Clark Proceedings of the 18th International Meshing Roundtable

1. Auflage 2009
ISBN: 978-3-642-04319-2
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 654 Seiten

This volume contains the articles presented at the 18th International Meshing Roundtable (IMR) organized, in part, by Sandia National Laboratories and held October 25-28, 2009 in Salt Lake City, Utah, USA. The volume presents recent results of mesh generation and adaptation which has applications to finite element simulation. It introduces theoretical and novel ideas with practical potential.

Autoren/Hrsg.

Clark, Brett W.

Inhaltsverzeichnis

1;Preface;4
2;Organization;5
3;Reviewers;7
4;Contents;9
5;Session 1A: Surface Meshing;9
5.1;Size Function Smoothing Using an Element Area Gradient;13
5.1.1;Introduction;13
5.1.2;Size Function Smoothing Using an Element Area Gradient;14
5.1.3;Size Function Smoothing Tools in SMS;21
5.1.4;Examples;22
5.1.5;Conclusion;23
5.1.6;References;23
5.2;Removing Self Intersections of a Triangular Mesh by Edge Swapping, Edge Hammering, and Face Lifting;25
5.2.1;Introduction;25
5.2.2;Previous Work;26
5.2.3;Typical Sources of a Self Intersection;27
5.2.4;Detail of the Proposed Method;30
5.2.4.1;Improvement Criteria;30
5.2.4.2;Edge Swapping;31
5.2.4.3;Edge Hammering;31
5.2.4.4;Face Lifting;33
5.2.4.5;Order-Dependency Issues;35
5.2.5;Potential Expansions and Discussions;36
5.2.5.1;Expanding the Proposed Method for a Quadrilateral Mesh;36
5.2.5.2;Locally Adjusting the Clearance Requirement;37
5.2.5.3;Boundary Fidelity Issue;38
5.2.6;Examples;38
5.2.7;Conclusions;40
5.2.8;References;40
5.3;Conformal Refinement of Unstructured Quadrilateral Meshes;42
5.3.1;Introduction;42
5.3.2;Previous Work;43
5.3.3;Description of Mesh Refinement/Coarsening Algorithm;45
5.3.3.1;Overview;45
5.3.3.2;Subdivision Templates;45
5.3.3.3;Coarsening;47
5.3.3.4;Refinement;48
5.3.4;Remapping or Solution Transfer;50
5.3.5;Results;50
5.3.6;Discussion;52
5.3.7;References;54
5.4;Guaranteed-Quality All-Quadrilateral Mesh Generation with Feature Preservation;56
5.4.1;Introduction;56
5.4.2;Previous Work;57
5.4.3;Guaranteed-Quality Meshing of Point Cloud;59
5.4.4;Guaranteed-Quality Meshing of Smooth Curves;61
5.4.4.1;Curve Decomposition;61
5.4.4.2;Adaptive Quadtree Construction;61
5.4.4.3;Buffer Zone Clearance;62
5.4.4.4;Template Implementation;63
5.4.4.5;First Buffer Layer Construction;66
5.4.4.6;Second Buffer Layer Construction;69
5.4.5;Sharp Feature and Boundary Layer;69
5.4.6;Results;71
5.4.7;Conclusion and Future Work;73
5.4.8;References;73
6;Session 1B: Hexahedral Meshing;9
6.1;Advances in Octree-Based All-Hexahedral Mesh Generation: Handling Sharp Features;75
6.1.1;Introduction;75
6.1.1.1;Motivation;75
6.1.1.2;Limitations of Available Octree-Based Software;76
6.1.1.3;Objective;76
6.1.2;Octree Building;77
6.1.2.1;An Almost Regular Octree;77
6.1.2.2;Subdivision Criteria;77
6.1.2.3;Balancing and Pairing Rules;78
6.1.3;Polyhedral Cutting;79
6.1.3.1;Connecting Hanging Vertices;79
