Page 7 - Basic Structured Grid Generation

vi Contents

4. Structured grid generation – algebraic methods 76
4.1 Co-ordinate transformations 76
4.2 Unidirectional interpolation 80
4.2.1 Polynomial interpolation 80
4.2.2 Hermite interpolation polynomials 85
4.2.3 Cubic splines 87
4.3 Multidirectional interpolation and TFI 92
4.3.1 Projectors and bilinear mapping in two dimensions 92
4.3.2 Numerical implementation of TFI 94
4.3.3 Three-dimensional TFI 96
4.4 Stretching transformations 98
4.5 Two-boundary and multisurface methods 103
4.5.1 Two-boundary technique 103
4.5.2 Multisurface transformation 104
4.5.3 Numerical implementation 106
4.6 Website programs 108
4.6.1 Subdirectory: Book/univariate.gds 109
4.6.2 Subdirectory: Book/Algebra 109
4.6.3 Subdirectory: Book/bilinear.gds 112
4.6.4 Subdirectory: Book/tfi.gds 114
4.6.5 Subdirectory: Book/analytic.gds 115

5. Differential models for grid generation 116
5.1 The direct and inverse problems 116
5.2 Control functions 119
5.3 Univariate stretching functions 120
5.3.1 Orthogonality considerations 121
5.4 Conformal and quasi-conformal mapping 122
5.5 Numerical techniques 125
5.5.1 The Thomas Algorithm 125
5.5.2 Jacobi, Gauss-Seidel, SOR methods 127
5.5.3 The conjugate gradient method 129
5.6 Numerical solutions of Winslow equations 131
5.6.1 Thomas Algorithm 132
5.6.2 Orthogonality 134
5.7 One-dimensional grids 136
5.7.1 Grid control 136
5.7.2 Numerical aspects 139
5.8 Three-dimensional grid generation 140
5.9 Surface-grid generation model 141
5.10 Hyperbolic grid generation 142
5.11 Solving the hosted equations 143
5.11.1 An example 143
5.11.2 More general steady-state equation 145
5.12 Multiblock grid generation 146
5.13 Website programs 148
5.13.1 Subdirectory: Book/Winslow.gds 148