. The procedural steps are as

follows:

1. Compute only the diagonal coe�cients mii of the consistent mass matrix.

2. Compute the total mass of the element, m.

3. Compute a number s by adding the diagonal coe�cients mii associated with

translational d.o.f. (but not rotational d.o.f., if any) that are mutually parallel

and in the same direction. The number of these translational d.o.f. is denoted

by x.

4. Scale all the diagonal coe�cients by multiplying them by the ratio xm=s, thus

preserving the total mass of the element.

This HRZ lumping procedure is used for all the elements in ODESSY when generating

lumped mass matrices.

A.5 Element Initial Stress Sti�ness Matrix

In the derivation of element initial stress sti�ness matrices it is convenient to reorder nodal

degrees of freedom by introducing the element displacement vector d�, where translational

d.o.f. are reordered so that �rst all x-direction d.o.f. are given, then y, and then z as

follows

d� = fu1 u2 : : : ui : : : un v1 v2 : : : vi : : : vn w1 w2 : : : wi : : : wngT (A.18)

Relating d.o.f. to the reordered element vector d� the element initial stress sti�ness matrix


k� =ZGT SG jJj d (A.19)


coordinates for the element, see Fig. A.1, G a matrix obtained by appropriate di�erentia-

tion of shape functions Ni, S a matrix of initial stresses, and jJj is the determinant of theJacobian matrix J.


G =

2664g 0 0

0 g 0

0 0 g

3775 (A.20)

204 A.6. Thermal Element \Sti�ness Matrix"


g =

2664Ni;x

Ni;y

Ni;z

3775 ; i = 1; : : : ; n (A.21)


S =

2664s 0 0

0 s 0

0 0 s

3775 (A.22)

and each submatrix s is de�ned as

s =

2664�x �xy �xz

�xy �y �yz

�xz �yz �z

3775 (A.23)


A.6 Thermal Element \Sti�ness Matrix"


matrix kth given by

kth =ZBthT

�Bth jJj d (A.24)

Here, is the domain of the �nite element in its local coordinate system, see Fig. A.1,

Bth a matrix obtained by appropriate di�erentiation of shape functions Ni, � the thermal

conductivity matrix, and jJj is the determinant of the Jacobian matrix J. If the material

is isotropic, � can simply be replaced by the scalar �, the conductivity coe�cient.



i(A.25)


has the form

bthi =

2664Ni;x

Ni;y

Ni;z

3775 ; i = 1; : : : ; n (A.26)



h =Z!2

NT hN jJj d! (A.27)

Here, !2 is the surface of the �nite element described in curvilinear, non-dimensional ��,� � �, or � � � coordinates for the element, for which the convection boundary condition

is applied. N contains shape functions Ni, h is the convection coe�cient speci�ed, and jJjis the determinant of the Jacobian matrix J for the surface !2.


A.7 Consistent Thermal Nodal Flux Vector


q =Z!1

NT qS jJj d! +Z!2

NT h Te jJj d! (A.28)

where the �rst term derives from speci�ed ux at the surface !1 and the latter term from

a speci�ed convection boundary condition at surface !2. The surfaces !1, !2 are described

in curvilinear, non-dimensional � � �, � � �, or � � � coordinates for the element. The

scalar qS is prescribed ux normal to the surface !1, N contains shape functions Ni that

are evaluated on the surface !, jJj the determinant of the Jacobian matrix for the surface

!, h the convection coe�cient speci�ed, and Te is the environmental temperature speci�ed

for the convection boundary condition.

A.8 Gauss Quadrature

The Gauss integration rules used for the 3D isoparametric �nite elements are given in

Table A.3.

206 A.8. Gauss Quadrature

Table A.3: Gauss quadrature used for 3D solid isoparametric �nite elements.

Integration rule:

Element type k m k� kth ";�

8-node 8-point 8-point 8-point 8-point 8-point


Appendix

B

2D Solid Isoparametric Finite Elements

Five different 2D solid isoparametric �nite elements have been implemented in

ODESSY. These isoparametric �nite elements are formulated in a uni�ed way for

both plane stress, plane strain, and axisymmetric situations. The elements have 3-, 4-, 6-,

8-, and 9-nodes, respectively, and are shown in Fig. B.1. The 6-, 8-, and 9-node elements

can have straight as well as curved boundaries.

(r,u)x,u

y,v(z,w)

axis of rotationalsymmetry( )

ξξ ξ

ξ

ξξ

ξ ξ

ξ

η

ηη

2

1

3

2 3

1

Figure B.1: Domain, node numbering, and nodal degrees of freedom of 3-, 4-, 6-, 8-, and 9-

node 2D isoparametric �nite elements. Local, non-dimensional coordinates �, �

and area coordinates �1, �2, �3 are shown for the quadrilateral and triangular

elements, respectively. Text in parantheses refer to standard notations for

problems with rotational symmetry.

207

208 B.1. Shape Functions for 2D Solid Isoparametric Finite Elements.

B.1 Shape Functions for 2D Solid Isoparametric Fi-

nite Elements.

Within the isoparametric formulation of a �nite element with an arbitrary number n of

nodal points, the same set of shape functions

Ni = Ni(�; �); i = 1; : : : ; n (B.1)

is used for interpolation of global x, y coordinates from nodal values xi, yi and of displace-

ments functions u, v from nodal values ui, vi, i.e.,

x =nXi=1

Nixi; y =nXi=1

Niyi (B.2)

u(x; y) =nXi=1

Niui; v(x; y) =nXi=1

Nivi (B.3)

Shape functions for the 4-, 8-, and 9-node elements are given in Tables B.1, B.2, and B.3,

respectively.

Table B.1: Shape functions for 4-node 2D solid isoparametric element.