6.1.3.2;2-Dimensional Case;80
6.1.3.3;3-Dimensional Case;80
6.1.4;Dual-Mesh Generation;82
6.1.4.1;2-Dimensional Case;82
6.1.4.2;3-Dimensional Case;82
6.1.5;Subdomains Coloring;84
6.1.6;Geometry and Features Detection;85
6.1.7;Buffer-Layer Insertion;86
6.1.8;Smoothing;87
6.1.9;Conclusion;89
6.1.10;Some Results;90
6.1.11;References;94
6.2;Conforming Hexahedral Mesh Generation via Geometric Capture Methods;95
6.2.1;Introduction;95
6.2.2;Background;96
6.2.2.1;Dual Sheets and Columns;96
6.2.2.2;Sheet Operations;97
6.2.2.3;Fundamental Hexahedral Meshes;98
6.2.3;Theory and Assertions;99
6.2.3.1;Non-conforming Meshes to Fundamental Meshes;100
6.2.3.2;Assertion on Hexahedral Quality;101
6.2.4;Algorithm;102
6.2.4.1;Establishing 2-Manifolds;102
6.2.4.2;Capturing Geometric Topology;103
6.2.4.3;Fundamental Conversions;105
6.2.4.4;Mesh Optimization;106
6.2.5;Examples;107
6.2.6;Alternate Methods;109
6.2.6.1;Tri-valent Vertices;109
6.2.6.2;Regularizing Mesh Near Boundaries;110
6.2.7;Conclusion;110
6.2.8;References;111
6.3;Efficient Hexahedral Mesh Generation for Complex Geometries Using an Improved Set of Refinement Templates;113
6.3.1;Introduction;113
6.3.2;Refinement Templates;114
6.3.3;Options for Hexahedral Mesh Generation;117
6.3.3.1;Temporal Rotation;117
6.3.3.2;Temporal Local Inflation;117
6.3.4;Applications;120
6.3.4.1;Ramses Model;123
6.3.4.2;Pediatric Brain Model;123
6.3.4.3;Three-Year-Old Child Model;123
6.3.5;Conclusions;123
6.3.6;References;124
6.4;Embedding Features in a Cartesian Grid;126
6.4.1;Background;126
6.4.2;Feature Embedding;129
6.4.3;Embedding Procedures;132
6.4.3.1;Embedding Vertices;133
6.4.3.2;Embedding Curves;138
6.4.3.3;Embedding Surfaces;141
6.4.3.4;Embedding Volumes;143
6.4.4;Completing the Mesh;144
6.4.5;Conclusion;145
6.4.6;References;146
7;Session 2A: Optimization;10
7.1;Label-Invariant Mesh Quality Metrics;148
7.1.1;Measuring Quality $Within$ Mesh Elements;148
7.1.2;Label-Invariance of Quality Metrics;152
7.1.3;Linear Planar Triangles;154
7.1.3.1;The Linear Map;154
7.1.3.2;The Reference Element;154
7.1.3.3;Label-Invariance;154
7.1.4;Quadratic Planar Triangles;156
7.1.4.1;The Quadratic Map;156
7.1.4.2;Symmetry Points for Maps to Triangular Elements;157
7.1.4.3;Symmetry Relation for Jacobian of the Quadratic Map;157
7.1.4.4;Label-Invariance;158
7.1.4.5;The Shape Quality of Quadratic Triangles;161
7.1.5;Summary;162
7.1.6;References;163
7.2;Perturbing Slivers in 3D Delaunay Meshes;165
7.2.1;Introduction;165
7.2.1.1;Slivers;165
7.2.1.2;Tetrahedron Quality;166
7.2.1.3;Previous Work;167
7.2.1.4;Contribution;170
7.2.2;Algorithm;171
7.2.2.1;Circumsphere Radius;173
7.2.2.2;Volume;174
7.2.2.3;Random Perturbation;175
7.2.3;Experiments and Results;175
7.2.4;Conclusion and Discussion;179
7.2.5;References;180
7.3;Mesh Smoothing Algorithms for Complex Geometric Domains;182
7.3.1;Introduction;182
7.3.2;Review of Smoothing Algorithms;183
7.3.2.1;Laplacian Smoothing;184
7.3.2.2;Smart Laplacian Smoothing;185
