Liseikin Grid Generation Methods

3rd Auflage 2017
ISBN: 978-3-319-57846-0
Verlag: Springer Nature Switzerland
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 541 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 Third Edition;6
2;Contents;13
3;1 General Considerations;21
3.1;1.1 Introduction;21
3.2;1.2 General Concepts Related to Grids;22
3.2.1;1.2.1 Grid Cells;23
3.2.2;1.2.2 Requirements Imposed on Grids;26
3.2.3;1.2.3 Grid Classes;33
3.3;1.3 Methods for Grid Generation;47
3.3.1;1.3.1 Mapping Methods;48
3.3.2;1.3.2 Methods for Unstructured Grids;56
3.4;1.4 Big Codes;57
3.4.1;1.4.1 Interactive Systems;58
3.4.2;1.4.2 New Techniques;59
3.5;1.5 Comments;60
3.6;References;62
4;2 Coordinate Transformations;67
4.1;2.1 Introduction;67
4.2;2.2 General Notions and Relations;68
4.2.1;2.2.1 Jacobi Matrix;68
4.2.2;2.2.2 Tangential Vectors;69
4.2.3;2.2.3 Normal Vectors;71
4.2.4;2.2.4 Representation of Vectors Through the Base Vectors;73
4.2.5;2.2.5 Metric Tensors;75
4.2.6;2.2.6 Cross Product;78
4.3;2.3 Relations Concerning Second Derivatives;81
4.3.1;2.3.1 Christoffel Symbols;82
4.3.2;2.3.2 Differentiation of the Jacobian;83
4.3.3;2.3.3 Basic Identity;84
4.4;2.4 Conservation Laws;87
4.4.1;2.4.1 Scalar Conservation Laws;87
4.4.2;2.4.2 Vector Conservation Laws;89
4.5;2.5 Time-Dependent Transformations;93
4.5.1;2.5.1 Reformulation of Time-Dependent Transformations;94
4.5.2;2.5.2 Basic Relations;95
4.5.3;2.5.3 Equations in the Form of Scalar Conservation Laws;97
4.5.4;2.5.4 Equations in the Form of Vector Conservation Laws;101
4.6;2.6 Comments;105
4.7;References;105
5;3 Grid Quality Measures;106
5.1;3.1 Introduction;106
5.2;3.2 Curve Geometry;107
5.2.1;3.2.1 Basic Curve Vectors;107
5.2.2;3.2.2 Curvature;109
5.2.3;3.2.3 Torsion;110
5.3;3.3 Surface Geometry;111
5.3.1;3.3.1 Surface Base Vectors;112
5.3.2;3.3.2 Metric Tensors;113
5.3.3;3.3.3 Second Fundamental Form;115
5.3.4;3.3.4 Surface Curvatures;116
5.3.5;3.3.5 Curvatures of Discrete Surfaces;118
5.4;3.4 Metric-Tensor Invariants;120
5.4.1;3.4.1 Algebraic Expressions for the Invariants;120
5.4.2;3.4.2 Geometric Interpretation;121
5.5;3.5 Characteristics of Grid Lines;123
5.5.1;3.5.1 Sum of Squares of Cell Edge Lengths;123
5.5.2;3.5.2 Eccentricity;124
5.5.3;3.5.3 Curvature;124
5.5.4;3.5.4 Measure of Coordinate Line Torsion;128
5.6;3.6 Characteristics of Faces of Three-Dimensional Cells;128
5.6.1;3.6.1 Cell Face Skewness;128
5.6.2;3.6.2 Face Aspect-Ratio;129
5.6.3;3.6.3 Cell Face Area Squared;130
5.6.4;3.6.4 Cell Face Warping;130
5.7;3.7 Characteristics of Grid Cells;132
5.7.1;3.7.1 Cell Aspect-Ratio;132
5.7.2;3.7.2 Square of Cell Volume;132
5.7.3;3.7.3 Cell Area Squared;133
5.7.4;3.7.4 Cell Skewness;133
5.7.5;3.7.5 Characteristics of Nonorthogonality;134
5.7.6;3.7.6 Grid Density;135
5.7.7;3.7.7 Characteristics of Deviation from Conformality;136
5.7.8;3.7.8 Grid Eccentricity;140
5.7.9;3.7.9 Measures of Grid Warping and Grid Torsion;141
5.7.10;3.7.10 Quality Measures of Simplexes;141
5.8;3.8 Comments;142
5.9;References;143
6;4 Stretching Method;145
6.1;4.1 Introduction;145
6.2;4.2 Formulation of the Method;147
6.3;4.3 Theoretical Foundation;148