N1 =14(1� �)(1� �) N2 =

14(1 + �)(1� �)

N3 =14(1 + �)(1 + �) N4 =

14(1� �)(1 + �)


N5 =12(1� �2)(1� �) N6 =

12(1 + �)(1� �2)

N7 =12(1� �2)(1 + �) N8 =

12(1� �)(1� �2)

N1 =14(1� �)(1� �)� 1

2(N5 +N8)

N2 =14(1 + �)(1� �)� 1

2(N5 +N6)

N3 =14(1 + �)(1 + �)� 1

2(N6 +N7)

N4 =14(1� �)(1 + �)� 1

2(N7 +N8)

In case of triangular isoparametric elements, the interpolation functions are de�ned con-

veniently in terms of non-dimensional area coordinates �1, �2, and �3 within the element

as shown in Fig. B.1. Only two of the three dimensionless area coordinates are mutually

independent, due to the area constraint relation

�1 + �2 + �3 = 1 (B.4)

Appendix B. 2D Solid Isoparametric Finite Elements 209


N9 = (1� �2)(1� �2)

N5 =12(1� �2)(1� �)� 1

2N9 N6 =

12(1 + �)(1� �2)� 1

2N9

N7 =12(1� �2)(1 + �)� 1

2N9 N8 =

12(1� �)(1� �2)� 1

2N9

N1 =14(1� �)(1� �)� 1

2(N5 +N8)� 1

4N9

N2 =14(1 + �)(1� �)� 1

2(N5 +N6)� 1

4N9

N3 =14(1 + �)(1 + �)� 1

2(N6 +N7)� 1

4N9

N4 =14(1� �)(1 + �)� 1

2(N7 +N8)� 1

4N9

If �1 and �2 are selected as independent coordinates the following relations are obtained

�1 = �; �2 = �; �3 = 1� � � � (B.5)

Derivatives with respect to � and �, with the constraint in Eq. B.4 taken into account,

can be found as@

@�=

@

@�1� @

@�3;

@

@�=

@

@�2� @

@�3(B.6)

The element matrices for the triangular elements can be formulated similarly to the quadri-

lateral elements by using Eqs. B.4 - B.6. The shape functions for the 3- and 6-node elements

are given in Tables B.4 and B.5.


N1 = �1 N2 = �2 N3 = �3


N1 = �1(2 �1 � 1) N2 = �2(2 �2 � 1) N3 = �3(2 �3 � 1)

N4 = 4 �1 �2 N5 = 4 �2 �3 N3 = 4 �1 �3

210 B.2. Element Sti�ness Matrix

B.2 Element Sti�ness Matrix


k =Z!

BT EB jJj d! (B.7)

Here, ! is the domain of the �nite element in its local coordinate system, see Fig. B.1, and

jJj is the determinant of the Jacobian matrix J which at each point de�nes the transfor-

mation of di�erentials d� and d� into dx and dy. Like J, the strain-displacement matrix

B depends on coordinates of the nodal points, whereas the constitutive matrix E depends

only on the constitutive parameters of the assumed linearly elastic material which can be

either isotropic or anisotropic in the implementation in ODESSY.

For the plane stress and strain situations, d! = t d� d�, where t is the thickness of the

�nite element, and for rotationel symmetry we have d! = 2� r d� d�, where r is the radius.

The axis of rotational symmetry is assumed to be parallel with the y-axis as shown in Fig.

B.1.

The expressions for the Jacobian J and for the strain-displacement matrix B can be found

in a uniform way for plane stress, plane strain and axisymmetric �nite elements, if the

strain vector " is de�ned in the following way

"(x; y) = f"x "y xy "zgT (B.8)

In order to have the relations between standard notations for axisymmetric problems and

the notations used here, it should be noted that r = x, z = y, w = v, "r = "x, "z = "y,

rz = xy, and "� = "z as indicated on Fig. B.1.

The element nodal vector d containing nodal displacements is given by

d = fdT1 dT2 : : :dTi : : : dTngT (B.9)

where the vector di of nodal degrees of freedom is

di = fui vigT ; i = 1; : : : ; n (B.10)

The relation between the strain vector " and the displacement vector d is

" = B d (B.11)

The strain-displacement matrix B can be found to have the following form

B = [ b1 b2 : : : bi : : : bn ] (B.12)


has the form

bi =

26666664

Ni;x 0

0 Ni;y

Ni;y Ni;x

Ni

r0

37777775 ; i = 1; : : : ; n (B.13)


The fourth row in bi is used only in case of an axisymmetric problem. The derivatives of

the shape functions with respect to x and y are given by

(Ni;x

Ni;y

)=

"�;x �;x

�;y �;y

#(Ni;�

Ni;�

)= �

(Ni;�

Ni;�

); i = 1; : : : ; n (B.14)


� = J�1 (B.15)

of the Jacobian

J =

"x;� y;�

x;� y;�

#=

nXi=1

"Ni;�xi Ni;�yi

Ni;�xi Ni;�yi

#(B.16)

B.3 Consistent Nodal Load Vector


f =Z!

NT FB jJj d! +Z�

NT FS jJj d� (B.17)


body forces, � the boundary described in curvilinear, non-dimensional � or � coordinates for

the element at which boundary forces FS are applied, and N contains shape functions Ni.

In the boundary integral, N and jJj are evaluated on �. For the plane stress and strain

situations, d� = t d�, where t is the thickness of the �nite element, and for rotationel

symmetry we have d� = 2� r d�, where r is the radius.



f th =Z!

BT E "th jJj d! (B.18)


"th =

n"thx "thy thxy "

thz

oT=nT T 0 T

oT�; T = T � T0 (B.19)



strains. These thermal strains are based on Gauss points values for obtaining the highest

accuracy.

B.4 Consistent and Lumped Mass Matrices


m =Z!

%NT N jJj d! (B.20)

212 B.5. Element Initial Stress Sti�ness Matrix

Here, ! is the domain of the �nite element described in curvilinear, non-dimensional �� coordinates for the element, see Fig. B.1, % the mass density, N contains shape functions

Ni, and jJj is the determinant of the Jacobian matrix J.

Lumped mass matrices for these elements are also available. These are generated using

the HRZ lumping scheme as described in Appendix A.

B.5 Element Initial Stress Sti�ness Matrix

In the derivation of element initial stress sti�ness matrices it is convenient to reorder nodal

degrees of freedom by introducing the element displacement vector d�, where translational

d.o.f. are reordered so that �rst all x-direction d.o.f. are given, and then y as follows

d� = fu1 u2 : : : ui : : : un v1 v2 : : : vi : : : vngT (B.21)

Relating d.o.f. to the reordered element vector d�, the element initial stress sti�ness matrix


k� =Z!

