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Finite Element Method


Gouri Dhatt Gilbert Touzot

Emmanuel Lefranccedilois

Series Editor Piotr Breitkopf

First published 2012 in Great Britain and the United States by ISTE Ltd and John Wiley amp Sons Inc

Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address

ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA

wwwistecouk wwwwileycom

copy ISTE Ltd 2012 The rights of Gouri Dhatt Gilbert Touzot and Emmanuel Lefranccedilois to be identified as the author of this work have been asserted by them in accordance with the Copyright Designs and Patents Act 1988

Library of Congress Control Number 2012946444 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-368-5

Printed and bound in Great Britain by CPI Group (UK) Ltd Croydon Surrey CR0 4YY

2 ndash DIVERS TYPES DEacuteLEacuteMENTS v

TABLE OF CONTENTS

Introduction 101 The finite element method 1

011 General remarks 1012 Historical evolution of the method 2013 State of the art 3

02 Object and organization of the book 3021 Teaching the finite element method 3022 Objectives of the book 4023 Organization of the book 4

03 Numerical modeling approach 6031 General aspects 6032 Physical model 7033 Mathematical model 9034 Numerical model 10035 Computer model 13

Bibliography 16Conference proceedings 17Monographs 18Periodicals 19

Chapter 1 Approximations with finite elements 2110 Introduction 2111 General remarks 21

111 Nodal approximation 21112 Approximations with finite elements 28

12 Geometrical definition of the elements 33121 Geometrical nodes 33122 Rules for the partition of a domain into elements 33123 Shapes of some classical elements 35124 Reference elements 37125 Shapes of some classical reference elements 41126 Node and element definition tables 44

13 Approximation based on a reference element 45131 Expression of the approximate function u(x) 45132 Properties of approximate function u(x) 49

vi FINITE ELEMENT METHOD

14 Construction of functions N ( )ξ and N ( )ξ 54141 General method of construction 54142 Algebraic properties of functions N and N 59

15 Transformation of derivation operators 61151 General remarks 61152 First derivatives 62153 Second derivatives 65154 Singularity of the Jacobian matrix 68

16 Computation of functions N their derivatives and the Jacobian matrix 72161 General remarks 72162 Explicit forms for N 73

17 Approximation errors on an element 75171 Notions of approximation errors 75172 Error evaluation technique 80173 Improving the precision of approximation 83

18 Example of application rainfall problem 89Bibliography 95

Chapter 2 Various types of elements 9720 Introduction 9721 List of the elements presented in this chapter 9722 One-dimensional elements 99

221 Linear element (two nodes C0) 99222 High-precision Lagrangian elements (continuity C0) 101223 High-precision Hermite elements 105224 General elements 109

23 Triangular elements (two dimensions) 111231 Systems of coordinates 111232 Linear element (triangle three nodes C0) 113233 High-precision Lagrangian elements (continuity C0) 115234 High-precision Hermite elements 123

24 Quadrilateral elements (two dimensions) 127241 Systems of coordinates 127242 Bilinear element (quadrilateral 4 nodes C0) 128243 High-precision Lagrangian elements 129244 High-precision Hermite element 134

25 Tetrahedral elements (three dimensions) 137251 Systems of coordinates 137252 Linear element (tetrahedron four nodes C0) 139253 High-precision Lagrangian elements (continuity C0) 140254 High-precision Hermite elements 142

26 Hexahedric elements (three dimensions) 143261 Trilinear element (hexahedron eight nodes C0) 143262 High-precision Lagrangian elements (continuity C0) 144263 High-precision Hermite elements 150

TABLE OF CONTENTS vii

27 Prismatic elements (three dimensions) 150271 Element with six nodes (prism six nodes C0) 150272 Element with 15 nodes (prism 15 nodes C0) 151

28 Pyramidal element (three dimensions) 152281 Element with five nodes 152

29 Other elements 153291 Approximation of vectorial values 153292 Modifications of the elements 155293 Elements with a variable number of nodes 156294 Superparametric elements 158295 Infinite elements 158

Bibliography 160

Chapter 3 Integral formulation 16130 Introduction 16131 Classification of physical systems 163

311 Discrete and continuous systems 163312 Equilibrium eigenvalue and propagation problems 164

32 Weighted residual method 172321 Residuals 172322 Integral forms 173

33 Integral transformations 174331 Integration by parts 174332 Weak integral form 177333 Construction of additional integral forms 179

34 Functionals 182341 First variation 182342 Functional associated with an integral form 183343 Stationarity principle 187344 Lagrange multipliers and additional functionals 188

35 Discretization of integral forms 194351 Discretization of W 194352 Approximation of the functions u 197353 Choice of the weighting functions ψ 198354 Discretization of a functional (Ritz method) 205355 Properties of the systems of equations 208

36 List of PDEs and weak expressions 209361 Scalar field problems 210362 Solid mechanics 213363 Fluid mechanics 217

Bibliography 229

Chapter 4 Matrix presentation of the finite element method 23140 Introduction 23141 The finite element method 231

411 Finite element approach 231412 Conditions for convergence of the solution 243

viii FINITE ELEMENT METHOD

413 Patch test 25642 Discretized elementary integral forms We 264

421 Matrix expression of We 264422 Case of a nonlinear operator LL 267423 Integral form We on the reference element 269424 A few classic forms of We and of elementary matrices 274

43 Techniques for calculating elementary matrices 274431 Explicit calculation for a triangular element (Poissonrsquos equation) 274432 Explicit calculation for a quadrangular element (Poissonrsquos equation) 279433 Organization of the calculation of the elementary matrices by

numerical integration 280434 Calculation of the elementary matrices linear problems 282

44 Assembly of the global discretized form W 297441 Assembly by expansion of the elementary matrices 298442 Assembly in structural mechanics 303

45 Technique of assembly 305451 Stages of assembly 305452 Rules of assembly 305453 Example of a subprogram for assembly 307454 Construction of the localization table LOCE 308

46 Properties of global matrices 310461 Band structure 310462 Symmetry 314463 Storage methods 314

47 Global system of equations 318471 Expression of the system of equations 318472 Introduction of the boundary conditions 318473 Reactions 321474 Transformation of variables 321475 Linear relations between variables 323

48 Example of application Poissonrsquos equation 32449 Some concepts about convergence stability and error calculation 329

491 Notations 329492 Properties of the exact solution 330493 Properties of the solution obtained by the finite element method 331494 Stability and locking 334495 One-dimensional exact finite elements 337

Bibliography 343

Chapter 5 Numerical Methods 34550 Introduction 34551 Numerical integration 346

511 Introduction 346512 One-dimensional numerical integration 348513 Two-dimensional numerical integration 360514 Numerical integration in three dimensions 368

TABLE OF CONTENTS ix

515 Precision of integration 372516 Choice of number of integration points 375517 Numerical integration codes 379

52 Solving systems of linear equations 384521 Introduction 384522 Gaussian elimination method 385523 Decomposition 391524 Adaptation of algorithm (544) to the case of a matrix stored by

the skyline method 39953 Solution of nonlinear systems 404

531 Introduction 404532 Substitution method 407533 NewtonndashRaphson method 411534 Incremental (or step-by-step) method 420535 Changing of independent variables 421536 Solution strategy 424537 Convergence of an iterative method 426

54 Resolution of unsteady systems 429541 Introduction 429542 Direct integration methods for first-order systems 431543 Modal superposition method for first-order systems 463544 Methods for direct integration of second-order systems 466545 Modal superposition method for second-order systems 476

55 Methods for calculating the eigenvalues and eigenvectors 480551 Introduction 480552 Recap of some properties of eigenvalue problems 481553 Methods for calculating the eigenvalues 488

Bibliography 502

Chapter 6 Programming technique 50560 Introduction 50561 Functional blocks of a finite element program 50662 Description of a typical problem 50763 General programs 508

631 Possibilities of general programs 508632 Modularity 511

64 Description of the finite element code 512641 Introduction 512642 General organization 513643 Description of tables and variables 517

65 Library of elementary finite element method programs 521651 Functional blocks 521652 List of thermal elements 530653 List of elastic elements 538654 List of elements for fluid mechanics 545

x FINITE ELEMENT METHOD

66 Examples of application 549661 Heat transfer problems 550662 Planar elastic problems 558663 Fluid flow problems 566

Appendix Sparse solver 577Sofiane HADJI

70 Introduction 57771 Methodology of the sparse solver 578

711 Assembly of matrices in sparse form row-by-row format 579712 Permutation using the ldquominimum degreerdquo algorithm 584713 Modified columnndashcolumn storage format 587714 Symbolic factorization 589715 Numerical factorization 590716 Solution of the system by descentascent 592

72 Numerical examples 593Bibliography 595

Index 597

2 ndash DIVERS TYPES DEacuteLEacuteMENTS 1

Introduction

01 The finite element method

011 GENERAL REMARKS

Modern technological advances challenge engineers to carry out increasinglycomplex and costly projects which are subject to severe reliability and safetyconstraintsThese projects cover domains such as space travel aeronautics andnuclear applicationswhere reliability and safety are of crucial importanceOtherprojects are connected with the protection of the environment for examplecontrol of thermal acoustic or chemical pollutionwater course managementmanagement of groundwater and weather forecasting For a properunderstanding analysts need mathematical models that allow them to simulatethe behavior of complex physical systemsThese models are then used duringthe design phase of the projects

Engineering sciences (mechanics of solids and fluids thermodynamics etc) areused to describe the behavior of physical systems in the form of partial differentialequations Today the finite element method has become one of the mostfrequently used methods for solving such equationsThis method requires intensiveuse of a computer and can be applied to solve almost all problems encountered inpractice steady or transient problems in linear and nonlinear regions for one-two- and three-dimensional domains Moreover it can be successfully adaptedfor use in the heterogeneous environments and domains of complex forms oftenencountered in practice by engineers

The finite element method consists of using a simple approximation of unknownvariables to transform partial differential equations into algebraic equations Itdraws on the following three disciplines

mdash engineering sciences to describe physical laws (partial differential equations)mdash numerical methods for the elaboration and solution of algebraic equationsmdash computing tools to carry out the necessary calculations efficiently using a

computer

2 FINITE ELEMENT METHOD

012 HISTORICAL EVOLUTION OF THE METHOD

Structural mechanics allows us to analyze frames and trussesThe behavior ofeach truss or beam element is represented by an elementary stiffness matrixconstructed using knowledge of the strength of materialsUsing these matriceswe are able to construct a system of algebraic equations verifying the conditionsof displacement continuity and balance of forces at the nodesThe solution ofthe system of equations corresponding to applied loads leads to the displacementsof all nodes in the structure The emergence of computers and the requirementsof the aeronautical industry led to rapid developments in the field of structuralmechanics in the 1950sThe concept of finite elements was introduced byTurnerCloughMartin and Topp [TUR 56] in 1956they represented an elastictwo-dimensional domain by an assembly of tr iangular panels across whichdisplacements are presumed to vary in a linear manner The behavior of eachpanel is represented by an elementary stiffness matrix Structural mechanics toolsare then employed to obtain nodal displacements under different applied loadsand boundary conditions

