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Series in Information and Computational Science·Collector's Edition(54):Geometric Partial Differential Equation Methods in Computational Geometry
Since the 1970s, Science Press has published more than thirty volumes in its series Monographs in Computational Methods. This series was established and led by the late academician, Feng Kang, the founding director of the Computing Center of the Chinese Academy of Sciences. The monograph series has provided timely information of the frontier directions and latest research results in computational mathematics. It has had great impact on young scientists and the entire research community, and has played a very important role in the development of computational mathematics in China.
To cope with these new scientific developments, the Ministry of Education of the People's Republic of China in 1998 combined several subjects, such as computational mathematics, numerical algorithms, information science, and operations research and optimal control, into a new discipline called Information and Computational Science. As a result, Science Press also reorganized the editorial board of the monograph series and changed its name to Series in Information and Computational Science. The first editorial board meeting was held in Beijing in September 2004, and it discussed the new objectives, and the directions and contents of the new monograph series.
The aim of the new series is to present the state of the art in Information and Computational Science to serior undergraduate and graduate students, as well as to scientists working in these fields. Hence, the series will provide concrete and systematic expositions of the advances in information and computational science, encompassing also related interdisciplinary developments.
I would like to thank the previous editorial board members and assistants, and all the mathematicians who have contributed significantly to the monograph series on Computational Methods. As a result of their contributions the monograph series achieved an outstanding reputation in the community. I sincerely wish that we will extend this support to the new Series in Information and Computational Science, so that the new series can equally enhance the scientific development in information and computational science in this century.
Table of Contents
Preface
Acronyms
Chapter 1 Elementary Differential Geometry
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 ofEuler-Lagrange Operators
2．5．2 Euler-Lagrange Operator for Surfaces with Graph Representation
2．6 GradientFlow
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
……
Chapter 5 Discrete Surface Design by Quasi Finite Difference Method
Chapter 6 Spline Surface Design by Quasi Finite Difference Method and FiniteElement Method
Chapter 7 Subdivision Surface Dqesign by Finite Element Methods
Chapter 8 Level-Set Method for Surface Design and Its Applications
Chapter 9 Quality Meshing with Geometric Flows