GT S G jJj d! (B.22)

Here, ! is the domain of the �nite element in its local coordinate system, see Fig. B.1,

G a matrix obtained by appropriate di�erentiation of shape functions Ni, S a matrix of

initial stresses, and jJj is the determinant of the Jacobian matrix J.


G =

"g 0

0 g

#(B.23)


g =

"Ni;x

Ni;y

#; i = 1; : : : ; n (B.24)


S =

"s 0

0 s

#(B.25)


s =

"�x �xy

�xy �y

#(B.26)


B.6 Thermal Element \Sti�ness Matrix"


matrix kth given by

kth =Z!

BthT�Bth jJj d! (B.27)


Here, ! is the domain of the �nite element described in curvilinear, non-dimensional �� coordinates for the element, see Fig. B.1, Bth a matrix obtained by appropriate di�erenti-

ation of shape functions Ni, � the thermal conductivity matrix, and jJj is the determinantof the Jacobian matrix J. If the material is isotropic, � can simply be replaced by the

scalar �, the conductivity coe�cient.



i(B.28)


has the form

bthi =

"Ni;x

Ni;y

#; i = 1; : : : ; n (B.29)



h =Z�2

NT hN jJj d� (B.30)

Here, �2 is the boundary of the �nite element described in curvilinear, non-dimensional �

or � coordinates for the element, for which the convection boundary condition is applied.

N contains shape functions Ni, h is the convection coe�cient speci�ed, and jJj is thedeterminant of the Jacobian matrix J for the boundary �2.

B.7 Consistent Thermal Nodal Flux Vector


q =Z�1

NT qS jJj d�+Z�2

NT h Te jJj d� (B.31)

where the �rst term derives from speci�ed ux at the boundary �1 and the latter term

from a speci�ed convection boundary condition at boundary �2. The boundaries �1, �2are described in curvilinear, non-dimensional � or � coordinates for the element. The

scalar qS is prescribed ux normal to the boundary �1, N contains shape functions Ni that

are evaluated on the boundary �, jJj the determinant of the Jacobian matrix J for the

boundary �, h the convection coe�cient speci�ed, and Te is the environmental temperature

speci�ed for the convection boundary condition.

B.8 Gauss Quadrature

The Gauss integration rules used for the 2D isoparametric �nite elements are given in

Table B.6.

It should be noted that the 4-point integration rule can be employed everywhere in the

numerical integration for the 9-node Lagrange element with nearly as good results as the

computationally more expensive 9-point integration.

214 B.8. Gauss Quadrature

Table B.6: Gauss quadrature used for 2D solid isoparametric �nite elements.

Integration rule:

Element type k m k� kth ";�






Appendix

C

Isoparametric Mindlin Plate and Shell

Finite Elements

Six different isoparametric Mindlin plate �nite elements have been implemented

in ODESSY, and all these plate elements can also be used as at shell elements. The

element matrices are �rst described for Mindlin plate elements with in-plane membrane

capability, where the elements are formed by combining a plane membrane element, i.e.,

the 2D solid isoparametric elements, with a standard Mindlin plate bending element.

In Section C.10 it is shown how these elements are transformed into at shell elements.

This is an easy way to formulate shell elements and the elements pass patch tests and

do not exhibit strain under rigid body motion. However, although membrane-bending

coupling is present throughout an actual curved shell, it is absent in individual at shell

elements and this might lead to inaccurate results, especially if a small number of elements

are used to model a curved surface. This must be taken into account when generating

�nite element models using these at shell elements.

The elements have 3-, 4-, 6-, 8-, and 9-nodes, respectively, and are shown in Fig. C.1. The

6-, 8-, and 9-node elements can have straight as well as curved boundaries in the element

plane. Both a 9-node \Lagrange" and a 9-node \heterosis" Mindlin plate �nite element

have been implemented, see Section C.9.

It is important to notice that all element matrices for the Mindlin elements are given in

terms of standard right-hand-rule rotations �x, �y as illustrated in Figs. C.1 and C.2.

When the Mindlin plate theory is derived it is normally done using the rotations �1, �2which are de�ned as

�1 = ��y�2 = �x

(C.1)

The use of rotations �1, �2 greatly simplify the algebra when developing the theory but as

the aim of this description of the implemented Mindlin plate elements is to give an overview

of the element matrices, standard right-hand-rule rotations �x, �y are used everywhere.

Furthermore, in the transformation of these at plate elements to at shell elements with

215

216

θ

θyi

zx

y

iw

z u x

y

vξξξ

ξ

ξξ

ξ

ξξ

η η

η

θxi i

i

i

32

1

21

3

Figure C.1: Domain, node numbering, and nodal degrees of freedom of 3-, 4-, 6-, 8-, and

9-node isoparametric Mindlin plate �nite elements. Local, non-dimensional

coordinates �, � and area coordinates �1, �2, �3 are shown for the quadrilateral

and triangular elements, respectively.

x

y

z

1

2

θθ

θ

θ

x

y

Figure C.2: Sign convections for rotations.

arbitrary orientation it is necessary to use right-hand-rule rotations �x, �y.

The Mindlin plate theory is based on the four following main assumptions:

1. The domain is of the following special form:

=�(x; y; z) 2 <3 j z 2

��t2;t

2

�; (x; y) 2 ! � <2

�

where t is the plate thickness and ! its area. The boundary of ! is denoted by

�.

2. �z = 0

3. u(x; y; z) = �z �1(x; y) = z �y(x; y) and

v(x; y; z) = �z �2(x; y) = �z �x(x; y)

Appendix C. Isoparametric Mindlin Plate and Shell Finite Elements 217

4. w(x; y; z) = w(x; y)

In Assumption 1, we may take the plate thickness t to be a function of x and y, if desired.

Assumption 2 is the plane stress hypothesis. It contradicts Assumption 4 but ultimately

causes no problems. It should be noted that no plate theory is completely consistent with

3D theory and, at the same time, both simple and useful. Assumption 3 implies that plane

sections remain plane. A rotation is interpreted as the rotation of a �ber initially normal

to the plate midsurface (i.e., z = 0). Furthermore, from Assumption 3 the convenience of

using the rotations �1, �2 can be seen. By Assumption 4, the transverse displacement w

does not vary through the thickness. From these four assumptions it is also seen that only

C0 continuity is required of displacement and rotation variables. In the classical Kirchho�

plate theory C1 continuity is required which is a signi�cant impediment to the derivation

of elements. Within the Mindlin theory, the isoparametric formulation is very well suited

and has therefore been chosen.

