Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-24T15:58:59.208Z Has data issue: false hasContentIssue false
  • English
  • Français

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:[email protected] (L. Chen), [email protected] (P. C. Song), [email protected] (T. N. Yang), [email protected] (Y. Z. Ji), [email protected] (Z. Wang), [email protected] (Z. G. Yang), [email protected] (L. Q. Chen)
*Corresponding author. Email addresses:[email protected] (L. Chen), [email protected] (P. C. Song), [email protected] (T. N. Yang), [email protected] (Y. Z. Ji), [email protected] (Z. Wang), [email protected] (Z. G. Yang), [email protected] (L. Q. Chen)
*Corresponding author. Email addresses:[email protected] (L. Chen), [email protected] (P. C. Song), [email protected] (T. N. Yang), [email protected] (Y. Z. Ji), [email protected] (Z. Wang), [email protected] (Z. G. Yang), [email protected] (L. Q. Chen)
*Corresponding author. Email addresses:[email protected] (L. Chen), [email protected] (P. C. Song), [email protected] (T. N. Yang), [email protected] (Y. Z. Ji), [email protected] (Z. Wang), [email protected] (Z. G. Yang), [email protected] (L. Q. Chen)
*Corresponding author. Email addresses:[email protected] (L. Chen), [email protected] (P. C. Song), [email protected] (T. N. Yang), [email protected] (Y. Z. Ji), [email protected] (Z. Wang), [email protected] (Z. G. Yang), [email protected] (L. Q. Chen)
*Corresponding author. Email addresses:[email protected] (L. Chen), [email protected] (P. C. Song), [email protected] (T. N. Yang), [email protected] (Y. Z. Ji), [email protected] (Z. Wang), [email protected] (Z. G. Yang), [email protected] (L. Q. Chen)
*Corresponding author. Email addresses:[email protected] (L. Chen), [email protected] (P. C. Song), [email protected] (T. N. Yang), [email protected] (Y. Z. Ji), [email protected] (Z. Wang), [email protected] (Z. G. Yang), [email protected] (L. Q. Chen)
Get access

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

The authors contribute equally.

