Nonconvex Energy Minimization and Relaxation in Computational Material Science

Abstract

The mathematical modeling of microstructures has important applications in material science, e.g. advanced materials or phase transitions, micromagnetism, homogenization, and optimization. It is typical in those models that the continuous minimization problem (M) is nonconvex and lacks classical solutions. There exist infimising sequences in (M) that have a weak limit, but non-(quasi-)convexity implies, in general, that the weak limit u is not a solution of the problem (M). In physical and in numerical experiments, we observe oscillations of strains which form a macroscopic or averaged quantity Du. The efficient numerical simulation on the macroscopic level aims to compute the weak limit u as a solution of a related Relaxed Problem. This is in contrast to microscopic mechanisms which are directly approached by a finite element minimization of (M). The two relaxations discussed are based on Young measures (G) and on quasiconvexification (Q). The presentation discusses adapted finite element strategies for relaxed problems such as the 2-well problem in one-dimension, in higher dimensions, in linearized elasticity (with hysteresis) or in related topics in micromagnetism and homogenization problems.