6.3.1;4.3.1 Model Problems;150
6.3.2;4.3.2 Basic Majorants;153
6.4;4.4 Basic Intermediate Transformations;168
6.4.1;4.4.1 Basic Local Stretching Functions;168
6.4.2;4.4.2 Basic Boundary Contraction Functions;172
6.4.3;4.4.3 Other Univariate Transformations;178
6.4.4;4.4.4 Construction of Basic Intermediate Transformations;180
6.4.5;4.4.5 Multidirectional Equidistribution;183
6.5;4.5 Comments;185
6.6;References;188
7;5 Algebraic Grid Generation;192
7.1;5.1 Introduction;192
7.2;5.2 Transfinite Interpolation;192
7.2.1;5.2.1 Unidirectional Interpolation;193
7.2.2;5.2.2 Tensor Product;194
7.2.3;5.2.3 Boolean Summation;195
7.3;5.3 Algebraic Coordinate Transformations;198
7.3.1;5.3.1 Formulation of Algebraic Coordinate Transformation;198
7.3.2;5.3.2 General Algebraic Transformations;200
7.4;5.4 Lagrange and Hermite Interpolations;202
7.4.1;5.4.1 Coordinate Transformations Based on Lagrange Interpolation;203
7.4.2;5.4.2 Transformations Based on Hermite Interpolation;208
7.5;5.5 Control Techniques;211
7.6;5.6 Transfinite Interpolation from Triangles and Tetrahedrons;213
7.7;5.7 Drag and Sweeping Methods;216
7.8;5.8 Comments;216
7.9;References;217
8;6 Grid Generation Through Differential Systems;218
8.1;6.1 Introduction;218
8.2;6.2 Elliptic Equations;218
8.2.1;6.2.1 Laplace Systems;220
8.2.2;6.2.2 Poisson Systems;229
8.2.3;6.2.3 Other Elliptic Equations;248
8.3;6.3 Biharmonic Equations;248
8.3.1;6.3.1 Formulation of the Approach;249
8.3.2;6.3.2 Transformed Equations;249
8.4;6.4 Orthogonal Systems;250
8.4.1;6.4.1 Derivation from the Condition of Orthogonality;250
8.4.2;6.4.2 Multidimensional Equations;251
8.5;6.5 Hyperbolic and Parabolic Systems;252
8.5.1;6.5.1 Specification of Aspect Ratio;253
8.5.2;6.5.2 Specification of Jacobian;256
8.5.3;6.5.3 Parabolic Equations;259
8.5.4;6.5.4 Hybrid Grid Generation Scheme;259
8.6;6.6 Grid Equations for Nonstationary Problems;260
8.6.1;6.6.1 Method of Lines;261
8.6.2;6.6.2 Moving-Grid Techniques;261
8.6.3;6.6.3 Time-Dependent Deformation Method;263
8.7;6.7 Comments;264
8.8;References;267
9;7 Variational Methods;271
9.1;7.1 Introduction;271
9.2;7.2 Calculus of Variations;271
9.2.1;7.2.1 General Formulation;272
9.2.2;7.2.2 Euler--Lagrange Equations;273
9.2.3;7.2.3 Convexity Condition;276
9.2.4;7.2.4 Functionals Dependent on Metric Elements;277
9.2.5;7.2.5 Functionals Dependent on Tensor Invariants;278
9.3;7.3 Integral Grid Characteristics;281
9.3.1;7.3.1 Dimensionless Functionals;281
9.3.2;7.3.2 Dimensionally Heterogeneous Functionals;285
9.3.3;7.3.3 Functionals Dependent on Second Derivatives;287
9.4;7.4 Adaptation Functionals;288
9.4.1;7.4.1 One-Dimensional Functionals;289
9.4.2;7.4.2 Multidimensional Approaches;291
9.5;7.5 Functionals of Attraction;295
9.5.1;7.5.1 Lagrangian Coordinates;296
9.5.2;7.5.2 Attraction to a Vector Field;297
9.5.3;7.5.3 Jacobian-Weighted Functional;298
9.6;7.6 Energy Functionals of Harmonic Function Theory;300
9.6.1;7.6.1 General Formulation of Harmonic Maps;300
9.6.2;7.6.2 Application to Grid Generation;301
9.6.3;7.6.3 Relation to Other Functionals;302
9.7;7.7 Combinations of Functionals;303
