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A Comparison of Fourier Spectral Iterative Perturbation Method and Finite Element Method in Solving Phase-Field Equilibrium Equations

Part of: Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  27 March 2017

Pengcheng Song ,
Tiannan Yang ,
Yanzhou Ji ,
Zhuo Wang ,
Zhigang Yang ,
Longqing Chen  and
Lei Chen
Pengcheng Song*
Affiliation:
Science and Technology on Reactor Fuel and Materials Laboratory, Nuclear Power Institute of China, Chengdu 610041, China Key Laboratory for Advanced Materials of Ministry of Education, Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, China Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA
Tiannan Yang*
Affiliation:
Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA
Yanzhou Ji*
Affiliation:
Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA
Zhuo Wang*
Affiliation:
Department of Mechanical Engineering, Mississippi State University, MS 39762, USA
Zhigang Yang*
Affiliation:
Key Laboratory for Advanced Materials of Ministry of Education, Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, China
Longqing Chen*
Affiliation:
Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA
Lei Chen*
Affiliation:
Department of Mechanical Engineering, Mississippi State University, MS 39762, USA
*
*Corresponding author. Email addresses:chen@me.msstate.edu (L. Chen), 15828622005@163.com (P. C. Song), tuy123@psu.edu (T. N. Yang), yxj135@psu.edu (Y. Z. Ji), zw352@msstate.edu (Z. Wang), zgyang@tsinghua.edu.cn (Z. G. Yang), lqc3@psu.edu (L. Q. Chen)
*Corresponding author. Email addresses:chen@me.msstate.edu (L. Chen), 15828622005@163.com (P. C. Song), tuy123@psu.edu (T. N. Yang), yxj135@psu.edu (Y. Z. Ji), zw352@msstate.edu (Z. Wang), zgyang@tsinghua.edu.cn (Z. G. Yang), lqc3@psu.edu (L. Q. Chen)
*Corresponding author. Email addresses:chen@me.msstate.edu (L. Chen), 15828622005@163.com (P. C. Song), tuy123@psu.edu (T. N. Yang), yxj135@psu.edu (Y. Z. Ji), zw352@msstate.edu (Z. Wang), zgyang@tsinghua.edu.cn (Z. G. Yang), lqc3@psu.edu (L. Q. Chen)
*Corresponding author. Email addresses:chen@me.msstate.edu (L. Chen), 15828622005@163.com (P. C. Song), tuy123@psu.edu (T. N. Yang), yxj135@psu.edu (Y. Z. Ji), zw352@msstate.edu (Z. Wang), zgyang@tsinghua.edu.cn (Z. G. Yang), lqc3@psu.edu (L. Q. Chen)
*Corresponding author. Email addresses:chen@me.msstate.edu (L. Chen), 15828622005@163.com (P. C. Song), tuy123@psu.edu (T. N. Yang), yxj135@psu.edu (Y. Z. Ji), zw352@msstate.edu (Z. Wang), zgyang@tsinghua.edu.cn (Z. G. Yang), lqc3@psu.edu (L. Q. Chen)
*Corresponding author. Email addresses:chen@me.msstate.edu (L. Chen), 15828622005@163.com (P. C. Song), tuy123@psu.edu (T. N. Yang), yxj135@psu.edu (Y. Z. Ji), zw352@msstate.edu (Z. Wang), zgyang@tsinghua.edu.cn (Z. G. Yang), lqc3@psu.edu (L. Q. Chen)
*Corresponding author. Email addresses:chen@me.msstate.edu (L. Chen), 15828622005@163.com (P. C. Song), tuy123@psu.edu (T. N. Yang), yxj135@psu.edu (Y. Z. Ji), zw352@msstate.edu (Z. Wang), zgyang@tsinghua.edu.cn (Z. G. Yang), lqc3@psu.edu (L. Q. Chen)
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Abstract

This paper systematically compares the numerical implementation and computational cost between the Fourier spectral iterative perturbation method (FSIPM) and the finite element method (FEM) in solving partial differential equilibrium equations with inhomogeneous material coefficients and eigen-fields (e.g., stress-free strain and spontaneous electric polarization) involved in phase-field models. Four benchmark numerical examples, including inhomogeneous elastic, electrostatic, and steady-state heat conduction problems demonstrate that (1) the FSIPM rigorously requires uniform hexahedral (3D) and quadrilateral (2D) mesh and periodic boundary conditions for numerical implementation while the FEM permits arbitrary mesh and boundary conditions; (2) the FSIPM solutions are comparable to their FEM counterparts, and both of them agree with the analytic solutions, (3) the FSIPM is much faster in solving equilibrium equations than the FEM to achieve the accurate solutions, thus exhibiting a greater potential for large-scale 3D computations.

Keywords

Phase-fieldfourier spectral iterative perturbation method (FSIPM)finite element method (FEM)computational costnumerical implementation

MSC classification

Secondary: 35P05: General topics in linear spectral theory
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 5 , May 2017 , pp. 1325 - 1349
Copyright
Copyright © Global-Science Press 2017 

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Footnotes

†

The authors contribute equally.

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