We should also highlight the work carried out byArgyris and Kelsey [ARG 60]who employed the notion of energy in structural analysis The basic ideasinvolved in the finite element method however appeared earlier in an articlepublished by Courant in 1943 [COU 43]

From 1960 onward finite element method developed rapidly in a number ofdirections

mdash The method was reformulated based on energetic and var iationalconsiderations in the general form of weighted residuals or weakformulations [ZIE 65 GRE 69 FIN 75 ARA 68]

mdash A number of authors created high-precision elements [FEL 66] curvedelements and isoparametric elements [ERG 68 IRO 68]

mdash The finite element method was recognized as a general method of solutionfor partial differential equations It thus came into use in solving nonlinearand transient problems of structures as well as in other fields such as soiland rock mechanics (geomechanics) fluid mechanics and thermodynamics[PRO 01ndashPRO 13]

mdash A mathematical basis for finite element method was established usingconcepts of functional analysis [PRO 14ndashPRO 15]

Since 1967 many books have been published on the finite element method[MON 01ndashMON 29]We wish to highlight in particular the three editionsof the book by Zienkiewicz [MON 02] which are available throughoutthe world We may refer to Pironneau Geacuteradin Imbert Batoz-Dhatt and

INTRODUCTION 3

Dhatt-Touzot among others for books available in FrenchAn exhaustive listof reference works in the domain of finite elements may be easily obtained usingan Internet search engine

013 STATE OF THE ART

The finite element method (FEM) is nowadays widely used in industr ialapplications including aeronautical aerospace automobile naval and nuclearconstruction fieldsand in applications of fluid mechanicsincluding tidal studiessedimentary transportation the study of thermal or chemical pollutionphenomena and fluid-structure interactions A number of general-purposecomputer codes are available for industrial users of the finite element methodsuch as IDEAS SAMCEF NASTRAN ABAQUS FIDAP MARC ANSYSADINA LSDYNA ASTER and CASTEM

In order for the finite element method to be effective in industrial applicationscomputer codes must be used to assist in the preparation of data and theinterpretation of results These pre- and post-processing tools are usuallyintegrated into more general computer-aided design (CAD) software packagessuch as IDEAS CASTOR or CATIA

02 Object and organization of the book

021 TEACHING THE FINITE ELEMENT METHOD

The finite element method is now widely taught at both undergraduate andpostgraduate level The teaching of the finite element method requires amultidisciplinary approach involving different aspects

mdash understanding of the physical problem and intuitive knowledge of the natureof the solution being sought

mdash representation of the physical phenomenon in the form of partial differentialequations and weak ldquovariationalrdquo or ldquointegralrdquo formulations

mdash discretization techniques to produce a discrete or algebraic modelmdash matricial organization of datamdash numerical methods for integration of functions with several variables

solution of linear and nonlinear algebraic equationsmdash computer programming tools for handling massive data files and for creating

user friendly graphic interface


It is hard to conceive a balanced formation in all these diverse disciplinesMoreover suitable teaching software must be used that includes certaincharacteristics of general industr ial codes Finally the practical aspects ofimplementing the finite element method in computer codes must not beoverlooked

022 OBJECTIVES OF THE BOOK

This book attempts to simplify the teaching of the finite element method bysmoothing out certain difficulties It has been developed by engineers to solveengineering problemsThusthe presentation of the book is primarily addressedto this audienceThe mathematical knowledge required is limited to the domainsof differential and matrix calculus

This book is intended for readers who wish to understand the finite elementmethod and apply it to solve engineering problems using a computerMoreoverit should be of use to students and researchers in applied sciences and as well asto practicing engineers who wish to go beyond the basic level of knowledgeimplied by the use of ldquoblack boxrdquo programs

023 ORGANIZATION OF THE BOOK

This volume is organized into six chapters each providing a relativelyindependent presentation of various concepts of the finite element method aswell as the corresponding numerical and programming techniques Examplesare provided for illustrative purposesaccompanied by simple programs writtenusing Matlabcopy

Chapter 1

This chapter presents the approximation of continuous functions over subdomainsin terms of nodal values and introduces the concepts of interpolation functionsreference elements geometrical transformations and approximation error

Chapter 2

This chapter presents the interpolation functions for classical elements in onetwo and three dimensions

INTRODUCTION 5

Chapter 3

This chapter gives a description of the weighted residual method that allows usto obtain weak formulations (known as integral formulations) associated withpartial differential equations (known as strong formulations)

Chapter 4

This chapter presents the matrix formulation of the finite element methodwhichconsists of discretizing the integral formulation from Chapter 3 using theapproximations of unknown functions from Chapters 1 and 2 We introducenotions of elementary matrices and vectors assembly and global matrices andvectors

Chapter 5

This chapter describes the numerical methods needed to construct and solvethe systems of algebraic equations formed in Chapter 4 numerical methodsof integration methods for the solution of linear and nonlinear algebraicsystems domainmethods for integrating first- and second-order propagationsystems in time domain and methods for calculating the eigenvalues and vectors

Chapter 6

This chapter provides a brief overview of the finite difference and finite volumemethods as well as the computing techniques that are characteristic of the finiteelement method using a simple program written in Matlabcopy

Figure 01 shows the logical sequence of these chaptersNote that Chapters 13 and 4 are devoted to the fundamental concepts underlying the finite elementmethod while Chapters 2 and 5 are intended as reference chapters and finallyChapter 6 presents a simple program for illustrative purposes


Chapters 1 and 2

Approximation of theunknowns

Chapter 3

Transformation ofequations

(Integral formulation)

Chapter 4

Discretization(Matrix formulation)

Chapter 5

Numericalmethods

Chapter 6

Computerimplementation

(Solution)

Figure 01 Logical flow of the chapters

03 Numerical modeling approach

031 GENERAL ASPECTS

This section gives a brief introduction to the different concepts to be coveredin the following chapters

Numerical modeling is used to simulate the behavior of physical systems usingcomputers This involves

mdash description in engineering terms of the physical system in question andthe problem under study (physical model )

mdash translation of the engineer ing problem into a mathematical form(mathematical model)

mdash construction of a numerical model (or algebraic model) that can be solvedusing a computer and which uses discretization methods such as the finiteelement method

mdash development of a computer code to simulate the behavior of the physicalsystem (computer model)

INTRODUCTION 7

A variety of errors may be introduced into different models or during the passagefrom one model to another Three main types of errors are encountered

mdash Errors in the choice of the mathematical modelrepresenting the differencebetween the exact solution to the mathematical model and the real behaviorof the physical system

mdash Discretization errorsrepresenting the difference between the exact solutionto the mathematical model and the exact solution to the numerical model

mdash Computer-based errors due to the limited precision of the calculationscarried out by the computer and potentially programming errors

Modeling specialists should be able to master these errors so that the solutionprovided by the software is reasonably close to the real behavior of the physicalsystem under study (a priori unknown) In practice it is often necessary to carryout the steps described above more than once until a satisfactory solution isproduced

032 PHYSICAL MODEL

The description of a physical system includes

mdash a representation of its geometrymdash the selection of unknown variables for which we wish to evaluate spatio-

temporal variations

mdash the physical laws governing the systemrsquos behavior

mdash values for the physical properties that are assumed to be known

mdash applied forcesboundary conditions andwhere applicableinitial conditionsfor transient problems

EXAMPLE 01 Thermal equilibrium of a truss (physical model)

mdash Geometry rectilinear truss of length L and rectangular section A oriented in thedirection x (Figure 02)


Section A

x

x = L

x = 0

Figure 02 Rectilinear truss of constant section

mdash Unknown variables

- the temperature (in degrees Celsius or Kelvin) T x( )

- the thermal flux component as a function of x (Wm2) q x( )

The problem represents steady state flowvariables are thus independent of time

mdash Physical laws

- conservation of thermal flow as a function of x

- Fourierrsquos law of thermal behavior relating the temperature gradient tothe flow

mdash Physical properties thermal conductivity k (WdegC-m)

mdash Thermal load from the Joule effect (electrical current) f (Wm3)

mdash Boundary conditions

- Imposed temperature T T( ) 0 0=

- Imposed flux q L qL( ) = Wm2

Objectives

Obtain T(x) and q(x) that verify the physical laws and boundary conditionsOne possible application would be to estimate heat loss through the wallof a dwelling for which we wish to improve the insulation ie limit thethermal flow

INTRODUCTION 9

033 MATHEMATICAL MODEL

This model is obtained by expressing the laws of conservation and constitutivelaws in the form of partial differential equationsAs this problem is to be solvedusing the finite element method it will also be necessary to give an integralformulation (or weak formulation)

EXAMPLE 02 Application to the thermal equilibrium of a truss (mathematicalmodel)

mdash Law of conservation of thermal flow written as a function of x for Example 01

d qAdx

f A f( )

minus = gt0 00 0 source of volumetric heat

mdash Fourierrsquos constitutive law q x kdT x

dx( )

( )= minus


T x T

q x L k dTdx

qx L

L

( )

( )

= =

= = minus = gt=

0

0

0

(loss)

Dirichlet

Neumann

These relations may also be written in the form (for a constant section A)

ddx

kdT x

dxf

( )

⎛⎝⎜

⎞⎠⎟

+ =0 0

The exact solution to the mathematical model in the present case (where k isconstant) is

T x Tqk

xfk

Lx xL( ) = minus + minus⎛⎝

⎞⎠0

0 2

2

The integral form obtained using the weighted residual method is written as

W d Tdx

k dTdx

dx W T f dxL

Neu

L= + minus =int intδ δ

00

00

with W T L qNeu L= δ ( ) and T x = =δ 0 0( )

where δT x( ) is any test function

The solution to the problem is the function T(x)such thatW = 0 for all test functions


034 NUMERICAL MODEL

The numerical model associated with the mathematical model is obtained usinga discretization method such as

mdash the finite difference method or

mdash the finite element method

In this case we will illustrate the numerical model using the finite differencemethod

EXAMPLE 03 Application to the thermal equilibrium of a truss (numericalmodel based on the finite difference method)

For cases where ldquokrdquo is constant the differential equation governing the thermalequilibrium of the truss is written as

kd T x

dxf

2

2 0 0( )

+ =

T x 0 0( )= = and k dTdx

qL

L= minus

Let us take a set of equidistant discretization points (known as nodes) across thedomainThis may be illustrated using three equidistant nodes

∆ x

x = 0 x = Lx = 05L x = 15L

1 2 3 4 X

This set of nodes is known as a meshA fourthldquofictionalrdquo node has been added inorder to give the same level of spatial precision for the boundary condition at x = L

The equilibrium relationship is applied at each nodeldquoirdquo

kd T x

dxf i

i

2

2 0 0 1 2 3( )

+ = =

Let us associate an unknown variable with each node in the mesh so that

T x T T x x T

T x L T T x L x T

( ) ( )

( ) ( )1 1 2 2

3 3 4 4

0= = = == = = + =

∆∆

where ∆x L= 2 is the distance between two successive nodes

INTRODUCTION 11

The finite difference method consists of rewriting the derivatives in discrete form sothat

d Tdx

d Tdx

T T Tx

dTdx

dTdx

x i

i i i

x

2

2

2

21 1

2

2= asymp

minus +

=

+ minus

∆

ii

i iT Tx

asympminus+ minus1 1

2∆

We thus obtain the discrete form of the thermal equilibrium equation at node ldquoirdquo

kT T T

xf ii i i+ minusminus + + = =1 1

2 02

0 2 3∆

Its application to nodes 2 and 3 associated with the boundary condition on node 1translates as

