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14
Finite element analysis of microstructure for the cubic to tetragonal transformation
 SIAM J. Numer. Anal
, 1998
"... Abstract. Martensitic crystals which can undergo a cubic to tetragonal phase transformation have a nonconvex energy density with three symmetryrelated, rotationally invariant energy wells. We give estimates for the numerical approximation of a firstorder laminate for such martensitic crystals. We ..."
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Cited by 19 (13 self)
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Abstract. Martensitic crystals which can undergo a cubic to tetragonal phase transformation have a nonconvex energy density with three symmetryrelated, rotationally invariant energy wells. We give estimates for the numerical approximation of a firstorder laminate for such martensitic crystals. We give bounds for the L 2 convergence of directional derivatives in the “twin ” plane, for the L 2 convergence of the deformation, for the weak convergence of the deformation gradient, for the convergence of the microstructure, and for the convergence of nonlinear integrals of the deformation gradient.
The simply laminated microstructure in martensitic crystals that undergo a cubic to orthorhombic phase transformation
, 1999
"... Abstract. We study simply laminated microstructures of a martensitic crystal capable of undergoing a cubic to orthorhombic transformation of type P (432) → P (222)′. The free energy density modeling such a crystal is minimized on six energy wells that are pairwise rankone connected. We consider th ..."
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Cited by 14 (9 self)
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Abstract. We study simply laminated microstructures of a martensitic crystal capable of undergoing a cubic to orthorhombic transformation of type P (432) → P (222)′. The free energy density modeling such a crystal is minimized on six energy wells that are pairwise rankone 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.
Approximation of a martensitic laminate with varying volume fractions
 Mathematical Modelling and Numerical Analysis M2AN
, 1999
"... Abstract. We give results for the approximation of a laminate with varying volume fractions for multiwell energy minimization problems modeling martensitic crystals that can undergo either an orthorhombic to monoclinic or a cubic to tetragonal transformation. We construct energy minimizing sequence ..."
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Cited by 11 (7 self)
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Abstract. We give results for the approximation of a laminate with varying volume fractions for multiwell 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. Nous considérons des problèmes de minimisation d’énergie avec multiples puits de potentiel. De tels problèmes modélisent, pour des cristaux martensitiques, des transitions de phase d’un réseau orthorhombique à monoclinique, ou cubique à tétragonal, par exemple. Des résultats d’approximation des structures laminaires correspondantes, avec fractions volumiques variables, sont donnés. Des suites minimisantes, avec déformations compatibles aux conditions au bord, sont construites et permettent l’obtention de plusieurs estimations d’erreur concernant l’approximation de la déformation macroscopique limite en fonction de l’énergie élastique. Finalement, nous décrivons des résultats concernant l’approximation par éléments finis de la structure laminaire avec fractions volumiques variables. 1.
A Discontinuous Finite Element Method For Solving A Multiwell Problem
 SIAM J. Numer. Anal
, 1999
"... 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 an ..."
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Cited by 4 (3 self)
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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.
An Adaptive Finite Element Method For Solving A Double Well Problem Describing Crystalline Microstructure
 Math. Model. Numer. Anal
, 1998
"... . 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 EricksenJames energy can be associated to deformations in martensitic materials that are observed in ex ..."
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Cited by 3 (2 self)
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. 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 EricksenJames 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, nonconvex minimization, multiwell problem, microstructure, multiscale, nonline...
Stability of microstructures for some martensitic transformations
 MATHEMATICAL AND COMPUTER MODELLING
, 2000
"... 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 cert ..."
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Cited by 3 (2 self)
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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.
Stability of Microstructure for Tetragonal to Monoclinic Martensitic Transformations
, 1999
"... 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 var ..."
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Cited by 3 (3 self)
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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.
Nonconforming elements in leastsquares mixed finite element methods
 Math. Comp
"... Abstract. In this paper we analyze the finite element discretization for the firstorder system least squares mixed model for the secondorder elliptic problem by means of using nonconforming and conforming elements to approximate displacement and stress, respectively. Moreover, on arbitrary regular ..."
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Cited by 1 (1 self)
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Abstract. In this paper we analyze the finite element discretization for the firstorder system least squares mixed model for the secondorder 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 lowestorder RaviartThomas element. 1.
On the Stability of Microstructure for General Martensitic Transformations
 LECTURES ON APPLIED MATHEMATICS
, 2000
"... 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 tr ..."
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Cited by 1 (1 self)
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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 Comparison Of Classical And New Finite Element Methods For The Computation Of Laminate Microstructure
, 2001
"... A geometrically nonlinear continuum theory has been developed for the equilibria of martensitic crystals based on elastic energy minimization. For these nonconvex functionals, typically no classical solutions exist, and minimizing sequences involving Young measures are studied. Direct minimizations ..."
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Cited by 1 (1 self)
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A geometrically nonlinear continuum theory has been developed for the equilibria of martensitic crystals based on elastic energy minimization. For these nonconvex functionals, typically no classical solutions exist, and minimizing sequences involving Young measures are studied. Direct minimizations using discretization based on conforming, nonconforming, 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. Onedimensional prototype problems due to Bolza and Tartar and a twodimensional numerical model of the EricksenJames 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.