7.3.2.3;Centroidal Voronoi Tessellation (CVT)-Based Smoothing;185
7.3.2.4;Optimal Delaunay Triangulation (ODT)-Based Smoothing;186
7.3.2.5;Angle-Based Smoothing (AB);187
7.3.2.6;Well-Centered Triangulation (WCT) Smoothing;188
7.3.3;New Smoothing Algorithms;189
7.3.3.1;Modifying Existing Algorithms for Complex Geometry;189
7.3.3.2;Centroid of Off-Centers (CO);191
7.3.3.3;Distance Weighted Centroid of Incenters (WCI);191
7.3.3.4;Sliced-Petal Smoothing;192
7.3.4;Results and Discussions;192
7.3.4.1;Implementation and Data Sets;193
7.3.4.2;Experiments;193
7.3.4.3;Future Work;197
7.3.5;References;198
7.4;A Novel Method for Surface Mesh Smoothing: Applications in Biomedical Modeling;201
7.4.1;Introduction;201
7.4.2;Surface Mesh Smoothing Algorithm;203
7.4.2.1;Initial Mesh Smoothing;203
7.4.2.2;Curvature Estimation and Labeling;206
7.4.2.3;Adaptive Mesh Smoothing;207
7.4.2.4;Mesh Quality Improvement;208
7.4.3;Implementation and Results;211
7.4.4;Conclusions;214
7.4.5;References;215
7.5;Quality Improvement of Non-manifold Hexahedral Meshes for Critical Feature Determination of Microstructure Materials;217
7.5.1;Introduction;217
7.5.2;Previous Work;219
7.5.3;Data Acquisition and All-Hex Mesh Generation;221
7.5.4;Vertex Classification;221
7.5.5;Quality Improvement of Non-manifold Hex Meshes;223
7.5.5.1;Modified Pillowing for Non-manifold Boundaries;223
7.5.5.2;Fairing and Regularization for Curves;224
7.5.5.3;Geometric Flow and Optimization;227
7.5.6;Finite Element Analysis and Results;228
7.5.7;Conclusion and Future Work;233
7.5.8;References;234
8;Session 2B: Geometry;10
8.1;Automatic CAD Models Comparison and Re-meshing in the Context of Mechanical Design Optimization;237
8.1.1;Introduction;237
8.1.2;Comparison between CAD Models;239
8.1.2.1;Vectorial Space, Metric Tensor and Initia Tensor;239
8.1.2.2;Comparison between CAD Models;241
8.1.3;Automatic Remeshing;243
8.1.3.1;Algorithm;243
8.1.3.2;Results;248
8.1.4;Conclusion;250
8.1.5;References;250
8.2;Distance Solutions for Medial Axis Transform;252
8.2.1;Introduction;252
8.2.2;Solutions of Eikonal Equation;253
8.2.2.1;H-J/Eikonal Equation for $d$;253
8.2.2.2;Domain and Initialization;254
8.2.2.3;Distance Solution Examples;255
8.2.3;Medial Axis Transform via Distance Field: $d$ MAT;256
8.2.3.1;Feature Detection Criteria;257
8.2.3.2;Thinning and Representation;259
8.2.3.3;Solution Superposition;263
8.2.4;Conclusion;265
8.2.5;A Finite Difference Solution Procedures;266
8.2.5.1;A.1 Fast-Marching Method;266
8.2.5.2;A.2 Fast-Sweeping Method;267
8.2.6;References;268
8.3;Automatic Non-manifold Topology Recovery and Geometry Noise Removal;271
8.3.1;Introduction;271
8.3.2;Algorithm Description;273
8.3.2.1;Pre-requisites;273
8.3.2.2;The Optional e Tolerance Parameter;274
8.3.2.3;Pre-processing;274
8.3.2.4;Vertex Association and Projection;274
8.3.2.5;Edge Association;275
8.3.2.6;Topology Post-processing;276
8.3.3;Applications;279
8.3.4;Conclusion;282
8.3.5;References;282
8.4;A New Procedure to Compute Imprints in Multi-sweeping Algorithms;284
