Author: Xu Guoliang
Language: English
97870303676482013-05; Soft Cover; 242x166mm; 392 pages
Language: English
97870303676482013-05; Soft Cover; 242x166mm; 392 pages
Preface
Acronyms
Chapter 1 Elementary DifferentiaIGeometry
1.1 Parametric Representation of Surfaces
1.2 Curvatures of Surfaces
1.3 The Fundamental Equations and the Fundamental Theorem of Surfaces
1.4 Gauss-Bonnet Theorem
1.5 Differential Operators on Surfaces
1.6 Basic Properties of Differential Operators
1.7 Differential Operators Acting on Surface and Normal Vector
1.8 Some Global Properties of Surfaces
1.8.1 Green's Formulas
1.8.2 Integral Formulas of Surfaces
1.9 Differential Geometry of Implicit Surfaces
Chapter 2 Construction of Geometric Partial Differential Equations for Parametric Surfaces
2.1 Variation of Functionals for Parametric Surfaces
2.2 The Second-order Euler-Lagrange Operator
2.3 The Fourth-order Euler-Lagrange Operator
2.4 The Sixth-order Euler-Lagrange Operator
2.5 Other Euler-Lagrange Operators
2.5.1 Additivity of Euler-Lagrange Operators
2.5.2 Euler-Lagrange Operator for Surfaces with Graph Representation
2.6 Gradient Flow
2.6.1 L2-Gradient Flow for Parametric Surfaces
2.6.2 H-1 Gradient Flow for Parametric Surfaces
2.7 Other Geometric Flows
2.7.1 Area-Preserving or Volume-Preserving Second-order Geometric Flows
2.7.2 Other Sixth-order Geometric Flows
2.7.3 Geometric Flow for Surfaces with Graph Representation
2.8 Notes
2.9 Related Works
2.9.1 The Choice of Energy Functionals
2.9.2 About Geometric Flows
Chapter 3 Construction of Geometric Partial Differential Equations for Level-Set Surfaces
3.1 Variation of Functionals on Level-Set Surfaces
3.2 The Second-order Euler-Lagrange Operator
3.3 The Fourth-order Euler-Lagrange Operator
3.4 The Sixth-order Euler-Lagrange Operator
3.5 L2_Gradient Flows for Level Sets
3.6 H-1-Gradient Flow for Level Sets
3.7 Construction of Geometric Flows from Operator Conversion
3.8 Relationship Among Three Construction Methods of the Geometric Flows
Chapter 4 Discretization of Differential Geometric Operators and Curvatures
4.1 Discretization of the Laplace-Beltrami Operator over Triangular Meshes
4.1.1 Discretization of the Laplace-Beltrami Operator over Triangular Meshes
4.1.2 Convergence Test of Different Discretization Schemes of'the LB Operator
4.1.3 Convergence of the Discrete LB Operator over Triangular Meshes
4.1.4 Proof of the Convergence Results
4.2 Discretization of the Laplace-Beltrami Operator over Quadrilateral Meshes and Its Convergence Analysis
4.2.1 Discretization of LB Operator over Quadrilateral Meshes
4.2.2 Convergence Property of the Discrete LB Operator
4.2.3 Simplified Integration Rule
4.2.4 Numerical Experiments
4.3 Discretization of the Gaussian Curvature over Triangular Meshes
4.3.1 Discretization of the Gaussian Curvature over Triangular Meshes
4.3.2 Numerical Experiments
4.3.3 Convergence Properties of the Discrete Gaussian Curvatures
4.3.4 Modified Gauss-Bonnet Schemes and Their Convergence
4.3.5 A Counterexample for the Regular Vertex with Valence 4
4.4 Discretization of the Gaussian Curvature over Quadrilateral Meshes and Its Convergence Analysis
4.4.1 Discretization of the Gaussian Curvature over Quadrilateral Meshes
4.4.2 Convergence Property of the Discrete Gaussian Curvature
4.5 Consistent Approximations of Some Geometric Differential Operators
4.5.1 Consistent Discretizations of Differential Geometric Operators and Curvatures Based on the Quadratic Fitting of Surfaces
4.5.2 Convergence Property of Discrete Differential Operators
4.5.3 Consistent Discretization of Differential Operators Based on Biquadratic Interpolation
4.6 Related Work on the Discretization of the Gaussian Curvature
Chapter 5 Discrete Surface Design by Quasi Finite Difference Method
5.1 Introduction
5.2 2k-th Order Geometric Partial Differential Equations of Special Forms
