Grid Generation Methods (eBook)

Preface to the Third Edition 6
Contents 13
1 General Considerations 21
1.1 Introduction 21
1.2 General Concepts Related to Grids 22
1.2.1 Grid Cells 23
1.2.2 Requirements Imposed on Grids 26
1.2.3 Grid Classes 33
1.3 Methods for Grid Generation 47
1.3.1 Mapping Methods 48
1.3.2 Methods for Unstructured Grids 56
1.4 Big Codes 57
1.4.1 Interactive Systems 58
1.4.2 New Techniques 59
1.5 Comments 60
References 62
2 Coordinate Transformations 67
2.1 Introduction 67
2.2 General Notions and Relations 68
2.2.1 Jacobi Matrix 68
2.2.2 Tangential Vectors 69
2.2.3 Normal Vectors 71
2.2.4 Representation of Vectors Through the Base Vectors 73
2.2.5 Metric Tensors 75
2.2.6 Cross Product 78
2.3 Relations Concerning Second Derivatives 81
2.3.1 Christoffel Symbols 82
2.3.2 Differentiation of the Jacobian 83
2.3.3 Basic Identity 84
2.4 Conservation Laws 87
2.4.1 Scalar Conservation Laws 87
2.4.2 Vector Conservation Laws 89
2.5 Time-Dependent Transformations 93
2.5.1 Reformulation of Time-Dependent Transformations 94
2.5.2 Basic Relations 95
2.5.3 Equations in the Form of Scalar Conservation Laws 97
2.5.4 Equations in the Form of Vector Conservation Laws 101
2.6 Comments 105
References 105
3 Grid Quality Measures 106
3.1 Introduction 106
3.2 Curve Geometry 107
3.2.1 Basic Curve Vectors 107
3.2.2 Curvature 109
3.2.3 Torsion 110
3.3 Surface Geometry 111
3.3.1 Surface Base Vectors 112
3.3.2 Metric Tensors 113
3.3.3 Second Fundamental Form 115
3.3.4 Surface Curvatures 116
3.3.5 Curvatures of Discrete Surfaces 118
3.4 Metric-Tensor Invariants 120
3.4.1 Algebraic Expressions for the Invariants 120
3.4.2 Geometric Interpretation 121
3.5 Characteristics of Grid Lines 123
3.5.1 Sum of Squares of Cell Edge Lengths 123
3.5.2 Eccentricity 124
3.5.3 Curvature 124
3.5.4 Measure of Coordinate Line Torsion 128
3.6 Characteristics of Faces of Three-Dimensional Cells 128
3.6.1 Cell Face Skewness 128
3.6.2 Face Aspect-Ratio 129
3.6.3 Cell Face Area Squared 130
3.6.4 Cell Face Warping 130
3.7 Characteristics of Grid Cells 132
3.7.1 Cell Aspect-Ratio 132
3.7.2 Square of Cell Volume 132
3.7.3 Cell Area Squared 133
3.7.4 Cell Skewness 133
3.7.5 Characteristics of Nonorthogonality 134
3.7.6 Grid Density 135
3.7.7 Characteristics of Deviation from Conformality 136
3.7.8 Grid Eccentricity 140
3.7.9 Measures of Grid Warping and Grid Torsion 141
3.7.10 Quality Measures of Simplexes 141
3.8 Comments 142
References 143
4 Stretching Method 145
4.1 Introduction 145
4.2 Formulation of the Method 147
4.3 Theoretical Foundation 148
4.3.1 Model Problems 150
4.3.2 Basic Majorants 153
4.4 Basic Intermediate Transformations 168
4.4.1 Basic Local Stretching Functions 168
4.4.2 Basic Boundary Contraction Functions 172
4.4.3 Other Univariate Transformations 178
4.4.4 Construction of Basic Intermediate Transformations 180
4.4.5 Multidirectional Equidistribution 183
4.5 Comments 185
References 188
5 Algebraic Grid Generation 192
5.1 Introduction 192
5.2 Transfinite Interpolation 192
5.2.1 Unidirectional Interpolation 193
5.2.2 Tensor Product 194
5.2.3 Boolean Summation 195
5.3 Algebraic Coordinate Transformations 198
5.3.1 Formulation of Algebraic Coordinate Transformation 198
5.3.2 General Algebraic Transformations 200
5.4 Lagrange and Hermite Interpolations 202
5.4.1 Coordinate Transformations Based on Lagrange Interpolation 203
5.4.2 Transformations Based on Hermite Interpolation 208
5.5 Control Techniques 211
5.6 Transfinite Interpolation from Triangles and Tetrahedrons 213
5.7 Drag and Sweeping Methods 216
5.8 Comments 216
