A TREATISE ON THE APPLICATION OF GENERALISED COORDINATES TO THE KINETICS OF A MATERIAL SYSTEM
CONTENTS SECTION I THEORETICAL FOUNDATIONS CHAPTER 1 ?BARYCENTRIC COORDINATES AND THEIR PROPERTIES INTRODUCTION BARYCENTRIC COORDINATES FOR SIMPLICES GENERALIZED BARYCENTRIC COORDINATES 2D COORDINATES WACHSPRESS COORDINATES DISCRETE HARMONIC COORDINATES MEAN VALUE COORDINATES COMPLETE FAMILY OF COORDINATES METRIC COORDINATES POISSON COORDINATES GORDON–WIXOM COORDINATES HARMONIC COORDINATES MAXIMUM ENTROPY COORDINATES 1 2 10 LOCAL COORDINATES 1 2 11 A?NE COORDINATES 1 2 12 SIBSON COORDINATES 1 2 13 LAPLACE COORDINATES 1 2 14 HERMITE COORDINATES 1 2 15 COMPLEX COORDINATES 1 2 16 COMPARISON 3D COORDINATES OTHER COORDINATES CHAPTER 2 ?SHAPE QUALITY FOR GENERALIZED BARYCENTRIC INTERPOLATION DECONSTRUCTING THE A PRIORI ERROR ESTIMATE SHAPE QUALITY METRICS FOR SIMPLICES SHAPE QUALITY METRICS FOR POLYGONS AND POLYHEDRA INTERPOLATION ERROR ESTIMATES ON POLYGONS TRIANGULATION COORDINATES INTERPOLATION ERROR ESTIMATES ON POLYHEDRA AND POLYTOPES HARMONIC COORDINATES IN 3D AND HIGHER WACHSPRESS COORDINATES IN 3D AND HIGHER EXTENSIONS AND FUTURE DIRECTIONS CHAPTER 3 ?TRANS?NITE BARYCENTRIC COORDINATES WEIGHTED MEAN VALUE INTERPOLATION GENERAL CONSTRUCTION TRANS?NITE THREE-POINT COORDINATES TRANS?NITE LAPLACE COORDINATES TRANS?NITE WACHSPRESS COORDINATES TRANS?NITE LAPLACE AND WACHSPRESS COORDINATES COINCIDE FOR A DISK GORDON–WIXOM INTERPOLATION LAGRANGE-TYPE GORDON–WIXOM INTERPOLATION HERMITE-TYPE GORDON–WIXOM INTERPOLATION MODI?ED HERMITE-TYPE GORDON–WIXOM INTERPOLATION MODI?ED GORDON–WIXOM FOR POLYHARMONIC INTERPOLATION GENERALIZED MEAN VALUE POTENTIALS AND DISTANCE FUNCTION APPROXIMATIONS GENERALIZED MEAN VALUE POTENTIALS FOR SMOOTH DOMAINS GENERALIZED POTENTIALS FOR POLYGONS CHAPTER 4 ?BARYCENTRIC MAPPINGS CONVEX POLYGONS ARBITRARY POLYGONS BIJECTIVE BARYCENTRIC MAPPING PERTURBED TARGET POLYGONS BIJECTIVE COMPOSITE BARYCENTRIC MAPPING LIMIT OF COMPOSITE BARYCENTRIC MAPPINGS EXTENSIONS CLOSED PLANAR CURVES POLYHEDRA PRACTICAL CONSIDERATIONS CHOOSING THE COORDINATES CHOOSING THE VERTEX PATHS CHAPTER 5 ?A PRIMER ON LAPLACIANS BASIC PROPERTIES LAPLACIANS ON RIEMANNIAN MANIFOLDS EXTERIOR CALCULUS HODGE DECOMPOSITION THE SPECTRUM DISCRETE LAPLACIANS LAPLACIANS ON GRAPHS LAPLACIANS ON SIMPLICIAL MANIFOLDS STRONGLY AND WEAKLY DE?NED LAPLACIANS THE COTAN LAPLACIAN AND BEYOND DISCRETE VERSUS SMOOTH LAPLACIANS SECTION II APPLICATIONS IN COMPUTER GRAPHICS CHAPTER 6 ?MESH PARAMETERIZATION