C.1 Shape Functions for Isoparametric Mindlin Plate

Finite Elements.

The set of shape functions Ni depend only on the two local coordinates � and �, i.e.

Ni = Ni(�; �); i = 1; : : : ; n (C.2)

where n is the number of nodal points for the plate �nite element.

Within the isoparametric formulation of a Mindlin plate �nite element, the same set of

shape functions Ni is used for interpolation of global x, y coordinates from nodal values xi,

yi, of displacements functions u, v, and w from nodal values ui, vi, and wi and of rotation

functions �x, �y from nodal values �xi, �yi, i.e.,

x =nXi=1

Nixi; y =nXi=1

Niyi (C.3)

u(x; y) =nXi=1

Niui; v(x; y) =nXi=1

Nivi; w(x; y) =nXi=1

Niwi (C.4)

�x(x; y) =nXi=1

Ni�xi; �y(x; y) =nXi=1

Ni�yi (C.5)

Shape functions for the 3-, 4-, 6-, 8-, and 9-node elements are similar to those used for the

2D solid elements described in Appendix B, i.e., the shape functions are given in Tables

B.4, B.1, B.5, B.2, and B.3, respectively.

C.2 Element Sti�ness Matrix

The element sti�ness matrix k for a Mindlin plate �nite element is given by

k =Z!

BT DM B jJj d! (C.6)

218 C.2. Element Sti�ness Matrix

where DM is the elasticity matrix, B the generalized strain-displacement matrix, jJj thedeterminant of the Jacobian matrix J, and ! is the domain of the �nite element described

in curvilinear, non-dimensional �� coordinates for the element as shown in Fig. C.1. Thematrices J and B depend on the coordinates of the nodal points, whereas DM depends

only on the thickness t of the plate and on the constitutive parameters of the assumed

linearly elastic material which can be either isotropic or anisotropic. The elasticity matrix

DM is given by

DM =

26666666666666666664

0 0 0 0 0

Et 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 DK 0 0

0 0 0 0 0

0 0 0 0 0 0Gyzt

1:20

0 0 0 0 0 0 0Gxzt

1:2

37777777777777777775

(C.7)

where E is the plane stress elasticity matrix, DK the Kirchho� plate bending elasticity

matrix, Gyzt and Gxzt represent the lateral shear sti�nesses of the Mindlin plate element,

here modi�ed by a Cowper factor of 1:2, see Cook, Malkus & Plesha (1989) and Zienkiewicz

& Taylor (1989). The symbol t denotes the element plate thickness, which in the imple-

mentation in ODESSY is assumed to be constant within each element.

For an isotropic material the plane stress elasticity matrix E is given as

E =1

E

2664

1 �� 0

�� 1 0

0 0 2(1 + �)

3775 (C.8)

and the Kirchho� plate bending elasticity matrix DK is

DK =

26664D �D 0

�D D 0

0 0(1� �)D

2

37775 (C.9)

where E is Young's modulus, � Poisson's ratio, and D = Et3=12(1��2) is the plate exuralrigidity.

In terms of the vector di of nodal degrees of freedom, cf. Figs. C.1 and C.2

di = fui vi wi �xi �yigT ; i = 1; : : : ; n (C.10)

the relationship between the vector d of generalized displacements of element nodal points

d = fdT1 dT2 : : :dTi : : : dTngT (C.11)

and the generalized strain vector " takes the standard form " = B d where B is given by

B = [ b1 b2 : : : bi : : : bn ] (C.12)


If the generalized strain vector " is de�ned as

"(x; y) =

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

u;x

v;y

u;y +v;x�1;x

�2;y

�1;y + �2;x

�2 � w;y

�1 � w;x

9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>;

=

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

u;x

v;y

u;y +v;x��y;x�x;y

��y;y + �x;x

�x � w;y

��y � w;x

9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>;

(C.13)

then the submatrix bi, which is associated with the nodal point i of the �nite element, has

the form

bi =

26666666666666664

Ni;x 0 0 0 0

0 Ni;y 0 0 0

Ni;y Ni;x 0 0 0

0 0 0 0 �Ni;x

0 0 0 Ni;y 0

0 0 0 Ni;x �Ni;y

0 0 �Ni;y Ni 0

0 0 �Ni;x 0 �Ni

37777777777777775

; i = 1; : : : ; n (C.14)

It is seen that the �rst three rows in bi originate from the membrane part, the next three

from the bending terms and the last two rows from the shear part. The submatrix bi is

divided into these three parts, and the evaluation of the sti�ness matrix k is carried out

for each part, possibly using di�erent orders of Gauss quadrature as will be described in

Section C.8.

The derivatives of the shape functions Ni with respect to x and y must be calculated when

evaluating the submatrix bi, and these are given by(Ni;x

Ni;y

)=

"�;x �;x

�;y �;y

#(Ni;�

Ni;�

)= �

(Ni;�

Ni;�

); i = 1; : : : ; n (C.15)

where the matrix � is the inverse of the Jacobian

J =

"x;� y;�

x;� y;�

#=

nXi=1

"Ni;�xi Ni;�yi

Ni;�xi Ni;�yi

#(C.16)

C.3 Consistent Nodal Load Vector


f =Z!

NT FB jJj d! +Z!

NT FS jJj d! +Z�

NT FS jJj t d� (C.17)

where ! is the domain of the �nite element in its local coordinate system, t the thickness

of the element, FB represents body forces, � the boundary described in curvilinear, non-

dimensional � or � coordinates for the element, FS represents surface forces, andN contains

220 C.4. Consistent and Lumped Mass Matrices

shape functions Ni. In the surface and boundary integrals, N and jJj are evaluated on !

and �, respectively.



f th =Z!

BT E "th jJj d! (C.18)


"th =

n"thx "thy thxy

thyz

thxz

oT=nT T 0 0 0

oT�; T = T � T0 (C.19)



strains. These thermally strains are based on Gauss points values for obtaining the highest

accuracy.

C.4 Consistent and Lumped Mass Matrices


m =Z!

%NT N jJj d! (C.20)

Here, ! is the domain of the �nite element in its local coordinate system, % the mass

density, N contains shape functions Ni, and jJj is the determinant of the Jacobian matrix

J.

