Journal
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
Volume 110, Issue 3, Pages 279-300Publisher
WILEY
DOI: 10.1002/nme.5372
Keywords
Phase-field models; Dendritic Crystal Growth; Unconditional Energy Stability; Linear Elliptic Equations; Second Order; Invariant Energy Quadratization
Funding
- ASPIRE grant from the Office of the Vice President for Research at the University of South Carolina
- U.S. National Science Foundation [DMS-1200487, DMS-1418898]
- SC EPSCOR/IDEA award
- [NSF-DMS-1200487]
- [DMS-1517347]
- [NIH-2R01GM078994-05A1]
- [NSFC-11571032]
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1517347] Funding Source: National Science Foundation
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1418898, 1200487] Funding Source: National Science Foundation
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We present two accurate and efficient numerical schemes for a phase field dendritic crystal growth model, which is derived from the variation of a free-energy functional, consisting of a temperature dependent bulk potential and a conformational entropy with a gradient-dependent anisotropic coefficient. We introduce a novel Invariant Energy Quadratization approach to transform the free-energy functional into a quadratic form by introducing new variables to substitute the nonlinear transformations. Based on the reformulated equivalent governing system, we develop a first and a second order semi-discretized scheme in time for the system, in which all nonlinear terms are treated semi-explicitly. The resulting semi-discretized equations consist of a linear elliptic equation system at each time step, where the coefficient matrix operator is positive definite and thus, the semi-discrete system can be solved efficiently. We further prove that the proposed schemes are unconditionally energy stable. Convergence test together with 2D and 3D numerical simulations for dendritic crystal growth are presented after the semi-discrete schemes are fully discretized in space using the finite difference method to demonstrate the stability and the accuracy of the proposed schemes. Copyright (C) 2016 John Wiley & Sons, Ltd.
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