Hunter Peter - Finite Element Method - Boundary Element Method.pdf

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fembemnotes.dvi
FEM/BEM NOTES
Professor Peter Hunter
p.hunter@auckland.ac.nz
Associate Professor Andrew Pullan
a.pullan@auckland.ac.nz
Department of Engineering Science
The University of Auckland
New Zealand
February 21, 2001
c
Copyright 1997 : Department of Engineering Science
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Contents
1 Finite Element Basis Functions 1
1.1 RepresentingaOne-DimensionalField........................ 1
1.2 LinearBasisFunctions ................................ 2
1.3 BasisFunctionsasWeightingFunctions ....................... 4
1.4 QuadraticBasisFunctions .............................. 7
1.5 Two- and Three-Dimensional Elements . . . . . .................. 7
1.6 HigherOrderContinuity ............................... 10
1.7 Triangular Elements . . ................................ 14
1.8 Curvilinear Coordinate Systems . . ......................... 16
1.9 CMISSExamples................................... 19
2 Steady-State Heat Conduction 21
2.1 One-Dimensional Steady-State Heat Conduction .................. 21
2.1.1 Integralequation ............................... 22
2.1.2 Integrationbyparts.............................. 22
2.1.3 Finiteelementapproximation ........................ 23
2.1.4 Elementintegrals............................... 24
2.1.5 Assembly................................... 25
2.1.6 Boundary conditions . . . . ......................... 27
2.1.7 Solution.................................... 27
2.1.8 Fluxes..................................... 27
2.2 An x -DependentSourceTerm ............................ 28
2.3 TheGalerkinWeightFunctionRevisited....................... 29
2.4 Two and Three-Dimensional Steady-State Heat Conduction . . . . . . ....... 30
2.5 BasisFunctions-ElementDiscretisation....................... 32
2.6 Integration....................................... 34
2.7 AssembleGlobalEquations.............................. 35
2.8 Gaussian Quadrature . ................................ 37
2.9 CMISSExamples................................... 40
3 The Boundary Element Method 41
3.1 Introduction . . . . . . ................................ 41
3.2 The Dirac-Delta Function and Fundamental Solutions . . .............. 41
3.2.1 Dirac-Delta function . . . . ......................... 41
3.2.2 Fundamental solutions . . . ......................... 43
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CONTENTS
3.3 The Two-Dimensional Boundary Element Method .................. 46
3.4 Numerical Solution Procedures for the Boundary Integral Equation . . . . . .... 51
3.5 NumericalEvaluationofCoefficientIntegrals .................... 53
3.6 The Three-Dimensional Boundary Element Method . . . . . . ........... 55
3.7 A Comparison of the FE and BE Methods ...................... 56
3.8 MoreonNumericalIntegration............................ 58
3.8.1 Logarithmicquadratureandotherspecialschemes ............. 58
3.8.2 Specialsolutions ............................... 59
3.9 The Boundary Element Method Applied to other Elliptic PDEs ........... 59
3.10SolutionofMatrixEquations............................. 59
3.11CouplingtheFEandBEtechniques ......................... 60
3.12OtherBEMtechniques ................................ 62
3.12.1 Trefftzmethod ................................ 62
3.12.2 RegularBEM................................. 62
3.13Symmetry....................................... 63
3.14AxisymmetricProblems ............................... 65
3.15InfiniteRegions.................................... 67
3.16 Appendix: Common Fundamental Solutions . . . .................. 70
3.16.1 Two-Dimensionalequations ......................... 70
3.16.2 Three-Dimensional equations . . ...................... 70
3.16.3 Axisymmetricproblems ........................... 71
3.17CMISSExamples................................... 71
4 Linear Elasticity 73
4.1 Introduction . . .................................... 73
4.2 TrussElements .................................... 74
4.3 BeamElements .................................... 77
4.4 PlaneStressElements................................. 79
4.5 Navier’sEquation................................... 81
4.6 NoteonCalculatingNodalLoads........................... 83
4.7 Three-Dimensional Elasticity ............................. 84
4.8 IntegralEquation ................................... 86
4.9 Linear Elasticity with Boundary Elements ...................... 86
4.10 Fundamental Solutions . . . ............................. 89
4.11 Boundary Integral Equation . ............................. 90
4.12 Body Forces (and Domain Integrals in General) . .................. 93
4.13CMISSExamples................................... 95
5 Transient Heat Conduction 97
5.1 Introduction . . .................................... 97
5.2 Finite Differences . . . . . . ............................. 97
5.2.1 Explicit Transient Finite Differences . . . .................. 97
5.2.2 Von Neumann Stability Analysis . ...................... 99
5.2.3 HigherOrderApproximations ........................100
5.3 TheTransientAdvection-DiffusionEquation ....................101
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CONTENTS
iii
5.4 Masslumping.....................................104
5.5 CMISSExamples...................................106
6 Modal Analysis 109
6.1 Introduction . . . . . . ................................109
6.2 FreeVibrationModes.................................109
6.3 AnAnalyticExample.................................111
6.4 Proportional Damping ................................112
6.5 CMISSExamples...................................113
7 Domain Integrals in the BEM 115
7.1 Achieving a Boundary Integral Formulation . . . ..................115
7.2 Removing Domain Integrals due to Inhomogeneous Terms . . . . . . .......116
7.2.1 TheGalerkinVectortechnique........................116
7.2.2 TheMonteCarlomethod...........................117
7.2.3 ComplementaryFunction-ParticularIntegralmethod ............118
7.3 DomainIntegralsInvolvingtheDependentVariable.................118
7.3.1 The Perturbation Boundary Element Method . . ..............119
7.3.2 The Multiple Reciprocity Method . . . . ..................120
7.3.3 The Dual Reciprocity Boundary Element Method ..............122
8 The BEM for Parabolic PDES 133
8.1 Time-Stepping Methods . . . . . . .........................133
8.1.1 Coupled Finite Difference - Boundary Element Method . . . . .......133
8.1.2 Direct Time-Integration Method . . . . . ..................135
8.2 LaplaceTransformMethod..............................136
8.3 TheDR-BEMForTransientProblems ........................137
8.4 TheMRMforTransientProblems ..........................138
Bibliography
141
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