8.4.1;Introduction;284
8.4.2;The Multi-sweeping Method;286
8.4.3;Basic Definitions;287
8.4.3.1;Sweep Node;287
8.4.3.2;Loop Geometry Engine;287
8.4.3.3;Control Loop;288
8.4.3.4;Computational Domain;289
8.4.4;Loop Face Projection and Imprinting;290
8.4.4.1;Loop Face Projection;290
8.4.4.2;Loop Face Imprinting Pre-process;291
8.4.4.3;Mapping of Sweep Nodes from the Computational Domain to a Source Surface;291
8.4.4.4;Loop Face Imprinting;293
8.4.4.5;Final Location of Inner Sweep Nodes;295
8.4.5;Loop Edge Meshing and Volume Decomposition;297
8.4.6;Examples;298
8.4.7;Conclusions;300
8.4.8;References;301
8.5;Defeaturing CAD Models Using a Geometry-Based Size Field and Facet-Based Reduction Operators;303
8.5.1;Introduction;303
8.5.2;Background;305
8.5.3;Defeaturing Algorithm;308
8.5.3.1;Obtaining a Discrete Model;310
8.5.3.2;Detection of Features for Suppression;310
8.5.3.3;Suppressing Features;315
8.5.3.4;Meshing the Defeatured Model;316
8.5.4;Results;316
8.5.5;Conclusion;319
8.5.6;References;319
9;Session 3A: Parallel & Hybrid;11
9.1;Towards Exascale Parallel Delaunay Mesh Generation;321
9.1.1;Introduction;321
9.1.2;Background;323
9.1.2.1;Related Work in Parallel Mesh Generation;323
9.1.2.2;Related Work in Parallel Runtime Systems;324
9.1.3;Multi-layered Runtime System;325
9.1.4;Multi-layered Parallel Mesh Generation;327
9.1.4.1;Domain Decomposition Step;329
9.1.4.2;Parallel Delaunay Mesh Generation Step;329
9.1.5;Putting It All Together;331
9.1.5.1;Preliminary Data;331
9.1.6;Conclusions;335
9.1.7;References;336
9.2;On the Use of Space Filling Curves for Parallel Anisotropic Mesh Adaptation;339
9.2.1;Introduction;339
9.2.2;A Brief Overview of the Mesh Adaptation Platform;341
9.2.3;The Considered Test Cases;342
9.2.4;The Hilbert Space Filling Curve;343
9.2.4.1;Mesh Entities Renumbering;343
9.2.5;Exploiting Space Filling Curves for Efficient Shared-Memory Multi-threaded Parallelization;346
9.2.5.1;A Shared Memory Multi-threaded Parallelization;346
9.2.5.2;Parallelizing the Flow Solver;348
9.2.5.3;Parallelizing the Error Estimate;350
9.2.6;Exploiting Space Filling Curves for Efficient Out-of-Core Parallelization;351
9.2.6.1;A Fast Mesh Partitioning Algorithm;351
9.2.6.2;Parallelizing the Local Adaptive Remesher;353
9.2.6.3;Parallelizing the Solution Interpolation;355
9.2.7;Conclusion;356
9.2.8;References;357
9.3;Mesh Insertion of Hybrid Meshes;360
9.3.1;Introduction;360
9.3.2;An Optimal Non-manifold Hybrid Mesh Data Structure (NHMD);362
9.3.3;Mesh Insertion Algorithm;365
9.3.3.1;Detection of the Undesired Entities of the Target Mesh;366
9.3.3.2;Connecting One-Dimensional and Two-Dimensional Entities to the Tool Mesh;368
9.3.3.3;Connecting Three-Dimensional Entities to the Tool Mesh;369
9.3.4;Examples;373
9.3.5;Conclusion and Future Work;374
9.3.6;References;375
9.4;Tensor-Guided Hex-Dominant Mesh Generation with Targeted All-Hex Regions;377
9.4.1;Introduction;377
9.4.2;Related Work;380
