计算几何中的几何偏微分方程方法（英文版）

    Guoliang Xu和Qin
Zhang编写的这本《计算几何中的几何偏微分方程方法》的主要内容包括几何偏微分方程的构造方法、各种微分几何算子的离散化方法及其离散格式的收敛性、几何偏微分方程数值求解的有限差分法、有限元法以及水平集方法，还包括几何偏微分方程在曲而平滑、曲面拼接、N边洞填补、自由曲面设计、曲面重构、曲而恢复、分子曲面构造以及三维实体几何形变中的应用。
本书内容新颖、文字简练、可读性强，可作为理工科院校的应用数学、计算数学、计算几何、计算机辅助设计以及计算机图形学等专业本科生和研究生的教材，也可作为在上述领域中从事研究工作的广大科技工作者的参考书。

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