A Survey of Classical and New Finite Element Methods for the Computation of Crystalline Microstructure

Recently, a geometrically nonlinear continuum theory has been developed for the equilibria of martensitic crystals based on elastic energy minimization. For these non-convex functionals, typically no classical solution exists, and minimizing sequences involving Young measures are studied. This paper presents an extensive computational survey of nite-element discretizations designed for this non-convex minimization problem supporting theoretical results previously obtained by the authors. Case studies for non-convex prototype problems are shown that compare the performance of three nite elements: conforming, classical non-conforming, and discontinuous nite elements. Both classical elements yield solutions that depend heavily on the underlying numerical mesh. The discontinuous nite element method overcomes this problem and shows optimal convergence behavior independent of the numerical mesh.