Lumped mass matrices are generated using the HRZ lumping scheme as described in

Appendix A.

C.5 Element Initial Stress Sti�ness Matrix

In the derivation of element initial stress sti�ness matrices it is convenient to reorder

and omit some nodal degrees of freedom by introducing the reordered, condensed element

displacement vector d� that only contains translational degrees of freedom. These trans-

lational d.o.f. are reordered so that �rst all x-direction d.o.f. are given, then y, and then

z as follows

d� = fu1 u2 : : : ui : : : un v1 v2 : : : vi : : : vn w1 w2 : : : wi : : : wngT (C.21)

Relating d.o.f. to the reordered, condensed element vector d�, the element initial stress

sti�ness matrix k� for an isoparametric Mindlin plate �nite element is given by

k� =Z!

GT S G jJj d! (C.22)

where ! is the domain of the �nite element described in curvilinear, non-dimensional ��coordinates for the element, G a matrix obtained by appropriate di�erentiation of shape


functions Ni, S a matrix of initial membrane stresses, and jJj is the determinant of theJacobian matrix J.


G =

2664g 0 0

0 g 0

0 0 g

3775 (C.23)


g =

"Ni;x

Ni;y

#; i = 1; : : : ; n (C.24)


S =

2664s 0 0

0 s 0

0 0 s

3775 (C.25)


s =

"�x �xy

�xy �y

#(C.26)

Here �x, �xy, etc., are membrane stresses in the plate found by an initial static stress

analysis.

C.6 Thermal Element \Sti�ness Matrix"

The thermal element sti�ness matrix consists of contributions from the heat conduction

matrix kth given by

kth =Z!

BthT�Bth jJj d! (C.27)

Here, ! is the domain of the �nite element in its local coordinate system, Bth a matrix

obtained by appropriate di�erentiation of shape functions Ni, � the thermal conductivity

matrix, and jJj is the determinant of the Jacobian matrix J. If the material is isotropic,

� can simply be replaced by the scalar �, the conductivity coe�cient.



i(C.28)


has the form

bthi =

"Ni;x

Ni;y

#; i = 1; : : : ; n (C.29)



h =Z!2

NT hN jJj d! +Z�2

NT hN jJj t d� (C.30)

222 C.7. Consistent Thermal Nodal Flux Vector

The convection boundary condition is applied to either the surface !2 or the boundary �2of the �nite element, both described in local coordinates. N contains shape functions Ni,

t is the thickness, h the convection coe�cient speci�ed, and jJj is the determinant of theJacobian matrix J for the surface !2 or the boundary �2.

C.7 Consistent Thermal Nodal Flux Vector


q =Z!1

NT qS jJj d! +Z�1

NT qS jJj t d�

+Z!2

NT h Te jJj d! +Z�2

NT h Te jJj t d� (C.31)

where the �rst two terms derive from speci�ed ux at either surface !1 or boundary �1and the two latter terms from speci�ed convection boundary conditions at surface !2 or

boundary �2. The surfaces !1, !2 and boundaries �1, �2 are described in curvilinear,

non-dimensional coordinates for the element and t is the thickness of the element. The

scalar qS is prescribed ux normal to the surface !1 or the boundary �1, N contains shape

functions Ni that are evaluated on the surface ! or the boundary �, jJj the determinantof the Jacobian matrix J for the surface ! or the boundary �, h the convection coe�cient

speci�ed, and Te is the environmental temperature speci�ed for the convection boundary

condition.

C.8 Gauss Quadrature

Before the Gauss integration rules used for element matrices of the Mindlin plate elements

are described, a few general remarks about these elements are given.

The main advantage of isoparametric Mindlin plate elements is the simple theory as only C0

continuity is required of displacement and rotation variables. The basis is a \thick" plate

theory in which transverse shear strains are accounted for, but the elements can also be used

for \thin" plates. However, all the implemented elements will su�er from shear \locking"

in the thin plate limit, i.e., when t ! 0. In order to understand this locking behaviour

the sti�ness matrix k can be regarded as being composed of a membrane sti�ness matrix

km, a bending sti�ness matrix kb, and a shear sti�ness matrix ks. The strain-displacement

matrix B is similarly divided into these three parts, i.e., B = Bm+Bb +Bs. Here, Bm is

obtained by using the �rst three rows in B in Eq. C.12 associated with in-plane membrane

strains "x, "y, and �xy and setting all other rows in B to zero. Similarly, Bb is associated

with in-plane bending strains "x, "y, and �xy and is obtained by using row 4, 5, and 6 of

B. The matrix Bs is associated with transverse shear strains �yz and �xz, and is obtained

by using the 2 last rows in B. In this way the element sti�ness matrix can be written as

k =Z!

BTm DM Bm jJj d!| {z }+

Z!

BTb DM Bb jJj d!| {z }+

Z!

BTs DM Bs jJj d!| {z } (C.32)

km kb ks


When Gauss quadrature is used for the numerical integration of ks, it brings two constraints

to a Mindlin plate element for each integration point, one associated with yz and the other

with xz as described by, e.g., Cook, Malkus & Plesha (1989). Too many integration points

may therefore lead to shear locking for a thin plate.

This locking phenomenon can be avoided by adopting a reduced or selective integration

rule to generate k as described by Hughes, Cohen & Haroun (1978), and their guidelines

for using selective integration rules have therefore been followed. Alternatively, one could

rede�ne the transverse shear interpolation as described by Hughes (1987). The Gauss

quadrature used is given in Table C.1.

Table C.1: Gauss quadrature used for isoparametric Mindlin plate �nite elements.

Integration rule:

Element type: km kb ks m k� kth ";�

3-node 1-point 1-point 1-point 1-point 1-point 1-point 1-point

4-node 4-point 4-point 1-point 4-point 4-point 4-point 1-,4-,1-point




The use of selective integration, however, leads to the unpleasant property that the ele-

ments can have zero-energy modes, i.e., mechanisms. This is avoided when full integration

rules are used but then shear locking appears. Using the selective integration rules given

in Table C.1, the 4-node bilinear element has two possible zero-energy modes, where only

one of them is communicable between adjacent elements. The 8-node serendipity element

has one mechanism which is not communicable between adjacent elements, and the 9-node

Lagrange element has one mechanism. The two triangular elements implemented have no

mechanisms but su�er from shear locking.

