Liseikin Grid Generation Methods

2. Auflage 2010
ISBN: 978-90-481-2912-6
Verlag: Springer Netherlands
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 390 Seiten

Reihe: Scientific Computation

This text is an introduction to methods of grid generation technology in scientific computing. Special attention is given to methods developed by the author for the treatment of singularly-perturbed equations, e.g. in modeling high Reynolds number flows. Functionals of conformality, orthogonality, energy and alignment are discussed.

Autoren/Hrsg.

Liseikin, Vladimir D.

Inhaltsverzeichnis

1;Preface to the Second Edition;6
2;Contents;12
3;1 General Considerations;19
3.1;Introduction;19
3.2;General Concepts Related to Grids;20
3.2.1;Grid Cells;21
3.2.2;Requirements Imposed on Grids;23
3.2.2.1;Grid Size and Cell Size;23
3.2.2.2;Grid Organization;24
3.2.2.3;Cell and Grid Deformation;24
3.2.2.4;Consistency with Geometry;25
3.2.2.5;Consistency with Solution;25
3.2.2.6;Compatibility with Numerical Methods;27
3.3;Grid Classes;28
3.3.1;Structured Grids Generated by Mapping Approach;28
3.3.1.1;Realization of Grid Requirements;29
3.3.1.2;Coordinate Grids;30
3.3.1.3;Boundary-Conforming Grids;31
3.3.1.4;Shape of Computational Domains;32
3.3.2;Unstructured Grids;33
3.3.3;Block-Structured Grids;34
3.3.3.1;Communication of Adjacent Coordinate Lines;35
3.3.3.2;Topology of the Grid;35
3.3.3.3;Conditions Imposed on Grids in Blocks;37
3.3.4;Overset Grids;38
3.3.5;Hybrid Grids;39
3.4;Approaches to Grid Generation;39
3.4.1;Methods for Structured Grids;40
3.4.1.1;Algebraic Methods;40
3.4.1.2;Differential Methods;40
3.4.1.3;Variational Methods;41
3.4.2;Methods for Unstructured Grids;41
3.4.2.1;Octree Approach;41
3.4.2.2;Delaunay Approach;42
3.4.2.3;Advancing-Front Techniques;42
3.5;Big Codes;43
3.5.1;Interactive Systems;44
3.5.2;New Techniques;45
3.5.2.1;Domain Decomposition;45
3.5.2.2;New Methods;45
3.6;Comments;46
4;2 Coordinate Transformations;48
4.1;Introduction;48
4.2;General Notions and Relations;49
4.2.1;Jacobi Matrix;49
4.2.2;Tangential Vectors;50
4.2.3;Normal Vectors;52
4.2.4;Representation of Vectors Through the Base Vectors;53
4.2.5;Metric Tensors;55
4.2.5.1;Covariant Metric Tensor;55
4.2.5.2;Contravariant Metric Tensor;56
4.2.5.2.1;Geometric Interpretation;56
4.2.5.3;Relations Between Covariant and Contravariant Elements;57
4.2.6;Cross Product;58
4.2.6.1;Geometric Meaning;58
4.2.6.2;Relation to Volumes;59
4.2.6.3;Relation to Base Vectors;60
4.3;Relations Concerning Second Derivatives;61
4.3.1;Christoffel Symbols;61
4.3.2;Differentiation of the Jacobian;63
4.3.3;Basic Identity;64
4.4;Conservation Laws;66
4.4.1;Scalar Conservation Laws;66
4.4.1.1;Mass Conservation Law;67
4.4.1.2;Convection-Diffusion Equation;67
4.4.1.3;Laplace Equation;68
4.4.2;Vector Conservation Laws;68
4.4.2.1;Example;70
4.5;Time-Dependent Transformations;72
4.5.1;Reformulation of Time-Dependent Transformations;72
4.5.2;Basic Relations;73
4.5.2.1;Velocity of Grid Movement;73
4.5.2.2;Derivatives of the Jacobian;74
4.5.2.3;Basic Identity;75
4.5.3;Equations in the Form of Scalar Conservation Laws;75
4.5.3.1;Examples of Scalar Conservation-Law Equations;76
4.5.3.1.1;Parabolic Equation;76
4.5.3.1.2;Mass Conservation Law;77
4.5.3.1.3;Convection-Diffusion Equation;77
4.5.3.1.4;Energy Conservation Law;77