9.4.3;Technical Approach and Preliminaries;381
9.4.3.1;Metric Field Representation, Support, and Operations;382
9.4.4;Generating Metric Tensor Fields;384
9.4.4.1;Boundary Field Form-Fitting;384
9.4.4.2;Interior Form-Fitting;385
9.4.5;Hex-Dominant Mesh Generation;386
9.4.5.1;Topological Insertion Operators and Face Groups;386
9.4.5.2;Planning and Scheduling Insertions;388
9.4.5.3;Element Shaping;389
9.4.5.4;Boundary Conformity;390
9.4.5.5;Hex-Dominant Mesh Finalization;390
9.4.6;Results;390
9.4.7;Conclusion;393
9.4.8;References;394
10;Session 3B: Applications;11
10.1;Using Parameterization and Springs to Determine Aneurysm Wall Thickness;397
10.1.1;Introduction;397
10.1.2;Previous Work;398
10.1.3;Wall-Thickness Estimation;400
10.1.3.1;Deformation through Parameterization;401
10.1.3.2;Relaxation Using Springs;404
10.1.3.3;Thickness Calculation;405
10.1.4;Boundary-Layer Meshing;406
10.1.5;Results;407
10.1.6;Conclusion;412
10.1.7;References;412
10.2;Hybrid Mesh Generation for Reservoir Flow Simulation in CPG Grids;415
10.2.1;Problem Statement;416
10.2.1.1;Mathematical Model of Flow Simulation;416
10.2.1.2;Numerical Schemes;417
10.2.1.3;Meshes Overview;417
10.2.2;Hybrid Meshes and Non Cartesian Grids;418
10.2.2.1;General Methodology and Its Restriction;418
10.2.2.2;A Solution Using Grid Deformation;418
10.2.3;Hybrid Meshes for Real Grids with High Deformation;419
10.2.3.1;Mapping the Cavity Space;419
10.2.3.2;Correction of the Mapping;421
10.2.4;Numerical Example;424
10.2.5;Conclusions;426
10.2.6;References;427
10.3;VECTIS Mesher – A 3D Cartesian Approach Employing Marching Cubes;428
10.3.1;Introduction;428
10.3.2;Context of the Mesher in VECTIS-MAX System;430
10.3.3;Requirements for Mesh Quality;431
10.3.4;General Approach;431
10.3.4.1;Scaling;432
10.3.4.2;Dual Levels;432
10.3.4.3;Shoeboxes;433
10.3.4.4;In/Out Status;433
10.3.4.5;Box Generation;433
10.3.4.6;Types of Refinement;435
10.3.4.7;Generation of Patches;435
10.3.4.8;Tying of Patches;437
10.3.4.9;Generation of Inner Faces;438
10.3.4.10;Removal of Small Cells;438
10.3.4.11;Problem of Concave Cells;439
10.3.5;Examples of Generated Meshes;440
10.3.6;Description of the Meshing Algorithm;441
10.3.7;Tools Helping to Decrease Time and Memory Demands;443
10.3.7.1;Features Improving Time Efficiency;443
10.3.7.2;Features Improving Memory Efficiency;444
10.3.8;References;445
10.4;Shape Operator Metric for Surface Normal Approximation;446
10.4.1;Introduction;446
10.4.2;Framework;447
10.4.2.1;Surface Approximation;448
10.4.2.2;Normal Approximation;448
10.4.2.3;Behavior;449
10.4.3;Shape Operator Metric for Normal Approximation;450
10.4.4;Implementation of SOM;451
10.4.5;Results;452
10.4.5.1;Numerical Validation;454
10.4.6;Summary and Conclusion;457
10.4.7;References;457
10.4.8;Appendix;458
10.4.8.1;Appendix A: Distance Tightness Bounds;458
10.4.8.2;Appendix B: Limit Approximation Efficiency;459
10.4.8.3;Appendix C: Shape Operator Metric and Aspect Ratio;460
11;Session 4: Tetrahedral Meshing;11
