Table Of ContentAdvanced Numerical Simulation Methods
Advanced Numerical
Simulation Methods
From CAD Data Directly to Simulation Results
Gernot Beer
Institute for Structural Analysis,Graz University ofTechnology,
Graz,Austria
ThecoverisadisplayoftheCADmodelandthesimulationresults(displacementcontours)
fortwotunnelswithacrosspassage(copyrightG.Beer).
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LibraryofCongressCataloging-in-PublicationData
Beer,G.(Gernot)
Advancednumericalsimulationmethods:fromCADdatadirectlyto
simulationresults/GernotBeer.
pagescm
Includesbibliographicalreferencesandindex.
ISBN978-1-138-02634-6(hardcover:alk.paper)–ISBN978-1-315-76631-7
(ebook:alk.paper) 1. Computersimulation. I.Title.
QA76.9.C65B44 2015
(cid:1)
003.3–dc23
2015023012
Publishedby: CRCPress/Balkema
P.O.Box11320,2301EHLeiden,TheNetherlands
e-mail:[email protected]
www.crcpress.com–www.taylorandfrancis.com
ISBN:978-1-138-02634-6(Hbk)
ISBN:978-1-315-76631-7(eBookPDF)
Table of contents
Preface xi
Abouttheauthor xv
1 Introduction 1
1 Abriefhistoryofsimulation 1
1.1 Theworld’sfirstsimulation 1
1.2 Emergenceofmathematicsandmechanics 3
1.3 Computerage 4
2 Basicstepsinsimulation 9
2.1 Geometrydescription 9
2.2 Approximationoftheunknown 11
2.3 Solution 12
2.4 Recoveryoftheresults 12
3 Achangeofparadigm:towardsamoreefficientand
accuratesimulation 12
4 Organizationofthetext 13
2 Stage1:Basisfunctions 17
1 One-dimensionalbasisfunctions 17
1.1 LagrangeandSerendipityfunctions 17
1.2 FromB-splinestoNURBS 22
2 Two-dimensionalbasisfunctions 33
2.1 LagrangeandSerendipityfunctions 33
2.2 B-splines 38
2.3 NURBS 38
2.4 T-splines 40
3 Programming 44
4 TheNURBStoolkit 45
5 Summaryandconclusions 51
3 Stage2:Geometry 55
1 Coordinatesystems 55
1.1 Coordinatetransformation 55
2 Curves 56
2.1 MappingwithSerendipity/Lagrangebasisfunctions 56
2.2 MappingwithNURBS 57
vi Table of contents
3 Programming 58
3.1 NURBStoolkit 58
3.2 Geometryfunctions 59
3.3 Examples 63
3.4 Example1:Circulararc 63
3.5 Example2:Horseshoetunnel 65
3.6 Example3:Platewithhole 67
4 Surfaces 68
4.1 MappingwithSerendipity/Lagrangebasisfunctions 69
4.2 MappingwithNURBSbasisfunctions 70
4.3 Programming 71
5 Surfaceofrevolution 73
5.1 Example1:Cylindricalsurface 74
5.2 Example2:Sphericalsurface 75
5.3 Example3:Bellshapedsurface 77
6 Loftedsurfaces 79
7 NURBSsurfaceswithcutouts 81
7.1 AnalysissuitabletrimmedNURBSsurfaces 82
8 InfiniteNURBSpatch 87
8.1 Example 92
9 Summaryandconclusions 93
4 Stage3:ComputerAidedDesign 95
1 Introduction 95
2 IGESdatastructure 98
3 HowCADprogramsdescribegeometry–entitytypes 100
3.1 Lineentity(110) 100
3.2 Surfaceofrevolutionentity(120) 100
3.3 RationalB-splineentity(126) 101
3.4 RationalB-splinesurfaceentity(128) 101
3.5 Boundaryentity(141) 101
4 NURBSsurfaces 102
5 TrimmedNURBSsurfaces 104
6 Summaryandconclusions 113
5 Stage4:Introductiontonumericalsimulation 117
1 One-dimensionalsimulation 117
1.1 Ritzmethod 119
1.2 Approximation 121
2 Stepsinthesimulation 126
2.1 Descriptionofthegeometry 126
2.2 Descriptionofknownvalues 126
2.3 Convergencetests 127
2.4 Approximationofunknown 128
2.5 P-refinementororderelevation 128
2.6 H-refinement,theFiniteElementMethod 128
2.7 Knotinsertion,isogeometricmethod 132
Table of contents vii
2.8 K-refinement 132
2.9 Summaryandconclusions 133
3 2-Dsimulation,planestressandplanestrain 135
3.1 BoundaryConditions(BC) 138
3.2 UsingoneNURBSpatch 140
3.3 ComparisonwithclassicalFEM 141
3.4 Example 142
3.5 MultipleNURBSpatches 143
3.6 Bezièrelements 147
3.7 TrimmedNURBSpatches 148
3.8 Convergencetest 150