T T

T T Tx fk

T T Tx fk

1 0

1 2 3

20

2 3 4

20

2 0

2 0

=

minus + + =

minus + + =

∆

∆

The boundary condition for x = L gives

T Tx

qk

T Tx qk

L L4 24 22

2minus = minus rArr = minus∆∆

Organizing these relations in a matrix form leads to

1 0 0

1 2 1

0 1 1 0 5

1

2

3

0

02

02

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪=

minus

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

T

T

T

T

f x k

f x k q x kL

∆∆ ∆

which gives

T

T

T f x k q x k

T f x k q x kL

L

2

3

0 02

0 02

1 5

2 2

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆


Remark

In this case (where f0 is constant) we observe that the solution to thenumerical model coincides with that of the mathematical model at the nodes

EXAMPLE 04 Approach based on the finite element method

The finite element method consists of constructing a discrete representation of the integralform W of Example 02 To do thiswe first select a set of two elements as illustratedbelow

1 2

x = 0 x = Lx = 05L

1 2 3 X

ElementElement

The integral form is written as

W W WeNeu

e

= + ==sum 0

1

2

where Wd T x

dxk

dT x

dxdx T x f dxe

x

x

x

x

i

i

i

i

=( ) ( )

minus ( )int int+ +

0

1 1

δ

δ and W T L qNeu L= ( ) δ

For each element we choose a linear approximation of the solution function T(x)and the test functionδ δT x J T x( ) ( ) with = =0 0

For element 1 L x x Le = minus =2 1 2

T xx x

LT

x xL

T x x

T xx x

LT

x xL

T

e e

e e

( )

( )

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

= =

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

21

12 1 2

21

12

0

δ δ δ

L2

where T T1 0= and T1 0=δ

The elementary integral form associated with element 1 is then written as

W T T kL

T

Tf L

e

e1

1 21

20

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ

INTRODUCTION 13


The approach taken is equivalent to that used for element 1 The elementaryintegral form is written as

W T T kL

T

Tf L

e

e2

2 32

30

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ

The flow term is expressed as W T qNeu L= δ 3

After assembly the integral form W is written as

W T T T kL

T

T

T

f L

qL

=lt gtminus

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

minus⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=δ δ δ1 2 3

1

2

3

02

1 1 0

1 2 1

0 1 14

1

2

1

0

0 0

Introducing the boundary condition at node 1 we obtain the following system

2 2 1

1 1 42

1

22

30

0kL

T

Tf L kT L

qL

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

=⎧⎨⎩

⎫⎬⎭

minusminus⎧

⎨⎩

⎫⎬⎭

⎧ ⎫

leading to1 5

2 22

3

0 02

0 02

T

T

T f x k q x k

T f x k q x kL

L

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆

Remark

In this particular case the finite difference and finite element methodsprovide exactly identical solutions at the nodes

035 COMPUTER MODEL

The two programs given in Figures 03 and 04 are written in the Matlabcopy

programming language and cover Examples 03 and 04 respectively


----- initializationclear all

----- geometryL=1 length (m)nnt=20 number of nodesdx=L(nnt-1) discretization dx

----- propertieskd=2 thermal conductivityf0=50 heat production per unit of length

----- boundary conditionsT0=30 Dirichlet at node 1qL=10 Neumann at node nnt

----- construction of system of equationsvkg=zeros(nntnnt) initialization of the vkg global matrixvfg=zeros(nnt1) initialization of the vfg global vector----- node loopif(nntgt2)

for i=2nnt-1vfg(i)=f0dx^2kdvkg(i[i-1 i i+1])=[-1 2 -1]

endend----- Dirichlet boundary condition (x=0)vkg(11)=1 vfg(1)=T0

----- Neumann boundary condition (x=L)vkg(nnt[nnt-1 nnt])=[-1 1] vfg(nnt)=05f0dx^2kd-qLdxkd----- solutionvsol=vkgvfg----- display numerical solutionplot([0nnt-1]dxvsol)

Figure 03 1D program using the finite difference method (Example 03)

INTRODUCTION 15

----- initializationclear all----- geometryL=1 length (m)nnt=20 number of nodesLe=L(nnt-1) discretization dx=Le----- propertieskd=2 thermal conductivityf0=50 heat production per unit of length----- boundary conditionsT0=30 Dirichlet at node 1qL=10 Neumann at node nnt----- construction of system of equationsvkg=zeros(nntnnt) initialization of vkgvfg=zeros(nnt1) initialization of vfgc=kdLe----- elementary matrix and vectorvke=[c -c -c c]vfe=f0Le2[1 1]----- element loopfor ie=1nelt

vfg([ie ie+1])= vfg([ie ie+1])+vfevkg([ie ie+1] [ie ie+1])= vkg([ie ie+1] [ie ie+1])+vke

end----- Dirichlet boundary condition (x=0)vkg(1)=zeros(1nnt) vkg(1 1)=1 vfg(1)=T0----- Neumann boundary condition (x=L)vkg(nnt)= vfg(nnt ndashqL----- solutionvsol=vkgvfg----- display numerical solution and exact solutionplot([0nnt-1]Levsol)hold onx=0L100Lsolexact=-05f0kdx^2+(f0L-qL)kdxplot(xsolexact)

Figure 04 1D program using the finite element method (Example 04)


0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

2 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

4 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

6 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

8 nodes

Figure 05 Exact solutions and solutions obtained using the finiteelement method

Figure 05 shows the results of the program based on the finite element methodfor different numbers of nodes Each illustration shows the exact solution tothe problem (continuous curve) and the numerical solution (dotted line)

Bibliography

[ARA 68] DE ARANTES E OLIVEIRA ERldquoTheoretical foundations of the finite elementmethodrdquo International Journal of Solids and Structures vol 4 pp 929ndash952 1968

[ARG 60] ARGYRIS JH KELSEY S Energy Theorems and Structural Analysis ButterworthLondon 1960

[COU 43] COURANT RldquoVariational methods for the solution of problems of equilibriumand vibrationsrdquoBulletin of theAmerican Mathematical Society vol49pp1ndash231943

[ERG 68] ERGATOUDIS JG IRONS BM ZIENKIEWICZ OCldquoThree-dimensional analysisofArch Dams and their foundationsrdquoSymposium onArch Dams Institution of CivilEngineers London March 1968

[FEL 66] FELIPPA CA Refined finite element analysis of linear and non-linear two-dimensional structuresReport UC SESM 66-22Department of Civil EngineeringUniversity of California Berkeley CA October 1966

[FIN 75] FINLAYSON BAldquoWeighted residual methods and their relation to finite elementmethods in flow problemsrdquo Finite Elements in Fluids vol 2 pp 1ndash31 1975

[GRE 69] GREENE RE JONES REMCLAY RWSTROME DRldquoGeneralized variationalpr inciples in the finite-element method American Institute of Aeronautics andAstronautics Journal vol 7 no 7 pp 1254ndash1260 1969

INTRODUCTION 17

[HEN 43] MCHENRY DldquoA lattice analogy of the solution of plane stress problemsrdquo Journalof Institution of Civil Engineers vol 21 pp 59ndash82 1943

[HOF 56] HOFF NJ Analysis of StructuresWiley NewYork 1956[HRE 41] HRENNIKOFFAldquoSolutions of problems in elasticity by the framework Methodrdquo

Journal of Applied Mechanics vol 8A169ndashA175 1942[IRO 68] IRONS BMZIENKIEWICZ OCldquoThe isoparametric finite element system ndash a new

concept in finite element analysisrdquoProceedingsConference on RecentAdvances in StressAnalysis Royal Aeronautical Society London 1968

[TUR 56] TURNER MJ CLOUGH RW MARTIN HCTOPP LJldquoStiffness and deflectionanalysis of complex structuresrdquo Journal ofAeronautical Science vol23pp805ndash8231956

[ZIE 65] ZIENKIEWICZ OC HOLISTER GS Stress Analysis Wiley New York 1965

Conference proceedings

[PRO 01] Proceedings of the 1st 2nd and 3rd Conferences on Matrix Methods in StructuralMechanicsWright-Patterson AFB Ohio 1965 1968 1971

[PRO 02] HOLLAND I BELL K (eds) Finite Element Methods in Stress Analysis TapirTrondheim Norway 1969

[PRO 03] Proceedings of the 1st 2nd3rd and 4th Conferences on Structural Mechanics in ReactorTechnology 1971 1973 1975 1977

[PRO 04] Symposium on Applied Finite Element Methods in Civil Engineering VanderbiltUniversity NashvilleASCE 1969

[PRO 05] GALLAGHER RHYAMADAY ODEN JT (eds)Recent Advances in Matrix Methodsof Structural Analysis and Design University of Alabama Press Huntsville 1971

[PRO 06] DE VEUBEKE BF (ed) High Speed Computing of Elastic Structures University ofLiegravege 1971

[PRO 07] BREBBIA CA TOTTENHAM H (eds) Variational Methods in Engineer ingSouthampton University 1973

[PRO 08] FENVES SJ PERRONE N ROBINSON J SCHNOBRICH WC (eds) Numerical andComputational Methods in Structural MechanicsAcademic Press NewYork 1973

[PRO 09] GALLAGHER RH ODEN JTTAYLOR C ZIENKIEWICZ OC (eds) InternationalSymposium on Finite Element Methods in Flow Problems Wiley 1974

[PRO 10] BATHE KJ ODEN JT WUNDERLICH W (eds) Formulation and ComputationalAlgorithms in Finite Element Analysis (US Germany Symposium) MIT Press 1977

[PRO 11] GRAY WG PINDER GF BREBBIA CA (eds) Finite Elements inWater ResourcesPentech Press London 1977

[PRO 12] ROBINSON J (ed)Finite Element Methods in Commercial EnvironmentsRobinsonand Associates Dorset England 1978

[PRO 13] GLOWINSKI R RODIN EY ZIENKIEWICZ OC (eds) Energy Methods in FiniteElements AnalysisWiley 1979

[PRO 14] AZIZ AK (ed) The Mathematical Foundations of the Finite Element Method withApplications to Partial Differential EquationsAcademic Press New York 1972

[PRO 15] WHITEMAN JR(ed)The Mathematics of Finite Elements and Applications AcademicPress London 1973


Monographs

[MON 01] PRZEMIENIECKI JSTheory of Matrix StructuralAnalysis McGraw-HillNew York1968

[MON 02] ZIENKIEWICZ OCThe Finite Element MethodThe Basis (Vol1) Solid Mechanics(Vol 2) amp Fluid Mechanics (Vol 3) 5th ed Butterworth Heinermann 2000

[MON 03] DESAI CS ABEL JF Introduction to the Finite Element Method Van NostrandReinhold NewYork 1972

[MON 04] ODEN JTFinite Elements of Non-Linear Continua McGraw-HillNew York1972[MON 05] MARTIN HCCAREY GF Introduction to Finite ElementAnalysisMcGraw-Hill

NewYork 1973[MON 06] NORRIE DJ DEVRIES GThe Finite Element Method Academic PressNewYork