5.2.1 Numerical Solving Methods
5.2.2 Comparative Results and Application Examples
5.3 Fourth-order Geometric Partial Differential Equations of General Forms
5.3.1 Numerical Solving of Fourth-order Geometric Partial Differential Equation of General Forms
5.3.2 Comparative Results and Application Examples
5.4 Minimal Mean Curvature Variation Flow
5.4.1 Numerical Solving of the Minimal Mean Curvature Variation Flow
5.4.2 Application Examples
5.5 A Note About the Convergence
5.5.1 Fully Discrete Scheme of the Boundary Conditions
5.5.2 Semi-Discretization of Boundary Conditions
Chapter 6 Spline Surface Design by Quasi Finite Difference Method and Finite Element Method
6.1 Spline Surface Construction by Quasi Finite Difference Method
6.1.1 B-spline Surface
6.1.2 Construction of Geometric Partial Differential Equation B-spline Surface
6.1.3 Minimal B-spline Surface
6.1.4 Numerical Experiments of Convergence
6.2 Spline Surface Construction by Finite Element Methods
6.2.1 GPDEs and Their Mixed Variational Forms
6.2.2 Construction Steps of GPDE Spline Surfaces
6.2.3 Numerical Examples of Convergence
6.3 Regularization of Spline Surfaces
6.3.1 L2-Gradient Flows
6.3.2 Numerical Solutions of the L2-Gradient Flows
6.3.3 Regularization of B-Spline Curves
6.4 About Finite Difference Method and Finite Element Method
6.5 Numerical Integration
6.6 Related Work
6.6.1 Bezier and B-spline Curves and Surfaces
6.6.2 Differential Equation Surfaces
6.6.3 Geometric Differential Equation Surfaces
Chapter 7 Subdivision Surface Design by Finite Element Methods
7.1 Sobolev Spaces on Surfaces
7.2 Finite Element Spaces
7.2.1 Loop's Subdivision Scheme
7.2.2 The Limit Surface Corresponding to Vertices
7.2.3 Evaluation of Regular Surface Patches
7.2.4 Evaluation of Irregular Surface Patches
7.2.5 Basis Functions and Classifications of Surface Patches
7.2.6 Parametric Representation and Isoparametric Elements
7.3 Mean Curvature Flow and Surface Modeling
7.3.1 The Background of Surface Modeling
7.3.2 A Variant of the Mean Curvature Flow
7.3.3 Numerical Solutions
7.4 Fourth-order Geometric Partial Differential Equations
7.4.1 Variational Form of the Fourth-order Equation
7.4.2 Discretization of Fourth-order Equations
7.4.3 Applications and Examples of Fourth-order Equations
7.5 Sixth-order Geometric Partial Differential Equations
7.5.1 Weak Forms
7.5.2 Disretization of the Sixth-order Equations
7.6 Subdivision Surfaces with Boundaries
7.6.1 Extended Loop's Subdivision Surfaces
7.6.2 Minimal Surface Construction
7.6.3 Gl Surface Construction
7.7 Related Work on Subdivision Surfaces
Chapter 8 Level-Set Method for Surface Design and Its Applications..
8.1 Introduction
8.2 Preliminaries
8.2.1 Cubic B-spline Interpolation
8.2.2 Runge-Kutta Method with Variable Time Step-Size
8.2.3 ENO Interpolation
8.2.4 Upwind Scheme
8.3 Local Level-Set Method
8.3.1 Algorithm Outline
8.3.2 Calculation of the Global Distance Function
8.3.3 Thin Shell of a Level Set of a Cubic Spline Function
8.3.4 Initialization
8.3.5 Evolution
8.3.6 Re-initialization
8.4 Applications of the Level-Set Method in Geometric Design
8.4.1 3D Surface Reconstruction from Scattered Data Set
8.4.2 Biomolecular Surface Construction
8.4.3 Surface Metamorphosis
8.4.4 Surface Restoration
Chapter 9 Quality Meshing with Geometric Flows
9.1 Introduction
9.2 Single-domain Triangular and Tetrahedral Quality Meshing
9.3 Single-domain Quadrilateral and Hexahedral Quality Meshing
9.4 Multi-domain Tetrahedral Quality Meshing
9.4.1 Quality Improvement Algorithm and Implementation
9.4.2 Application Examples
9.5 Multi-domain Hexahedral Quality Meshing
9.5.1 Quality Improvement Algorithm and Implementation
9.5.2 Application Examples and Discussion
9.6 Multi-domain Triangular Quality Meshing with Gaps
9.6.1 Problem Background