9.7.1;7.7.1 Natural Boundary Conditions;304
9.8;7.8 Comments;304
9.9;References;305
10;8 Curve and Surface Grid Methods;308
10.1;8.1 Introduction;308
10.2;8.2 Grids on Curves;309
10.2.1;8.2.1 Formulation of Grids on Curves;309
10.2.2;8.2.2 Grid Methods;311
10.3;8.3 Formulation of Surface Grid Methods;313
10.3.1;8.3.1 Mapping Approach;314
10.3.2;8.3.2 Associated Metric Relations;315
10.4;8.4 Beltramian System;317
10.4.1;8.4.1 Beltramian Operator;317
10.4.2;8.4.2 Surface Grid System;318
10.5;8.5 Interpretations of the Beltramian System;320
10.5.1;8.5.1 Variational Formulation;320
10.5.2;8.5.2 Harmonic-Mapping Interpretation;321
10.5.3;8.5.3 Formulation Through Invariants;322
10.5.4;8.5.4 Formulation Through the Surface Christoffel Symbols;323
10.6;8.6 Control of Surface Grids;328
10.6.1;8.6.1 Control Functions;328
10.6.2;8.6.2 Monitor Approach;329
10.6.3;8.6.3 Control Through Variational Methods;330
10.6.4;8.6.4 Orthogonal Grid Generation;333
10.7;8.7 Hyperbolic Method;334
10.7.1;8.7.1 Hyperbolic Governing Equations;335
10.8;8.8 Comments;335
10.9;References;337
11;9 Comprehensive Method;339
11.1;9.1 Introduction;339
11.2;9.2 Hypersurface Geometry and Grid Formulation;341
11.2.1;9.2.1 Hypersurface Grid Formulation;341
11.2.2;9.2.2 Monitor Hypersurfaces;342
11.2.3;9.2.3 Metric Tensors;344
11.2.4;9.2.4 Relations Between Metric Elements;345
11.2.5;9.2.5 Christoffel Symbols;347
11.3;9.3 Functional of Smoothness;348
11.3.1;9.3.1 Formulation of the Functional;348
11.3.2;9.3.2 Geometric Interpretation;350
11.3.3;9.3.3 Euler--Lagrange Equations;353
11.3.4;9.3.4 Equivalent Forms;355
11.3.5;9.3.5 Inverted Beltrami Equations;358
11.4;9.4 Role of the Mean Curvature;360
11.4.1;9.4.1 Mean Curvature and Inverted Beltrami Grid Equations;360
11.4.2;9.4.2 Mean Curvature and Rate of Grid Clustering;363
11.4.3;9.4.3 Diffusion Functional;372
11.4.4;9.4.4 Dimensionless Functionals;374
11.5;9.5 Formulation of Comprehensive Grid Generator;376
11.5.1;9.5.1 Formulation of Control Metrics;377
11.5.2;9.5.2 Energy and Diffusion Functionals;379
11.5.3;9.5.3 Beltrami and Diffusion Equations;380
11.5.4;9.5.4 Inverted Beltrami and Diffusion Equations;383
11.5.5;9.5.5 Specification of Individual Control Metrics;386
11.5.6;9.5.6 Control Metrics for Generating Grids with Balanced Properties;393
11.6;9.6 Comments;395
11.7;References;396
12;10 Numerical Implementations of Comprehensive Grid Generators;398
12.1;10.1 One-Dimensional Equation;398
12.1.1;10.1.1 Numerical Algorithm;399
12.2;10.2 Multidimensional Finite Difference Algorithms;401
12.2.1;10.2.1 Parabolic Simulation;401
12.2.2;10.2.2 Two-Dimensional Equations;404
12.2.3;10.2.3 Three--Dimensional Problem;409
12.3;10.3 Spectral Element Algorithm;412
12.4;10.4 Finite Element Method;415
12.5;10.5 Inverse Matrix Method;417
12.6;10.6 Method of Minimization of Energy Functional;418
12.6.1;10.6.1 Generation of Fixed Grids;419
12.6.2;10.6.2 Adaptive Grid Generation;425
12.7;10.7 Parallel Mesh Generation;430
12.8;References;431
13;11 Control of Grid Properties;433
13.1;11.1 Grid Adaptation to Function Values;433
13.1.1;11.1.1 Control Operator;433
13.1.2;11.1.2 Grid Equations;436
