CDS/CIMMS Lunchtime Seminar: Discrete-Mechanics Approximation Schemes with Guaranteed Optimal Convergence in Incompressible Elasticity

Dr. Michael Ortiz, Caltech
Aeronautics and Mechanical Engineering

We present a finite-element discretization scheme for the compressible and incompressible elasticity problems that possesses  the following properties:

i) The discretization scheme is defined on a triangulation of the domain;

ii) the discretization scheme is defined---and is identical---in all spatial
    dimensions;

iii) the displacement field converges optimally with mesh refinement;

iv) the {\it $\inf$-$\sup$} condition is automatically satisfied.

The discretization scheme is motivated both by considerations of topology and analysis, and it consists of the combination of a certain mesh pattern and a choice of interpolation that guarantee optimal convergence of displacements and pressures. Rigorous proofs of the satisfaction of the {\it $\inf$-$\sup$} condition are  presented for the problem of linearized incompressible elasticity. We additionally show that the discretization schemes can be given a compelling interpretation in terms of discrete differential operators. In particular, we develop a discrete analog of the classical tensor differential complex in terms of which the discrete and continuous boundary-value problems are formally identical. We also present numerical tests that demonstrate the dimension-independent scope of the scheme and its good performance in problems of finite elasticity.