E-Book, Englisch, Band 90, 342 Seiten
Beer / Marussig / Duenser The Isogeometric Boundary Element Method
1. Auflage 2019
ISBN: 978-3-030-23339-6
Verlag: Springer International Publishing
Format: PDF
Kopierschutz: 1 - PDF Watermark
E-Book, Englisch, Band 90, 342 Seiten
Reihe: Lecture Notes in Applied and Computational Mechanics
ISBN: 978-3-030-23339-6
Verlag: Springer International Publishing
Format: PDF
Kopierschutz: 1 - PDF Watermark
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).
Autoren/Hrsg.
Weitere Infos & Material
1;Preface;6
2;Acknowledgements;8
3;Contents;10
4;1 Introduction;16
4.1;1.1 Motivation;16
4.2;1.2 A Short History of the Boundary Element Method;17
4.3;1.3 Notation;22
4.3.1;1.3.1 Cartesian Coordinates and Vector Components;22
4.3.2;1.3.2 Indices and Summation Convention;22
4.3.3;1.3.3 Tensor Algebra;23
4.3.4;1.3.4 Notes on Naming Convention;24
4.4;1.4 Glossary of Terms;24
4.5;1.5 Organisation of the Text;26
5;2 The Boundary Integral Equations;28
5.1;2.1 Potential Problems;31
5.1.1;2.1.1 Steady State Flow;31
5.1.2;2.1.2 Transient Flow;34
5.1.3;2.1.3 Scalar Wave Problems, Acoustics;36
5.2;2.2 Elasto-Statics – Navier–Cauchy Equations;37
5.2.1;2.2.1 Isotropic Material;37
5.2.2;2.2.2 Anisotropic Material;41
5.3;2.3 Stokes Flow;47
5.3.1;2.3.1 Fundamental Solutions;48
5.4;2.4 Summary;49
6;3 Basis Functions, B-splines;50
6.1;3.1 Introduction;50
6.2;3.2 Lagrange Polynomials;51
6.3;3.3 B-splines;53
6.3.1;3.3.1 Increasing the Order;56
6.3.2;3.3.2 Control over the Continuity;56
6.3.3;3.3.3 Representation of Functions;58
6.4;3.4 Refinement Options;60
6.4.1;3.4.1 Order Elevation;60
6.4.2;3.4.2 Knot Insertion;62
6.4.3;3.4.3 k-Refinement;63
6.4.4;3.4.4 Comparison;64
6.5;3.5 Tensor Product B-splines;71
6.6;3.6 Refinement of Tensor Product B-splines;72
6.6.1;3.6.1 General Refinement;72
6.6.2;3.6.2 Local Refinement Options;73
6.6.3;3.6.3 Hierarchical B-splines;73
6.6.4;3.6.4 Truncated Hierarchical B-splines;82
6.7;3.7 Summary and Conclusions;86
7;4 Description of the Geometry;87
7.1;4.1 NURBS Curves;87
7.1.1;4.1.1 Definition;87
7.1.2;4.1.2 Shape Control by the Control Polygon;90
7.1.3;4.1.3 Refinement;92
7.2;4.2 NURBS Surfaces;94
7.2.1;4.2.1 Definition;94
7.2.2;4.2.2 Constructing Surfaces by Boundary Curves;96
7.2.3;4.2.3 Trimmed Surfaces;97
7.3;4.3 Solid Models;101
7.4;4.4 NURBS in the Context of Approximation;102
7.5;4.5 Unified Notation for the Remainder of This Book;105
7.6;4.6 Summary and Conclusions;106
8;5 Getting Geometry Information from CAD Programs;107
8.1;5.1 General Considerations;107
8.2;5.2 Neutral Format Translators;109
8.2.1;5.2.1 STEP;109
8.3;5.3 Bézier Extraction;112
8.4;5.4 Summary and Conclusions;115
9;6 Numerical Treatment of Integral Equations;117
9.1;6.1 Introduction;117
9.2;6.2 Satisfaction of Integral Equations;118
