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• W2022429537 abstract "The minimization of nonconvex functionals naturally arises in materials sciences where deformation gradients in certain alloys exhibit microstructures. For example, minimizing sequences of the nonconvex Ericksen-James energy can be associated with deformations in martensitic materials that are observed in experiments (2, 3). | From the numerical point of view, classical conforming and nonconforming nite element discretizations have been observed to give minimizers with their quality being highly dependent on the underlying triangulation, see (8,24,26,27) for a survey. Recently, a new approach has been proposed and analyzed in (15, 16) that is based on discontinuous nite elements to reduce the pollution eect of a general triangulation on the computed minimizer. The goal of the present paper is to propose and analyze an adaptive method, giving a more accurate resolution of laminated microstructure on arbitrary grids. In modern materials sciences, alloys are the subject of research that exhibit a memory shape eect: when cooled beyond a certain critical temperature, the crystal structure changes rapidly, and a new conguration of the atoms with less symmetry properties (martensitic phase) can be observed, exhibiting microstructure. When heated, the original (austenite) phase in the alloy is taken again and no microstructure prevails any more. | A mathematical description that is based on the elastic energy minimization has been developed for the equilibria of certain martensitic crystals in (2, 3, 12{14). The invariance of the energy density with respect to symmetry-related states of the material and their rotational invariance are the reasons for the nonconvexity of the density and multiple energy wells. We e.g. refer to (27) for a survey of dierent phase transformations that can be observed in experiments and can be described by this model. Mathematically, for a large class of boundary conditions, the gradients of energy-minimizing sequences of deformations must oscillate between the energy wells to allow the energy to converge to the lowest possible value. It is because of the non-quasiconvex character of the energy that the weak limit of these minimizing sequences is no minimizer of the problem, in general." @default.
• W2022429537 created "2016-06-24" @default.
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• W2022429537 date "1999-07-01" @default.
• W2022429537 modified "2023-09-23" @default.
• W2022429537 title "An adaptive finite element method for solving a double well problem describing crystalline microstructure" @default.
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• W2022429537 doi "https://doi.org/10.1051/m2an:1999163" @default.
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