The 9-node Lagrange element, in general, gives the most accurate results, see Hughes,

Cohen & Haroun (1978) and Cook, Malkus & Plesha (1989), but it is not a \foolproof"

element as it has a spurious zero-energy mode. However, an improved 9-node element

called the \heterosis" element has been implemented in ODESSY, and the theory for this

element is given in the following.

C.9 The \Heterosis" Plate Element

The \heterosis" Mindlin plate element was developed by Hughes & Cohen (1978). The

element is a 9-node element which employs serendipity shape functions for the transverse

displacement w, and Lagrange shape functions for displacements u, v and rotations �x, �y.

224 C.10. Generating Flat Shell Elements

That is, the transverse displacement w is omitted for the 9-th node which is situated in

the center of the element, see Fig. C.1. The zero-energy mode observed for the 9-node

Lagrange element is thereby eliminated and the \heterosis" element has no mechanisms.

Selective integration as described for the 9-node element in Table C.1 is used for this

element. Using the notations N8i and Ni for shape functions for the 8-node serendipity

element and 9-node Lagrange element, respectively, the submatrix bi, see Eq. C.14, for

the Heterosis �nite element is given by

bi =

26666666666666664

Ni;x 0 0 0 0

0 Ni;y 0 0 0

Ni;y Ni;x 0 0 0

0 0 0 0 �Ni;x

0 0 0 Ni;y 0

0 0 0 Ni;x �Ni;y

0 0 �N8i;y Ni 0

0 0 �N8i;x 0 �Ni

37777777777777775

; i = 1; : : : ; 8 (C.33)

and

b9 =

26666666666666664

N9;x 0 0 0

0 N9;y 0 0

N9;y N9;x 0 0

0 0 0 �N9;x

0 0 N9;y 0

0 0 N9;x �N9;y

0 0 N9 0

0 0 0 �N9

37777777777777775

(C.34)

When the consistent nodal load vector f is calculated, see Eq. C.17, the serendipity shape

functions N8i must be used for interpolating loads in the z-direction.

The \heterosis" element has proven to be a very good plate element which can be used for

both thin and thick plates, and it has no mechanisms.

C.10 Generating Flat Shell Elements

The six di�erent Mindlin plate �nite elements can also be used as shell elements with 6 d.o.f.

per node. The six Mindlin plate elements described above have no \drilling freedoms" �zincluded in the formulation, and an element rotation �z is thus not measured and gives

no contribution to the strain energy stored in the element. For analysis of a at or folded

plate structure the rotation �z is not necessary, and furthermore, for a slightly curved shell

structure the sti�ness corresponding to a �z degree of freedom is small. However, for a

strongly curved shell structure, where the curvature can be measured as the angle between

the element planes of two adjacent elements, this contribution to the sti�ness of the element

may be large.

I have chosen to use the approximation of transforming the Mindlin plate elements into

general at shell elements, but in order to avoid an ill-conditioned or even singular sti�-


ness matrix, the at shell elements are given a small sti�ness for \drilling rotations" �zaccording to Bathe (1982). The element sti�ness matrix is expanded from 5 to 6 d.o.f.,

and the sti�ness coe�cients in the diagonal of the element sti�ness matrix corresponding

to rotations �z are set to 1=1000 of the smallest diagonal element in the original sti�ness

matrix. These added sti�nesses must be large enough to enable the accurate solution of the

�nite element equilibrium equations and small enough not to a�ect the system response

signi�cantly.

This expanded element matrix k� is then transformed from the element plane to the global

coordinate system using the standard transformation

k = TT k� T (C.35)

where T is the transformation matrix between the local and global element degrees of

freedom. All other element vectors and matrices for these at shell elements are calculated

in a somewhat similar way; they are established in their local coordinate system and then

transformed to the global system. In this way the computer code for the plate elements

can be used in all operations for the shell elements.

These at shell elements pass patch tests and do not exhibit strain under rigid body

motion. However, although membrane-bending coupling is present throughout an actual

curved shell, it is absent in individual at shell elements and this might lead to inaccurate

results, especially if a small number of elements are used to model a curved surface. This

must be taken into account when generating �nite element models using these at shell

elements.

226 C.10. Generating Flat Shell Elements

Appendix

D

Necessary Optimality Conditions for

Eigenvalue Problems

In this appendix necessary optimality conditions for eigenvalue optimization problems are

derived in case of a multiple optimum eigenvalue. These derivations were mainly performed

by Alexander P. Seyranian in our joint work on problems involving multiple eigenvalues,

and they can also be found in Seyranian, Lund & Olho� (1994). In case of a simple

optimum fundamental eigenvalue, the necessary optimality conditions are given in Section

8.4.1.

The optimization problem considered concerns maximization of the lowest of the eigenval-

ues �j; j = 1; : : : ; n, subject to a constant volume constraint as de�ned by Eqs. 8.14 and

8.15, i.e.,

Maximize min �j; j = 1; : : : ; n (D.1)

a1; : : : ; aI

Subject to

F (a1; : : : ; aI) = 0 (D.2)

The necessary optimality conditions �rst are derived for a double optimum fundamental

eigenvalue and then for any multiplicity N of the lowest eigenvalue and they give much

insight in the di�culties in maximizing a lowest multiple eigenvalue.

D.1 Double Optimum Fundamental Eigenvalue

Let us consider the case when the optimum is achieved at the double lowest eigenvalue

�1 = �2, where �1 = �2 < �3 � : : :. This is a non-di�erentiable case, and we have to use

directional derivatives as described in Section 4.9.

227

228 D.1. Double Optimum Fundamental Eigenvalue

Taking the vector of varied design variables in the form a+ "e, kek = 1, according to Eq.

4.46 we obtain the directional derivatives �1 and �2 from

det

�� fT11e� � fT12e

fT12e fT22e� �

�� = 0 (D.3)

This is a quadratic equation in �. Solving it we obtain for any direction e

�1;2 =fT11e + fT22e�

q(fT11e� fT22e)

2+ 4 (fT12e)

2

2(D.4)

The necessary optimality condition for a maximum is

min(�1; �2) � 0 (D.5)

for any direction e satisfying the condition fT0 e = 0.

