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(xué)
講座時間：2018年11月16日17時0分
講座地點：數(shù)學(xué)學(xué)院340
主辦單位：煙臺大學(xué)數(shù)學(xué)與信息科學(xué)學(xué)院
內(nèi)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(xué)數(shù)學(xué)科學(xué)學(xué)院教授、黨委書記， 國家杰出青年基金獲得者。 主要從事非標(biāo)準(zhǔn)有限元方法，特別是彈性力學(xué)問題及相關(guān)問題的非標(biāo)準(zhǔn)有限元方法的構(gòu)造、數(shù)值分析及自適應(yīng)有限元方法等方面的研究。發(fā)表相關(guān)領(lǐng)域的論文60余篇，曾獲中國計算數(shù)學(xué)學(xué)會的“首屆青年創(chuàng)新獎”，全國百篇優(yōu)秀博士學(xué)位論文和德國洪堡研究獎學(xué)金等榮譽。 現(xiàn)任三個國際期刊的編委和北京計算數(shù)學(xué)學(xué)會理事長。
