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37
On the computation of crystalline microstructure
 Acta Numerica
, 1996
"... Microstructure is a feature of crystals with multiple symmetryrelated energyminimizing states. Continuum models have been developed explaining microstructure as the mixture of these symmetryrelated states on a fine scale to minimize energy. This article is a review of numerical methods and the num ..."
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Cited by 56 (17 self)
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Microstructure is a feature of crystals with multiple symmetryrelated energyminimizing states. Continuum models have been developed explaining microstructure as the mixture of these symmetryrelated states on a fine scale to minimize energy. This article is a review of numerical methods and the numerical analysis for the computation of crystalline microstructure.
Singularities and Computation of Minimizers for Variational Problems
"... Various issues are addressed related to the computation of minimizers for variational problems. Special attention is paid (i) to problems with singular minimizers, which natural numerical schemes may fail to detect, and the role of the choice of function space for such problems, and (ii) to problems ..."
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Cited by 32 (2 self)
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Various issues are addressed related to the computation of minimizers for variational problems. Special attention is paid (i) to problems with singular minimizers, which natural numerical schemes may fail to detect, and the role of the choice of function space for such problems, and (ii) to problems for which there is no minimizer, which lead to dicult numerical questions such as the computation of microstructure for elastic materials that undergo phase transformations involving a change of shape.
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 25 (14 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.
Nonconforming finite element approximation of crystalline microstructure
 Math. Comp
, 1998
"... Abstract. We consider a class of nonconforming finite element approximations of a simply laminated microstructure which minimizes the nonconvex variational problem for the deformation of martensitic crystals which can undergo either an orthorhombic to monoclinic (double well) or a cubic to tetragona ..."
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Cited by 22 (9 self)
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Abstract. We consider a class of nonconforming finite element approximations of a simply laminated microstructure which minimizes the nonconvex variational problem for the deformation of martensitic crystals which can undergo either an orthorhombic to monoclinic (double well) or a cubic to tetragonal (triple well) transformation. We first establish a series of error bounds in terms of elastic energies for the L 2 approximation of derivatives of the deformation in the direction tangential to parallel layers of the laminate, for the L 2 approximation of the deformation, for the weak approximation of the deformation gradient, for the approximation of volume fractions of deformation gradients, and for the approximation of nonlinear integrals of the deformation gradient. We then use these bounds to give corresponding convergence rates for quasioptimal finite element approximations. 1.
Convergence of adaptive finite element methods in computational mechanics
 Proceedings of the Sixth World Congress on Computational Mechanics
, 2004
"... Abstract. The boundary value problem representing one time step of the primal formulation of elastoplasticity with positive hardening leads to a variational inequality of the second kind with some nondifferentiable functional. This paper establishes an adaptive finite element algorithm for the solu ..."
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Cited by 17 (6 self)
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Abstract. The boundary value problem representing one time step of the primal formulation of elastoplasticity with positive hardening leads to a variational inequality of the second kind with some nondifferentiable functional. This paper establishes an adaptive finite element algorithm for the solution of this variational inequality that yields the energy reduction and, up to higher order terms, the R−linear convergence of the stresses with respect to the number of loops. Applications include several plasticity models: linear isotropickinematic hardening, linear kinematic hardening, and multisurface plasticity as model for nonlinear hardening laws. For perfect plasticity the adaptive algorithm yields strong convergence of the stresses. Numerical examples confirm an improved linear convergence rate and study the performance of the algorithm in comparison with the more frequently applied maximum refinement rule. 1.
A constrained sequentiallamination algorithm for the simulation of subgrid microstructure in martensitic materials
 Computer Methods in Applied Mechanics and Engineering
, 2003
"... We present a practical algorithm for partially relaxing multiwell energy densities such as pertain to materials undergoing martensitic phase transitions. The algorithm is based on sequential lamination, but the evolution of the microstructure during a deformation process is required to satisfy a con ..."
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Cited by 16 (3 self)
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We present a practical algorithm for partially relaxing multiwell energy densities such as pertain to materials undergoing martensitic phase transitions. The algorithm is based on sequential lamination, but the evolution of the microstructure during a deformation process is required to satisfy a continuity constraint, in the sense that the new microstructure should be reachable from the preceding one by a combination of branching and pruning operations. All microstructures generated by the algorithm are in static and configurational equilibrium. Owing to the continuity constrained imposed upon the microstructural evolution, the predicted material behavior may be pathdependent and exhibit hysteresis. In cases in which there is a strict separation of micro and macrostructural lengthscales, the proposed relaxation algorithm may effectively be integrated into macroscopic finiteelement calculations at the subgrid level. We demonstrate this aspect of the algorithm by means of a numerical example concerned with the indentation of an CuAlNi shape memory alloy by a spherical indenter. Key words martensitic phase transitions, relaxation, rankone convexity, microstructure, sequential lamination, shapememory alloys, indentation. 1
Local stress regularity in scalar nonconvex variational problems
 In preparation
"... Local stress regularity in scalar nonconvex variational problems by ..."
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Cited by 15 (6 self)
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Local stress regularity in scalar nonconvex variational problems by
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 14 (8 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 mesh transformation method for computing microstructures
 Numer. Math
"... Abstract. A numerical method is established to solve the problem of minimizing a nonquasiconvex potential energy. Convergence of the method is proved both in the case on its own and in the case when it is combined with a weak boundary condition. Numerical examples are given to show that the method, ..."
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Cited by 9 (9 self)
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Abstract. A numerical method is established to solve the problem of minimizing a nonquasiconvex potential energy. Convergence of the method is proved both in the case on its own and in the case when it is combined with a weak boundary condition. Numerical examples are given to show that the method, especially when applied together with a continuation method and some other numerical techniques, is not only successful and efficient in solving problems with laminated microstructures but also capable of computing more complicated microstructures. 1.
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 7 (2 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.