A comparison of classical and new finite element methods for the computation of laminate microstructure

A geometrically nonlinear continuum theory has been developed for the equil ibria of martensitic crystals based on elastic energy minimization. For the se non-convex functionals, typically no classical solutions exist, and mini mizing sequences involving Young measures are studied. Direct minimizations using discretization based on conforming, non-conforming, and discontinuou s elements have been proposed for the numerical approximation of this probl em. Theoretical results predict the superiority of the discontinuous finite element. Detailed numerical studies of the available finite element discre tizations in this paper validate the theory. One-dimensional prototype prob lems due to Bolza and Tartar and a tno-dimensional numerical model of the E ricksen-James energy are presented. Both classical elements yield solutions that possess suboptimal convergence rates and depend heavily on the underl ying numerical mesh. The discontinuous finite element method overcomes this problem and shows optimal convergence behavior independent of the numerica l mesh. (C) 2001 IMACS. Published by Elsevier Science B.V. All rights reser ved.