4.5.3.1.5;Linear Wave Equation;78
4.5.3.2;Lagrangian Coordinates;79
4.5.4;Equations in the Form of Vector Conservation Laws;79
4.6;Comments;83
5;3 Grid Quality Measures;84
5.1;Introduction;84
5.2;Curve Geometry;84
5.2.1;Basic Curve Vectors;85
5.2.1.1;Tangent Vector;85
5.2.1.2;Curves in Three-Dimensional Space;86
5.2.2;Curvature;87
5.2.3;Torsion;88
5.3;Surface Geometry;89
5.3.1;Surface Base Vectors;89
5.3.2;Metric Tensors;90
5.3.2.1;Covariant Metric Tensor;90
5.3.2.2;Contravariant Metric Tensor;91
5.3.3;Second Fundamental Form;92
5.3.4;Surface Curvatures;92
5.3.4.1;Principal Curvatures;92
5.3.4.2;Mean Curvature;93
5.3.4.3;Gaussian Curvature;94
5.4;Metric-Tensor Invariants;94
5.4.1;Algebraic Expressions for the Invariants;95
5.4.2;Geometric Interpretation;96
5.5;Characteristics of Grid Lines;97
5.5.1;Sum of Squares of Cell Edge Lengths;98
5.5.2;Eccentricity;98
5.5.3;Curvature;99
5.5.3.1;Local Straightness of the Coordinate Line;99
5.5.3.2;Expansion of the Curvature Vector in the Normal Vectors;100
5.5.3.3;Measure of Coordinate Line Curvature;101
5.5.4;Measure of Coordinate Line Torsion;102
5.6;Characteristics of Faces of Three-Dimensional Grids;102
5.6.1;Cell Face Skewness;102
5.6.2;Face Aspect-Ratio;103
5.6.3;Cell Face Area Squared;103
5.6.4;Cell Face Warping;104
5.6.4.1;Mean Curvature of the Coordinate Surface;104
5.6.4.2;Gaussian Curvature of the Coordinate Surface;105
5.6.4.3;Measures of Face Warping;105
5.7;Characteristics of Grid Cells;105
5.7.1;Cell Aspect-Ratio;106
5.7.2;Square of Cell Volume;106
5.7.3;Cell Area Squared;106
5.7.4;Cell Skewness;106
5.7.5;Characteristics of Nonorthogonality;107
5.7.6;Grid Density;108
5.7.7;Characteristics of Deviation from Conformality;109
5.7.7.1;Two-Dimensional Space;110
5.7.7.1.1;Evaluation of the Cell Angles;110
5.7.7.1.2;Evaluation of the Cell Aspect Ratio;111
5.7.7.2;Three-Dimensional Space;112
5.7.7.3;Generalization to Arbitrary Dimensions;113
5.7.8;Grid Eccentricity;113
5.7.9;Measures of Grid Warping and Grid Torsion;113
5.7.10;Quality Measures of Simplexes;114
5.8;Comments;115
6;4 Stretching Method;117
6.1;Introduction;117
6.2;Formulation of the Method;118
6.3;Theoretical Foundation;120
6.3.1;Model Problems;121
6.3.2;Basic Majorants;124
6.3.2.1;Relation Between Optimal Univariate Transformations and Majorants of the First Derivative;124
6.3.2.2;Exponential Functions;126
6.3.2.3;Power Singularities;127
6.3.2.4;Logarithmic Function;128
6.3.2.5;Relations Among Basic Majorants;128
6.3.2.6;Interior Layers;128
6.3.2.7;Estimates of the Higher Derivatives;130
6.3.2.8;Invariants of Equations;131
6.4;Basic Intermediate Transformations;132
6.4.1;Basic Local Stretching Functions;132
6.4.1.1;Width of Boundary Layers;134
6.4.2;Basic Boundary Contraction Functions;136
6.4.2.1;Basic Univariate Transformations;137
6.4.2.1.1;Continuous Mappings;138
6.4.2.1.2;Smooth Mappings;138
6.4.3;Other Univariate Transformations;141
6.4.3.1;Eriksson Function;141
6.4.3.2;Tangent Function;141
6.4.3.3;Procedure for the Construction of Local Contraction Functions;142
6.4.4;Construction of Basic Intermediate Transformations;143
6.4.4.1;Functions with Boundary Contraction;144
6.4.4.2;Functions with Interior Contraction;144
6.4.4.3;Clustering near Arbitrary Surfaces;145
6.4.4.4;Nonuniform Clustering;146
6.5;Comments;146
7;5 Algebraic Grid Generation;148