From Eq. D.3 we see that if we take the direction as �e, then both �1 and �2 will change

their signs to the opposite ones. This means that if for some direction e both derivatives

�1, �2 are negative then the design point is not a maximum, since a change in sign of the

direction e leads to �1 > 0; �2 > 0, i.e., a better design. This means that the necessary

optimality condition in the bimodal case is

�1�2 � 0 (D.6)

for any admissible direction e, i.e., direction that satis�es the condition

fT0 e = 0 (D.7)

The optimality condition in Eq. D.6 is fundamentally di�erent from that of the di�eren-

tiable case due to its non-linear nature. The condition was �rst formulated by Masur &

Mr�oz (1979, 1980).

Using Eqs. D.3 and D.4 we can express the necessary optimality condition of Eq. D.6 in

the form

(fT11e)(fT22e)� (fT12e)

2 � 0 (D.8)

for any arbitrary direction e satisfying the condition in Eq. D.7.

D.1.1 Lemma 1: Existence of Improving Direction in Bimodal

Case

Let us formulate the Lemma for existence of an improving direction e:

Lemma 1: If the vectors f11; f12; f22; f0 are linearly independent then there exists an

improving direction e for which �1 > 0; �2 > 0.

Appendix D. Necessary Optimality Conditions for Eigenvalue Problems 229

D.1.2 Proof of Lemma 1

To prove Lemma 1 consider the following system of linear algebraic equations of the vari-

ables e1; : : : ; eI

fT11e = �01 > 0

fT12e = 0 (D.9)

fT22e = �02 > 0

fT0 e = 0

If the vectors f11; f12; f22; f0 are linearly independent, then a solution e to the system in

Eq. D.9 exists for arbitrary values of �01 and �02 . The vector ~e =

e

kek is then an improving

direction since from Eqs. D.3 and D.9 we have

�1 = fT11~e =�01kek > 0 (D.10)

�2 = fT22~e =�02kek > 0

which proves Lemma 1.

D.1.3 Theorem 1: Necessary Optimality Conditions for Bimodal

Case

Now let us formulate the necessary optimality conditions for the bimodal case

Theorem 1: If the vector of design variables a constitutes the solution of the optimization

problem, Eqs. D.1 and D.2, with the double eigenvalue �1 = �2 < �3 � : : :,

then the vectors f11; f12; f22; f0 are linearly dependent

11f11 + 2 12f12 + 22f22 � 0f0 = 0 (D.11)

with the coe�cients sk satisfying the inequality

11 22 � 212 (D.12)

i.e., satisfying conditions of positive semi-de�niteness of the symmetric ma-

trix sk; s; k = 1; 2.

Here it is assumed that the rank of the matrix consisting of the vectors f11; f12; f22; f0 is

equal to 3. Note that the linear independence of the four vectors mentioned above is

possible only when the dimension I of the vector of design variables a is greater than 3.

230 D.2. N-fold Optimum Fundamental Eigenvalue

D.1.4 Proof of Theorem 1

At the optimum point, linear dependence of the vectors fsk, f0 in Eq. D.11 is a consequence

of Lemma 1, and to prove Eq. D.12 we express, for example, f22 from Eq. D.11

f22 = � 11 22

f11 � 2 12

22f12 +

0

22f0 (D.13)

and substitute this expression into Eq. D.8. Using Eq. D.7 we get

11

22

�fT11e

�2+ 2

12

22

�fT12e

� �fT11e

�+�fT12e

�2 � 0 (D.14)

This quadratic form of fT11e and fT12e is positive semi-de�nite only if its coe�cients satisfy

the inequality in Eq. D.12.

Lemma 1 and Theorem 1 were formulated and proved for the �rst time by Bratus &

Seyranian (1983).

D.2 N-fold Optimum Fundamental Eigenvalue

Consider next the general case when in the optimization problem, Eqs. D.1 and D.2, the

maximum is attained at an N -fold multiple lowest eigenvalue �1 = �2 = : : : = �N <

�N+1 � : : :.

In this case for any admissible direction e, i.e., direction satisfying the condition in Eq.

D.7, we �nd directional derivatives �j; j = 1; : : : ; N , from Eq. 4.46, i.e.,


�� = 0; s; k = 1; : : : ; N (D.15)

If the maximum is attained then there must be no admissible direction e for which all

� = �j; j = 1; : : : ; N , are of the same sign. This is an obvious generalization of the

necessary optimality condition in Eq. D.6.

The Lemma for existence of improving directions and the Theorem for necessary optimality

conditions in the general case when maximum is attained at an N -fold multiple lowest

eigenvalue are given in the following.

D.2.1 Lemma 2: Existence of Improving Direction in the Gen-

eral Case

The Lemma for existence of an improving direction e can be formulated as

Lemma 2: If the vectors f0; fsk, s; k = 1; : : : ; N , k � s (the total number of these

vectors is equal to (N + 1)N=2 + 1) are linearly independent, then there

exists an improving direction e for which �j > 0; j = 1; : : : ; N .

Note that the linear independence of the vectors is only possible if I � (N + 1)N=2 + 1,

where I is the dimension of the vector a of design variables.


D.2.2 Proof of Lemma 2

To prove Lemma 2 we consider the system of linear equations in e1; : : : ; eI

fTske = �sk�0s ; s; k = 1; : : : ; N; k � s (D.16)

fT0 e = 0

where �0s are given positive constants. If the vectors f0; fsk are linearly independent, a

solution to Eq. D.16 exists for any �0s , in particular when �0s > 0. Suppose the vector e is

a solution to the system in Eq. D.16 and let us normalize this vector as ~e =e

kek . Thenwe obtain from Eqs. D.16 and D.15

�j = fTjj~e =�0j

kek > 0; j = 1; : : : ; N (D.17)

which implies existence of an improving direction. This proves Lemma 2, see also Seyranian

(1987).

When I < (N + 1)N=2 + 1, the vectors fsk; f0 are always linearly dependent and hence an

improving direction may not exist.