References 217
6 Grid Generation Through Differential Systems 218
6.1 Introduction 218
6.2 Elliptic Equations 218
6.2.1 Laplace Systems 220
6.2.2 Poisson Systems 229
6.2.3 Other Elliptic Equations 248
6.3 Biharmonic Equations 248
6.3.1 Formulation of the Approach 249
6.3.2 Transformed Equations 249
6.4 Orthogonal Systems 250
6.4.1 Derivation from the Condition of Orthogonality 250
6.4.2 Multidimensional Equations 251
6.5 Hyperbolic and Parabolic Systems 252
6.5.1 Specification of Aspect Ratio 253
6.5.2 Specification of Jacobian 256
6.5.3 Parabolic Equations 259
6.5.4 Hybrid Grid Generation Scheme 259
6.6 Grid Equations for Nonstationary Problems 260
6.6.1 Method of Lines 261
6.6.2 Moving-Grid Techniques 261
6.6.3 Time-Dependent Deformation Method 263
6.7 Comments 264
References 267
7 Variational Methods 271
7.1 Introduction 271
7.2 Calculus of Variations 271
7.2.1 General Formulation 272
7.2.2 Euler--Lagrange Equations 273
7.2.3 Convexity Condition 276
7.2.4 Functionals Dependent on Metric Elements 277
7.2.5 Functionals Dependent on Tensor Invariants 278
7.3 Integral Grid Characteristics 281
7.3.1 Dimensionless Functionals 281
7.3.2 Dimensionally Heterogeneous Functionals 285
7.3.3 Functionals Dependent on Second Derivatives 287
7.4 Adaptation Functionals 288
7.4.1 One-Dimensional Functionals 289
7.4.2 Multidimensional Approaches 291
7.5 Functionals of Attraction 295
7.5.1 Lagrangian Coordinates 296
7.5.2 Attraction to a Vector Field 297
7.5.3 Jacobian-Weighted Functional 298
7.6 Energy Functionals of Harmonic Function Theory 300
7.6.1 General Formulation of Harmonic Maps 300
7.6.2 Application to Grid Generation 301
7.6.3 Relation to Other Functionals 302
7.7 Combinations of Functionals 303
7.7.1 Natural Boundary Conditions 304
7.8 Comments 304
References 305
8 Curve and Surface Grid Methods 308
8.1 Introduction 308
8.2 Grids on Curves 309
8.2.1 Formulation of Grids on Curves 309
8.2.2 Grid Methods 311
8.3 Formulation of Surface Grid Methods 313
8.3.1 Mapping Approach 314
8.3.2 Associated Metric Relations 315
8.4 Beltramian System 317
8.4.1 Beltramian Operator 317
8.4.2 Surface Grid System 318
8.5 Interpretations of the Beltramian System 320
8.5.1 Variational Formulation 320
8.5.2 Harmonic-Mapping Interpretation 321
8.5.3 Formulation Through Invariants 322
8.5.4 Formulation Through the Surface Christoffel Symbols 323
8.6 Control of Surface Grids 328
8.6.1 Control Functions 328
8.6.2 Monitor Approach 329
8.6.3 Control Through Variational Methods 330
8.6.4 Orthogonal Grid Generation 333
8.7 Hyperbolic Method 334
8.7.1 Hyperbolic Governing Equations 335
8.8 Comments 335
References 337
9 Comprehensive Method 339
9.1 Introduction 339
9.2 Hypersurface Geometry and Grid Formulation 341
9.2.1 Hypersurface Grid Formulation 341
9.2.2 Monitor Hypersurfaces 342
9.2.3 Metric Tensors 344
9.2.4 Relations Between Metric Elements 345
9.2.5 Christoffel Symbols 347
9.3 Functional of Smoothness 348
9.3.1 Formulation of the Functional 348
9.3.2 Geometric Interpretation 350
9.3.3 Euler--Lagrange Equations 353
9.3.4 Equivalent Forms 355
9.3.5 Inverted Beltrami Equations 358
9.4 Role of the Mean Curvature 360
9.4.1 Mean Curvature and Inverted Beltrami Grid Equations 360
9.4.2 Mean Curvature and Rate of Grid Clustering 363
9.4.3 Diffusion Functional 372
9.4.4 Dimensionless Functionals 374
9.5 Formulation of Comprehensive Grid Generator 376
9.5.1 Formulation of Control Metrics 377
9.5.2 Energy and Diffusion Functionals 379
9.5.3 Beltrami and Diffusion Equations 380
9.5.4 Inverted Beltrami and Diffusion Equations 383