11.1;The Meccano Method for Automatic Tetrahedral Mesh Generation of Complex Genus-Zero Solids;461
11.1.1;Introduction;461
11.1.2;Meccano Technique Algorithm;462
11.1.3;Meccano Technique for a Complex Genus-Zero Solid;463
11.1.3.1;Meccano;464
11.1.3.2;Mapping from Cube Faces to Solid Surface Patches;465
11.1.3.3;Coarse Tetrahedral Mesh of the Meccano;467
11.1.3.4;Local Refined Tetrahedral Mesh of the Meccano;467
11.1.3.5;External Node Mapping on Solid Boundary;470
11.1.3.6;Relocation of Inner Nodes;470
11.1.3.7;Solid Mesh Optimization: Untangling and Smoothing;471
11.1.4;Test Examples;472
11.1.4.1;Example 1: Bust;472
11.1.4.2;Example 2: Bunny;474
11.1.5;Conclusions and Future Research;476
11.1.6;References;477
11.2;Collars and Intestines: Practical Conforming Delaunay Refinement;479
11.2.1;Introduction;479
11.2.2;Preliminaries;480
11.2.2.1;Definitions;480
11.2.2.2;Generic Delaunay Refinement Algorithm;481
11.2.3;Delaunay Refinement in 2D;481
11.2.3.1;Collar Protection Region;482
11.2.3.2;Intestine Protection Region;484
11.2.4;Delaunay Refinement in 3D;485
11.2.4.1;Collar Protection Region;485
11.2.4.2;Intestine Protection Region;488
11.2.5;Implementation Details and Examples;492
11.2.6;References;494
11.3;An Analysis of Shewchuk’s Delaunay Refinement Algorithm;496
11.3.1;Introduction;496
11.3.2;The Algorithm;498
11.3.3;Proof of Termination;499
11.3.4;Parent Sequences;504
11.3.5;Output Edge Lengths;508
11.3.6;Vertex Degrees;509
11.3.7;Output Mesh Size;511
11.3.8;Conclusions;514
11.3.9;References;514
11.4;Hexagonal Delaunay Triangulation;516
11.4.1;Introduction;516
11.4.2;Previous Work;517
11.4.3;Hexagonal Subdivision;518
11.4.3.1;Subdividing a Hexagon;518
11.4.3.2;Reverse Operation;518
11.4.3.3;Adaptive Refinement;519
11.4.4;Simple Refinement;519
11.4.4.1;Refinement Rules;519
11.4.4.2;Extracting the Dual Mesh;520
11.4.4.3;Proving Angles;520
11.4.5;Extended Refinement;521
11.4.5.1;Classification of Hexagons;522
11.4.5.2;Extended Refinement Rules;523
11.4.5.3;Local Coarsening;523
11.4.5.4;Modifications to the Dual Mesh;524
11.4.6;Properties;528
11.4.6.1;Angle Bounds;528
11.4.6.2;Aspect Ratio;529
11.4.6.3;Minimum Triangle Size;529
11.4.6.4;Delaunay Property;530
11.4.7;Triangulating Planar Straight-Line Graphs;530
11.4.7.1;Input Line Segment;530
11.4.7.2;Extended Refinement Rules;530
11.4.7.3;Line intersections;531
11.4.7.4;Adding Triangles;532
11.4.8;Results;532
11.4.9;Conclusion and Future Work;534
11.4.10;References;535
11.5;Tetrahedral Mesh Improvement Using Multi-face Retriangulation;536
11.5.1;Introduction and Motivation;536
11.5.2;Related Work;538
11.5.2.1;Mesh Smoothing;539
11.5.2.2;Topological Operations;540
11.5.2.3;Vertex Insertion;540
11.5.3;Tetrahedral Mesh Quality Improvement;541
11.5.3.1;Multi-face Retriangulation;542
11.5.3.2;Edge Collapse;543
11.5.3.3;Quality Measures;543
11.5.4;Implementation;544
11.5.5;Tests and Results;546
11.5.6;Discussion and Future Work;550
11.5.7;References;551
12;Session 5: Adaptivity;12
