Adaptive Hierarchical Isogeometric Finite Element Methods

Preface......Page 4
Acknowledgment......Page 5
Contents......Page 7
List of Figures......Page 10
List of Tables......Page 14
List of Algorithms......Page 15
Chapter 1 Introduction......Page 16
 State of the Art......Page 18
 Notational Conventions......Page 19
Chapter 2 Prerequisites from Applied Geometry and Spline Theory......Page 21
 2.1 Parametric Curves, Surfaces and Volumes......Page 22
  2.2.1 Polynomial Spaces......Page 23
  2.2.3 B-splines......Page 24
  2.2.4 Non-Uniform Rational B-splines......Page 27
 2.3 Multivariate Tensor-product Splines......Page 29
  2.4.1 Evaluations......Page 30
  2.4.2 Refinement......Page 31
Chapter 3 Mathematical Modelling and Finite Element Analysis......Page 33
  3.1.2 Continuum Mechanics......Page 34
  3.2.1 Abstract Setting......Page 38
  3.2.2 Application to Models......Page 39
  3.3.1 Galerkin Projection......Page 41
  3.3.2 Finite Element Function Spaces......Page 42
  3.3.3 Reference Elements......Page 44
  3.3.4 Convergence Analysis......Page 45
  3.4.1 Some Finite Element Spaces......Page 46
  3.4.2 Isoparametric Finite Elements......Page 48
  3.4.3 Mesh Refinement......Page 50
  3.4.4 Adaptive Finite Elements and Error Estimation......Page 51
 3.5 Implementation Issues......Page 53
  3.5.1 Data Representation......Page 54
  3.5.2 Assembly of System Matrices......Page 55
  3.5.4 Postprocessing......Page 58
Chapter 4 Isogeometric Analysis......Page 60
 4.1 Basic Idea and Fundamentals......Page 61
  4.2.1 B-splines and NURBS as Basis Functions......Page 63
  4.2.2 Isogeometric Mesh and Relationship to Finite Element Methods......Page 64
  4.2.3 Geometry Mapping – the Relation to CAGD......Page 67
  4.2.4 Shared Concepts between the Components......Page 69
 4.3 Convergence Analysis......Page 70
  4.4.1 Isogeometric Reference Element......Page 72
  4.4.2 B´ezier Extraction......Page 74
 4.5 Uniform Refinement......Page 75
 4.6 Implementation Issues......Page 76
  4.6.2 Element Structure and Enumeration......Page 77
  4.6.3 Matrix Assembly......Page 79
  4.6.4 Boundary Conditions......Page 81
  4.6.5 Visualization......Page 82
  4.7.1 Heat Conduction on a Half-Disk......Page 83
  4.7.2 Heat Conduction on an L-shape......Page 84
  4.7.3 Plate with a Hole......Page 85
  4.7.4 Turbine Blade......Page 87
  4.7.5 Wheel Disk......Page 88
 4.8 Conclusion and Applications......Page 90
Chapter 5 Local Refinement for Isogeometric Analysis......Page 92
  5.1.1 Aspects of Refinement in Isogeometric Analysis......Page 93
  5.1.2 Isogeometric Analysis Refinement Techniques......Page 94
  5.2.1 Basic Definitions......Page 97
  5.2.2 Properties of Hierarchical B-Splines......Page 100
  5.3.1 Hierarchical Structures......Page 101
  5.3.2 Hierarchical Refinement......Page 106
  5.3.3 Hierarchical Reference Element......Page 110
  5.3.4 Refinement Criterium and A Posteriori Error Estimation......Page 112
 5.4 Implementation Issues......Page 113
  5.4.2 Hierarchical Refinement......Page 114
  5.4.3 Matrix Assembly......Page 116
  5.4.4 Treatment of Boundary Conditions......Page 117
  5.5.1 Advection-Diffusion......Page 118
  5.5.2 Heat Conduction of a Half-Disk......Page 120
  5.5.3 Plate with Hole......Page 121
  5.5.4 Circular Plate......Page 122
  5.5.5 Heat Conduction on an L-shape......Page 123
Chapter 6 Conclusions......Page 128
Appendix A Software......Page 129
Appendix B Geometric Data......Page 130
Bibliography......Page 132