Numerical Geometry, Grid Generation and Scientific Computing: Proceedings of the 10th International Conference, NUMGRID 2020 / Delaunay 130, ... Computational Science and Engineering, 143)

The focus of these conference proceedings is on research, development, and applications in the fields of numerical geometry, scientific computing and numerical simulation, particularly in mesh generation and related problems. In addition, this year’s special focus is on Delaunay triangulations and their applications, celebrating the 130th birthday of Boris Delaunay. In terms of content, the book strikes a balance between engineering algorithms and mathematical foundations. It presents an overview of recent advances in numerical geometry, grid generation and adaptation in terms of mathematical foundations, algorithm and software development and applications. The specific topics covered include: quasi-conformal and quasi-isometric mappings, hyperelastic deformations, multidimensional generalisations of the equidistribution principle, discrete differential geometry, spatial and metric encodings, Voronoi-Delaunay theory for tilings and partitions, duality in mathematical programming and numerical geometry, mesh-based optimisation and optimal control methods. Further aspects examined include iterative solvers for variational problems and algorithm and software development. The applications of the methods discussed are multidisciplinary and include problems from mathematics, physics, biology, chemistry, material science, and engineering. Foreword Preface Contents Conference Organization Organizers Organizing Committee Program Committee Web Site Reviewers Part I Delaunay-Voronoi Theory and Applications Local Groups in Delone Sets 1 Introduction and Basic Definitions 2 Main Result and Conjectures 3 Proof of the Main Result 4 Concluding Remarks References Manifolds of Triangulations, Braid Groups of Manifolds, and the Groups Gamma_n^k 1 Introduction 2 The Manifold of Triangulations 2.1 Geometrical Manifold of Triangulations 2.2 Topological Manifold of Triangulations 2.3 Combinatorial Manifold of Triangulations 3 The Groups Gamma_n^k 3.1 The Case d=3 3.2 General Case 4 Main Results 4.1 Gamma_n^k-Valued Invariant of Braids on Manifolds 4.2 Triangulations of Polyhedra and Groups Gamma_n^k 5 Other Developments: The Groups G̃amma_n^k References A Proof of the Invariant-Based Formula for the Linking Number and Its Asymptotic Behaviour 1 The Gauss Integral for the Linking Number of Curves 2 The Outline of the Invariant-Based Formula and Consequences 3 Past Results about the Gauss Integral for the Linking Number 4 Six Isometry Invariants of Skewed Line Segments in 3-Space 5 Invariant-Based Formula for the Linking Number of Segments 6 The Asymptotic Behaviour of the Linking Number of Segments 7 Example Computations of the Linking Number and a Discussion Appendix 1: Proofs of Lemmas about Isometry Invariants Appendix 2: Proofs of Lemmas for the lk Formula in Theorem 2 Appendix 3: Proofs of Corollaries of Main Theorem 2 References The Singularity Set of Optimal Transportation Maps 1 Introduction 2 Generative Models and Optimal Transportation 3 Convex Geometric View of Optimal Transportation 3.1 Optimal Transportation Map 3.2 Alexandrov Solution 3.3 Minkowski and Alexandrov Theorems 3.4 Secondary Polytope and Secondary Power Diagram 4 Singularity Set of Optimal Transportation Map 4.1 Singularity Set 4.2 Algorithm for Singularity Set 4.3 Singularity Set Homotopy Equivalence 5 Conclusion References Polygonal and Polyhedral Delaunay Meshing 1 Introduction 2 Power Diagram