A new finite element method for computing crystalline microstructure

Prof. Matthias Gobbert

Department of Mathematics, University of Maryland at Baltimore County

For many crystalline materials, temperature changes result in microstructure. In the phenomenon of twinning, the microstructure is formed by microscopic bands of material, where each band, called laminate, consists of material in one of the several possible crystal phases, called martensites. Mathematical models have been developed based on the minimization of an elastic energy functional. Due to the different variants of martensite, this functional is non-convex and has several minimizers. Numerical computations have been performed using conforming as well as non-conforming finite elements. Both methods suffer from suboptimal convergence rates (half order in the mesh parameter) and a strong dependence on the alignment between the numerical grid and the physical laminates. This talk presents a discontinuous finite element, for which an optimal convergence estimate for the energy (second order in the mesh parameter) has been shown. Computational results demonstrate both the convergence rate and the independence from mesh alignment. This work is joint with Andreas Prohl, Universitaet Kiel, Germany