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Method of Finite Elements I

• Held by Prof. Dr. E. Chatzi, Dr. P. Steffen• Assistants: Adrian Egger (HIL E 13.3), Harry Mylonas (HIL H33.1),

Konstantinos Tatsis (HIL H33.1)• Lectures homepage: http://www.chatzi.ibk.ethz.ch/education/method-

of-finite-elements-i.html• Course book: “Finite Element Procedures” by K.J. Bathe• Performance assessment

Method of Finite Elements I

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Method of Finite Elements I

A piece of Computational Mechanics

Method of Finite Elements I

Computational Mechanics

Nano-mechanics

Micro-mechanics

Continuum Mechanics Systems

Source: http://www.colorado.edu/MCEN/MCEN4173/chap_01.pdf

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Method of Finite Elements I

A piece of Computational Mechanics

Method of Finite Elements I©

sd.ru

b.de

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Method of Finite Elements I

A piece of Computational Mechanics

Method of Finite Elements I

Computational Mechanics

Nano-mechanics

Micro-mechanics

Continuum Mechanics

Solids & Structures Fluids Multiphysics

Systems

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Method of Finite Elements I

A piece of Computational Mechanics

Method of Finite Elements I

Continuum Mechanics

Statics

Time Invariant

Quasi-static

Dynamics

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Method of Finite Elements I

A piece of Computational Mechanics

Method of Finite Elements I

Continuum Mechanics

Statics

Linear Nonlinear

Dynamics

Linear Nonlinear

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Method of Finite Elements I

Course Overview

• 20.02.2016 – Introductory Concepts• 27.02.2016– The Direct Stiffness Method• 06.03.2016 – Dynamics Considerations• 13.03.2016 – The Variational Formulation.• 20.03.2016 – Computer Lab 1• 27.04.2016 – 1D Elements (truss/beam)• 03.04.2016 – 2D Elements (plane stress/strain)

*all lecture sessions come with a Demo in the last part of the session

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Method of Finite Elements I

• 10.04.2016 – Practical application of the MFEPractical Considerations

• 08.05.2016 – Results Interpretation• 15.05.2016 - Demo Session: Integration/Conditioning/Error

Estimators• 22.05.2016 – Computer Lab 2• 29.05.2016 – Final Project Presentations

*all lecture sessions come with a Demo in the last part of the session

Course Overview

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Method of Finite Elements I

• An overview of the MFE I course• MFE development• Introduction to the use of Finite Elements• Modelling the physical problem• Finite elements as a tool for computer-aided design and assessment• Example Demonstrators on various computation platforms

Today’s lecture

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Method of Finite Elements I

Chapter 1: FE Analysis in brief…

Object: A Solid with known mechanical properties. (a skyscraper; a shaft; bio tissue …)

Main Features

• Acting Loads• Boundary: The surface enclosing the geometry • Solid: Interior + Boundary • Boundary conditions: prescribed displacements/tractions on the boundary

FEA was originally developed for solid mechanics applications.

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Method of Finite Elements I

FE Analysis in brief…Problem Statement

undeformed deformed?

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Method of Finite Elements I

How does it work?

Physical ModelDescribe the problem: Simplify a real engineering problem into a

problem that can be solved by FEA

FE modelDiscretize/mesh the solid, define material properties, apply boundary conditions

TheoryChoose approximate functions, formulate linear equations, and

solve equations

ResultsObtain, visualize and explain

the results

Pre-processor

Post-processor Solver

Source: http://www.colorado.edu/MCEN/MCEN4173/chap_01.pdf

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Method of Finite Elements I

The MFE is the confluence of three ingredients: matrix structural analysis, variational approach and a computer

Theoretical Formulation1. “Lösung von Variationsproblemen” by W. Ritz in 19082.“Weak formulation” by B. Galerkin in 19153.“Mathematical foundation” by R. Courant ca. 1943

Formulation & First Applications (1950s and 1960s)1. 1950s, M.J. Turner at Boeing (aerospace industry in general):

Direct Stiffness Method 2. Matrix formulation of structural analysis by Agyris in 19543. Term ‚Finite Element‘ coined by Clough in 19604. First book on EM by Zienkiewicz and Cheung in 1967

