含弹性应变能的各向异性相场模型的指数时间差分方法

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

数据与计算发展前沿 ›› 2020, Vol. 2 ›› Issue (3): 113-125.

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

所属专题：下一代互联网络技术与应用

含弹性应变能的各向异性相场模型的指数时间差分方法

王姗姗^1,²(),张鉴^1,^*()

1. 中国科学院计算机网络信息中心,北京 100190
2. 中国科学院大学,北京 100049

收稿日期:2020-03-02 出版日期:2020-06-20 发布日期:2020-08-19
通讯作者: 张鉴
作者简介:王姗姗,中国科学院计算机网络信息中心,中国科学院大学,硕士研究生,主要研究方向为高性能计算、相场模拟。
本文承担工作为：指数时间差分方法的求解格式与能量稳定性的证明,以及数值实验的实现。
Wang Shanshan is a graduate student at Computer Network Information Center of the Chinese Academy of Sciences, University of Chinese Academy of Sciences. Her main research directions are high performance calculation and phase field simulation.
In this paper, she undertakes the following work: the solution scheme of the Exponential Time Difference method, the proof of the energy stability, and the realization of the numerical experiment.
E-mail: wangshanshan@cnic.cn|张鉴,中国科学院计算机网络信息中心,研究员,博士生导师,主要研究方向为高性能计算、偏微分方程数值算法和并行算法研究。
本文承担工作为：指数时间差分方法以及相场数值模拟实验的指导。
Zhang Jian is a research fellow and the Ph.D. supervisor at Computer Network Information Center of the Chinese Academy of Sciences. His main research fields are high performance computing, PDE numerical algorithm and parallel algorithm.
In this paper, he undertakes the following tasks: guidance of the Exponential Time Difference method and the phase field numerical simulation experiment.
E-mail: zhangjian@sccas.cn
基金资助:
国家重点研发计划项目(2016YFB0201100);国家重点研发计划项目(2017YFB0202803);国家自然科学基金资助项目(11871454);国家自然科学基金资助项目(91630204);国家自然科学基金资助项目(61531166003);中国科学院战略性先导科技专项(B类)(XDB22020102);中国科学院信息化专项(XXH13506-204)

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

Wang Shanshan^1,²(),Zhang Jian^1,^*()

1. Computer Network Information Center,Chinese Academy of Sciences, Beijing 100190, China
2. University of Chinese Academy of Sciences, Beijing 100049, China

Received:2020-03-02 Online:2020-06-20 Published:2020-08-19
Contact: Zhang Jian

摘要/Abstract

摘要：

【目的】材料微结构中界面能异性和弹性应变能是产生各向异性的主要因素,本文主要研究含弹性应变能的各向异性相场模型的紧致指数时间差分方法。【方法】在紧致指数时间差分方法框架下引入了界面能异性和弹性应变能的计算,将界面能异性和弹性应变能归于指数时间差分方法的非线性项统一处理,为二者设计了算子分裂格式。【结果】从数学上证明了算子分裂格式能够保证能量稳定,并进行了镍基合金以及Zr的氢化物的材料腐蚀相场模型的数值实验,验证了含弹性应变能的各向异性相场模型的指数时间差分方法的能量稳定性。【局限】本文仅得到了指数时间差分方法的一阶和二阶求解格式,更高阶的求解格式有待进一步探索。【结论】设计了能量稳定的含弹性应变能的各向异性相场模型的指数时间差分方法。

关键词: 相场模型, 紧致指数时间差分方法, 界面能异性, 弹性应变能

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

王姗姗,张鉴. 含弹性应变能的各向异性相场模型的指数时间差分方法[J]. 数据与计算发展前沿, 2020, 2(3): 113-125.

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.

图/表 10

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表1

指数时间差分方法一阶求解格式的误差"

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	-

表1

表2

指数时间差分方法二阶求解格式的误差"

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	-

表2

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含弹性应变能的各向异性相场模型的指数时间差分方法

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

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