and Lifting Procedure 3 Ball Movement as a Transformation of Dual Polyhedra 4 Dirichlet Functional for Power Function 5 Numerical Experiments 6 Discussion References On Decomposition of Embedded Prismatoids in R^3Without Additional Points 1 Introduction 2 Preliminaries 2.1 Prisms, Antiprisms, and Prismatoids 2.2 S_n,m-Prismatoids 2.3 Twisted Prisms 2.4 Decompositions of S_n,m-Prismatoids 3 New Results on Decomposition of S_n,m-Prismatoids 3.1 Outline of the Proof 3.2 Proof of Lemma 2 References Out-of-core Constrained Delaunay Tetrahedralizations for LargeScenes 1 Introduction 2 Related Works 3 Algorithm 3.1 Overview 3.2 Expected Memory Calculation 3.3 Subdivision Stage 3.4 Repairing Stage 3.5 Merging Stage 3.5.1 Spatial Hashing 3.5.2 Merging Time Complexity 4 Experimental Results 4.1 Runtime and Memory Results 4.2 Quality Results 5 Discussion and Future Work References Part II Adaptive Meshing Size Gradation Control for Anisotropic Hybrid Meshes 1 Introduction 2 Metric and Mesh Generation 3 Size Gradation Control for Anisotropic Meshes 4 Numerical Results 4.1 Line 4.2 Circle 4.3 Inviscid Flow Simulation 4.4 Turbulent Flow Simulation 5 Conclusion References Adjoint Computation on Anisotropic Meshes in High-fidelity RANS Simulations 1 Introduction 2 Flow Solver 3 Adjoint System 4 Numerical Implementation 5 Numerical Results 6 Conclusions References Moving Deforming Mesh Generation Based on the Quasi-Isometric Functional 1 Introduction 2 Quasi-Isometric Functional 3 Preconditioned Minimization Algorithm and the Moving Mesh Interpolation Strategy 4 Mesh Stretching and Size Gradation Control 5 MPI-Based Parallel Implementation 6 Numerical Experiments 7 Conclusions and Discussion References Adaptive Grids for Non-monotone Waves and Instabilities in a Non-equilibrium PDE Model 1 Introduction 2 The Non-equilibrium PDE Model 2.1 The One-Dimensional Case 2.2 Travelling Waves: A Bifurcation Diagram 2.3 The Adaptive Moving Grid in 1D 2.4 Numerical Results 3 The Two-Dimensional Case 3.1 Non-monotone Waves and Instabilities 3.2 The Adaptive Moving Grid in 2D 3.3 Numerical Results References RBF-VerBSS Hybrid Method for Mesh Deformation 1 Introduction 2 Related Work 2.1 RBF Interpolation Method 2.2 VerBSS Method 2.3 Robustness of the Delaunay Graph and the MSA Method 3 RBF-VerBSS Hybrid Mesh Deformation Method 4 Results and Discussions 4.1 NACA 0012 Airfoil Rotation 4.2 3D Moving Aircraft 5 Conclusions References A Uniform Convergence Analysis for a Bakhvalov-Type Mesh with an Explicitly Defined Transition Point 1 Introduction 2 Preliminaries 3 A Bakhvalov-Type Mesh with an Explicitly Defined Transition Point 4 The Barrier-Function Analysis of Roos and Linß for Shishkin-Type Meshes 5 The Error Analysis 6 Numerical Results for 2D Problems 7 Conclusion References On a Comprehensive Grid for Solving Problems Having Exponential or Power-of-First-Type Layers 1 Introduction 2 Transformation for Eliminating High-Order Singularities 3 Numerical Experiments 3.1 Numerical Experiments with High-Order Schemes 3.2 Experiments with Autonomous Equations 3.3 Numerical Experiments with a Two-Dimensional Problem of a Viscous-Gas Flow over a Flat Plate 4 Conclusions References Preserved Structure Constants for Red Refinements of ProductElements 1 Introduction and Basic Definitions 2 Red Refinement Strategy and Its Properties 3 Main Results References Part III Meshing and CAD Global Parametrization Based on Ginzburg-Landau Functional 1 Introduction and Related Work 2 Periodic Global Parametrization 2.1 PGP Basics 2.2 Ginzburg-Landau to the Rescue 2.3 Newton Method 2.4 