9.6.2 Sketch of Multi-domain Meshing Algorithm
9.6.3 Algorithm Details
9.6.4 Results
References
Index
Chapter 1 Elementary DifferentiaIGeometry
1.1 Parametric Representation of Surfaces
1.2 Curvatures of Surfaces
1.3 The Fundamental Equations and the Fundamental Theorem of Surfaces
1.4 Gauss-Bonnet Theorem
1.5 Differential Operators on Surfaces
1.6 Basic Properties of Differential Operators
1.7 Differential Operators Acting on Surface and Normal Vector
1.8 Some Global Properties of Surfaces
1.8.1 Green's Formulas
1.8.2 Integral Formulas of Surfaces
1.9 Differential Geometry of Implicit Surfaces
Chapter 2 Construction of Geometric Partial Differential Equations for Parametric Surfaces
2.1 Variation of Functionals for Parametric Surfaces
2.2 The Second-order Euler-Lagrange Operator
2.3 The Fourth-order Euler-Lagrange Operator
2.4 The Sixth-order Euler-Lagrange Operator
2.5 Other Euler-Lagrange Operators
2.5.1 Additivity of Euler-Lagrange Operators
2.5.2 Euler-Lagrange Operator for Surfaces with Graph Representation
2.6 Gradient Flow
2.6.1 L2-Gradient Flow for Parametric Surfaces
2.6.2 H-1 Gradient Flow for Parametric Surfaces
2.7 Other Geometric Flows
2.7.1 Area-Preserving or Volume-Preserving Second-order Geometric Flows
2.7.2 Other Sixth-order Geometric Flows
2.7.3 Geometric Flow for Surfaces with Graph Representation
2.8 Notes
2.9 Related Works
2.9.1 The Choice of Energy Functionals
2.9.2 About Geometric Flows
Chapter 3 Construction of Geometric Partial Differential Equations for Level-Set Surfaces
3.1 Variation of Functionals on Level-Set Surfaces
3.2 The Second-order Euler-Lagrange Operator
3.3 The Fourth-order Euler-Lagrange Operator
3.4 The Sixth-order Euler-Lagrange Operator
3.5 L2_Gradient Flows for Level Sets
3.6 H-1-Gradient Flow for Level Sets
3.7 Construction of Geometric Flows from Operator Conversion
3.8 Relationship Among Three Construction Methods of the Geometric Flows
Chapter 4 Discretization of Differential Geometric Operators and Curvatures
4.1 Discretization of the Laplace-Beltrami Operator over Triangular Meshes
4.1.1 Discretization of the Laplace-Beltrami Operator over Triangular Meshes
4.1.2 Convergence Test of Different Discretization Schemes of'the LB Operator
4.1.3 Convergence of the Discrete LB Operator over Triangular Meshes
4.1.4 Proof of the Convergence Results
4.2 Discretization of the Laplace-Beltrami Operator over Quadrilateral Meshes and Its Convergence Analysis
4.2.1 Discretization of LB Operator over Quadrilateral Meshes
4.2.2 Convergence Property of the Discrete LB Operator
4.2.3 Simplified Integration Rule
4.2.4 Numerical Experiments
4.3 Discretization of the Gaussian Curvature over Triangular Meshes
4.3.1 Discretization of the Gaussian Curvature over Triangular Meshes
4.3.2 Numerical Experiments
4.3.3 Convergence Properties of the Discrete Gaussian Curvatures
4.3.4 Modified Gauss-Bonnet Schemes and Their Convergence
4.3.5 A Counterexample for the Regular Vertex with Valence 4
4.4 Discretization of the Gaussian Curvature over Quadrilateral Meshes and Its Convergence Analysis
4.4.1 Discretization of the Gaussian Curvature over Quadrilateral Meshes
4.4.2 Convergence Property of the Discrete Gaussian Curvature
4.5 Consistent Approximations of Some Geometric Differential Operators
4.5.1 Consistent Discretizations of Differential Geometric Operators and Curvatures Based on the Quadratic Fitting of Surfaces
4.5.2 Convergence Property of Discrete Differential Operators
4.5.3 Consistent Discretization of Differential Operators Based on Biquadratic Interpolation
4.6 Related Work on the Discretization of the Gaussian Curvature
Chapter 5 Discrete Surface Design by Quasi Finite Difference Method
5.1 Introduction
5.2 2k-th Order Geometric Partial Differential Equations of Special Forms
5.2.1 Numerical Solving Methods
5.2.2 Comparative Results and Application Examples
