. We study simply laminated microstructures of a martensitic crystal capable of undergoing a cubic to orthorhombic transformation of type P (432) ! P (222) 0 . The free energy density modeling such a crystal is minimized on six energy wells that are pairwise rank-one connected. We consider the energy minimization problem with Dirichlet boundary data compatible with an arbitrary but fixed simple laminate. We first show that for all but a few isolated values of transformation strains, this problem has a unique Young measure solution solely characterized by the boundary data that represents the simply laminated microstructure. We then present a theory of stability for such a microstructure, and apply it to the conforming finite element approximation to obtain the corresponding error estimates for the finite element energy minimizers. 1. Introduction One of the most frequently observed microstructures of a martensitic crystal is a fine-scale twin or a simple laminate which is an arra...

We give results for the approximation of a laminate with varying volume fractions for multi-well energy minimization problems modeling martensitic crystals that can undergo either an orthorhombic to monoclinic or a cubic to tetragonal transformation. We construct energy minimizing sequences of deformations which satisfy the corresponding boundary condition, and we establish a series of error bounds in terms of the elastic energy for the approximation of the limiting macroscopic deformation and the simply laminated microstructure. Finally, we give results for the corresponding finite element approximation of the laminate with varying volume fractions.

Many physical materials of practical relevance can attain several variants of crystalline microstructure. The appropriate energy functional is necessarily nonconvex, and the minimization of the functional becomes a challenging problem. A new numerical method based on discontinuous finite elements and a scaled energy functional is proposed. It exhibits excellent con- vergence behavior for the energy (second order) as well as other crucial quantities of interest for general spatial meshes, contrary to standard (non-)conforming methods. Both theoretical analyses and numerical test calculations are presented and contrasted to other current finite element methods for this problem.

. The minimization of nonconvex functionals naturally arises in material sciences where deformation of certain alloys exhibit microstructures. As an example, minimizing sequences of the nonconvex Ericksen-James energy can be associated to deformations in martensitic materials that are observed in experiments, [1, 2]. --- From the numerical point of view, classical conforming and nonconforming finite element discretizations have been observed to give minimizers with their quality being highly dependent on the underlying triangulation, see [7, 8, 21, 23, 24] for a survey. Recently, a new approach based on discontinuous finite elements has been proposed and analyzed in [12, 13]. The present paper is devoted to propose and analyze an adaptive method to resolve microstructures on arbitrary grids, giving a more accurate resolution of laminate microstructure. Key words: Adaptive algorithm, finite element method, non-convex minimization, multi-well problem, microstructure, multiscale, nonline...

We analyze the stability of laminated microstructure for martensitic crystals that undergo cubic to trigonal, orthorhombic to triclinic, and trigonal to monoclinic transformations. We show that the microstructure is unique and stable for all laminates except when the lattice parameters satisfy certain identities.

We give an analysis of the stability and uniqueness of the simply laminated microstructure for all three tetragonal to monoclinic martensitic transformations. The energy density for tetragonal to monoclinic transformations has four rotationally invariant wells since the transformation has four variants. One of these tetragonal to monoclinic martensitic transformations corresponds to the shearing of the rectangular side, one corresponds to the shearing of the square base, and one corresponds to the shearing of the plane orthogonal to a diagonal in the square base. We show that the simply laminated microstructure is stable except for a class of special material parameters. In each case that the microstructure is stable, we derive error estimates for the finite element approximation.

Abstract. In this paper we analyze the finite element discretization for the first-order system least squares mixed model for the second-order elliptic problem by means of using nonconforming and conforming elements to approximate displacement and stress, respectively. Moreover, on arbitrary regular quadrilaterals, we propose new variants of both the rotated Q1 nonconforming element and the lowest-order Raviart-Thomas element. 1.

We describe a general theory for the stability of the laminated microstructure for martensitic crystals. Our theory has been applied to the orthorhombic to monoclinic transformation, the cubic to tetragonal transformation, the tetragonal to monoclinic transformation, and the cubic to orthorhombic transformation.

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 solutions exist, and minimizing sequences involving Young measures are studied. Direct minimizations using discretization based on conforming, non-conforming, and discontinuous elements have been proposed for the numerical approximation of this problem. Theoretical results predict the superiority of the discontinuous finite element. Detailed numerical studies of the available finite element discretizations in this paper validate the theory. One-dimensional prototype problems due to Bolza and Tartar and a two-dimensional numerical model of the Ericksen--James energy are presented. Both classical elements yield solutions that possess suboptimal convergence rates and 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. 2001 IMACS. Published by Elsevier Science B.V. All rights reserved.

Abstract. 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. Key words. Ericksen-James energy density, nite element method, non-convex minimization, nonlinear conjugate gradients, multi-well problem, nonlinear elasticity, materials science. AMS subject classi cations. 49M07, 65K10, 65N30, 73C50, 73S10. 1 Introduction. Many new materials of interest in materials science and structural mechanics have been found to exhibit microstructure under certain ambient conditions. For example, certain alloys show laminate microstructure that can be observed in laboratory