Grid generation methods / by Vladimir D. Liseĭkin.

Format:

Book

Author/Creator:

Liseĭkin, V. D.

Series:

Scientific computation

Language:

English

Subjects (All):

Numerical grid generation (Numerical analysis).

Physical Description:

x, 390 pages: illustrations ; 24 cm.

Edition:

Second edition.

Place of Publication:

Dordrecht ; London : Springer, 2009.

Summary:

This book is an introduction to structured and unstructured grid methods in scientific computing, addressing graduate students, scientists as well as practitioners. Basic local and integral grid quality measures are formulated and new approaches to mesh generation are reviewed. In addition to the content of the successful first edition, a more detailed and practice oriented description of monitor metrics in Beltrami and diffusion equations is given for generating adaptive numerical grids. Also, new techniques developed by the author are presented, in particular a technique based on the inverted form of Beltrami's partial differential equations with respect to control metrics. This technique allows the generation of adaptive grids for a wide variety of applied problems, including grid clustering to given function values and gradients, grid alignment with given vector fields, and combinations thereof. Applications of geometric methods to the analysis of numerical grid behavior as well as grid generation based on the minimization of functionals of smoothness, conformality, orthogonality, energy, and alignment complete the second edition of this outstanding compendium on grid generation methods.

Contents:

