Numerical Methods for Linear Elasticity

                    Zhiqiang Cai

               Department of Mathematics
                   Purdue University


                      Abstract

In this talk, I will give a brief review on existing numerical methods
for linear elasticity. There are two types of approaches: one based on
the displacement formulation and the other on the stress-displacement
formulation. The latter is preferable to the former for some important
practical problems: e.g., modelling of nearly incompressible or
incompressible materials; modelling of plastic materials where the
elimination of the stress tensor is difficult. In addition, the stress
are usually physical quantities of primary interest.

A mixed finite element method is to discretize the weak form of the
stress-displacement form and requires a stable combination of finite
element spaces to approximate these variables. Stress-displacement
finite elements are extremely difficult to construct. This is caused
by symmetry constraint of the stress tensor. In this talk, I will study
a stable pair that has the minimal degrees of freedom.

The second part of the talk is to develop an appraoch that stabilize
the stress-displacement system through the L2 norm least-squares
principle. The principal attractions of our least-squares method include
freedom in the choice of finite element spaces, fast multigrid solver
for the resulting algebraic equations, and free practical and sharp
a posteriori error indicator for adaptive mesh refinements. Numerical
results for a benchmark test problem of planar elasticity will be
presented.