Special Numerical Analysis Seminar
SPEAKER: Douglas Arnold,
University of Minnesota
TITLE:
Mixed Finite Element Methods for Elasticity
DATE: Wednesday, February 20, 2008
TIME: 3:00 P.M.
PLACE: 2405 Siebel Center
201 N. Goodwin Ave., Urbana, IL
ABSTRACT
The most natural formulation for the equations of elasticity is as a
first order system, reflecting the very different nature of the
equilibrium equation and the constitutive equation. Moreover this
system applies more widely than second order formulations, for example
to incompressible, plastic, or viscoelastic materials. The first-order
system is captured variationally in the Hellinger-Reissner variational
principle, which characterizes the symmetric stress tensor field and
the displacement vector field as a saddle-point of a suitable
functional. However it has proven extremely difficult to develop
stable and effective finite element discretizations of this
formulation—so called mixed finite elements for elasticity. Efforts
to develop such methods go back to the earliest days of the finite
element method. However, stable mixed elasticity elements using
polynomial shape functions have been developed only recently using the
theory of finite element exterior calculus (FEEC). This talk will
review the subject and especially recent progress connected to FEEC,
which has led to very simple stable elements in two and three dimensions.
BIOGRAPHY
Douglas Arnold is Director of the Institute for Mathematics and its
Applications and Professor of Mathematics at the University of
Minnesota. He will serve a two-year term as president of SIAM starting
in 2009. In 2002 he was a plenary lecturer at the International
Congress of Mathematicians in Beijing and in 2006 a member of the
Program Committee for the ICM in Madrid. Prof. Arnold received his
Ph.D. in Mathematics from the University of Chicago in 1979 and has
been on the faculties of the University of Maryland and Penn State
University. Prof. Arnold's research interests include numerical
analysis, partial differential equations, mechanics, and in particular,
the interplay among these fields. The development of finite element
exterior calculus is a major direction of his current research work.