4 Summaryandconclusions 150
6 Stage5:Platesandshells 153
1 Kirchhoffplate 153
1.1 Plates 154
1.2 Examples 158
2 Kirchhoffshells 161
2.1 Example1:Scordelisroof 162
2.2 Example2:TrimmedScordelisroof 164
2.3 Example3:ArchedScordelisroof 166
3 Multiplepatches 168
3.1 Assembly 168
3.2 Example 169
4 Summaryandconclusions 172
7 Stage6:Integralequations 175
1 Introduction 175
2 Trefftzmethod 176
2.1 Example 180
2.2 Conclusions 183
3 Integralequations 183
3.1 TheoremofBetti 187
3.2 Rigidbodytrick 191
3.3 Conclusions 193
4 Numericalsolutionofintegralequations 193
4.1 Nyströmmethod 193
4.2 Galerkinmethod 196
4.3 Collocation 196
4.4 Discretisation 197
5 Summaryandconclusions 199
8 Stage7:Theboundaryelementmethodforplaneproblems 201
1 Introduction 201
2 Classicalisoparametricapproach 201
2.1 Numericalevaluationofintegrals 203
viii Table of contents
3 NURBSbasedapproach 206
3.1 Boundaryconditions 210
4 Assemblyofmultiplepatches 211
4.1 PureNeumannproblem 211
4.2 MixedNeumann/Dirichletproblem 211
4.3 Symmetry 212
5 Postprocessing 213
5.1 Resultsontheboundary 213
5.2 Resultsinsidethedomain 214
6 Programming 216
7 Examples 227
7.1 Potentialproblem:Flowpastisolator 227
7.2 Elasticityproblem:Circularexcavationininfinitedomain 229
7.3 Practicalexample:Horseshoetunnel 231
8 Conclusions 233
9 Stage8:Theboundaryelementmethod
forthree-dimensionalproblems 235
1 Introduction 235
2 Numericalintegration 236
2.1 Regularintegration 236
2.2 DeterminationoftheoptimalnumberofGausspoints 237
2.3 Regularintegration 238
2.4 Nearlysingularintegration 239
2.5 Weaklysingularintegration 243
2.6 Infinitepatches 246
3 Symmetry 247
4 Multiplepatches 249
5 Postprocessing 249
5.1 Stressrecovery 249
5.2 Internalstresscomputation 251
6 Testexamples 252
6.1 Infinitetunnel 252
6.2 Loadingoninfinitehalf-space 253
7 Examples 253
7.1 Infinitetunnelininfinitedomainneartunnelface 253
7.2 Finitetunnelinasemi-infinitedomain 254
7.3 Branchedtunnel 255
8 Conclusions 260
10 Stage9:Theboundaryelementmethodwithvolumeeffects 263
1 Introduction 263
2 Effectofbodyforcesandinitialstrain 263
2.1 Bodyforces 264
2.2 Effectofinitialstrain 265
2.3 Solution 266
Table of contents ix
3 Implementationforplaneproblems 266
3.1 Geometrydefinition 266
3.2 Computationofthevolumeintegral 268
4 Implementationfor3-Dproblems 268
4.1 Geometrydefinition 269
4.2 Computationofthevolumeintegral 270
5 Iterativesolutionalgorithm 270
6 Inclusions 271
6.1 Computationofbodyforce 273
6.2 Stepsintheanalysis 275
7 Inelasticbehavior 276
7.1 Yieldconditions 277
7.2 Determinationofplasticstrainincrement 278
8 Implementationofplasticityforplaneproblems 280
8.1 Determinationofplasticzone 280
8.2 Computationofthevolumeterm 284
8.3 Numericalintegration 286
8.4 Internalstresscomputation 287
8.5 Extensionofplasticzoneduringiteration 287
9 Implementationfor3-Dproblems 288
9.1 Determinationoftheplasticzone 288
9.2 Computationofthevolumeterm 289
9.3 Numericalintegration 289
10 Programming 289
11 Example 293
12 Summaryandconclusions 295
11 Stage10:Thetimedomain 297
1 Introduction 297
1.1 Bernoullibeamwithmass 297
2 Solutionsinthefrequencydomain 298
2.1 Numericalsolution 299
3 Solutionsinthetimedomain 301
3.1 Finitedifferencemethod 301
3.2 Newmarkmethod 302
4 Programming 306
5 Summaryandconclusions 309
Appendix: Fundamentalsolutions 311
1 Stresssolution(cid:1)(x,y) 312
2 DerivedsolutionfordisplacementS(x,y) 313
3 DerivedsolutionfortractionR(x,y) 314
4 DerivedsolutionfordisplacementS(x,y) 316
5 DerivedsolutionfortractionR(x,y) 317
6 DerivativesofkernelS(x,y) 319
7 DerivativesofkernelR(x,y) 320
Subjectindex 325
