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The Isogeometric Boundary Element Method - Gernot Beer, Benjamin Marussig, Christian Duenser Blick ins Buch

The Isogeometric Boundary Element Method (eBook)

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eBook Download: PDF
2019
XIV, 335 Seiten
Springer International Publishing (Verlag)
978-3-030-23339-6 (ISBN)

Lese- und Medienproben

The Isogeometric Boundary Element Method - Gernot Beer, Benjamin Marussig, Christian Duenser
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This book discusses the introduction of isogeometric technology to the boundary element method (BEM) in order to establish an improved link between simulation and computer aided design (CAD) that does not require mesh generation. In the isogeometric BEM, non-uniform rational B-splines replace the Lagrange polynomials used in conventional BEM. This may seem a trivial exercise, but if implemented rigorously, it has profound implications for the programming, resulting in software that is extremely user friendly and efficient. The BEM is ideally suited for linking with CAD, as both rely on the definition of objects by boundary representation. The book shows how the isogeometric philosophy can be implemented and how its benefits can be maximised with a minimum of user effort. Using several examples, ranging from potential problems to elasticity, it demonstrates that the isogeometric approach results in a drastic reduction in the number of unknowns and an increase in the quality of the results. In some cases even exact solutions without refinement are possible. The book also presents a number of practical applications, demonstrating that the development is not only of academic interest. It then elegantly addresses heterogeneous and non-linear problems using isogeometric concepts, and tests them on several examples, including a severely non-linear problem in viscous flow. The book makes a significant contribution towards a seamless integration of CAD and simulation, which eliminates the need for tedious mesh generation and provides high-quality results with minimum user intervention and computing.

Gernot Beer is emeritus Professor at Graz University of Technology, Austria. He has been very active in promoting the Boundary Element Method and wrote 3 books on the topic, one published with Springer: The Boundary Element Method with Programming (2008).