9.3;6.3 Singular Integrals;119
9.4;6.4 Solving the Cauchy Principal Value Problem;119
9.5;6.5 Discretisation of the Integral Equations;121
9.5.1;6.5.1 Geometry Discretisation;121
9.5.2;6.5.2 Approximation of the Physical Values;125
9.5.3;6.5.3 Refinement Philosophy (Geometry Independent Field Approximation);126
9.5.4;6.5.4 Location of Collocation Points;126
9.5.5;6.5.5 System of Equations;127
9.6;6.6 Solution of System of Equations;132
9.6.1;6.6.1 Properties of BEM Matrices;132
9.6.2;6.6.2 Cross Approximation;132
9.6.3;6.6.3 Adaptive Cross-Approximation (ACA);133
9.6.4;6.6.4 Hierarchical Matrices;135
9.6.5;6.6.5 Iterative Solvers;138
9.7;6.7 Post-processing;138
9.8;6.8 Summary and Conclusions;140
10;7 Numerical Integration;142
10.1;7.1 Numerical Integration Methods;143
10.1.1;7.1.1 Gauss Quadrature;143
10.1.2;7.1.2 Gauss–Laguerre;147
10.2;7.2 Computation of the Shortest Distance of a Point to a Patch;148
10.3;7.3 One-Dimensional Integration;149
10.3.1;7.3.1 Regular Integration;149
10.3.2;7.3.2 Singular Integration;151
10.4;7.4 Two-Dimensional Integration;154
10.4.1;7.4.1 Regular Integration;154
10.4.2;7.4.2 Singular Integration;158
10.5;7.5 Summary and Conclusions;161
11;8 Steady State Potential Problems;163
11.1;8.1 Introduction;163
11.2;8.2 Post-processing;164
11.3;8.3 Computation of Boundary Values;166
11.3.1;8.3.1 Plane Problems;166
11.3.2;8.3.2 3-D Problems;167
11.4;8.4 Confined Flow Problems;169
11.4.1;8.4.1 Example 1: Flow Around Obstacle;170
11.4.2;8.4.2 Flow Beneath a Dam;172
11.4.3;8.4.3 Flow Through Pipe;174
11.5;8.5 Unconfined Flow Problems;176
11.5.1;8.5.1 Seepage Through Dam;176
11.6;8.6 Summary and Conclusions;180
12;9 Static Linear Solid Mechanics;181
12.1;9.1 Introduction;181
12.2;9.2 Post-processing;183
12.3;9.3 Computation of Boundary Values;186
12.3.1;9.3.1 Plane Problems;186
12.3.2;9.3.2 3-D Problems;187
12.4;9.4 Examples;189
12.4.1;9.4.1 Circular Excavation in Infinite Domain;189
12.4.2;9.4.2 Cantilever in Three Dimensions;192
12.4.3;9.4.3 Practical Example: Crankshaft;193
12.5;9.5 Summary and Conclusions;195
13;10 Simulation with Trimmed Models;196
13.1;10.1 Double Mapping Method – A Global Approach;197
13.1.1;10.1.1 Numerical Example;198
13.1.2;10.1.2 Concluding Remarks;199
13.2;10.2 Local Approaches;201
13.2.1;10.2.1 Element Detection;201
13.2.2;10.2.2 Integration Schemes;204
13.2.3;10.2.3 Assembly of Multiple Patches;210
13.2.4;10.2.4 Stabilisation;210
13.2.5;10.2.5 Numerical Results;220
13.3;10.3 Summary and Conclusions;226
14;11 Body Force Effects;228
14.1;11.1 Introduction;228
14.2;11.2 Constant Body Forces;230
14.3;11.3 General Body Forces;231
14.3.1;11.3.1 Types of Body Forces;231
14.3.2;11.3.2 Internal Cell Method;232
14.3.3;11.3.3 Isogeometric Mapping Method;232
14.4;11.4 Summary and Conclusions;239