12.1;Automatic All Quadrilateral Mesh Adaption through Refinement and Coarsening;553
12.1.1;Introduction;553
12.1.2;Background;554
12.1.2.1;Current Methods;554
12.1.2.2;Concurrent Refinement and Coarsening;555
12.1.3;Automated Mesh Adaptation;555
12.1.3.1;Sizing Functions;555
12.1.3.2;Tools and Requirements;556
12.1.3.3;Algorithm;558
12.1.3.4;Algorithm Example;558
12.1.4;Examples;564
12.1.4.1;Nosecone;564
12.1.4.2;Plate with Hole in Tension;565
12.1.5;Conclusion;567
12.1.5.1;Further Research;568
12.1.6;References;569
12.2;Optimal 3D Highly Anisotropic Mesh Adaptation Based on the Continuous Mesh Framework;571
12.2.1;Continuous Mesh Framework;574
12.2.2;Interpolation Error: Discrete-Continuous Duality;576
12.2.3;Optimal Control of the Interpolation Error in L$^{p}$ Norm;580
12.2.4;3D Numerical Validations;582
12.2.5;References;589
12.3;Anisotropic Mesh Adaptation for Solution of Finite Element Problems Using Hierarchical Edge-Based Error Estimates;591
12.3.1;Introduction;591
12.3.2;Interpolation Error Analysis;592
12.3.2.1;Edge-Based Error Estimates and a Tensor Metric;592
12.3.2.2;Metrics for the $L^{p}$-norm of Error and Its Gradient;596
12.3.2.3;Extension to General Functions;597
12.3.2.4;On Selection of a$_{k}$;598
12.3.2.5;Error Estimates as Functions of N$_{h}$;598
12.3.3;Mesh Adaptation Algorithm;599
12.3.4;Numerical Results;599
12.3.4.1;Interpolation Problems;599
12.3.4.2;Applications to PDEs;601
12.3.5;Conclusion;605
12.3.6;References;605
12.4;On 3D Anisotropic Local Remeshing for Surface, Volume and Boundary Layers;607
12.4.1;Introduction;607
12.4.2;Metric-Based Anisotropic Local Remeshing;608
12.4.2.1;Metric Tensors in Mesh Adaptation;608
12.4.2.2;Adaptive CFD Simulations;610
12.4.3;Metric-Based Estimates;611
12.4.3.1;Techniques for Enhancing Robustness and Performance;611
12.4.3.2;L$^{p}$ Norm Interpolation Error;613
12.4.3.3;Geometric Estimate for Surfaces;614
12.4.3.4;Boundary Layers Metric;616
12.4.4;Quality-Driven Local Mesh Operators;618
12.4.4.1;Insertion and Collapse;618
12.4.4.2;Using the Boundary Layers Metric;622
12.4.5;Conclusions and Future Work;624
12.4.6;References;624
12.5;A Comparison of Gradient- and Hessian-Based Optimization Methods for Tetrahedral Mesh Quality Improvement;627
12.5.1;Introduction;627
12.5.2;Problem Statement;628
12.5.2.1;Element and Mesh Quality;628
12.5.2.2;Aspect Ratio Quality Metric;629
12.5.2.3;Inverse Mean Ratio Quality Metric;629
12.5.2.4;Vertex Condition Number Quality Metric;630
12.5.2.5;Quality Improvement Problem;630
12.5.3;Improvement Algorithms;630
12.5.3.1;Steepest Descent Method;631
12.5.3.2;Conjugate Gradient Method;631
12.5.3.3;Quasi-Newton Method;631
12.5.3.4;Trust-Region Method;631
12.5.3.5;Feasible Newton Method;632
12.5.4;Numerical Experiments;632
12.5.4.1;Increasing Problem Size;633
12.5.4.2;Initial Mesh Configuration;636
12.5.4.3;Graded Meshes;640
12.5.4.4;Mesh Quality Metric;640
12.5.5;Future Work;642
12.5.6;References;643
13;Author Index;645
14;Index by Affiliation;647