1 General Considerations 1

1.1 Introduction 1

1.2 General Concepts Related to Grids 2

1.2.1 Grid Cells 3

1.2.2 Requirements Imposed on Grids 5

1.3 Grid Classes 10

1.3.1 Structured Grids Generated by Mapping Approach 10

1.3.2 Unstructured Grids 15

1.3.3 Block-Structured Grids 16

1.3.4 Overset Grids 20

1.3.5 Hybrid Grids 21

1.4 Approaches to Grid Generation 21

1.4.1 Methods for Structured Grids 22

1.4.2 Methods for Unstructured Grids 23

1.5 Big Codes 25

1.5.1 Interactive Systems 26

1.5.2 New Techniques 27

1.6 Comments 28

2 Coordinate Transformations 31

2.1 Introduction 31

2.2 General Notions and Relations 32

2.2.1 Jacobi Matrix 32

2.2.2 Tangential Vectors 33

2.2.3 Normal Vectors 35

2.2.4 Representation of Vectors Through the Base Vectors 36

2.2.5 Metric Tensors 38

2.2.6 Cross Product 41

2.3 Relations Concerning Second Derivatives 44

2.3.1 Christoffel Symbols 44

2.3.2 Differentiation of the Jacobian 46

2.3.3 Basic Identity 47

2.4 Conservation Laws 49

2.4.1 Scalar Conservation Laws 49

2.4.2 Vector Conservation Laws 51

2.5 Time-Dependent Transformations 55

2.5.1 Reformulation of Time-Dependent Transformations 55

2.5.2 Basic Relations 56

2.5.3 Equations in the Form of Scalar Conservation Laws 58

2.5.4 Equations in the Form of Vector Conservation Laws 62

2.6 Comments 66

3 Grid Quality Measures 67

3.1 Introduction 67

3.2 Curve Geometry 67

3.2.1 Basic Curve Vectors 68

3.2.2 Curvature 70

3.2.3 Torsion 71

3.3 Surface Geometry 72

3.3.1 Surface Base Vectors 72

3.3.2 Metric Tensors 73

3.3.3 Second Fundamental Form 75

3.3.4 Surface Curvatures 75

3.4 Metric-Tensor Invariants 77

3.4.1 Algebraic Expressions for the Invariants 78

3.4.2 Geometric Interpretation 79

3.5 Characteristics of Grid Lines 80

3.5.1 Sum of Squares of Cell Edge Lengths 81

3.5.2 Eccentricity 81

3.5.3 Curvature 82

3.5.4 Measure of Coordinate Line Torsion 85

3.6 Characteristics of Faces of Three-Dimensional Grids 85

3.6.1 Cell Face Skewness 85

3.6.2 Face Aspect-Ratio 86

3.6.3 Cell Face Area Squared 86

3.6.4 Cell Face Warping 87

3.7 Characteristics of Grid Cells 88

3.7.1 Cell Aspect-Ratio 89

3.7.2 Square of Cell Volume 89

3.7.3 Cell Area Squared 89

3.7.4 Cell Skewness 89

3.7.5 Characteristics of Nonorthogonality 90

3.7.6 Grid Density 91

3.7.7 Characteristics of Deviation from Conformality 92

3.7.8 Grid Eccentricity 96

3.7.9 Measures of Grid Warping and Grid Torsion 96

3.7.10 Quality Measures of Simplexes 97

3.8 Comments 98

4 Stretching Method 101

4.1 Introduction 101

4.2 Formulation of the Method 102

4.3 Theoretical Foundation 104

4.3.1 Model Problems 105

4.3.2 Basic Majorants 108

4.4 Basic Intermediate Transformations 116

4.4.1 Basic Local Stretching Functions 116

4.4.2 Basic Boundary Contraction Functions 120

4.4.3 Other Univariate Transformations 125

4.4.4 Construction of Basic Intermediate Transformations 127

4.5 Comments 130

5 Algebraic Grid Generation 133

5.1 Introduction 133

5.2 Transfinite Interpolation 133

5.2.1 Unidirectional Interpolation 134

5.2.2 Tensor Product 135

5.2.3 Boolean Summation 136

5.3 Algebraic Coordinate Transformations 139

5.3.1 Formulation of Algebraic Coordinate Transformation 139

5.3.2 General Algebraic Transformations 141

5.4 Lagrange and Hermite Interpolations 143

5.4.1 Coordinate Transformations Based on Lagrange Interpolation 143

5.4.2 Transformations Based on Hermite Interpolation 147

5.5 Control Techniques 150

5.6 Transfinite Interpolation from Triangles and Tetrahedrons 151

5.7 Comments 153

6 Grid Generation Through Differential Systems 155

6.1 Introduction 155

6.2 Laplace Systems 157

6.2.1 Two-Dimensional Equations 158

6.2.2 Three-Dimensional Equations 161

6.3 Poisson Systems 164

6.3.1 Formulation of the System 165

6.3.2 Justification for the Poisson System 166

6.3.3 Equivalent Forms of the Poisson System 168

6.3.4 Orthogonality at Boundaries 170

6.3.5 Control of the Angle of Intersection 177

6.4 Biharmonic Equations 181

6.4.1 Formulation of the Approach 181

6.4.2 Transformed Equations 182

6.5 Orthogonal Systems 182

6.5.1 Derivation from the Condition of Orthogonality 183

6.5.2 Multidimensional Equations 184

6.6 Hyperbolic and Parabolic Systems 185

6.6.1 Specification of Aspect Ratio 186

6.6.2 Specification of Jacobian 188

6.6.3 Parabolic Equations 191

6.6.4 Hybrid Grid Generation Scheme 191

6.7 Comments 192

7 Dynamic Adaptation 195

7.1 Introduction 195

7.2 One-Dimensional Equidistribution 196

7.2.1 Example of an Equidistributed Grid 197

7.2.2 Original Formulation 199

7.2.3 Differential Formulation 200

7.2.4 Specification of Weight Functions 201