Discretization 3 Results 4 Conclusion References Parametrization of Plane Irregular Regions: A Semi-automatic Approach I 1 Introduction 2 Region Decomposition 2.1 Concavity Measures and Admissible Regions 2.2 Decomposition Method 3 Parametrization Construction 3.1 Compatible Parametrizations of the Boundaries 3.2 Parametrizations of the Subregions 4 Summary and Examples 5 Conclusions and Future Work References A Hybrid Approach to Fast Indirect Quadrilateral Mesh Generation 1 Introduction 2 Related Work 3 Hybrid Quad Mesh Generation 3.1 Segmentation of Boundary Regions 3.2 Greedy Triangle Merging 3.3 Merging of Remaining Triangles 3.3.1 Merging Nodes 3.3.2 Splitting Nodes 3.3.3 Complexity of Triangle Merging 4 Post-Processing 5 Results 6 Conclusion References Hexahedral Mesh Generation Using Voxel Field Recovery 1 Introduction 2 Initial Data Representation 3 Hexahedral Mesh Generation 4 Mesh Quality Improvement 5 Test Examples 6 Conclusions References Generation of Boundary Layer Meshes by the Enhanced Jump-and-Walk Method with a Fast Collision Detecting Algorithm 1 Introduction 2 Overview 3 Computation and Smoothing of Marching Directions and Distances 4 Validity Check for Elements 5 Construction of the Background Grid with Medial Surface Mesh (MSM) 6 The Enhanced Jump-and-Walk Method (EJW) 6.1 Risky Cliff Points 6.2 Algorithm 6.3 Time Complexity 7 Generation of Isotropic Meshes 8 Work Examples and Industrial Applications 8.1 Hybrid Mesh 8.2 Comparison of JWM and EJW 9 Conclusions and Discussions References Part IV Numerical Geometry and Applications An Improved Algorithm for Scattered Data Interpolation Using Quartic Triangular Bézier Surfaces 1 Introduction 2 Preliminaries 2.1 Nielson's MNN 2.2 The G^1-Continuity Conditions 2.3 The Vertex Enclosure Problem 3 Outline of Our Algorithms 3.1 Determining the Set T_s 3.1.1 Solving the Vertex Enclosure Problem 3.2 Computing the Control Points of the Sub-patches 4 Examples References On Integral-Based (Transfinite) Laplace Coordinates 1 Introduction 2 Introduction to Integral-Based Barycentric Coordinates 3 2D Integral-Based Barycentric Coordinates as a Limit Case of Their Discrete Counterparts 4 2D Integral-Based Coordinates and Dirichlet Energy Minimization 5 Potential Application to Surface Generation 6 3D Integral-Based Coordinates and Dirichlet Energy Minimization 7 Conclusion References Part V Numerical Methods Fully-Implicit Collocated Finite-Volume Method for the Unsteady Incompressible Navier–Stokes Problem 1 Introduction 2 The Interface Flux Approximation 3 Analysis of Eigenvalues of Matrix Coefficients 4 The Flux Continuity on Internal Face 5 Boundary Conditions 6 Gradient Reconstruction 7 Problem Solution 8 Numerical Tests 9 Conclusion References Efficient Steady Flow Computations with Exponential Multigrid Methods 1 Introduction 2 Exponential Multigrid Frame 2.1 Exponential Time Integration 2.2 Realization of EXP1 with the Krylov Method 2.3 Preconditioned Runge-Kutta Method 2.4 The V-Cycle p-Multigrid Framework 3 Spatial Discretization 3.1 Modal Discontinuous Galerkin Method 4 Time-Stepping Strategy 5 Numerical Results 5.1 Flow Over a Circular Cylinder in Quasi-2D 5.2 Flow Over a Sphere in 3D 6 Conclusions References Exponential Time Integrators for Unsteady Advection–Diffusion Problems on Refined Meshes 1 Introduction 2 Problem Formulation and Methods 2.1 Exponential Time Integrators 2.2 ROS2 Method: Beyond Splitting 3 Numerical Experiments 3.1 Test 1: Time Dependent Source and Boundary Conditions 3.2 Test 2: Time Dependent Boundary Conditions 4 Conclusions References Author Index