APPLICATIONS OF MESH PARAMETERIZATION NOTIONS OF TOPOLOGY TUTTE’S BARYCENTRIC MAPPING THEOREM SOLVING THE LINEAR SYSTEMS CHOOSING THE WEIGHTS CHAPTER 7 ?PLANAR SHAPE DEFORMATION COMPLEX BARYCENTRIC COORDINATES HOLOMORPHIC FUNCTIONS GENERAL CONSTRUCTION OF COMPLEX BARYCENTRIC COORDINATES MAGIC COORDINATES VARIATIONAL BARYCENTRIC COORDINATES POINT-BASED BARYCENTRIC MAPS POINT-TO-POINT BARYCENTRIC COORDINATES CONFORMAL MAPS LOG DERIVATIVE CONSTRUCTION SHAPE INTERPOLATION VARIATIONAL CONFORMAL MAPS IMPLEMENTATION DETAILS VISUALIZING PLANAR MAPS CHAPTER 8 ?MULTI-SIDED PATCHES VIA BARYCENTRIC COORDINATES BÉZIER FORM OF CURVES EVALUATION DEGREE ELEVATION MULTISIDED BÉZIER PATCHES IN HIGHER DIMENSIONS APPLICATIONS SURFACE PATCHES SPATIAL DEFORMATION CHAPTER 9 ?GENERALIZED TRIANGULATIONS GENERALIZED PRIMAL-DUAL TRIANGULATIONS SOME CLASSICAL EXAMPLES PRIMAL-DUAL TRIANGULATIONS (PDT) A CHARACTERIZATION THEOREM COMBINATORIALLY REGULAR TRIANGULATIONS (CRT) EQUIVALENCE BETWEEN PDT AND CRT PARAMETRIZATION OF PRIMAL-DUAL TRIANGULATIONS DISCRETE REPRESENTATION USING GENERALIZED TRIANGULATIONS DISCRETE EXTERIOR CALCULUS FRAMEWORK APPLICATIONS IN MESH OPTIMIZATION CHAPTER 10 ?SELF-SUPPORTING SURFACES 10 1 INTRODUCTION AND HISTORICAL OVERVIEW 10 1 1 WHAT IS A MASONRY STRUCTURE? 10 1 2 HEYMAN’S SAFE THEOREM 10 1 3 GAUDÍ AND HANGING NETS 10 1 4 MAXWELL’S RECIPROCAL DIAGRAMS 10 2 SMOOTH THEORY 10 2 1 EQUILIBRIUM EQUATIONS 10 2 2 AIRY STRESS POTENTIAL 10 2 3 RELATIVE CURVATURES 10 2 4 CURVATURE INTERPRETATION OF EQUILIBRIUM 10 3 DISCRETE THEORY 10 3 1 THRUST NETWORK ANALYSIS 10 3 2 THRUST NETWORKS AS BLOCK NETWORKS 10 3 3 DISCRETE AIRY STRESS POTENTIAL 10 3 4 FEM DISCRETIZATION OF AIRY STRESS 10 3 5 DISCRETE CURVATURE INTERPRETATION OF EQUILIBRIUM 10 4 OPTIMIZING FOR STABILITY 10 4 1 ALTERNATING OPTIMIZATION 10 4 2 DUAL FORMULATION AS VERTEX WEIGHTS 10 4 3 PERFECT LAPLACIAN OPTIMIZATION 10 4 4 RELATIVE-CURVATURE-BASED SMOOTHING 10 4 5 STEEL-GLASS STRUCTURES AND PQ FACES 10 4 6 BLOCK LAYOUTS FROM STABLE SURFACES 10 5 CONCLUSION AND OPEN PROBLEMS 10 5 1 SENSITIVITY ANALYSIS OF MASONRY STRUCTURES 10 5 2 PROGRESSIVE STABLE STRUCTURES SECTION III APPLICATIONS IN COMPUTATIONAL MECHANICS CHAPTER 11 ?APPLICATIONS OF POLYHEDRAL FINITE ELEMENTS IN SOLID MECHANICS 11 1 INTRODUCTION 11 2 GOVERNING EQUATIONS OF SOLID MECHANICS 11 3 POLYHEDRAL FINITE ELEMENT FORMULATION 11 3 1 WEAK FORM OF GOVERNING EQUATIONS 11 3 2 SHAPE FUNCTIONS 11 3 3 ELEMENT INTEGRATION 11 4 RAPID ENGINEERING