7.3 Equidistribution in Multidimensional Space 209

7.3.1 One-Directional Equidistribution 209

7.3.2 Multidirectional Equidistribution 210

7.3.3 Control of Grid Quality 211

7.3.4 Equidistribution over Cell Volume 213

7.4 Adaptation Through Control Functions 216

7.4.1 Specification of the Control Functions in Elliptic Systems 216

7.4.2 Hyperbolic Equations 218

7.5 Grids for Nonstationary Problems 218

7.5.1 Method of Lines 219

7.5.2 Moving-Grid Techniques 219

7.5.3 Time-Dependent Deformation Method 221

7.6 Comments 223

8 Variational Methods 227

8.1 Introduction 227

8.2 Calculus of Variations 227

8.2.1 General Formulation 228

8.2.2 Euler-Lagrange Equations 229

8.2.3 Functional Dependent on Metric Elements 232

8.2.4 Functional Dependent on Tensor Invariants 233

8.2.5 Convexity Condition 235

8.3 Integral Grid Characteristics 236

8.3.1 Dimensionless Functionals 236

8.3.2 Dimensionally Heterogeneous Functionals 240

8.3.3 Functionals Dependent on Second Derivatives 242

8.4 Adaptation Functionals 243

8.4.1 One-Dimensional Functionals 244

8.4.2 Multidimensional Approaches 245

8.5 Functionals of Attraction 250

8.5.1 Lagrangian Coordinates 250

8.5.2 Attraction to a Vector Field 252

8.5.3 Jacobian-Weighted Functional 253

8.6 Energy Functionals of Harmonic Function Theory 255

8.6.1 General Formulation of Harmonic Maps 255

8.6.2 Application to Grid Generation 256

8.6.3 Relation to Other Functionals 256

8.7 Combinations of Functionals 257

8.7.1 Natural Boundary Conditions 258

8.8 Comments 258

9 Curve and Surface Grid Methods 261

9.1 Introduction 261

9.2 Grids on Curves 262

9.2.1 Formulation of Grids on Curves 262

9.2.2 Grid Methods 264

9.3 Formulation of Surface Grid Methods 266

9.3.1 Mapping Approach 266

9.3.2 Associated Metric Relations 268

9.4 Beltramian System 269

9.4.1 Beltramian Operator 269

9.4.2 Surface Grid System 270

9.5 Interpretations of the Beltramian System 272

9.5.1 Variational Formulation 272

9.5.2 Harmonic-Mapping Interpretation 273

9.5.3 Formulation Through Invariants 274

9.5.4 Formulation Through the Surface Christoffel Symbols 275

9.5.5 Relation to Conformal Mappings 280

9.5.6 Projection of the Laplace System 282

9.6 Control of Surface Grids 283

9.6.1 Control Functions 283

9.6.2 Projection on the Boundary Line 284

9.6.3 Monitor Approach 285

9.6.4 Control by Variational Methods 286

9.6.5 Orthogonal Grid Generation 289

9.7 Hyperbolic Method 290

9.7.1 Hyperbolic Governing Equations 291

9.8 Comments 291

10 Comprehensive Method 293

10.1 Introduction 293

10.2 Hypersurface Geometry and Grid Formulation 295

10.2.1 Hypersurface Grid Formulation 295

10.2.2 Monitor Hypersurfaces 296

10.2.3 Metric Tensors 297

10.2.4 Christoffel Symbols 298

10.2.5 Relations Between Metric Elements 300

10.3 Functional of Smoothness 301

10.3.1 Formulation of the Functional 301

10.3.2 Geometric Interpretation 302

10.3.3 Dimensionless Functionals 304

10.3.4 Euler-Lagrange Equations 305

10.3.5 Equivalent Forms 307

10.4 Hypersurface Grid Systems 309

10.4.1 Inverted Beltrami Equations 309

10.5 Formulation of Comprehensive Grid Generator 311

10.5.1 Energy and Diffusion Functionals 311

10.5.2 Relation to Harmonic Functions 312

10.5.3 Beltrami and Diffusion Equations 313

10.5.4 Inverted Beltrami and Diffusion Equations 315

10.6 Numerical Algorithms 317

10.6.1 Finite-Difference Algorithm 318

10.6.2 Spectral Element Algorithm 322

10.7 Formulation of Control Metrics 324

10.7.1 Specification of Individual Control Metrics 325

10.7.2 Control Metrics for Generating Grids with Balanced Properties 329

10.7.3 Application to Solution of Singularly-Perturbed Equations 330

10.8 Comments 331

11 Unstructured Methods 333

11.1 Introduction 333

11.2 Consistent Grids and Numerical Relations 334

11.2.1 Convex Cells 334

11.2.2 Consistent Grids 335

11.3 Methods Based on the Delaunay Criterion 337

11.3.1 Dirichlet Tessellation 339

11.3.2 Incremental Techniques 339

11.3.3 Approaches for Insertion of New Points 341

11.3.4 Two-Dimensional Approaches 342

11.3.5 Constrained Form of Delaunay Triangulation 346

11.3.6 Point Insertion Strategies 348

11.3.7 Surface Delaunay Triangulation 354

11.3.8 Three-Dimensional Delaunay Triangulation 354

11.4 Advancing-Front Methods 356

11.4.1 Procedure of Advancing-Front Method 356

11.4.2 Strategies to Select Out-of-Front Vertices 357

11.4.3 Grid Adaptation 358

11.4.4 Advancing-Front Delaunay Triangulation 358

11.4.5 Three-Dimensional Prismatic Grid Generation 359

11.5 Comments 360.

Notes:

Previous ed.: 1999.

ISBN:

9789048129119

9048129117

OCLC:

416286975