7.1;Introduction;148
7.2;Transfinite Interpolation;148
7.2.1;Unidirectional Interpolation;149
7.2.1.1;General Formulas;149
7.2.1.2;Two-Boundary Interpolation;150
7.2.2;Tensor Product;150
7.2.3;Boolean Summation;151
7.2.3.1;Bidirectional Interpolation;151
7.2.3.2;Three-Dimensional Interpolation;152
7.2.3.3;Recursive Form of Transfinite Interpolation;152
7.2.3.4;Outer Boundary Interpolation;153
7.2.3.5;Two-Dimensional Interpolation;153
7.3;Algebraic Coordinate Transformations;154
7.3.1;Formulation of Algebraic Coordinate Transformation;154
7.3.2;General Algebraic Transformations;156
7.4;Lagrange and Hermite Interpolations;158
7.4.1;Coordinate Transformations Based on Lagrange Interpolation;158
7.4.1.1;Lagrange Polynomials;159
7.4.1.2;Spline Functions;159
7.4.1.3;Construction Based on General Functions;160
7.4.1.4;Relations Between Blending Functions;161
7.4.2;Transformations Based on Hermite Interpolation;162
7.4.2.1;Construction of Blending Functions;163
7.4.2.2;Deficient Form of Hermite Interpolation;164
7.4.2.3;Specification of Normal Directions;164
7.5;Control Techniques;165
7.6;Transfinite Interpolation from Triangles and Tetrahedrons;166
7.7;Comments;168
8;6 Grid Generation Through Differential Systems;170
8.1;Introduction;170
8.2;Laplace Systems;172
8.2.1;Two-Dimensional Equations;173
8.2.2;Three-Dimensional Equations;176
8.3;Poisson Systems;179
8.3.1;Formulation of the System;180
8.3.2;Justification for the Poisson System;181
8.3.3;Equivalent Forms of the Poisson System;183
8.3.4;Orthogonality at Boundaries;185
8.3.4.1;Two-Dimensional Equations;186
8.3.4.1.1;Local Straightness at the Boundary;187
8.3.4.2;Three-Dimensional Equations;188
8.3.4.3;Projection of the Poisson System on the Boundary Curve;190
8.3.5;Control of the Angle of Intersection;192
8.4;Biharmonic Equations;196
8.4.1;Formulation of the Approach;196
8.4.2;Transformed Equations;197
8.5;Orthogonal Systems;197
8.5.1;Derivation from the Condition of Orthogonality;198
8.5.2;Multidimensional Equations;199
8.6;Hyperbolic and Parabolic Systems;200
8.6.1;Specification of Aspect Ratio;201
8.6.1.1;Initial-Value Problems;201
8.6.2;Specification of Jacobian;203
8.6.2.1;Orthogonal Grids in Two Dimensions;203
8.6.2.2;Two-Dimensional Nonorthogonal Grids;205
8.6.2.3;Three-Dimensional Version;205
8.6.3;Parabolic Equations;206
8.6.4;Hybrid Grid Generation Scheme;206
8.7;Comments;207
9;7 Dynamic Adaptation;209
9.1;Introduction;209
9.2;One-Dimensional Equidistribution;210
9.2.1;Example of an Equidistributed Grid;211
9.2.2;Original Formulation;213
9.2.3;Differential Formulation;214
9.2.4;Specification of Weight Functions;215
9.2.4.1;Optimally Distributed Grid;216
9.2.4.2;Equidistant Mesh;220
9.2.4.3;Utilization of the Second Derivative and Curvature;222
9.3;Equidistribution in Multidimensional Space;223
9.3.1;One-Directional Equidistribution;223
9.3.2;Multidirectional Equidistribution;224
9.3.2.1;Combination of One-Dimensional Equidistributions;224
9.3.2.2;Composition of Univariate Equidistributions;225
9.3.3;Control of Grid Quality;225
9.3.4;Equidistribution over Cell Volume;227
9.4;Adaptation Through Control Functions;230
9.4.1;Specification of the Control Functions in Elliptic Systems;230
9.4.1.1;Poisson System;230
9.4.1.2;Other Equations;231
9.4.2;Hyperbolic Equations;232
9.5;Grids for Nonstationary Problems;232