Preface 6
Acknowledgements 8
Contents 10
1 Introduction 16
1.1 Motivation 16
1.2 A Short History of the Boundary Element Method 17
1.3 Notation 22
1.3.1 Cartesian Coordinates and Vector Components 22
1.3.2 Indices and Summation Convention 22
1.3.3 Tensor Algebra 23
1.3.4 Notes on Naming Convention 24
1.4 Glossary of Terms 24
1.5 Organisation of the Text 26
2 The Boundary Integral Equations 28
2.1 Potential Problems 31
2.1.1 Steady State Flow 31
2.1.2 Transient Flow 34
2.1.3 Scalar Wave Problems, Acoustics 36
2.2 Elasto-Statics – Navier–Cauchy Equations 37
2.2.1 Isotropic Material 37
2.2.2 Anisotropic Material 41
2.3 Stokes Flow 47
2.3.1 Fundamental Solutions 48
2.4 Summary 49
3 Basis Functions, B-splines 50
3.1 Introduction 50
3.2 Lagrange Polynomials 51
3.3 B-splines 53
3.3.1 Increasing the Order 56
3.3.2 Control over the Continuity 56
3.3.3 Representation of Functions 58
3.4 Refinement Options 60
3.4.1 Order Elevation 60
3.4.2 Knot Insertion 62
3.4.3 k-Refinement 63
3.4.4 Comparison 64
3.5 Tensor Product B-splines 71
3.6 Refinement of Tensor Product B-splines 72
3.6.1 General Refinement 72
3.6.2 Local Refinement Options 73
3.6.3 Hierarchical B-splines 73
3.6.4 Truncated Hierarchical B-splines 82
3.7 Summary and Conclusions 86
4 Description of the Geometry 87
4.1 NURBS Curves 87
4.1.1 Definition 87
4.1.2 Shape Control by the Control Polygon 90
4.1.3 Refinement 92
4.2 NURBS Surfaces 94
4.2.1 Definition 94
4.2.2 Constructing Surfaces by Boundary Curves 96
4.2.3 Trimmed Surfaces 97
4.3 Solid Models 101
4.4 NURBS in the Context of Approximation 102
4.5 Unified Notation for the Remainder of This Book 105
4.6 Summary and Conclusions 106
5 Getting Geometry Information from CAD Programs 107
5.1 General Considerations 107
5.2 Neutral Format Translators 109
5.2.1 STEP 109
5.3 Bézier Extraction 112
5.4 Summary and Conclusions 115
6 Numerical Treatment of Integral Equations 117
6.1 Introduction 117
6.2 Satisfaction of Integral Equations 118
6.3 Singular Integrals 119
6.4 Solving the Cauchy Principal Value Problem 119
6.5 Discretisation of the Integral Equations 121
6.5.1 Geometry Discretisation 121
6.5.2 Approximation of the Physical Values 125
6.5.3 Refinement Philosophy (Geometry Independent Field Approximation) 126
6.5.4 Location of Collocation Points 126
6.5.5 System of Equations 127
6.6 Solution of System of Equations 132
6.6.1 Properties of BEM Matrices 132
6.6.2 Cross Approximation 132
6.6.3 Adaptive Cross-Approximation (ACA) 133
6.6.4 Hierarchical Matrices 135
6.6.5 Iterative Solvers 138
6.7 Post-processing 138
6.8 Summary and Conclusions 140
7 Numerical Integration 142
7.1 Numerical Integration Methods 143
7.1.1 Gauss Quadrature 143
7.1.2 Gauss–Laguerre 147
7.2 Computation of the Shortest Distance of a Point to a Patch 148
7.3 One-Dimensional Integration 149
7.3.1 Regular Integration 149
7.3.2 Singular Integration 151
7.4 Two-Dimensional Integration 154
7.4.1 Regular Integration 154
7.4.2 Singular Integration 158
7.5 Summary and Conclusions 161
8 Steady State Potential Problems 163
8.1 Introduction 163
8.2 Post-processing 164
8.3 Computation of Boundary Values 166
8.3.1 Plane Problems 166
8.3.2 3-D Problems 167
8.4 Confined Flow Problems 169
8.4.1 Example 1: Flow Around Obstacle 170
8.4.2 Flow Beneath a Dam 172
8.4.3 Flow Through Pipe 174
8.5 Unconfined Flow Problems 176
8.5.1 Seepage Through Dam 176
8.6 Summary and Conclusions 180
9 Static Linear Solid Mechanics 181
9.1 Introduction 181
9.2 Post-processing 183
9.3 Computation of Boundary Values 186
9.3.1 Plane Problems 186
9.3.2 3-D Problems 187
9.4 Examples 189
9.4.1 Circular Excavation in Infinite Domain 189
9.4.2 Cantilever in Three Dimensions 192
9.4.3 Practical Example: Crankshaft 193
9.5 Summary and Conclusions 195
10 Simulation with Trimmed Models 196
10.1 Double Mapping Method – A Global Approach 197
10.1.1 Numerical Example 198
10.1.2 Concluding Remarks 199
10.2 Local Approaches 201
10.2.1 Element Detection 201
10.2.2 Integration Schemes 204
10.2.3 Assembly of Multiple Patches 210
10.2.4 Stabilisation 210
10.2.5 Numerical Results 220
10.3 Summary and Conclusions 226
11 Body Force Effects 228
11.1 Introduction 228
11.2 Constant Body Forces 230
11.3 General Body Forces 231
11.3.1 Types of Body Forces 231
11.3.2 Internal Cell Method 232
11.3.3 Isogeometric Mapping Method 232
11.4 Summary and Conclusions 239
12 Treatment of Inhomogeneities/Inclusions 240
12.1 Introduction 240
12.2 Multi-region Approach 240
12.3 Inclusions 243
12.3.1 Evaluation of the Volume Integrals 244
12.3.2 Plane Problems 245
12.3.3 3-D Problems 251
12.4 Summary and Conclusions 258
13 Material Non-linear Behaviour 260
13.1 Introduction 260
13.2 Computation of Initial Flow/Stress 261
13.2.1 Solid Mechanics 261
13.3 Evaluation of the Volume Integrals 264
13.4 Plane Problems 264
13.4.1 Test Example 1: Cube with Inelastic Inclusion 264
13.4.2 Test Example 2: Hole in an Infinite Domain 265
13.5 3-D Problems 268
13.5.1 Test Example: Cantilever Beam with an Inelastic Inclusion 268
13.6 Summary and Conclusions 270
14 Applications in Geomechanics 271
14.1 Introduction 271
14.2 Simulation of the Excavation Process 271
14.3 Simulation of Ground Support 272
14.4 Geometry Definition 272
14.4.1 Tunnels 273
14.4.2 Caverns 273
14.5 Definition of Geology 274
14.6 Test Examples 275
14.6.1 Example 1 275
14.6.2 Test Example 2 276
14.7 Practical Examples 279
14.7.1 NATM Tunnel 279
14.7.2 Practical Example 2: Cavern of Hydroelectric Plant 282
14.7.3 Practical Example 3: Hudson River Crossing Project 286
14.8 Summary and Conclusions 289
15 Viscous Flow Problems 290
15.1 Introduction 290
15.2 Solution Process 292
15.2.1 Modified Newton–Raphson 292
15.2.2 Newton–Raphson 293
15.3 Plane Problems 294
15.3.1 Implementation 294
15.3.2 Flow in Cavity 295
15.3.3 Airfoil 300
15.4 3-D Problems 302
15.4.1 Implementation 302
15.4.2 Numerical Results 303
15.4.3 Truncated Geometry 303
15.4.4 Non-truncated Geometry 308
15.4.5 Practical Example 310
15.5 Summary and Conclusions 313
16 Time Dependent Problems 315
16.1 Transient Potential Problems 315
16.1.1 Time Marching Method 315
16.1.2 Laplace Domain 321
16.2 Acoustics 322
16.2.1 Pulsating Sphere Example 323
16.3 Summary 325
17 Summary and Outlook 326
Appendix A Fundamental Solutions 328
A.1 2-D Potential Flow 329
A.1.1 Isotropic Fundamental Solution 329
A.1.2 Anisotropic Fundamental Solutions 329
A.2 3-D Potential Flow 330
A.2.1 Isotropic Fundamental Solution 330
A.2.2 Anisotropic Fundamental Solutions 330
A.3 Transient Potential Flow 330
A.4 3-D Acoustics 330
A.5 Elasticity 331
A.5.1 Coordinate Transformation 332
A.5.2 Transversal Isotropy 332
A.5.3 Constant Body Force 333
A.6 2-D Stokes Flow 333
A.7 3-D Stokes Flow 333
Appendix References 334
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Erscheint lt. Verlag 21.9.2019
Reihe/Serie Lecture Notes in Applied and Computational Mechanics
Lecture Notes in Applied and Computational Mechanics
Zusatzinfo XIV, 335 p. 235 illus., 189 illus. in color.
Sprache englisch
Themenwelt Mathematik / Informatik ► Mathematik
Technik ► Maschinenbau
Schlagworte Boundary element method • Computer Aided Design • isogeometric analysis • time dependent problems • viscous flow problems
ISBN-10 3-030-23339-1 / 3030233391
ISBN-13 978-3-030-23339-6 / 9783030233396
Informationen gemäß Produktsicherheitsverordnung (GPSR)
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