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An adaptive finite element method for solving a double well problem describing crystalline microstructure

Published online by Cambridge University Press:  15 August 2002

Andreas Prohl
Andreas Prohl*
Affiliation:
Mathematisches Seminar, Christian-Albrechts-Universität Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany. apr@numerik.uni-kiel.de.
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Abstract

The minimization of nonconvex functionals naturally arises inmaterials sciences where deformation gradients in certain alloys exhibitmicrostructures. For example, minimizing sequences of the nonconvexEricksen-James energy can be associated with deformations in martensitic materials thatare observed in experiments[2,3]. — From the numericalpoint of view, classical conforming and nonconforming finite elementdiscretizations have been observed to give minimizerswith their quality being highlydependent on the underlying triangulation, see [8,24,26,27] for a survey. Recently, a newapproach has been proposed and analyzed in [15,16]that is based on discontinuous finite elements to reduce the pollution effectof a general triangulation on the computed minimizer.The goal of the present paper isto propose and analyzean adaptive method,giving a more accurate resolution of laminated microstructure on arbitrary grids.

Keywords

Adaptive algorithmfinite element methodnonconvex minimizationmulti-well problemmicrostructuremultiscalenonlinear elasticityshape-memory alloymaterials science.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 33 , Issue 4 , July 1999 , pp. 781 - 796
Copyright
© EDP Sciences, SMAI, 1999

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