Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach

• Published in: International journal for numerical methods in engineering Vol. 110; no. 3; pp. 279 - 300
• Main Authors: Zhao, Jia, Wang, Qi, Yang, Xiaofeng
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 20.04.2017; Wiley Subscription Services, Inc
• Subjects: Anisotropy; Dendritic Crystal Growth; Dendritic crystals; Entropy; Invariant Energy Quadratization; Invariants; Linear Elliptic Equations; Mathematical analysis; Mathematical models; Nonlinearity; Phase‐field models; Quadratic forms; Second Order; Unconditional Energy Stability
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.5372

Summary 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 © 2016 John Wiley & Sons, Ltd.