1973[MON 07] ROBINSON J IntegratedTheory of Finite Element Methods Wiley London 1973[MON 08] STRAND G FIX OJ Analysis of the Finite Element Methods Prentice-Hall

New Jersey 1973[MON 09] URAL OFinite Element MethodBasic Concepts and ApplicationsIntext Educational

Publishers 1973[MON 10] COOK RD Concepts and Applications of Finite Element Analysis Wiley 1974[MON 11] GALLAGHER RH Finite Element Analysis Fundamentals Prentice-Hall 1975[MON 12] HUEBNER KH The Finite Element Method for Engineers Wiley 1975[MON 13] WASHIZU KVariational Methods in Elasticity and Plasticity Pergamon Press1976[MON 14] BATHE KJWILSON ELNumerical Methods in Finite ElementAnalysis Prentice-

Hall 1976[MON 15] CHEUNG YK Finite Strip Method in Structural Analysis Pergamon Press 1976[MON 16] CONNOR JJBREBBIA CAFinite ElementTechnique for Fluid Flow Butterworth

Co 1976[MON 17] SEGERLIND LJ Applied Finite Element Analysis Wiley 1976[MON 18] MITCHELL AR WAIT R The Finite Element Method in Partial Differential

Equations Wiley 1977[MON 19] PINDER GF GRAY GW Finite Element Simulation in Surface and Sub-Surface

Hydrology Academic Press 1977[MON 20] TONG PROSSETOS JFinite Element MethodBasic Techniques and Implementation

MIT Press 1977[MON 21] CHUNG TJ Finite Element Analysis in Fluid Dynamics McGraw-Hill 1978[MON 22] CIARLET PG The Finite Element Method for Elliptic Problems North-Holland

1978[MON 23] IRONS BM AHMAD STechniques of Finite ElementsEllis HorwoodChichester

England 1978[MON 24] DESAI CS Elementary Finite Element Method Prentice-Hall 1979[MON 25] ZIENKIEWICZ OC La Meacutethode des Elements Finis (trans)PluralisFrance1976[MON 26] GALLAGHER RH Introduction aux Elements Finis (trans JLClaudon)Pluralis

France 1976[MON 27] ROCKEY KC EVANS HR GRIFFITHS DW Elements Finis (trans C Gomez)

Eyrolles France 1978[MON 28] ABSI E Meacutethode de calcul numeacuterique en eacutelasticiteacute Eyrolles 1979[MON 29] IMBERT JF Analyse des structures par eacuteleacutements finisCEPADUES EdFrance1979














TABLE OF CONTENTS


























Bibliography 160








Bibliography 229














Bibliography 343










Bibliography 502











Index 597


Introduction


011 GENERAL REMARKS





computer











INTRODUCTION 3




















Chapter 1


Chapter 2


INTRODUCTION 5

Chapter 3


Chapter 4


Chapter 5


Chapter 6




Chapters 1 and 2


Chapter 3



Chapter 4


Chapter 5

Numericalmethods

Chapter 6


(Solution)



031 GENERAL ASPECTS







INTRODUCTION 7






032 PHYSICAL MODEL



temporal variations







Section A

x

x = L

x = 0






mdash Physical laws








Objectives


INTRODUCTION 9





d qAdx

f A f( )



dx( )

( )= minus


T x T

q x L k dTdx

qx L

L

( )

( )

= =

= = minus = gt=

0

0

0

(loss)

Dirichlet

Neumann


ddx

kdT x

dxf

( )

⎛⎝⎜

⎞⎠⎟

+ =0 0


T x Tqk

xfk


⎞⎠0

0 2

2


W d Tdx

k dTdx

dx W T f dxL

Neu


00

00





034 NUMERICAL MODEL







kd T x

dxf

2

2 0 0( )

+ =


qL

L= minus


∆ x

x = 0 x = Lx = 05L x = 15L

1 2 3 4 X



kd T x

dxf i

i

2

2 0 0 1 2 3( )

+ = =


T x T T x x T

T x L T T x L x T

( ) ( )

( ) ( )1 1 2 2

3 3 4 4

0= = = == = = + =

∆∆


INTRODUCTION 11


d Tdx

d Tdx

T T Tx

dTdx

dTdx

x i

i i i

x

2

2

2

21 1

2

2= asymp

minus +

=

+ minus

∆

ii

i iT Tx


2∆


kT T T


2 02

0 2 3∆


T T

T T Tx fk

T T Tx fk

1 0

1 2 3

20

2 3 4

20

2 0

2 0

=

minus + + =

minus + + =

∆

∆


T Tx

qk

T Tx qk

L L4 24 22



1 0 0

1 2 1

0 1 1 0 5

1

2

3

0

02

02

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪=

minus

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

T

T

T

T

f x k

f x k q x kL

∆∆ ∆

which gives

T

T

T f x k q x k

T f x k q x kL

L

2

3

0 02

0 02

1 5

2 2

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆


Remark




1 2

x = 0 x = Lx = 05L

1 2 3 X

ElementElement


W W WeNeu

e

= + ==sum 0

1

2

where Wd T x

dxk

dT x

dxdx T x f dxe

x

x

x

x

i

i

i

i

=( ) ( )

minus ( )int int+ +

0

1 1

δ




T xx x

LT

x xL

T x x

T xx x

LT

x xL

T

e e

e e

( )

( )

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

= =

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

21

12 1 2

21

12

0

δ δ δ

L2



W T T kL

T

Tf L

e

e1

1 21

20

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ

INTRODUCTION 13



W T T kL

T

Tf L

e

e2

2 32

30

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ



W T T T kL

T

T

T

f L

qL

=lt gtminus

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

minus⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=δ δ δ1 2 3

1

2

3

02

1 1 0

1 2 1

0 1 14

1

2

1

0

0 0


2 2 1

1 1 42

1

22

30

0kL

T

Tf L kT L

qL

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

=⎧⎨⎩

⎫⎬⎭

minusminus⎧

⎨⎩

⎫⎬⎭

⎧ ⎫

leading to1 5

2 22

3

0 02

0 02

T

T

T f x k q x k

T f x k q x kL

L

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆

Remark


035 COMPUTER MODEL













INTRODUCTION 15






0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

2 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

4 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

6 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

8 nodes



Bibliography








INTRODUCTION 17
























Monographs


































Bibliography 160








Bibliography 229














Bibliography 343










Bibliography 502











Index 597


Introduction


011 GENERAL REMARKS





computer











INTRODUCTION 3




















Chapter 1


Chapter 2


INTRODUCTION 5

Chapter 3


Chapter 4


Chapter 5


Chapter 6




Chapters 1 and 2


Chapter 3



Chapter 4


Chapter 5

Numericalmethods

Chapter 6


(Solution)



031 GENERAL ASPECTS







INTRODUCTION 7






032 PHYSICAL MODEL



temporal variations







Section A

x

x = L

x = 0






mdash Physical laws








Objectives


INTRODUCTION 9





d qAdx

f A f( )



dx( )

( )= minus


T x T

q x L k dTdx

qx L

L

( )

( )

= =

= = minus = gt=

0

0

0

(loss)

Dirichlet

Neumann


ddx

kdT x

dxf

( )

⎛⎝⎜

⎞⎠⎟

+ =0 0


T x Tqk

xfk


⎞⎠0

0 2

2


W d Tdx

k dTdx

dx W T f dxL

Neu


00

00





034 NUMERICAL MODEL







kd T x

dxf

2

2 0 0( )

+ =


qL

L= minus


∆ x

x = 0 x = Lx = 05L x = 15L

1 2 3 4 X



kd T x

dxf i

i

2

2 0 0 1 2 3( )

+ = =


T x T T x x T

T x L T T x L x T

( ) ( )

( ) ( )1 1 2 2

3 3 4 4

0= = = == = = + =

∆∆


INTRODUCTION 11


d Tdx

d Tdx

T T Tx

dTdx

dTdx

x i

i i i

x

2

2

2

21 1

2

2= asymp

minus +

=

+ minus

∆

ii

i iT Tx


2∆


kT T T


2 02

0 2 3∆


T T

T T Tx fk

T T Tx fk

1 0

1 2 3

20

2 3 4

20

2 0

2 0

=

minus + + =

minus + + =

∆

∆


T Tx

qk

T Tx qk

L L4 24 22



1 0 0

1 2 1

0 1 1 0 5

1

2

3

0

02

02

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪=

minus

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

T

T

T

T

f x k

f x k q x kL

∆∆ ∆

which gives

T

T

T f x k q x k

T f x k q x kL

L

2

3

0 02

0 02

1 5

2 2

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆


Remark




1 2

x = 0 x = Lx = 05L

1 2 3 X

ElementElement


W W WeNeu

e

= + ==sum 0

1

2

where Wd T x

dxk

dT x

dxdx T x f dxe

x

x

x

x

i

i

i

i

=( ) ( )

minus ( )int int+ +

0

1 1

δ




T xx x

LT

x xL

T x x

T xx x

LT

x xL

T

e e

e e

( )

( )

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

= =

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

21

12 1 2

21

12

0

δ δ δ

L2



W T T kL

T

Tf L

e

e1

1 21

20

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ

INTRODUCTION 13



W T T kL

T

Tf L

e

e2

2 32

30

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ



W T T T kL

T

T

T

f L

qL

=lt gtminus

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

minus⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=δ δ δ1 2 3

1

2

3

02

1 1 0

1 2 1

0 1 14

1

2

1

0

0 0


2 2 1

1 1 42

1

22

30

0kL

T

Tf L kT L

qL

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

=⎧⎨⎩

⎫⎬⎭

minusminus⎧

⎨⎩

⎫⎬⎭

⎧ ⎫

leading to1 5

2 22

3

0 02

0 02

T

T

T f x k q x k

T f x k q x kL

L

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆

Remark


035 COMPUTER MODEL













INTRODUCTION 15






0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

2 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

4 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

6 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

8 nodes



Bibliography








INTRODUCTION 17
























Monographs





























Bibliography 343










Bibliography 502











Index 597


Introduction


011 GENERAL REMARKS





computer











INTRODUCTION 3




















Chapter 1


Chapter 2


INTRODUCTION 5

Chapter 3


Chapter 4


Chapter 5


Chapter 6




Chapters 1 and 2


Chapter 3



Chapter 4


Chapter 5

Numericalmethods

Chapter 6


(Solution)



031 GENERAL ASPECTS







INTRODUCTION 7






032 PHYSICAL MODEL



temporal variations







Section A

x

x = L

x = 0






mdash Physical laws








Objectives


INTRODUCTION 9





d qAdx

f A f( )



dx( )

( )= minus


T x T

q x L k dTdx

qx L

L

( )

( )

= =

= = minus = gt=

0

0

0

(loss)

Dirichlet

Neumann


ddx

kdT x

dxf

( )

⎛⎝⎜

⎞⎠⎟

+ =0 0


T x Tqk

xfk


⎞⎠0

0 2

2


W d Tdx

k dTdx

dx W T f dxL

Neu


00

00





034 NUMERICAL MODEL







kd T x

dxf

2

2 0 0( )

+ =


qL

L= minus


∆ x

x = 0 x = Lx = 05L x = 15L

1 2 3 4 X



kd T x

dxf i

i

2

2 0 0 1 2 3( )