13.2;11.2 Grid Generation with Node Clustering Near Isolated Points;437
13.3;11.3 Grids with Node Clustering Near Curves and Surfaces;441
13.4;11.4 Generation of Grids with Node Clustering in the Zones ƒ;445
13.4.1;11.4.1 Control Metric of a Monitor Surface;445
13.4.2;11.4.2 Spherical Control Metric;447
13.5;11.5 Application of Layer-Type Functions to Grid Codes;447
13.5.1;11.5.1 Specification of Basic Functions;448
13.5.2;11.5.2 Numerical Grids Aligned to Vector-Fields;449
13.5.3;11.5.3 Application to Grid Clustering;454
13.6;11.6 Generation of Multi-block Smooth Grids;456
13.6.1;11.6.1 Approaches to Smoothing Grids;456
13.6.2;11.6.2 Computation by Interpolation;458
13.7;References;460
14;12 Unstructured Methods;461
14.1;12.1 Introduction;461
14.2;12.2 Methods Based on the Delaunay Criterion;462
14.2.1;12.2.1 Dirichlet Tessellation;463
14.2.2;12.2.2 Incremental Techniques;465
14.2.3;12.2.3 Approaches for Insertion of New Points;466
14.2.4;12.2.4 Two-Dimensional Approaches;467
14.2.5;12.2.5 Constrained Form of Delaunay Triangulation;471
14.2.6;12.2.6 Point Insertion Strategies;473
14.2.7;12.2.7 Surface Delaunay Triangulation;479
14.2.8;12.2.8 Three-Dimensional Delaunay Triangulation;479
14.3;12.3 Advancing-Front Methods;481
14.3.1;12.3.1 Procedure of Advancing-Front Method;481
14.3.2;12.3.2 Strategies for Selecting Out-of-Front Vertices;482
14.3.3;12.3.3 Grid Adaptation;483
14.3.4;12.3.4 Advancing-Front Delaunay Triangulation;483
14.4;12.4 Meshing by Quadtree-Octree Decomposition;484
14.5;12.5 Three-Dimensional Prismatic Grid Generation;484
14.6;12.6 Comments;485
14.7;References;488
15;13 Applications of Adaptive Grids to Solution of Problems;492
15.1;13.1 Application to Unsteady Gas Dynamics Problems;492
15.1.1;13.1.1 Numerical Examples;498
15.2;13.2 Applications to Numerical Simulations of Tsunami Run-Up;499
15.2.1;13.2.1 Mathematical Model;499
15.2.2;13.2.2 Dynamically Adaptive Numerical Grid;500
15.2.3;13.2.3 Equations in Dynamic Curvilinear Coordinates;501
15.2.4;13.2.4 Numerical Algorithm;502
15.2.5;13.2.5 Some Results of Calculations;504
15.3;13.3 Application to Singularly-Perturbed Equations;506
15.3.1;13.3.1 Numerical Algorithm;506
15.4;13.4 Problem of Heat Transfer in Plasmas;508
15.4.1;13.4.1 The Tokamak Edge Region;510
15.4.2;13.4.2 Computations on Balanced Grids;511
15.5;13.5 Evaluations of Temperature-Profile Discrepancies;512
15.5.1;13.5.1 Mathematical Model for the Interaction of Heat Wave with Thermocouple;513
15.5.2;13.5.2 Generation of Adaptive Grid;515
15.5.3;13.5.3 Results of Numerical Experiments;517
15.6;13.6 Numerical Modeling of Nanopore Formation in Aluminium Oxide Films;521
15.6.1;13.6.1 Introduction;521
15.6.2;13.6.2 Mathematical Model;522
15.6.3;13.6.3 Numerical Approximation;526
15.6.4;13.6.4 Grid Generation;527
15.6.5;13.6.5 Numerical Experiments;528
15.7;13.7 Grids for Boundary Immersing Methods;529
15.7.1;13.7.1 Introduction;529
15.7.2;13.7.2 Formulation of the Method;530
15.7.3;13.7.3 Determination of Boundary Cells;533
15.7.4;13.7.4 Algorithm for Determining Interior Cells;534
15.7.5;13.7.5 Mesh Adaptation;535
15.8;References;536
16;Index;538