5.3 Fourth-order Geometric Partial Differential Equations of General Forms
5.3.1 Numerical Solving of Fourth-order Geometric Partial Differential Equation of General Forms
5.3.2 Comparative Results and Application Examples
5.4 Minimal Mean Curvature Variation Flow
5.4.1 Numerical Solving of the Minimal Mean Curvature Variation Flow
5.4.2 Application Examples
5.5 A Note About the Convergence
5.5.1 Fully Discrete Scheme of the Boundary Conditions
5.5.2 Semi-Discretization of Boundary Conditions
Chapter 6 Spline Surface Design by Quasi Finite Difference Method and Finite Element Method
6.1 Spline Surface Construction by Quasi Finite Difference Method
6.1.1 B-spline Surface
6.1.2 Construction of Geometric Partial Differential Equation B-spline Surface
6.1.3 Minimal B-spline Surface
6.1.4 Numerical Experiments of Convergence
6.2 Spline Surface Construction by Finite Element Methods
6.2.1 GPDEs and Their Mixed Variational Forms
6.2.2 Construction Steps of GPDE Spline Surfaces
6.2.3 Numerical Examples of Convergence
6.3 Regularization of Spline Surfaces
6.3.1 L2-Gradient Flows
6.3.2 Numerical Solutions of the L2-Gradient Flows
6.3.3 Regularization of B-Spline Curves
6.4 About Finite Difference Method and Finite Element Method
6.5 Numerical Integration
6.6 Related Work
6.6.1 Bezier and B-spline Curves and Surfaces
6.6.2 Differential Equation Surfaces
6.6.3 Geometric Differential Equation Surfaces
Chapter 7 Subdivision Surface Design by Finite Element Methods
7.1 Sobolev Spaces on Surfaces
7.2 Finite Element Spaces
7.2.1 Loop's Subdivision Scheme
7.2.2 The Limit Surface Corresponding to Vertices
7.2.3 Evaluation of Regular Surface Patches
7.2.4 Evaluation of Irregular Surface Patches
7.2.5 Basis Functions and Classifications of Surface Patches
7.2.6 Parametric Representation and Isoparametric Elements
7.3 Mean Curvature Flow and Surface Modeling
7.3.1 The Background of Surface Modeling
7.3.2 A Variant of the Mean Curvature Flow
7.3.3 Numerical Solutions
7.4 Fourth-order Geometric Partial Differential Equations
7.4.1 Variational Form of the Fourth-order Equation
7.4.2 Discretization of Fourth-order Equations
7.4.3 Applications and Examples of Fourth-order Equations
7.5 Sixth-order Geometric Partial Differential Equations
7.5.1 Weak Forms
7.5.2 Disretization of the Sixth-order Equations
7.6 Subdivision Surfaces with Boundaries
7.6.1 Extended Loop's Subdivision Surfaces
7.6.2 Minimal Surface Construction
7.6.3 Gl Surface Construction
7.7 Related Work on Subdivision Surfaces
Chapter 8 Level-Set Method for Surface Design and Its Applications..
8.1 Introduction
8.2 Preliminaries
8.2.1 Cubic B-spline Interpolation
8.2.2 Runge-Kutta Method with Variable Time Step-Size
8.2.3 ENO Interpolation
8.2.4 Upwind Scheme
8.3 Local Level-Set Method
8.3.1 Algorithm Outline
8.3.2 Calculation of the Global Distance Function
8.3.3 Thin Shell of a Level Set of a Cubic Spline Function
8.3.4 Initialization
8.3.5 Evolution
8.3.6 Re-initialization
8.4 Applications of the Level-Set Method in Geometric Design
8.4.1 3D Surface Reconstruction from Scattered Data Set
8.4.2 Biomolecular Surface Construction
8.4.3 Surface Metamorphosis
8.4.4 Surface Restoration
Chapter 9 Quality Meshing with Geometric Flows
9.1 Introduction
9.2 Single-domain Triangular and Tetrahedral Quality Meshing
9.3 Single-domain Quadrilateral and Hexahedral Quality Meshing
9.4 Multi-domain Tetrahedral Quality Meshing
9.4.1 Quality Improvement Algorithm and Implementation
9.4.2 Application Examples
9.5 Multi-domain Hexahedral Quality Meshing
9.5.1 Quality Improvement Algorithm and Implementation
9.5.2 Application Examples and Discussion
9.6 Multi-domain Triangular Quality Meshing with Gaps
9.6.1 Problem Background
9.6.2 Sketch of Multi-domain Meshing Algorithm
9.6.3 Algorithm Details
9.6.4 Results
References
Index
You May Also Like