+ = =


T x T T x x T

T x L T T x L x T

( ) ( )

( ) ( )1 1 2 2

3 3 4 4

0= = = == = = + =

∆∆


INTRODUCTION 11


d Tdx

d Tdx

T T Tx

dTdx

dTdx

x i

i i i

x

2

2

2

21 1

2

2= asymp

minus +

=

+ minus

∆

ii

i iT Tx


2∆


kT T T


2 02

0 2 3∆


T T

T T Tx fk

T T Tx fk

1 0

1 2 3

20

2 3 4

20

2 0

2 0

=

minus + + =

minus + + =

∆

∆


T Tx

qk

T Tx qk

L L4 24 22



1 0 0

1 2 1

0 1 1 0 5

1

2

3

0

02

02

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪=

minus

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

T

T

T

T

f x k

f x k q x kL

∆∆ ∆

which gives

T

T

T f x k q x k

T f x k q x kL

L

2

3

0 02

0 02

1 5

2 2

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆


Remark




1 2

x = 0 x = Lx = 05L

1 2 3 X

ElementElement


W W WeNeu

e

= + ==sum 0

1

2

where Wd T x

dxk

dT x

dxdx T x f dxe

x

x

x

x

i

i

i

i

=( ) ( )

minus ( )int int+ +

0

1 1

δ




T xx x

LT

x xL

T x x

T xx x

LT

x xL

T

e e

e e

( )

( )

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

= =

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

21

12 1 2

21

12

0

δ δ δ

L2



W T T kL

T

Tf L

e

e1

1 21

20

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ

INTRODUCTION 13



W T T kL

T

Tf L

e

e2

2 32

30

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ



W T T T kL

T

T

T

f L

qL

=lt gtminus

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

minus⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=δ δ δ1 2 3

1

2

3

02

1 1 0

1 2 1

0 1 14

1

2

1

0

0 0


2 2 1

1 1 42

1

22

30

0kL

T

Tf L kT L

qL

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

=⎧⎨⎩

⎫⎬⎭

minusminus⎧

⎨⎩

⎫⎬⎭

⎧ ⎫

leading to1 5

2 22

3

0 02

0 02

T

T

T f x k q x k

T f x k q x kL

L

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆

Remark


035 COMPUTER MODEL













INTRODUCTION 15






0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

2 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

4 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

6 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

8 nodes



Bibliography








INTRODUCTION 17
























Monographs

























Bibliography 502











Index 597


Introduction


011 GENERAL REMARKS





computer











INTRODUCTION 3




















Chapter 1


Chapter 2


INTRODUCTION 5

Chapter 3


Chapter 4


Chapter 5


Chapter 6




Chapters 1 and 2


Chapter 3



Chapter 4


Chapter 5

Numericalmethods

Chapter 6


(Solution)



031 GENERAL ASPECTS







INTRODUCTION 7






032 PHYSICAL MODEL



temporal variations







Section A

x

x = L

x = 0






mdash Physical laws








Objectives


INTRODUCTION 9





d qAdx

f A f( )



dx( )

( )= minus


T x T

q x L k dTdx

qx L

L

( )

( )

= =

= = minus = gt=

0

0

0

(loss)

Dirichlet

Neumann


ddx

kdT x

dxf

( )

⎛⎝⎜

⎞⎠⎟

+ =0 0


T x Tqk

xfk


⎞⎠0

0 2

2


W d Tdx

k dTdx

dx W T f dxL

Neu


00

00





034 NUMERICAL MODEL







kd T x

dxf

2

2 0 0( )

+ =


qL

L= minus


∆ x

x = 0 x = Lx = 05L x = 15L

1 2 3 4 X



kd T x

dxf i

i

2

2 0 0 1 2 3( )

+ = =


T x T T x x T

T x L T T x L x T

( ) ( )

( ) ( )1 1 2 2

3 3 4 4

0= = = == = = + =

∆∆


INTRODUCTION 11


d Tdx

d Tdx

T T Tx

dTdx

dTdx

x i

i i i

x

2

2

2

21 1

2

2= asymp

minus +

=

+ minus

∆

ii

i iT Tx


2∆


kT T T


2 02

0 2 3∆


T T

T T Tx fk

T T Tx fk

1 0

1 2 3

20

2 3 4

20

2 0

2 0

=

minus + + =

minus + + =

∆

∆


T Tx

qk

T Tx qk

L L4 24 22



1 0 0

1 2 1

0 1 1 0 5

1

2

3

0

02

02

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪=

minus

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

T

T

T

T

f x k

f x k q x kL

∆∆ ∆

which gives

T

T

T f x k q x k

T f x k q x kL

L

2

3

0 02

0 02

1 5

2 2

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆


Remark




1 2

x = 0 x = Lx = 05L

1 2 3 X

ElementElement


W W WeNeu

e

= + ==sum 0

1

2

where Wd T x

dxk

dT x

dxdx T x f dxe

x

x

x

x

i

i

i

i

=( ) ( )

minus ( )int int+ +

0

1 1

δ




T xx x

LT

x xL

T x x

T xx x

LT

x xL

T

e e

e e

( )

( )

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

= =

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

21

12 1 2

21

12

0

δ δ δ

L2



W T T kL

T

Tf L

e

e1

1 21

20

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ

INTRODUCTION 13



W T T kL

T

Tf L

e

e2

2 32

30

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ



W T T T kL

T

T

T

f L

qL

=lt gtminus

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

minus⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=δ δ δ1 2 3

1

2

3

02

1 1 0

1 2 1

0 1 14

1

2

1

0

0 0


2 2 1

1 1 42

1

22

30

0kL

T

Tf L kT L

qL

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

=⎧⎨⎩

⎫⎬⎭

minusminus⎧

⎨⎩

⎫⎬⎭

⎧ ⎫

leading to1 5

2 22

3

0 02

0 02

T

T

T f x k q x k

T f x k q x kL

L

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆

Remark


035 COMPUTER MODEL













INTRODUCTION 15






0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

2 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

4 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

6 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

8 nodes



Bibliography








INTRODUCTION 17
























Monographs
























Index 597


Introduction


011 GENERAL REMARKS





computer











INTRODUCTION 3




















Chapter 1


Chapter 2


INTRODUCTION 5

Chapter 3


Chapter 4


Chapter 5


Chapter 6




Chapters 1 and 2


Chapter 3



Chapter 4


Chapter 5

Numericalmethods

Chapter 6


(Solution)



031 GENERAL ASPECTS







INTRODUCTION 7






032 PHYSICAL MODEL



temporal variations







Section A

x

x = L

x = 0






mdash Physical laws








Objectives


INTRODUCTION 9





d qAdx

f A f( )



dx( )

( )= minus


T x T

q x L k dTdx

qx L

L

( )

( )

= =

= = minus = gt=

0

0

0

(loss)

Dirichlet

Neumann


ddx

kdT x

dxf

( )

⎛⎝⎜

⎞⎠⎟

+ =0 0


T x Tqk

xfk


⎞⎠0

0 2

2


W d Tdx

k dTdx

dx W T f dxL

Neu


00

00





034 NUMERICAL MODEL







kd T x

dxf

2

2 0 0( )

+ =


qL

L= minus


∆ x

x = 0 x = Lx = 05L x = 15L

1 2 3 4 X



kd T x

dxf i

i

2

2 0 0 1 2 3( )

+ = =


T x T T x x T

T x L T T x L x T

( ) ( )

( ) ( )1 1 2 2

3 3 4 4

0= = = == = = + =

∆∆


INTRODUCTION 11


d Tdx

d Tdx

T T Tx

dTdx

dTdx

x i

i i i

x

2

2

2

21 1

2

2= asymp

minus +

=

+ minus

∆

ii

i iT Tx


2∆


kT T T


2 02

0 2 3∆


T T

T T Tx fk

T T Tx fk

1 0

1 2 3

20

2 3 4

20

2 0

2 0

=

minus + + =

minus + + =

∆

∆


T Tx

qk

T Tx qk

L L4 24 22



1 0 0

1 2 1

0 1 1 0 5

1

2

3

0

02

02

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪=

minus

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

T

T

T

T

f x k

f x k q x kL

∆∆ ∆

which gives

T

T

T f x k q x k

T f x k q x kL

L

2

3

0 02

0 02

1 5

2 2

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆


Remark




1 2

x = 0 x = Lx = 05L

1 2 3 X

ElementElement


W W WeNeu

e

= + ==sum 0

1

2

where Wd T x

dxk

dT x

dxdx T x f dxe

x

x

x

x

i

i

i

i

=( ) ( )

minus ( )int int+ +

0

1 1

δ




T xx x

LT

x xL

T x x

T xx x

LT

x xL

T

e e

e e

( )

( )

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

= =

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

21

12 1 2

21

12

0

δ δ δ

L2



W T T kL

T

Tf L

e

e1

1 21

20

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ

INTRODUCTION 13



W T T kL

T

Tf L

e

e2

2 32

30

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ



W T T T kL

T

T

T

f L

qL

=lt gtminus

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

minus⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=δ δ δ1 2 3

1

2

3

02

1 1 0

1 2 1

0 1 14

1

2

1

0

0 0


2 2 1

1 1 42

1

22

30

0kL

T

Tf L kT L

qL

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

=⎧⎨⎩

⎫⎬⎭

minusminus⎧

⎨⎩

⎫⎬⎭

⎧ ⎫

leading to1 5

2 22

3

0 02

0 02

T

T

T f x k q x k

T f x k q x kL

L

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆

Remark


035 COMPUTER MODEL













INTRODUCTION 15






0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

2 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

4 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

6 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

8 nodes



Bibliography








INTRODUCTION 17
























Monographs



















Introduction


011 GENERAL REMARKS





computer











INTRODUCTION 3




















Chapter 1


Chapter 2


INTRODUCTION 5

Chapter 3


Chapter 4


Chapter 5


Chapter 6




Chapters 1 and 2


Chapter 3



Chapter 4


Chapter 5

Numericalmethods

Chapter 6


(Solution)



031 GENERAL ASPECTS







INTRODUCTION 7






032 PHYSICAL MODEL



temporal variations







Section A

x

x = L

x = 0






mdash Physical laws








Objectives


INTRODUCTION 9





d qAdx

f A f( )



dx( )

( )= minus


T x T

q x L k dTdx

qx L

L

( )

( )

= =

= = minus = gt=

0

0

0

(loss)

Dirichlet

Neumann


ddx

kdT x

dxf

( )

⎛⎝⎜

⎞⎠⎟

+ =0 0


T x Tqk

xfk


⎞⎠0

0 2

2


W d Tdx

k dTdx

dx W T f dxL

Neu


00

00





034 NUMERICAL MODEL







kd T x

dxf

2

2 0 0( )

+ =


qL

L= minus


∆ x

x = 0 x = Lx = 05L x = 15L

1 2 3 4 X



kd T x

dxf i

i

2

2 0 0 1 2 3( )

+ = =


T x T T x x T

T x L T T x L x T

( ) ( )