15;12 Treatment of Inhomogeneities/Inclusions;240
15.1;12.1 Introduction;240
15.2;12.2 Multi-region Approach;240
15.3;12.3 Inclusions;243
15.3.1;12.3.1 Evaluation of the Volume Integrals;244
15.3.2;12.3.2 Plane Problems;245
15.3.3;12.3.3 3-D Problems;251
15.4;12.4 Summary and Conclusions;258
16;13 Material Non-linear Behaviour;260
16.1;13.1 Introduction;260
16.2;13.2 Computation of Initial Flow/Stress;261
16.2.1;13.2.1 Solid Mechanics;261
16.3;13.3 Evaluation of the Volume Integrals;264
16.4;13.4 Plane Problems;264
16.4.1;13.4.1 Test Example 1: Cube with Inelastic Inclusion;264
16.4.2;13.4.2 Test Example 2: Hole in an Infinite Domain;265
16.5;13.5 3-D Problems;268
16.5.1;13.5.1 Test Example: Cantilever Beam with an Inelastic Inclusion;268
16.6;13.6 Summary and Conclusions;270
17;14 Applications in Geomechanics;271
17.1;14.1 Introduction;271
17.2;14.2 Simulation of the Excavation Process;271
17.3;14.3 Simulation of Ground Support;272
17.4;14.4 Geometry Definition;272
17.4.1;14.4.1 Tunnels;273
17.4.2;14.4.2 Caverns;273
17.5;14.5 Definition of Geology;274
17.6;14.6 Test Examples;275
17.6.1;14.6.1 Example 1;275
17.6.2;14.6.2 Test Example 2;276
17.7;14.7 Practical Examples;279
17.7.1;14.7.1 NATM Tunnel;279
17.7.2;14.7.2 Practical Example 2: Cavern of Hydroelectric Plant;282
17.7.3;14.7.3 Practical Example 3: Hudson River Crossing Project;286
17.8;14.8 Summary and Conclusions;289
18;15 Viscous Flow Problems;290
18.1;15.1 Introduction;290
18.2;15.2 Solution Process;292
18.2.1;15.2.1 Modified Newton–Raphson;292
18.2.2;15.2.2 Newton–Raphson;293
18.3;15.3 Plane Problems;294
18.3.1;15.3.1 Implementation;294
18.3.2;15.3.2 Flow in Cavity;295
18.3.3;15.3.3 Airfoil;300
18.4;15.4 3-D Problems;302
18.4.1;15.4.1 Implementation;302
18.4.2;15.4.2 Numerical Results;303
18.4.3;15.4.3 Truncated Geometry;303
18.4.4;15.4.4 Non-truncated Geometry;308
18.4.5;15.4.5 Practical Example;310
18.5;15.5 Summary and Conclusions;313
19;16 Time Dependent Problems;315
19.1;16.1 Transient Potential Problems;315
19.1.1;16.1.1 Time Marching Method;315
19.1.2;16.1.2 Laplace Domain;321
19.2;16.2 Acoustics;322
19.2.1;16.2.1 Pulsating Sphere Example;323
19.3;16.3 Summary;325
20;17 Summary and Outlook;326
21;Appendix A Fundamental Solutions;328
22;A.1 2-D Potential Flow;329
23;A.1.1 Isotropic Fundamental Solution;329
24;A.1.2 Anisotropic Fundamental Solutions;329
25;A.2 3-D Potential Flow;330
26;A.2.1 Isotropic Fundamental Solution;330
27;A.2.2 Anisotropic Fundamental Solutions;330
28;A.3 Transient Potential Flow;330
29;A.4 3-D Acoustics;330
30;A.5 Elasticity;331
31;A.5.1 Coordinate Transformation;332
32;A.5.2 Transversal Isotropy;332
33;A.5.3 Constant Body Force;333
34;A.6 2-D Stokes Flow;333
35;A.7 3-D Stokes Flow;333
36;Appendix References;334
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