Finite element analysis of nonconvex variational problems modeling crystalline microstructure

Prof. Bo Li

Department of Mathematics, University of California at Los Angeles

Crystalline solids capable of undergoing structural phase transformations can often be modeled by nonconvex 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. The numerical computation of microstructure faces many challenges due to the fine scale of oscillation and the presence of many local minima. In joint work with Luskin, we have developed an approximation theory for microstructure which identifies reliably computable quantities, validates many current numerical algorithms, and gives guidance for the development of improved computational methods. After a brief description of the background of the underlying problem, we will first give 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 the 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 nonconforming finite element approximation for the underlying variational problem.