《A hybrid extended finite element level set method for modeling phase transformations》.pdf

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2002; 54:1209–1233 (DOI: 10.1002/nme.468) A hybrid extended nite element=level set method for modeling phase transformations 1 2 1;∗;† H. Ji , D. Chopp and J. E. Dolbow 1 Department of Civil and Environmental Engineering; Duke University; Durham; NC 27708; U.S.A. 2Engineering Science and Applied Mathematics; Northwestern University; Evanston; IL 60208; U.S.A. SUMMARY A hybrid numerical method for modelling the evolution of sharp phase interfaces on xed grids is presented. We focus attention on two-dimensional solidication problems, where the temperature eld 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 0 () polynomials and enriched with a C () function. This approach leads to jumps in temperature gradient occurring only at the phase boundary, and is shown to signicantly improve estimates for the front velocity. Temporal derivatives of the temperature eld in the vicinity of the phase front are obtained with a projection that employs discontinuous enrichment. In conjunction with a ner nite dierence grid, the Level Set method is