講座主題：Mixed Finite Element Methods of Elasticity Problems
專家姓名：胡俊
工作單位：北京大學
講座時間：2018年11月16日17時0分
講座地點：數學學院340
主辦單位：煙臺大學數學與信息科學學院
內容摘要：
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.
主講人介紹：
胡俊， 北京大學數學科學學院教授、黨委書記， 國家杰出青年基金獲得者。 主要從事非標準有限元方法，特別是彈性力學問題及相關問題的非標準有限元方法的構造、數值分析及自適應有限元方法等方面的研究。發表相關領域的論文60余篇，曾獲中國計算數學學會的“首屆青年創新獎”，全國百篇優秀博士學位論文和德國洪堡研究獎學金等榮譽。 現任三個國際期刊的編委和北京計算數學學會理事長。
