Exponential Time Difference Method for Anisotropic Phase Field Model with Elastic Strain Energy

doi:10.11871/jfdc.issn.2096-742X.2020.03.010

Abstract

Abstract:

[Objective] The interface energy and elastic strain energy are the main factors of anisotropy in the microstructure of materials. In this paper, the compact Exponential Time Difference method of anisotropic phase field model with elastic strain energy is studied. [Methods] In the framework of the compact Exponential Time Difference method, the calculation of interface energy and elastic strain energy is introduced. The interface energy and elastic strain energy are treated as the nonlinear terms of the Exponential Time Difference method, and the operator splitting scheme is designed for them. [Results] It is mathematically proved that the operator splitting scheme can guarantee the energy stability, and the numerical experiments of the corrosion phase field model of Ni-based alloy and Zr-hydride are carried out, which verify the energy stability of the Exponential Time Difference method of the anisotropic phase field model with elastic strain energy. [Limitations] In this paper, only the first-order and second-order solutions of the Exponential Time Difference method are obtained, and the higher-order solutions need to be further explored. [Conclusions] The Exponential Time Difference method of energy stable anisotropic phase field model with elastic strain energy is designed.

Key words: phase field model, Compact Exponential Time Difference Method, interfacial anisotropy, elastic strain energy

Wang Shanshan,Zhang Jian. Exponential Time Difference Method for Anisotropic Phase Field Model with Elastic Strain Energy[J]. Frontiers of Data and Computing, 2020, 2(3): 113-125.

Figures/Tables 10

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Table 1

The error of the first order scheme of the Exponential Time Difference method"

时间步长	$δ$	$√ 2 δ$	$2 δ$	$2 √ 2 δ$	$4 δ$
$\eta $的 $L 2$ 误差	7.31E-02	1.07E-01	1.67E-01	2.70E-01	4.30E-01
$\eta $的收敛率	1.11	1.28	1.39	1.34	-
c的 $L 2$ 误差	5.05E-02	7.00E-02	1.08E-01	1.72E-01	2.71E-01
c的收敛率	0.95	1.25	1.34	1.31	-

Table 1

The error of the second order scheme of the Exponential Time Difference method"

时间步长	$δ$	$√ 2 δ$	$2 δ$	$2 √ 2 δ$	$4 δ$
$\eta $的 $L 2$ 误差	4.36E-03	7.49E-03	1.46E-02	3.39E-02	6.51E-02
$\eta $的收敛率	1.56	1.93	2.43	1.88	-
c的 $L 2$ 误差	2.39E-03	4.11E-03	8.10E-03	1.94E-02	3.81E-02
c的收敛率	1.56	1.96	2.52	1.94	-

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