10:00-11:00 Smooth Splines on Surfaces with General Topology
Bernard Mourrain 教授，法国国家信息与自动化研究所（INRIA）
11:00-12:00 A polygonal spline method for general 2nd-order elliptic equations and its applications
报告题目：Smooth Splines on Surfaces with General Topology
报告人：Bernard Mourrain 教授，法国国家信息与自动化研究所（INRIA）
报告摘要：In CAGD, a standard representation of shapes is by parameterized surfaces or volumes based on tensor product B-spline functions, which are the basis of the space of piecewise polynomial functions on a grid with a given regularity and degree. These pieces are trimmed and assembled to model complex objects with arbitrary topology. We would like to replace this type of constructions by structured spline representations adapted to the shape of a given model. We consider the space of piecewise polynomial differentiable functions on a quad mesh of general topology. This vector space of spline functions is characterized by glueing data across the shared edges. Using algebraic techniques, which involve the analysis of the module of syzygies of the glueing data, we give dimension formula for the space of geometrically smooth splines of degree k, for arbitrary topology when $k$ is big enough. We provide explicit constructions of basis functions attached respectively to vertices, edges and faces. Applications to Isogeometric Analysis are briefly discussed. This is a joint work with A. Blidia and N. Villamizar.
Bernard MOURRAIN教授长期从事在几何造型、计算机辅助几何设计、计算符号代数等方向的研究。符号计算领域Journal of Symbolic Computation、SIAM Journal on Applied Algebra and Geometry编委成员、计算几何领域Theoretical Computer Science, Computer Aided Geometric Design等特邀编辑。多个计算几何领域国际核心期刊，如Applicable Algebra in Engineering Communication and Computing, Discrete Applied Mathematics, Theoretical Computer Science, Computer Aided Geometric Design, Computer Aided Design, Math. Of Comp, Math, Review, CRAS的评审人以及ISSAC、Geometric Modeling and Processing、Symposium on Solid and Physical Modeling, Symbolic-Numeric Computation, MEGA, ACA, ADG, ACSM等计算几何与符号计算国际会议Program chair, program co-chair以及Member of program committees.
报告题目：A polygonal spline method for general 2nd-order ellipticequations and its applications
报告摘要：We explain how to use polygonal splines to numerically solve second-order elliptic partial differential equations. The convergence of the polygonal spline method will be studied. Also, we will use this approach to numerically study the solution of some mixed parabolic and hyperbolic partial differential equations. Comparison with standard bivariate spline method will be given to demonstrate that our polygonal splines have some better numerical performance.
Dr. Ming-Jun Lai is a professor of mathematics, University of Georgia, U.S.A. He has been a professor for more than 20 years. His research has been supported by U.S. National Science Foundation many times and he has published more than 120 papers with one monograph on multivariate splines.He has graduated 16 Ph.D. and has 3 Ph.D. students under his supervision now. He is an expert on multivariate splines and their applications in computer aided geometric design, numerical solution of partial differential equations, and wavelet analysis.