References

[1] Artemev, A., Jin, Y., Khachaturyan, A.G., Three-dimensional phase field model of proper martensitic transformation, Acta Mater. 49 (2001) 1165.CrossRefGoogle Scholar
[2] Bhate, D.N., Kumar, A., Bower, A.F., Diffuse interface model for electromigration and stress voiding, J. Appl. Phys. 87 (2000) 1712.Google Scholar
[3] Bhattacharyya, S., Heo, T.W., Chang, K., Chen, L.Q., A spectral iterative method for the computation of effective properties of elastically inhomogeneous polycrystals, Commun. Comput. Phys. 11 (2012) 726.Google Scholar
[4] Cahn, J.W., On spinodal decomposition, Acta Metall. 9 (1961) 795.CrossRefGoogle Scholar
[5] Cahn, J.W., Allen, S.M., A Microscopic Theory of Domain Wall Motion and Its Experimental Verification in Fe-Al Alloy Domain Growth Kinetics, J. de Physique 38 (1977) C7.Google Scholar
[6] Chen, L., Chen, J., Ji, Y.Z., Heo, T.W., Bhattacharyya, S., Lebensohn, R., Mathaudhu, S., Liu, Z.K., Chen, L.Q., An integrated fast Fourier transform-based phase-field and crystal plasticity approach for modelling grain growth of three dimensional polycrystals, Comput. Methods in Appl. Mech. Eng. 285 (2015) 829–848.Google Scholar
[7] Chen, L., Fan, F.F., Liang, H., Chen, J., Ji, Y.Z., Zhang, S.L., Zhu, T., Chen, L.Q., A phase-field model incorporating large elasto-plastic deformation: application to lithiated silicon electrodes, J. Electrochemical Soc. 161 (2014) F1–F9.Google Scholar
[8] Chen, L., Nguyen-Xuan, X., Nguyen-Thoi, T., Zeng, K.Y., Wu, S.C., Assessment of smoothed point interpolation methods for elastic mechanics, Int. J. Num. Med. Bio. Eng 89 (2010) 1635.Google Scholar
[9] Chen, L., Zhang, H.W., Liang, L.Y., Liu, Z., Qi, Y., Lu, P., Chen, J., Chen, L.Q., Modulation of dendritic patterns during electrodeposition: A nonlinear phase-field model, J. Power Sources 300 (2015) 376.CrossRefGoogle Scholar
[10] Chen, L.Q., Phase-field models for microstructure evolution, Ann. Rev. Mater. Res. 32 (2002) 113.Google Scholar
[11] Chen, L.Q., Yang, W., Computer simulation of the domain dynamics of a quenched system with a large number of nonconserved order parameters: The grain-growth kinetics, Phys. Rev. B 50 (1994) 15752.CrossRefGoogle ScholarPubMed
[12] Hu, S.Y., Chen, L.Q., A phase-field model for evolving microstructures with strong elastic inhomogeneity, Acta Mater. 49 (2001) 1879.Google Scholar
[13] Jin, Y.M., Artemev, A., Khachaturyan, A.G., Three-dimensional phase field model of low symmetry martensitic transformation in polycrystal: simulation of martensite in AuCd alloys, Acta Mater. 49 (2001) 2309.Google Scholar
[14] Khachaturyan, A.G., Theory of structural transformations in solids, New York: Wiley, 1983.Google Scholar
[15] Kobayashi, R., Warren, J.A., Craig Carter, W., A continuum model of grain boundaries, Phys. D 140 (2000) 141.Google Scholar
[16] Leo, P.H., Lowengrub, J.S., Jou, H.J., A diffuse interface model for microstructural evolution in elastically stressed solids, Acta Mater. 46 (1998) 2113.Google Scholar
[17] Li, D.Y., Chen, L.Q., Shape evolution and splitting of coherent particles under applied stresses, Acta Mater. 47 (1998) 247.CrossRefGoogle Scholar
[18] Liu, B., Raabe, D., Roters, F., Eisenlohr, P., Lebensohn, R.A., Comparison of finite element and fast Fourier transform crystal plasticity solvers for texture prediction, Model. Simul. Mater. Sci. Eng. 18 (8) (2010) 85005.Google Scholar
[19] Loginova, I., Amberg, G., Agren, J., Phase-field simulations of non-isothermal binary alloy solidification, Acta Mater. 49 (2001) 573.Google Scholar
[20] Murali, P., Bhandakkar, T.K., Li, W., Jhon, M.H., Gao, H.J., Ahluwalia, R., Role of modulus mismatch on crack propagation and toughness enhancement in bioinspired composites, Phys. Rev. E 84 (2011) 015102.Google Scholar
[21] Mahadevan, M., Bradley, R.M., Phase field model of surface electromigration in single crystal metal thin films, Phys. D 126 (1999) 201.Google Scholar
[22] Plapp, M., Karma, A., Eutectic colony formation: a stability analysis, Phys. Rev. E 60 (1999) 6865.Google Scholar
[23] Shen, J., Yang, X., Energy stable schemes for Cahn-Hilliard phase-field model of two-phase incompressible flows, Chinese Annals of Mathematics, Series B 31 (2010) 743.CrossRefGoogle Scholar
[24] Shen, J., Yang, X., Decoupled, energy stable schemes for phase-field models of two-phase incompressible flows, SIAM Journal on Numerical Analysis 53 (2015) 279.CrossRefGoogle Scholar
[25] Wang, B., Woo, C.H., Curie temperature and critical thickness of ferroelectric thin films, J. Appl. Phys. 97 (2005) 084109.CrossRefGoogle Scholar
[26] Wang, B., Woo, C.H., Curie-Weiss law in thin-film ferroelectrics, J. Appl. Phys. 100 (2006) 044114.Google Scholar
[27] Wang, J.J., Bhattacharyya, S., Li, Q., Heo, T.W., Ma, X.Q., Chen, L.Q., Elastic solutions with arbitrary elastic inhomogeneity and anisotropy, Phil. Mag. Let. 92 (2012) 327.Google Scholar
[28] Wang, J.J., Ma, X.Q., Li, Q., Britson, J., Chen, L.Q., Phase transitions and domain structures of ferroelectric nanoparticles: Phase field model incorporating strong elastic and dielectric inhomogeneity, Acta Mater. 61 (2013) 7591.Google Scholar
[29] Wang, Y., Khachaturyan, A.G., Three-dimensional field model and computer modeling of martensitic transformations, Acta Mater. 45 (1997) 759.Google Scholar
[30] Wang, Y.U., Jin, Y.M., Khachaturyan, A.G., Three-dimensional phase field microelasticity theory of a complex elastically inhomogeneous solid, Appl. Phys. Lett. 80 (2002) 4513.Google Scholar
[31] Woo, C.H., Zheng, Y., Depolarization in modeling nano-scale ferroelectrics using the Landau free energy functional, Appl. Phys. A 91 (2008) 59.Google Scholar
[32] Yu, P., Hu, W.Y., Chen, L.Q., Du, Q., An iterative-perturbation scheme for treating inhomogeneous elasticity in phase-field models, J. Comput. Phys. 208 (2005) 34.Google Scholar
[33] Zhu, J.Z., Chen, L.Q., Shen, J., Morphological evolution during phase separation and coarsening with strong inhomogeneous elasticity, Modell. Simul. Mater. Sci. Eng. 9 (2001) 499.Google Scholar
[34] Zou, W.N., He, Q.C., Zheng, Q.S., Inclusions in a finite elastic body, Int. J. Sol. and Struct. 49 (2012) 1627.CrossRefGoogle Scholar