( ) ( )1 1 2 2

3 3 4 4

0= = = == = = + =

∆∆


INTRODUCTION 11


d Tdx

d Tdx

T T Tx

dTdx

dTdx

x i

i i i

x

2

2

2

21 1

2

2= asymp

minus +

=

+ minus

∆

ii

i iT Tx


2∆


kT T T


2 02

0 2 3∆


T T

T T Tx fk

T T Tx fk

1 0

1 2 3

20

2 3 4

20

2 0

2 0

=

minus + + =

minus + + =

∆

∆


T Tx

qk

T Tx qk

L L4 24 22



1 0 0

1 2 1

0 1 1 0 5

1

2

3

0

02

02

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪=

minus

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

T

T

T

T

f x k

f x k q x kL

∆∆ ∆

which gives

T

T

T f x k q x k

T f x k q x kL

L

2

3

0 02

0 02

1 5

2 2

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆


Remark




1 2

x = 0 x = Lx = 05L

1 2 3 X

ElementElement


W W WeNeu

e

= + ==sum 0

1

2

where Wd T x

dxk

dT x

dxdx T x f dxe

x

x

x

x

i

i

i

i

=( ) ( )

minus ( )int int+ +

0

1 1

δ




T xx x

LT

x xL

T x x

T xx x

LT

x xL

T

e e

e e

( )

( )

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

= =

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

21

12 1 2

21

12

0

δ δ δ

L2



W T T kL

T

Tf L

e

e1

1 21

20

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ

INTRODUCTION 13



W T T kL

T

Tf L

e

e2

2 32

30

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ



W T T T kL

T

T

T

f L

qL

=lt gtminus

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

minus⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=δ δ δ1 2 3

1

2

3

02

1 1 0

1 2 1

0 1 14

1

2

1

0

0 0


2 2 1

1 1 42

1

22

30

0kL

T

Tf L kT L

qL

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

=⎧⎨⎩

⎫⎬⎭

minusminus⎧

⎨⎩

⎫⎬⎭

⎧ ⎫

leading to1 5

2 22

3

0 02

0 02

T

T

T f x k q x k

T f x k q x kL

L

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆

Remark


035 COMPUTER MODEL













INTRODUCTION 15






0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

2 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

4 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

6 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

8 nodes



Bibliography








INTRODUCTION 17
























Monographs




























INTRODUCTION 3




















Chapter 1


Chapter 2


INTRODUCTION 5

Chapter 3


Chapter 4


Chapter 5


Chapter 6




Chapters 1 and 2


Chapter 3



Chapter 4


Chapter 5

Numericalmethods

Chapter 6


(Solution)



031 GENERAL ASPECTS







INTRODUCTION 7






032 PHYSICAL MODEL



temporal variations







Section A

x

x = L

x = 0






mdash Physical laws








Objectives


INTRODUCTION 9





d qAdx

f A f( )



dx( )

( )= minus


T x T

q x L k dTdx

qx L

L

( )

( )

= =

= = minus = gt=

0

0

0

(loss)

Dirichlet

Neumann


ddx

kdT x

dxf

( )

⎛⎝⎜

⎞⎠⎟

+ =0 0


T x Tqk

xfk


⎞⎠0

0 2

2


W d Tdx

k dTdx

dx W T f dxL

Neu


00

00





034 NUMERICAL MODEL







kd T x

dxf

2

2 0 0( )

+ =


qL

L= minus


∆ x

x = 0 x = Lx = 05L x = 15L

1 2 3 4 X



kd T x

dxf i

i

2

2 0 0 1 2 3( )

+ = =


T x T T x x T

T x L T T x L x T

( ) ( )

( ) ( )1 1 2 2

3 3 4 4

0= = = == = = + =

∆∆


INTRODUCTION 11


d Tdx

d Tdx

T T Tx

dTdx

dTdx

x i

i i i

x

2

2

2

21 1

2

2= asymp

minus +

=

+ minus

∆

ii

i iT Tx


2∆


kT T T


2 02

0 2 3∆


T T

T T Tx fk

T T Tx fk

1 0

1 2 3

20

2 3 4

20

2 0

2 0

=

minus + + =

minus + + =

∆

∆


T Tx

qk

T Tx qk

L L4 24 22



1 0 0

1 2 1

0 1 1 0 5

1

2

3

0

02

02

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪=

minus

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

T

T

T

T

f x k

f x k q x kL

∆∆ ∆

which gives

T

T

T f x k q x k

T f x k q x kL

L

2

3

0 02

0 02

1 5

2 2

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆


Remark




1 2

x = 0 x = Lx = 05L

1 2 3 X

ElementElement


W W WeNeu

e

= + ==sum 0

1

2

where Wd T x

dxk

dT x

dxdx T x f dxe

x

x

x

x

i

i

i

i

=( ) ( )

minus ( )int int+ +

0

1 1

δ




T xx x

LT

x xL

T x x

T xx x

LT

x xL

T

e e

e e

( )

( )

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

= =

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

21

12 1 2

21

12

0

δ δ δ

L2



W T T kL

T

Tf L

e

e1

1 21

20

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ

INTRODUCTION 13



W T T kL

T

Tf L

e

e2

2 32

30

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ



W T T T kL

T

T

T

f L

qL

=lt gtminus

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

minus⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=δ δ δ1 2 3

1

2

3

02

1 1 0

1 2 1

0 1 14

1

2

1

0

0 0


2 2 1

1 1 42

1

22

30

0kL

T

Tf L kT L

qL

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

=⎧⎨⎩

⎫⎬⎭

minusminus⎧

⎨⎩

⎫⎬⎭

⎧ ⎫

leading to1 5

2 22

3

0 02

0 02

T

T

T f x k q x k

T f x k q x kL

L

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆

Remark


035 COMPUTER MODEL













INTRODUCTION 15






0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

2 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

4 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

6 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

8 nodes



Bibliography








INTRODUCTION 17
























Monographs

























Chapter 1


Chapter 2


INTRODUCTION 5

Chapter 3


Chapter 4


Chapter 5


Chapter 6




Chapters 1 and 2


Chapter 3



Chapter 4


Chapter 5

Numericalmethods

Chapter 6


(Solution)



031 GENERAL ASPECTS







INTRODUCTION 7






032 PHYSICAL MODEL



temporal variations







Section A

x

x = L

x = 0






mdash Physical laws








Objectives


INTRODUCTION 9





d qAdx

f A f( )



dx( )

( )= minus


T x T

q x L k dTdx

qx L

L

( )

( )

= =

= = minus = gt=

0

0

0

(loss)

Dirichlet

Neumann


ddx

kdT x

dxf

( )

⎛⎝⎜

⎞⎠⎟

+ =0 0


T x Tqk

xfk


⎞⎠0

0 2

2


W d Tdx

k dTdx

dx W T f dxL

Neu


00

00





034 NUMERICAL MODEL







kd T x

dxf

2

2 0 0( )

+ =


qL

L= minus


∆ x

x = 0 x = Lx = 05L x = 15L

1 2 3 4 X



kd T x

dxf i

i

2

2 0 0 1 2 3( )

+ = =


T x T T x x T

T x L T T x L x T

( ) ( )

( ) ( )1 1 2 2

3 3 4 4

0= = = == = = + =

∆∆


INTRODUCTION 11


d Tdx

d Tdx

T T Tx

dTdx

dTdx

x i

i i i

x

2

2

2

21 1

2

2= asymp

minus +

=

+ minus

∆

ii

i iT Tx


2∆


kT T T


2 02

0 2 3∆


T T

T T Tx fk

T T Tx fk

1 0

1 2 3

20

2 3 4

20

2 0

2 0

=

minus + + =

minus + + =

∆

∆


T Tx

qk

T Tx qk

L L4 24 22



1 0 0

1 2 1

0 1 1 0 5

1

2

3

0

02

02

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪=

minus

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

T

T

T

T

f x k

f x k q x kL

∆∆ ∆

which gives

T

T

T f x k q x k

T f x k q x kL

L

2

3

0 02

0 02

1 5

2 2

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆


Remark




1 2

x = 0 x = Lx = 05L

1 2 3 X

ElementElement


W W WeNeu

e

= + ==sum 0

1

2

where Wd T x

dxk

dT x

dxdx T x f dxe

x

x

x

x

i

i

i

i

=( ) ( )

minus ( )int int+ +

0

1 1

δ




T xx x

LT

x xL

T x x

T xx x

LT

x xL

T

e e

e e

( )

( )

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

= =

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

21

12 1 2

21

12

0

δ δ δ

L2



W T T kL

T

Tf L

e

e1

1 21

20

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ

INTRODUCTION 13



W T T kL

T

Tf L

e

e2

2 32

30

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ



W T T T kL

T

T

T

f L

qL

=lt gtminus

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

minus⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=δ δ δ1 2 3

1

2

3

02

1 1 0

1 2 1

0 1 14

1

2

1

0

0 0


2 2 1

1 1 42

1

22

30

0kL

T

Tf L kT L

qL

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

=⎧⎨⎩

⎫⎬⎭

minusminus⎧

⎨⎩

⎫⎬⎭

⎧ ⎫

leading to1 5

2 22

3

0 02

0 02

T

T

T f x k q x k

T f x k q x kL

L

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆

Remark


035 COMPUTER MODEL













INTRODUCTION 15






0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

2 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

4 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

6 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

8 nodes



Bibliography








INTRODUCTION 17
























Monographs


















INTRODUCTION 5

Chapter 3


Chapter 4


Chapter 5


Chapter 6




Chapters 1 and 2


Chapter 3



Chapter 4


Chapter 5

Numericalmethods

Chapter 6


(Solution)



031 GENERAL ASPECTS







INTRODUCTION 7






032 PHYSICAL MODEL



temporal variations







Section A

x

x = L

x = 0






mdash Physical laws








Objectives


INTRODUCTION 9





d qAdx

f A f( )



dx( )

( )= minus


T x T

q x L k dTdx

qx L

L

( )

( )

= =

= = minus = gt=

0

0

0

(loss)

Dirichlet

Neumann


ddx

kdT x

dxf

( )

⎛⎝⎜

⎞⎠⎟

+ =0 0


T x Tqk

xfk


⎞⎠0

0 2

2


W d Tdx

k dTdx

dx W T f dxL

Neu


00

00





034 NUMERICAL MODEL







kd T x

dxf

2

2 0 0( )

+ =


qL

L= minus


∆ x

x = 0 x = Lx = 05L x = 15L

1 2 3 4 X



kd T x

dxf i

i

2

2 0 0 1 2 3( )

+ = =


T x T T x x T

T x L T T x L x T

( ) ( )

( ) ( )1 1 2 2

3 3 4 4

0= = = == = = + =

∆∆


INTRODUCTION 11


d Tdx

d Tdx

T T Tx

dTdx

dTdx

x i

i i i

x

2

2

2

21 1

2

2= asymp

minus +

=

+ minus

∆

ii

i iT Tx


2∆


kT T T


2 02

0 2 3∆


T T

T T Tx fk

T T Tx fk

1 0

1 2 3

20

2 3 4

20

2 0

2 0

=

minus + + =

minus + + =

∆

∆


T Tx

qk

T Tx qk

L L4 24 22



1 0 0

1 2 1

0 1 1 0 5

1

2

3

0

02

02

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪=

minus

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

T

T

T

T

f x k

f x k q x kL

∆∆ ∆

which gives

T

T

T f x k q x k

T f x k q x kL

L

2

3

0 02

0 02

1 5

2 2

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆


Remark




1 2

x = 0 x = Lx = 05L

1 2 3 X

ElementElement


W W WeNeu

e

= + ==sum 0

1

2

where Wd T x

dxk

dT x

dxdx T x f dxe

x

x

x

x

i

i

i

i

=( ) ( )

minus ( )int int+ +

0

1 1

δ




T xx x

LT

x xL

T x x

T xx x

LT

x xL

T

e e

e e

( )

