Pin-pointing the phase boundary with the mesh transformation method in a relaxed double-well problem

  • Authors:

  • Zhiping Li;Dali Men

  • Affiliations:

  • LMAM & School of Mathematical Sciences, Peking University, Beijing, P.R. China;LMAM & School of Mathematical Sciences, Peking University, Beijing, P.R. China

  • Venue:
  • MAMECTIS'10 Proceedings of the 12th WSEAS international conference on Mathematical methods, computational techniques and intelligent systems
  • Year:
  • 2010

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Abstract

The mesh transformation method (MTM), which redistributes the mesh in favor of energy minimization, is applied to numerically solving a 1-dimensional relaxed double-well phase transition model problem. In piecewise affine continuous finite element function spaces with N mesh nodes, the convergence rate of the numerical solution is shown to be O(N-1) in H1-norm. The MTM solutions are shown to preserve some important properties of the problem. In particular, with MTM, the phase boundary, which separates the regions with or without undergoing austenitic-martensitic phase transition and is considered one of the most important physical features in the macroscopic phase transition model, is shown to be pin-pointed with an accuracy of order O(N-2).