D.2.3 Theorem 2: Necessary Optimality Conditions for the Gen-

eral Case

Let us formulate the theorem for the necessary optimality conditions

Theorem 2: If the vector of design variables a renders a lowest N -fold eigenvalue �1 =

�2 = : : : = �N a maximum, it is necessary that the vectors f0; fsk, s; k =

1; : : : ; N , k � s, are linearly dependent

NXs;k=1

skfsk � 0f0 = 0 (D.18)

with the coe�cients 0; sk satisfying conditions of positive semi-de�niteness

of the symmetric matrix sk, s; k = 1; : : : ; N .

Note that due to the symmetry skfsk = ksfks we have

NXs;k=1

skfsk =NXs=1

ssfss + 2NX

s;k=1

s>k

skfsk (D.19)

Nevertheless, we prefer the form of Eq. D.18 due to its convenience.

D.2.4 Proof of Theorem 2

Linear dependence of the vectors fsk; f0 at the optimum point is an obvious consequence of

Lemma 2 and we just need to prove the necessity of positive semi-de�niteness of the matrix


sk. To this end we choose a new basis of eigenvectors ~�1; : : : ;~�N for which the matrix

sk is diagonal, and show that if an optimum is attained then all ~ ss � 0, s = 1; : : : ; N .

Let us transform the eigenvectors

~�s =NXk=1

gks�k; s = 1; : : : ; N (D.20)

Here ~�s are transformed eigenvectors satisfying the orthonormality condition in Eq. 4.26,

and gks is the transformation matrix.

Using Eq. D.20 in Eq. 4.26 we obtain

~�T

i M~�j =

NXs=1

gsi�Ts

!M

NXk=1

gkj�k

!

=NX

s;k=1

gsigkj�TsM�k

=NX

s;k=1

gsigkj�sk (D.21)

=NXs=1

gsigsj

= �ij; i; j = 1; : : : ; N

In matrix form this equation is equivalent to

gTg = I and gT = g�1 (D.22)

where I is the unit matrix. The last equation means that the transformation matrix g is

an orthogonal matrix. Now let us express vectors �k from Eq. D.20 by ~�s. Due to Eq.

D.22 we have

�s =NXk=1

gsk ~�k; s = 1; : : : ; N (D.23)

Using the notation

rK� ~�rM =

@K

@a1� ~�

@M

@a1; : : : ;

@K

@aI� ~�

@M

@aI

!(D.24)

and Eq. 4.45, we obtain

NXk;s=1

ksfks =NX

k;s=1

ks�Tk

�rK� ~�rM

��s

=NX

k;s=1

ks

NXt=1

gkt ~�T

t

!�rK� ~�rM

� NXm=1

gsm ~�m

!

=NX

t;m=1

0@ NXk;s=1

gkt ksgsm

1A ~�

T

t

�rK� ~�rM

�~�m

=NX

t;m=1

~ tm~ftm

(D.25)


So, in the new basis the matrix ks takes the form

~ tm =NX

k;s=1

gkt ksgsm (D.26)

In matrix form we have

~ = gT g = g�1 g (D.27)

This means that there exists a basis in which the matrix ~ is diagonal. Then the optimality

condition in Eq. D.18 takes the form

NXs=1

~ ss~fss � 0f0 = 0 (D.28)

To show that the condition ~ ss > 0, s = 1; : : : ; N , is the necessary condition for optimality,

let us consider an admissible direction e, i.e., direction satisfying the condition Eq. D.7.

Multiplying Eq. D.28 by e we obtain

NXs=1

~ ss~fTsse = 0 (D.29)

Suppose that in Eqs. D.28 the j-th coe�cient ~ jj 6= 0. Let us take the admissible direction

e such that

~fTske = 0; s; k = 1; : : : ; N; k > s

~fTtte = �0t ; t = 1; : : : ; N; t 6= j (D.30)

fT0 e = 0

where �0t are arbitrary positive constants.

Such a direction e exists if we assume that the rank of the matrix consisting of the vectors

f0, ~fsk, s; k = 1; : : : ; N , k � s, is equal to (N + 1)N=2 � I. Normalizing this vector we get

~e =e

kek . Then according to Eq. D.15 for this direction ~e we obtain

�s = ~fTss~e =�0skek > 0; s = 1; : : : ; N; s 6= j

and (D.31)

�j = ~fTjj~e

So we can �nd �j from Eq. D.29 taking the direction ~e

�j = ~fTjj~e = � 1

~ jj

NXs=1

s6=j

~ ss~fTss~e

= � 1

kekNXs=1

s6=j

~ ss~ jj

!�0s

(D.32)

Here we have used Eq. D.31.


If maximum of the lowest N -fold eigenvalue is achieved, then for any admissible direction

e the sensitivities �k, k = 1; : : : ; N , must not be of the same sign. Since we have chosen ~e

such that all �s, s = 1; : : : ; N , s 6= j, are positive, then �j must be less than or equal to

zero, i.e., �j � 0. Using Eq. D.32 we get

NXs=1

s6=j

~ ss~ jj

!�0s � 0 (D.33)

for arbitrary choice of the positive constants �0s , s = 1; : : : ; N , s 6= j.

The inequality in Eq. D.33 can be satis�ed only if

~ ss~ jj

� 0; s = 1; : : : ; N; s 6= j (D.34)

since, otherwise, the constants �0s can be chosen such that the inequality in Eq. D.34 is

violated. This means that all ~ ss, s = 1; : : : ; N , must be of the same sign. Without loss of

generality, all ~ ss can be regarded as non-negative quantities. So, we have proved Lemma

2, i.e., that

~ ss � 0; s = 1; : : : ; N; (D.35)

which implies positive semi-de�niteness of the matrix of coe�cients sk, s; k = 1; : : : ; N ,

and this constitutes the necessary optimality condition in the general case of an N -fold

multiple lowest optimum eigenvalue.

Similar results for minimizing the maximum eigenvalue were obtained by Overton (1988).

Recently, Cox & Overton (1992), derived the necessary optimality conditions for discrete

and distributed eigenvalue problems using Clarke's generalized gradient, see Clarke (1990).

They also considered lower and upper bounds on design variables ai � ai � ai. These

results approve optimality conditions suggested by Olho� & Rasmussen (1977), and also

used by many others, see, e.g., Gajewski & Zyczkowski (1988).

Lund E. Finite Element Based Design Sensitivity Analysis and Optimization (1994)(en)(234s)

Documents

Transcript of Lund E. Finite Element Based Design Sensitivity Analysis and Optimization (1994)(en)(234s)