9.5.1;Method of Lines;233
9.5.2;Moving-Grid Techniques;233
9.5.2.1;Specification of Spatial Grid Distribution;233
9.5.2.2;Grid Movement Induced by Boundary Movement;234
9.5.2.3;Specification of Grid Speed;234
9.5.3;Time-Dependent Deformation Method;235
9.6;Comments;237
10;8 Variational Methods;241
10.1;Introduction;241
10.2;Calculus of Variations;241
10.2.1;General Formulation;242
10.2.2;Euler-Lagrange Equations;243
10.2.3;Functionals Dependent on Metric Elements;246
10.2.4;Functionals Dependent on Tensor Invariants;247
10.2.4.1;Two-Dimensional Tensor;247
10.2.4.2;Three-Dimensional Tensor;248
10.2.5;Convexity Condition;249
10.3;Integral Grid Characteristics;250
10.3.1;Dimensionless Functionals;250
10.3.1.1;Grid Skewness;250
10.3.1.2;Deviation from Orthogonality;251
10.3.1.3;Deviation from Conformality;252
10.3.2;Dimensionally Heterogeneous Functionals;254
10.3.2.1;Smoothness Functionals;254
10.3.2.2;Functionals of Orthogonality;254
10.3.3;Functionals Dependent on Second Derivatives;256
10.3.3.1;Functionals of Eccentricity;256
10.3.3.2;Functionals of Grid Warping and Grid Torsion;257
10.4;Adaptation Functionals;257
10.4.1;One-Dimensional Functionals;258
10.4.2;Multidimensional Approaches;259
10.4.2.1;Volume-Weighted Functional;260
10.4.2.2;Tangent-Length-Weigthed Functionals;261
10.4.2.3;Normal-Length-Weighted Functionals;261
10.4.2.4;Metric-Weighted Functionals;262
10.4.2.5;General Approach;263
10.4.2.6;Nonstationary Functionals;263
10.4.2.7;Weight Functions;264
10.5;Functionals of Attraction;264
10.5.1;Lagrangian Coordinates;264
10.5.2;Attraction to a Vector Field;266
10.5.3;Jacobian-Weighted Functional;267
10.6;Energy Functionals of Harmonic Function Theory;269
10.6.1;General Formulation of Harmonic Maps;269
10.6.2;Application to Grid Generation;270
10.6.3;Relation to Other Functionals;270
10.7;Combinations of Functionals;271
10.7.1;Natural Boundary Conditions;272
10.8;Comments;272
11;9 Curve and Surface Grid Methods;274
11.1;Introduction;274
11.2;Grids on Curves;275
11.2.1;Formulation of Grids on Curves;275
11.2.2;Grid Methods;277
11.2.2.1;Differential Approach;277
11.2.2.2;Variational Approach;278
11.2.2.3;Monitor Formulation;278
11.3;Formulation of Surface Grid Methods;279
11.3.1;Mapping Approach;279
11.3.2;Associated Metric Relations;281
11.4;Beltramian System;282
11.4.1;Beltramian Operator;282
11.4.2;Surface Grid System;283
11.5;Interpretations of the Beltramian System;285
11.5.1;Variational Formulation;285
11.5.2;Harmonic-Mapping Interpretation;286
11.5.3;Formulation Through Invariants;287
11.5.4;Formulation Through the Surface Christoffel Symbols;288
11.5.4.1;Surface Gauss Equations;288
11.5.4.2;Weingarten Equation;289
11.5.4.3;Mean Curvature;289
11.5.4.4;Relation Between Beltrami's Equation and Christoffel Symbols;290
11.5.5;Relation to Conformal Mappings;293
11.5.6;Projection of the Laplace System;295
11.6;Control of Surface Grids;296
11.6.1;Control Functions;296
11.6.2;Projection on the Boundary Line;297
11.6.3;Monitor Approach;298
11.6.4;Control by Variational Methods;299
11.6.4.1;Functionals Dependent on Invariants;300
11.6.4.2;Weight Skewness and Orthogonality Functionals;301
11.6.4.3;Weight Functions;301
11.6.5;Orthogonal Grid Generation;302
11.7;Hyperbolic Method;303
11.7.1;Hyperbolic Governing Equations;304
11.8;Comments;304
12;10 Comprehensive Method;306
