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

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 c...

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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
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Abstract	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.
AbstractList	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. 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. 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. 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.
Author	Wang, Qi Zhao, Jia Yang, Xiaofeng
Author_xml	– sequence: 1 givenname: Jia surname: Zhao fullname: Zhao, Jia organization: University of North Carolina – sequence: 2 givenname: Qi surname: Wang fullname: Wang, Qi organization: Beijing Computational Science Research Center – sequence: 3 givenname: Xiaofeng surname: Yang fullname: Yang, Xiaofeng email: xfyang@math.sc.edu organization: University of South Carolina
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Copyright	Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2017 John Wiley & Sons, Ltd.
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Snippet	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... 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... 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... 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...
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SubjectTerms	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
Title	Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach
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