报告题目： Mixed Finite Element Methods of Elasticity Problems
报告人: 胡俊教授 （北京大学）
内容摘要：The problems that are most frequently solved in scientific and engineering computing may probably be the elasticity equations. The finite element method (FEM) was invented in analyzing the stress of the elastic structures in the 1950s. The mixed FEM within the Hellinger-Reissner (H-R) principle for elasticity yields a direct stress approximation since it takes both the stress and displacement as an independent variable. The mixed FEM can be free of locking for nearly incompressible materials, and be applied to plastic materials, and approximate both the equilibrium and traction boundary conditions more accurate. However, the symmetry of the stress plus the stability conditions make the design of the mixed FEM for elasticity surprisingly hard. In fact, ``Four decades of searching for mixed finite elements for elasticity beginning in the 1960s did not yield any stable elements with polynomial shape functions" [D. N. Arnold, Proceedings of the ICM, Vol. I : Plenary Lectures and Ceremonies (2002)]. Since the 1960s, many mathematicians have worked on this problem but compromised to weakly symmetric elements, or composite elements. In 2002, using the elasticity complexes, Arnold and Winther designed the first family of symmetric mixed elements with polynomial shape functions on triangular grids in 2D.
The talk presents a new framework to design and analyze the mixed FEM of elasticity problems, which yields optimal symmetric mixed FEMs. In addition, those elements are very easy to implement since their basis functions, based on those of the scalar Lagrange elements, can been explicitly written down by hand. The main ingredients of this framework are a structure of the discrete stress space on both simplicial and product grids, two basic algebraic results, and a two-step stability analysis method.
报告人简介: 胡俊， 北京大学数学科学学院教授、党委书记， 国家杰出青年基金获得者。 主要从事非标准有限元方法，特别是弹性力学问题及相关问题的非标准有限元方法的构造、数值分析及自适应有限元方法等方面的研究。发表相关领域的论文60余篇，曾获中国计算数学学会的“首届青年创新奖”，全国百篇优秀博士学位论文和德国洪堡研究奖学金等荣誉。 现任三个国际期刊的编委和北京计算数学学会理事长。