12.1;Introduction;306
12.2;Hypersurface Geometry and Grid Formulation;308
12.2.1;Hypersurface Grid Formulation;308
12.2.2;Monitor Hypersurfaces;309
12.2.3;Metric Tensors;310
12.2.4;Christoffel Symbols;311
12.2.5;Relations Between Metric Elements;313
12.3;Functional of Smoothness;314
12.3.1;Formulation of the Functional;314
12.3.2;Geometric Interpretation;315
12.3.3;Dimensionless Functionals;317
12.3.4;Euler-Lagrange Equations;318
12.3.5;Equivalent Forms;320
12.4;Hypersurface Grid Systems;322
12.4.1;Inverted Beltrami Equations;322
12.5;Formulation of Comprehensive Grid Generator;324
12.5.1;Energy and Diffusion Functionals;324
12.5.2;Relation to Harmonic Functions;325
12.5.3;Beltrami and Diffusion Equations;326
12.5.4;Inverted Beltrami and Diffusion Equations;328
12.6;Numerical Algorithms;330
12.6.1;Finite-Difference Algorithm;331
12.6.1.1;One-Dimensional Equation;332
12.6.1.2;Two-Dimensional Equations;333
12.6.1.3;Three-Dimensional Equations;334
12.6.2;Spectral Element Algorithm;335
12.7;Formulation of Control Metrics;337
12.7.1;Specification of Individual Control Metrics;338
12.7.1.1;Control Metric for Generating Field-Aligned Grids;338
12.7.1.2;Control Metric for Generating Grids Adapting to the Values of a Function;339
12.7.1.3;Control Metrics for Generating Grids Adapting to the Gradient of a Function;341
12.7.2;Control Metrics for Generating Grids with Balanced Properties;342
12.7.3;Application to Solution of Singularly-Perturbed Equations;343
12.8;Comments;344
13;11 Unstructured Methods;346
13.1;Introduction;346
13.2;Consistent Grids and Numerical Relations;347
13.2.1;Convex Cells;347
13.2.1.1;Simplexes and Simplex Cells;348
13.2.2;Consistent Grids;348
13.2.2.1;Three-Dimensional Discretization;349
13.2.2.2;Discretization by Triangulation;350
13.3;Methods Based on the Delaunay Criterion;350
13.3.1;Dirichlet Tessellation;352
13.3.2;Incremental Techniques;352
13.3.2.1;A-Priori-Given Set of Points;353
13.3.2.2;Modernized Bowyer-Watson Technique;353
13.3.3;Approaches for Insertion of New Points;354
13.3.4;Two-Dimensional Approaches;355
13.3.4.1;Voronoi Diagram;355
13.3.4.2;Incremental Bowyer-Watson Algorithm;356
13.3.4.2.1;Properties of the Planar Delaunay Cavity;356
13.3.4.2.2;Initial Triangulation;358
13.3.4.3;Diagonal-Swapping Algorithm;358
13.3.5;Constrained Form of Delaunay Triangulation;359
13.3.5.1;Principal Component;359
13.3.5.2;Formulation of the Constrained Triangulation;360
13.3.5.3;Boundary-Conforming Triangulation;361
13.3.6;Point Insertion Strategies;361
13.3.6.1;Point Placement at the Circumcenter of the Maximum Triangle;362
13.3.6.1.1;Unconstrained Triangulation;362
13.3.6.1.2;Generalized Choice of the Insertion Triangles;364
13.3.6.2;Voronoi-Segment Point Insertion;364
13.3.6.2.1;Formulation of the Algorithm;364
13.3.6.2.2;Properties of the Triangulation;365
13.3.7;Surface Delaunay Triangulation;367
13.3.8;Three-Dimensional Delaunay Triangulation;367
13.3.8.1;Unconstrained Technique;368
13.3.8.2;Constrained Triangulation;368
13.4;Advancing-Front Methods;369
13.4.1;Procedure of Advancing-Front Method;369
13.4.2;Strategies to Select Out-of-Front Vertices;370
13.4.3;Grid Adaptation;371
13.4.4;Advancing-Front Delaunay Triangulation;371
13.4.5;Three-Dimensional Prismatic Grid Generation;372
13.5;Comments;373
14;References;376
15;Index;399