Finite element analysis of non-convex variational problems modeling crystalline solids

Bo Li
Department of Mathematics
University of California at Los Angeles

Abstract: Crystalline solids capable of undergoing structured phase transformations can often be modeled by non-convex variational problems of elastic energy functionals. Such functionals may admit no minimizers in sets of admissible deformations. Rather, energy minimizing sequences of deformations exhibiting fine microstructure can determine properties of the underlying crystal.

In this talk, we will be concerned with the finite element approximation of such a non-convex variational problem with a boundary condition consistent with a simply laminated microstructure.

After a brief description of the background of the underlying problem, we will first show a general theorem on the existence of finite element energy minimizers and provide an upper bound of the corresponding minimum energy. We will then present an approximation theory for the simply laminated microstructure. This theory consists of a series of estimates for the strong convergence of deformations, the weak convergence of deformation gradients, the approximation of volume fractions, and the approximation of nonlinear integrals. The approximation theory will then be applied to the conforming finite element approximation to obtain a series of the corresponding error estimates for the finite element energy minimizers. Finally, we will describe the parallel results for some non-conforming finite element approximation for the underlying variational problem.

This is joint work with Mitchell Luskin.