( )

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

= =

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

21

12 1 2

21

12

0

δ δ δ

L2



W T T kL

T

Tf L

e

e1

1 21

20

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ

INTRODUCTION 13



W T T kL

T

Tf L

e

e2

2 32

30

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ



W T T T kL

T

T

T

f L

qL

=lt gtminus

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

minus⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=δ δ δ1 2 3

1

2

3

02

1 1 0

1 2 1

0 1 14

1

2

1

0

0 0


2 2 1

1 1 42

1

22

30

0kL

T

Tf L kT L

qL

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

=⎧⎨⎩

⎫⎬⎭

minusminus⎧

⎨⎩

⎫⎬⎭

⎧ ⎫

leading to1 5

2 22

3

0 02

0 02

T

T

T f x k q x k

T f x k q x kL

L

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆

Remark


035 COMPUTER MODEL













INTRODUCTION 15






0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

2 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

4 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

6 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

8 nodes



Bibliography








INTRODUCTION 17
























Monographs



















Chapters 1 and 2


Chapter 3



Chapter 4


Chapter 5

Numericalmethods

Chapter 6


(Solution)



031 GENERAL ASPECTS







INTRODUCTION 7






032 PHYSICAL MODEL



temporal variations







Section A

x

x = L

x = 0






mdash Physical laws








Objectives


INTRODUCTION 9





d qAdx

f A f( )



dx( )

( )= minus


T x T

q x L k dTdx

qx L

L

( )

( )

= =

= = minus = gt=

0

0

0

(loss)

Dirichlet

Neumann


ddx

kdT x

dxf

( )

⎛⎝⎜

⎞⎠⎟

+ =0 0


T x Tqk

xfk


⎞⎠0

0 2

2


W d Tdx

k dTdx

dx W T f dxL

Neu


00

00





034 NUMERICAL MODEL







kd T x

dxf

2

2 0 0( )

+ =


qL

L= minus


∆ x

x = 0 x = Lx = 05L x = 15L

1 2 3 4 X



kd T x

dxf i

i

2

2 0 0 1 2 3( )

+ = =


T x T T x x T

T x L T T x L x T

( ) ( )

( ) ( )1 1 2 2

3 3 4 4

0= = = == = = + =

∆∆


INTRODUCTION 11


d Tdx

d Tdx

T T Tx

dTdx

dTdx

x i

i i i

x

2

2

2

21 1

2

2= asymp

minus +

=

+ minus

∆

ii

i iT Tx


2∆


kT T T


2 02

0 2 3∆


T T

T T Tx fk

T T Tx fk

1 0

1 2 3

20

2 3 4

20

2 0

2 0

=

minus + + =

minus + + =

∆

∆


T Tx

qk

T Tx qk

L L4 24 22



1 0 0

1 2 1

0 1 1 0 5

1

2

3

0

02

02

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪=

minus

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

T

T

T

T

f x k

f x k q x kL

∆∆ ∆

which gives

T

T

T f x k q x k

T f x k q x kL

L

2

3

0 02

0 02

1 5

2 2

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆


Remark




1 2

x = 0 x = Lx = 05L

1 2 3 X

ElementElement


W W WeNeu

e

= + ==sum 0

1

2

where Wd T x

dxk

dT x

dxdx T x f dxe

x

x

x

x

i

i

i

i

=( ) ( )

minus ( )int int+ +

0

1 1

δ




T xx x

LT

x xL

T x x

T xx x

LT

x xL

T

e e

e e

( )

( )

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

= =

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

21

12 1 2

21

12

0

δ δ δ

L2



W T T kL

T

Tf L

e

e1

1 21

20

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ

INTRODUCTION 13



W T T kL

T

Tf L

e

e2

2 32

30

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ



W T T T kL

T

T

T

f L

qL

=lt gtminus

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

minus⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=δ δ δ1 2 3

1

2

3

02

1 1 0

1 2 1

0 1 14

1

2

1

0

0 0


2 2 1

1 1 42

1

22

30

0kL

T

Tf L kT L

qL

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

=⎧⎨⎩

⎫⎬⎭

minusminus⎧

⎨⎩

⎫⎬⎭

⎧ ⎫

leading to1 5

2 22

3

0 02

0 02

T

T

T f x k q x k

T f x k q x kL

L

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆

Remark


035 COMPUTER MODEL













INTRODUCTION 15






0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

2 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

4 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

6 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

8 nodes



Bibliography








INTRODUCTION 17
























Monographs


















INTRODUCTION 7






032 PHYSICAL MODEL



temporal variations







Section A

x

x = L

x = 0






mdash Physical laws








Objectives


INTRODUCTION 9





d qAdx

f A f( )



dx( )

( )= minus


T x T

q x L k dTdx

qx L

L

( )

( )

= =

= = minus = gt=

0

0

0

(loss)

Dirichlet

Neumann


ddx

kdT x

dxf

( )

⎛⎝⎜

⎞⎠⎟

+ =0 0


T x Tqk

xfk


⎞⎠0

0 2

2


W d Tdx

k dTdx

dx W T f dxL

Neu


00

00





034 NUMERICAL MODEL







kd T x

dxf

2

2 0 0( )

+ =


qL

L= minus


∆ x

x = 0 x = Lx = 05L x = 15L

1 2 3 4 X



kd T x

dxf i

i

2

2 0 0 1 2 3( )

+ = =


T x T T x x T

T x L T T x L x T

( ) ( )

( ) ( )1 1 2 2

3 3 4 4

0= = = == = = + =

∆∆


INTRODUCTION 11


d Tdx

d Tdx

T T Tx

dTdx

dTdx

x i

i i i

x

2

2

2

21 1

2

2= asymp

minus +

=

+ minus

∆

ii

i iT Tx


2∆


kT T T


2 02

0 2 3∆


T T

T T Tx fk

T T Tx fk

1 0

1 2 3

20

2 3 4

20

2 0

2 0

=

minus + + =

minus + + =

∆

∆


T Tx

qk

T Tx qk

L L4 24 22



1 0 0

1 2 1

0 1 1 0 5

1

2

3

0

02

02

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪=

minus

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

T

T

T

T

f x k

f x k q x kL

∆∆ ∆

which gives

T

T

T f x k q x k

T f x k q x kL

L

2

3

0 02

0 02

1 5

2 2

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆


Remark




1 2

x = 0 x = Lx = 05L

1 2 3 X

ElementElement


W W WeNeu

e

= + ==sum 0

1

2

where Wd T x

dxk

dT x

dxdx T x f dxe

x

x

x

x

i

i

i

i

=( ) ( )

minus ( )int int+ +

0

1 1

δ




T xx x

LT

x xL

T x x

T xx x

LT

x xL

T

e e

e e

( )

( )

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

= =

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

21

12 1 2

21

12

0

δ δ δ

L2



W T T kL

T

Tf L

e

e1

1 21

20

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ

INTRODUCTION 13



W T T kL

T

Tf L

e

e2

2 32

30

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ



W T T T kL

T

T

T

f L

qL

=lt gtminus

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

minus⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=δ δ δ1 2 3

1

2

3

02

1 1 0

1 2 1

0 1 14

1

2

1

0

0 0


2 2 1

1 1 42

1

22

30

0kL

T

Tf L kT L

qL

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

=⎧⎨⎩

⎫⎬⎭

minusminus⎧

⎨⎩

⎫⎬⎭

⎧ ⎫

leading to1 5

2 22

3

0 02

0 02

T

T

T f x k q x k

T f x k q x kL

L

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆

Remark


035 COMPUTER MODEL













INTRODUCTION 15






0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

2 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

4 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

6 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

8 nodes



Bibliography








INTRODUCTION 17
























Monographs



















Section A

x

x = L

x = 0






mdash Physical laws








Objectives


INTRODUCTION 9





d qAdx

f A f( )



dx( )

( )= minus


T x T

q x L k dTdx

qx L

L

( )

( )

= =

= = minus = gt=

0

0

0

(loss)

Dirichlet

Neumann


ddx

kdT x

dxf

( )

⎛⎝⎜

⎞⎠⎟

+ =0 0


T x Tqk

xfk


⎞⎠0

0 2

2


W d Tdx

k dTdx

dx W T f dxL

Neu


00

00





034 NUMERICAL MODEL







kd T x

dxf

2

2 0 0( )

+ =


qL

L= minus


∆ x

x = 0 x = Lx = 05L x = 15L

1 2 3 4 X



kd T x

dxf i

i

2

2 0 0 1 2 3( )

+ = =


T x T T x x T

T x L T T x L x T

( ) ( )

( ) ( )1 1 2 2

3 3 4 4

0= = = == = = + =

∆∆


INTRODUCTION 11


d Tdx

d Tdx

T T Tx

dTdx

dTdx

x i

i i i

x

2

2

2

21 1

2

2= asymp

minus +

=

+ minus

∆

ii

i iT Tx


2∆


kT T T


2 02

0 2 3∆


T T

T T Tx fk

T T Tx fk

1 0

1 2 3

20

2 3 4

20

2 0

2 0

=

minus + + =

minus + + =

∆

∆


T Tx

qk

T Tx qk

L L4 24 22



1 0 0

1 2 1

0 1 1 0 5

1

2

3

0

02

02

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪=

minus

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

T

T

T

T

f x k

f x k q x kL

∆∆ ∆

which gives

T

T

T f x k q x k

T f x k q x kL

L

2

3

0 02

0 02

1 5

2 2

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆


Remark




1 2

x = 0 x = Lx = 05L

1 2 3 X

ElementElement


W W WeNeu

e

= + ==sum 0

1

2

where Wd T x

dxk

dT x

dxdx T x f dxe

x

x

x

x

i

i

i

i

=( ) ( )

minus ( )int int+ +

0

1 1

δ




T xx x

LT

x xL

T x x

T xx x

LT

x xL

T

e e

e e

( )

( )

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

= =

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

21

12 1 2

21

12

0

δ δ δ

L2



W T T kL

T

Tf L

e

e1

1 21

20

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ

INTRODUCTION 13



W T T kL

T

Tf L

e

e2

2 32

30

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ



W T T T kL

T

T

T

f L

qL

=lt gtminus

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

minus⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=δ δ δ1 2 3

1

2

3

02

1 1 0

1 2 1

0 1 14

1

2

1

0

0 0


2 2 1

1 1 42

1

22

30

0kL

T

Tf L kT L

qL

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

=⎧⎨⎩

⎫⎬⎭

minusminus⎧

⎨⎩

⎫⎬⎭

⎧ ⎫

leading to1 5

2 22

3

0 02

0 02

T

T

T f x k q x k

T f x k q x kL

L

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆

Remark


035 COMPUTER MODEL













INTRODUCTION 15






0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

2 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

4 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

6 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

8 nodes



Bibliography








INTRODUCTION 17
























Monographs


















INTRODUCTION 9





d qAdx

f A f( )



dx( )