ANALYSIS 11 5 FRAGMENTATION MODELING 11 5 1 FRACTURE METHODOLOGY 11 5 2 RANDOM VORONOI MESHES 11 5 3 FRAGMENTATION 11 6 SUMMARY AND OUTLOOK CHAPTER 12 ?EXTREMELY LARGE DEFORMATION WITH POLYGONAL AND POLYHEDRAL ELEMENTS 12 1 INTRODUCTION 12 2 FINITE ELASTICITY FORMULATIONS 12 2 1 DISPLACEMENT-BASED FORMULATION 12 2 2 A GENERAL TWO-?ELD MIXED VARIATIONAL FORMULATION 12 3 POLYGONAL AND POLYHEDRAL APPROXIMATIONS 12 3 1 DISPLACEMENT SPACE ON POLYGONS IN 2D 12 3 2 DISPLACEMENT SPACE ON POLYHEDRA IN 3D 12 3 3 PRESSURE SPACE ON POLYGONS IN 2D 12 4 QUADRATURE RULES AND ACCURACY REQUIREMENTS 12 5 GRADIENT CORRECTION SCHEME AND ITS PROPERTIES 12 5 1 GRADIENT CORRECTION FOR SCALAR PROBLEMS 12 5 2 GRADIENT CORRECTION FOR VECTORIAL PROBLEMS 12 6 CONFORMING GALERKIN APPROXIMATIONS 12 7 NUMERICAL EXAMPLES 12 7 1 DISPLACEMENT-BASED POLYGONAL AND POLYHEDRAL ELEMENTS 12 7 2 TWO-?ELD MIXED POLYGONAL ELEMENTS 12 8 APPLICATION TO THE STUDY OF FILLED ELASTOMERS 12 8 1 RESULTS FOR ?LLED NEO-HOOKEAN ELASTOMERS 12 8 2 RESULTS FOR A ?LLED SILICONE ELASTOMER CHAPTER 13 ?MAXIMUM-ENTROPY MESHFREE COORDINATES IN COMPUTATIONAL MECHANICS 13 1 INTRODUCTION 13 2 SELECTING BARYCENTRIC COORDINATES THROUGH ENTROPY MAXIMIZATION 13 3 INTRODUCING LOCALITY: LOCAL MAXIMUM-ENTROPY APPROXIMANTS 13 4 FURTHER EXTENSIONS 13 5 APPLICATIONS 13 5 1 HIGH-ORDER PARTIAL DI?ERENTIAL EQUATIONS 13 5 2 MANIFOLD APPROXIMATION 13 6 OUTLOOK CHAPTER 14 ?BEM-BASED FEM 14 1 INTRODUCTION 14 2 HIGH-ORDER BEM-BASED FEM IN 2D 14 2 1 CONSTRUCTION OF BASIS FUNCTIONS 14 2 2 FINITE ELEMENT METHOD 14 2 3 INTRODUCTION TO BOUNDARY ELEMENT METHODS 14 2 4 NUMERICAL EXAMPLES 14 3 ADAPTIVE BEM-BASED FEM IN 2D 14 3 1 ADAPTIVE FEM STRATEGY 14 3 2 RESIDUAL-BASED ERROR ESTIMATE FOR POLYGONAL MESHES 14 3 3 NUMERICAL EXAMPLES 14 4 DEVELOPMENTS AND OUTLOOK 14 4 1 HIERARCHICAL CONSTRUCTION FOR 3D PROBLEMS 14 4 2 CONVECTION-ADAPTED BASIS FUNCTIONS IN 3D CHAPTER 15 ?VIRTUAL ELEMENT METHODS FOR ELLIPTIC PROBLEMS ON POLYGONAL MESHES 15 1 INTRODUCTION 15 2 VIRTUAL ELEMENT SPACES AND GBC 15 2 1 GENERALITIES 15 2 2 LOWEST ORDER DISCRETE SPACE 15 2 3 GENERALIZATION TO ARBITRARY ORDER DISCRETE SPACES 15 2 4 THE NATURAL BASIS 15 2 5 A CONVENIENT BASIS 15 2 6 A LINK BETWEEN THE BASES 15 2 7 EXTENSION TO THREE DIMENSIONS 15 3 VIRTUAL ELEMENT METHOD FOR ELLIPTIC PDES 15 3 1 MODEL PROBLEM 15 3 2 OVERVIEW OF THE CONFORMING VEM 15 4 CONNECTION WITH OTHER METHODS 15 4 1 POLYGONAL AND POLYHEDRAL ?NITE ELEMENT METHOD 15 4 2 NODAL MFD METHOD 15 4 3 BEM-BASED FEM