A hybrid extended finite element/level set method for modeling phase transformations

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Abstract
A hybrid numerical method for modelling the evolution of sharp phase interfaces on fixed grids is presented. We focus attention on two‐dimensional solidification problems, where the temperature field evolves according to classical heat conduction in two subdomains separated by a moving freezing front. The enrichment strategies of the eXtended Finite Element Method (X‐FEM) are employed to represent the jump in the temperature gradient that governs the velocity of the phase boundary. A new approach with the X‐FEM is suggested for this class of problems whereby the partition of unity is constructed with C1(Ω) polynomials and enriched with a C0(Ω) function. This approach leads to jumps in temperature gradient occurring only at the phase boundary, and is shown to significantly improve estimates for the front velocity. Temporal derivatives of the temperature field in the vicinity of the phase front are obtained with a projection that employs discontinuous enrichment. In conjunction with a finer finite difference grid, the Level Set method is used to represent the evolution of the phase interface. An iterative procedure is adopted to satisfy the constraints on the temperature field on the phase boundary. The robustness and utility of the method is demonstrated with several benchmark problems of phase transformation. Copyright © 2002 John Wiley & Sons, Ltd.

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