( )= minus


T x T

q x L k dTdx

qx L

L

( )

( )

= =

= = minus = gt=

0

0

0

(loss)

Dirichlet

Neumann


ddx

kdT x

dxf

( )

⎛⎝⎜

⎞⎠⎟

+ =0 0


T x Tqk

xfk


⎞⎠0

0 2

2


W d Tdx

k dTdx

dx W T f dxL

Neu


00

00





034 NUMERICAL MODEL







kd T x

dxf

2

2 0 0( )

+ =


qL

L= minus


∆ x

x = 0 x = Lx = 05L x = 15L

1 2 3 4 X



kd T x

dxf i

i

2

2 0 0 1 2 3( )

+ = =


T x T T x x T

T x L T T x L x T

( ) ( )

( ) ( )1 1 2 2

3 3 4 4

0= = = == = = + =

∆∆


INTRODUCTION 11


d Tdx

d Tdx

T T Tx

dTdx

dTdx

x i

i i i

x

2

2

2

21 1

2

2= asymp

minus +

=

+ minus

∆

ii

i iT Tx


2∆


kT T T


2 02

0 2 3∆


T T

T T Tx fk

T T Tx fk

1 0

1 2 3

20

2 3 4

20

2 0

2 0

=

minus + + =

minus + + =

∆

∆


T Tx

qk

T Tx qk

L L4 24 22



1 0 0

1 2 1

0 1 1 0 5

1

2

3

0

02

02

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪=

minus

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

T

T

T

T

f x k

f x k q x kL

∆∆ ∆

which gives

T

T

T f x k q x k

T f x k q x kL

L

2

3

0 02

0 02

1 5

2 2

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆


Remark




1 2

x = 0 x = Lx = 05L

1 2 3 X

ElementElement


W W WeNeu

e

= + ==sum 0

1

2

where Wd T x

dxk

dT x

dxdx T x f dxe

x

x

x

x

i

i

i

i

=( ) ( )

minus ( )int int+ +

0

1 1

δ




T xx x

LT

x xL

T x x

T xx x

LT

x xL

T

e e

e e

( )

( )

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

= =

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

21

12 1 2

21

12

0

δ δ δ

L2



W T T kL

T

Tf L

e

e1

1 21

20

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ

INTRODUCTION 13



W T T kL

T

Tf L

e

e2

2 32

30

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ



W T T T kL

T

T

T

f L

qL

=lt gtminus

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

minus⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=δ δ δ1 2 3

1

2

3

02

1 1 0

1 2 1

0 1 14

1

2

1

0

0 0


2 2 1

1 1 42

1

22

30

0kL

T

Tf L kT L

qL

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

=⎧⎨⎩

⎫⎬⎭

minusminus⎧

⎨⎩

⎫⎬⎭

⎧ ⎫

leading to1 5

2 22

3

0 02

0 02

T

T

T f x k q x k

T f x k q x kL

L

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆

Remark


035 COMPUTER MODEL













INTRODUCTION 15






0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

2 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

4 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

6 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

8 nodes



Bibliography








INTRODUCTION 17
























Monographs



















034 NUMERICAL MODEL







kd T x

dxf

2

2 0 0( )

+ =


qL

L= minus


∆ x

x = 0 x = Lx = 05L x = 15L

1 2 3 4 X



kd T x

dxf i

i

2

2 0 0 1 2 3( )

+ = =


T x T T x x T

T x L T T x L x T

( ) ( )

( ) ( )1 1 2 2

3 3 4 4

0= = = == = = + =

∆∆


INTRODUCTION 11


d Tdx

d Tdx

T T Tx

dTdx

dTdx

x i

i i i

x

2

2

2

21 1

2

2= asymp

minus +

=

+ minus

∆

ii

i iT Tx


2∆


kT T T


2 02

0 2 3∆


T T

T T Tx fk

T T Tx fk

1 0

1 2 3

20

2 3 4

20

2 0

2 0

=

minus + + =

minus + + =

∆

∆


T Tx

qk

T Tx qk

L L4 24 22



1 0 0

1 2 1

0 1 1 0 5

1

2

3

0

02

02

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪=

minus

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

T

T

T

T

f x k

f x k q x kL

∆∆ ∆

which gives

T

T

T f x k q x k

T f x k q x kL

L

2

3

0 02

0 02

1 5

2 2

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆


Remark




1 2

x = 0 x = Lx = 05L

1 2 3 X

ElementElement


W W WeNeu

e

= + ==sum 0

1

2

where Wd T x

dxk

dT x

dxdx T x f dxe

x

x

x

x

i

i

i

i

=( ) ( )

minus ( )int int+ +

0

1 1

δ




T xx x

LT

x xL

T x x

T xx x

LT

x xL

T

e e

e e

( )

( )

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

= =

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

21

12 1 2

21

12

0

δ δ δ

L2



W T T kL

T

Tf L

e

e1

1 21

20

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ

INTRODUCTION 13



W T T kL

T

Tf L

e

e2

2 32

30

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ



W T T T kL

T

T

T

f L

qL

=lt gtminus

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

minus⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=δ δ δ1 2 3

1

2

3

02

1 1 0

1 2 1

0 1 14

1

2

1

0

0 0


2 2 1

1 1 42

1

22

30

0kL

T

Tf L kT L

qL

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

=⎧⎨⎩

⎫⎬⎭

minusminus⎧

⎨⎩

⎫⎬⎭

⎧ ⎫

leading to1 5

2 22

3

0 02

0 02

T

T

T f x k q x k

T f x k q x kL

L

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆

Remark


035 COMPUTER MODEL













INTRODUCTION 15






0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

2 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

4 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

6 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

8 nodes



Bibliography








INTRODUCTION 17
























Monographs


















INTRODUCTION 11


d Tdx

d Tdx

T T Tx

dTdx

dTdx

x i

i i i

x

2

2

2

21 1

2

2= asymp

minus +

=

+ minus

∆

ii

i iT Tx


2∆


kT T T


2 02

0 2 3∆


T T

T T Tx fk

T T Tx fk

1 0

1 2 3

20

2 3 4

20

2 0

2 0

=

minus + + =

minus + + =

∆

∆


T Tx

qk

T Tx qk

L L4 24 22



1 0 0

1 2 1

0 1 1 0 5

1

2

3

0

02

02

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪=

minus

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

T

T

T

T

f x k

f x k q x kL

∆∆ ∆

which gives

T

T

T f x k q x k

T f x k q x kL

L

2

3

0 02

0 02

1 5

2 2

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆


Remark




1 2

x = 0 x = Lx = 05L

1 2 3 X

ElementElement


W W WeNeu

e

= + ==sum 0

1

2

where Wd T x

dxk

dT x

dxdx T x f dxe

x

x

x

x

i

i

i

i

=( ) ( )

minus ( )int int+ +

0

1 1

δ




T xx x

LT

x xL

T x x

T xx x

LT

x xL

T

e e

e e

( )

( )

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

= =

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

21

12 1 2

21

12

0

δ δ δ

L2



W T T kL

T

Tf L

e

e1

1 21

20

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ

INTRODUCTION 13



W T T kL

T

Tf L

e

e2

2 32

30

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ



W T T T kL

T

T

T

f L

qL

=lt gtminus

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

minus⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=δ δ δ1 2 3

1

2

3

02

1 1 0

1 2 1

0 1 14

1

2

1

0

0 0


2 2 1

1 1 42

1

22

30

0kL

T

Tf L kT L

qL

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

=⎧⎨⎩

⎫⎬⎭

minusminus⎧

⎨⎩

⎫⎬⎭

⎧ ⎫

leading to1 5

2 22

3

0 02

0 02

T

T

T f x k q x k

T f x k q x kL

L

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆

Remark


035 COMPUTER MODEL













INTRODUCTION 15






0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

2 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

4 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

6 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

8 nodes



Bibliography








INTRODUCTION 17
























Monographs



















Remark




1 2

x = 0 x = Lx = 05L

1 2 3 X

ElementElement


W W WeNeu

e

= + ==sum 0

1

2

where Wd T x

dxk

dT x

dxdx T x f dxe

x

x

x

x

i

i

i

i

=( ) ( )

minus ( )int int+ +

0

1 1

δ




T xx x

LT

x xL

T x x

T xx x

LT

x xL

T

e e

e e

( )

( )

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

= =

= minus⎛⎝⎜

⎞⎠⎟

+ minus⎛⎝⎜

⎞⎠⎟

21

12 1 2

21

12

0

δ δ δ

L2



W T T kL

T

Tf L

e

e1

1 21

20

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ

INTRODUCTION 13



W T T kL

T

Tf L

e

e2

2 32

30

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ



W T T T kL

T

T

T

f L

qL

=lt gtminus

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

minus⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=δ δ δ1 2 3

1

2

3

02

1 1 0

1 2 1

0 1 14

1

2

1

0

0 0


2 2 1

1 1 42

1

22

30

0kL

T

Tf L kT L

qL

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

=⎧⎨⎩

⎫⎬⎭

minusminus⎧

⎨⎩

⎫⎬⎭

⎧ ⎫

leading to1 5

2 22

3

0 02

0 02

T

T

T f x k q x k

T f x k q x kL

L

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆

Remark


035 COMPUTER MODEL













INTRODUCTION 15






0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

2 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

4 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

6 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

8 nodes



Bibliography








INTRODUCTION 17
























Monographs


















INTRODUCTION 13



W T T kL

T

Tf L

e

e2

2 32

30

1 1

1 1 21

1=lt gt

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

minus⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎞

⎠⎟δ δ



W T T T kL

T

T

T

f L

qL

=lt gtminus

minus minusminus

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

minus⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=δ δ δ1 2 3

1

2

3

02

1 1 0

1 2 1

0 1 14

1

2

1

0

0 0


2 2 1

1 1 42

1

22

30

0kL

T

Tf L kT L

qL

minusminus

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

=⎧⎨⎩

⎫⎬⎭

minusminus⎧

⎨⎩

⎫⎬⎭

⎧ ⎫

leading to1 5

2 22

3

0 02

0 02

T

T

T f x k q x k

T f x k q x kL

L

⎧⎨⎩

⎫⎬⎭

=+ minus+ minus

⎧⎨⎩

⎫⎬⎭

∆ ∆∆ ∆

Remark


035 COMPUTER MODEL













INTRODUCTION 15






0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

2 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

4 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

6 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

8 nodes



Bibliography








INTRODUCTION 17
























Monographs




























INTRODUCTION 15






0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

2 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

4 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

6 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

8 nodes



Bibliography








INTRODUCTION 17
























Monographs



















0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

2 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

4 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

6 nodes

0 02 04 06 08 122

24

26

28

30

32

x [m]

Tem

pera

ture

[C]

8 nodes



Bibliography








INTRODUCTION 17
























Monographs


















INTRODUCTION 17
























Monographs



















Monographs


















Finite Element Method - download.e- · PDF fileFinite Element Method Gouri Dhatt Gilbert...

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Transcript of Finite Element Method - download.e- · PDF fileFinite Element Method Gouri Dhatt Gilbert...