MFE Development

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Method of Finite Elements I

Commercial Software (since 1970s)

1. General purpose packages for main frames (Abaqus..) in 1970s2. Special purpose software for PCs in 1980s

During this class, the following software packages will be used:ABAQUS, ANSYS, CUBUS, SAP2000

MFE Development

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Method of Finite Elements I

FEM is a big success story, because it…

1. can handle complex geometries2. can handle a wide variety of engineering problems

- mechanics of solids & fluids- dynamics/heat/electrostatic problems…

3. can handle complex restraints & loading4. is very well suited for computation

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Method of Finite Elements I

What is a Finite Element?

Source: http://www.colorado.edu/MCEN/MCEN4173/chap_01.pdf

Find the perimeter L of a circle of diameter dWe know that L = πd.

elements

nodes

Lij

Approximation via Inscribed PolygonThe element length is Lij = 2r sin(π/n). Since all elements have the same length, the polygon perimeter is Ln = n LijHence the approximation to π is πn = Ln/d = n sin(π/n).

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Method of Finite Elements I

What is a Finite Element?

Source: http://www.colorado.edu/MCEN/MCEN4173/chap_01.pdf

elements

nodes

Lij

n πn = n sin(π/n)1 0.0000000000000002 2.0000000000000004 2.8284271247461908 3.06146745892071816 3.12144515225805232 3.13654849054593964 3.140331156954753128 3.141277250932773256 3.141513801144301

Rectification of Circle by Inscribed Polygons (“Archimedes FEM”)

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Method of Finite Elements I

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Method of Finite Elements I

• Within the framework of continuum mechanics dependencies between geometrical and physical quantities are formulated on a differentially small element and then extended to the whole continuum

• As a result we obtain differential, partial differential or integral equations for which, generally, an analytical solution is not available – they have to be solved using some numerical procedure

• The MFE is based on the physical discretization of the observed domain, thus reducing the number of the degrees of freedom; moreover the governing equations are, in general, algebraic

Introduction to the Use of Finite Elements

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Method of Finite Elements I

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Method of Finite Elements I

Steps in the MFE• Discretization/MeshingThe continuum is discretized using a mesh of finite elements.

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Method of Finite Elements I

Steps in the MFEDiscretization: The continuum is discretized using a mesh of finite elements.

These elements are connected at nodes located on the element boundaries.

e

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Method of Finite Elements I

Steps in the MFE• The Shape FunctionsThe state of deformation, stresses, etc. in each element is approximated by a the set of corresponding values in the nodes; these nodal values are the basic unknowns of the MFE.Values between nodes, have to therefore result via interpolation.

The way in which these steps are approached has a great influence on the results of the calculations .

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Method of Finite Elements I

• The MFE is only a way of solving the mathematical model • The solution of the physical problem depends on the

quality of the mathematical model – the choice of the mathematical model is crucial

• The chosen mathematical model is reliable if the required response can be predicted within a given level of accuracy compared to the response of a very comprehensive (highly refined) mathematical model

• The most effective mathematical model for the analysis is the one that gives the required response with sufficient accuracy and at the lowest computational toll

Modelling of the Physical Problem

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Method of Finite Elements I

Complex physical problem modelled by a simple mathematical model

Simple Example

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Method of Finite Elements I

Detailed reference model – 2D plane stress model

Simple Example

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Method of Finite Elements I

• Choice of mathematical model must correspond to desired response• The most effective mathematical model delivers reliable answers

with the least amount of effort• Any solution (including MFE) of a mathematical model is limited

to information contained in or fed into the model: bad input – bad output (garbage in – garbage out)

• Assessment of accuracy is based on comparisons with the results from very comprehensive models – but in practice it has to be based on experience (experiments…)

• The engineer (user) should be able to judge the quality of the obtained results (i.e. for plausibility)

Considerations

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Method of Finite Elements I

Demonstrator 1: Seismic Analysis of a Concrete Gravity Dam in ABAQUS

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Method of Finite Elements I

Demonstrator 2: Analysis of a Wind Turbine Structure in ANSYS