98-15   Berichtsreihe des Mathematischen Seminars der Universität Kiel

Matthias K. Gobbert, Andreas Prohl:

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 finite-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 finite elements: conforming, classical non-conforming, and discontinuous finite elements. Both classical elements yield solutions that depend heavily on the underlying numerical mesh. The discontinuous finite element method overcomes this problem and shows optimal convergence behavior independent of the numerical mesh.

Mathematics Subject Classification (1991): 49M07, 65K10, 65N30, 73C50, 73S10

Keywords: Ericksen-James energy density, finite element method, non-convex minimization, non-linear conjugate gradients, multi-well problem, nonlinear elasticity, materials science.

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[Thu Feb 19 18:56:34 2009]
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