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Multiscale resolution in the computation of crystalline microstructure

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Summary.

This paper addresses the numerical approximation of microstructures in crystalline phase transitions without surface energy. It is shown that branching of different variants near interfaces of twinned martensite and austenite phases leads to reduced energies in finite element approximations. Such behavior of minimizing deformations is understood for an extended model that involves surface energies. Moreover, the closely related question of the role of different growth conditions of the employed bulk energy is discussed. By explicit construction of discrete deformations in lowest order finite element spaces we prove upper bounds for the energy and thereby clarify the question of the dependence of the convergence rate upon growth conditions and lamination orders. For first order laminates the estimates are optimal.

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Authors and Affiliations

  1. Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098, Kiel, Germany

    Sören Bartels

  2. Department of Mathematics, ETHZ, 8092, Zurich, Switzerland

    Andreas Prohl

Authors
  1. Sören Bartels
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  2. Andreas Prohl
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Correspondence to Andreas Prohl.

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Mathematics Subject Classification (2000): 65K10, 65M50, 65N30, 73C50, 73S10

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Bartels, S., Prohl, A. Multiscale resolution in the computation of crystalline microstructure. Numer. Math. 96, 641–660 (2004). https://doi.org/10.1007/s00211-003-0483-8

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  • DOI: https://doi.org/10.1007/s00211-003-0483-8

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