9.5.5 Specification of Individual Control Metrics 386
9.5.6 Control Metrics for Generating Grids with Balanced Properties 393
9.6 Comments 395
References 396
10 Numerical Implementations of Comprehensive Grid Generators 398
10.1 One-Dimensional Equation 398
10.1.1 Numerical Algorithm 399
10.2 Multidimensional Finite Difference Algorithms 401
10.2.1 Parabolic Simulation 401
10.2.2 Two-Dimensional Equations 404
10.2.3 Three--Dimensional Problem 409
10.3 Spectral Element Algorithm 412
10.4 Finite Element Method 415
10.5 Inverse Matrix Method 417
10.6 Method of Minimization of Energy Functional 418
10.6.1 Generation of Fixed Grids 419
10.6.2 Adaptive Grid Generation 425
10.7 Parallel Mesh Generation 430
References 431
11 Control of Grid Properties 433
11.1 Grid Adaptation to Function Values 433
11.1.1 Control Operator 433
11.1.2 Grid Equations 436
11.2 Grid Generation with Node Clustering Near Isolated Points 437
11.3 Grids with Node Clustering Near Curves and Surfaces 441
11.4 Generation of Grids with Node Clustering in the Zones ƒ 445
11.4.1 Control Metric of a Monitor Surface 445
11.4.2 Spherical Control Metric 447
11.5 Application of Layer-Type Functions to Grid Codes 447
11.5.1 Specification of Basic Functions 448
11.5.2 Numerical Grids Aligned to Vector-Fields 449
11.5.3 Application to Grid Clustering 454
11.6 Generation of Multi-block Smooth Grids 456
11.6.1 Approaches to Smoothing Grids 456
11.6.2 Computation by Interpolation 458
References 460
12 Unstructured Methods 461
12.1 Introduction 461
12.2 Methods Based on the Delaunay Criterion 462
12.2.1 Dirichlet Tessellation 463
12.2.2 Incremental Techniques 465
12.2.3 Approaches for Insertion of New Points 466
12.2.4 Two-Dimensional Approaches 467
12.2.5 Constrained Form of Delaunay Triangulation 471
12.2.6 Point Insertion Strategies 473
12.2.7 Surface Delaunay Triangulation 479
12.2.8 Three-Dimensional Delaunay Triangulation 479
12.3 Advancing-Front Methods 481
12.3.1 Procedure of Advancing-Front Method 481
12.3.2 Strategies for Selecting Out-of-Front Vertices 482
12.3.3 Grid Adaptation 483
12.3.4 Advancing-Front Delaunay Triangulation 483
12.4 Meshing by Quadtree-Octree Decomposition 484
12.5 Three-Dimensional Prismatic Grid Generation 484
12.6 Comments 485
References 488
13 Applications of Adaptive Grids to Solution of Problems 492
13.1 Application to Unsteady Gas Dynamics Problems 492
13.1.1 Numerical Examples 498
13.2 Applications to Numerical Simulations of Tsunami Run-Up 499
13.2.1 Mathematical Model 499
13.2.2 Dynamically Adaptive Numerical Grid 500
13.2.3 Equations in Dynamic Curvilinear Coordinates 501
13.2.4 Numerical Algorithm 502
13.2.5 Some Results of Calculations 504
13.3 Application to Singularly-Perturbed Equations 506
13.3.1 Numerical Algorithm 506
13.4 Problem of Heat Transfer in Plasmas 508
13.4.1 The Tokamak Edge Region 510
13.4.2 Computations on Balanced Grids 511
13.5 Evaluations of Temperature-Profile Discrepancies 512
13.5.1 Mathematical Model for the Interaction of Heat Wave with Thermocouple 513
13.5.2 Generation of Adaptive Grid 515
13.5.3 Results of Numerical Experiments 517
13.6 Numerical Modeling of Nanopore Formation in Aluminium Oxide Films 521
13.6.1 Introduction 521
13.6.2 Mathematical Model 522
13.6.3 Numerical Approximation 526
13.6.4 Grid Generation 527
13.6.5 Numerical Experiments 528
13.7 Grids for Boundary Immersing Methods 529
13.7.1 Introduction 529
13.7.2 Formulation of the Method 530
13.7.3 Determination of Boundary Cells 533
13.7.4 Algorithm for Determining Interior Cells 534
13.7.5 Mesh Adaptation 535
References 536
Index 538