×

Locally-synchronous, iterative solver for Fourier-based homogenization. (English) Zbl 1477.74115

Summary: We use the algebraic orthogonality of rotation-free and divergence-free fields in the Fourier space to derive the solution of a class of linear homogenization problems as the solution of a large linear system. The effective constitutive tensor constitutes only a small part of the solution vector. Therefore, we propose to use a synchronous and local iterative method that is capable to efficiently compute only a single component of the solution vector. If the convergence of the iterative solver is ensured, i.e., the system matrix is positive definite and diagonally dominant, it outperforms standard direct and iterative solvers that compute the complete solution. It has been found that for larger phase contrasts in the homogenization problem, the convergence is lost, and one needs to resort to other linear system solvers. Therefore, we discuss the linear system’s properties and the advantages as well as drawbacks of the presented homogenization approach.

MSC:

74S25 Spectral and related methods applied to problems in solid mechanics
74Q99 Homogenization, determination of effective properties in solid mechanics
65F10 Iterative numerical methods for linear systems

Software:

FFTHomPy
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Pierard O, Friebel C, Doghri I (2004) Mean-field homogenization of multi-phase thermo-elastic composites: a general framework and its validation. Composites Science and Technology 64(10):1587-1603 doi:10.1016/j.compscitech.2003.11.009. http://www.sciencedirect.com/science/article/pii/S0266353803004494
[2] Moulinec, H.; Suquet, P.; Pyrz, R., A fft-based numerical method for computing the mechanical properties of composites from images of their microstructures, IUTAM Symposium on Microstructure-Property Interactions in Composite Materials, 235-246 (1995), Netherlands, Dordrecht: Springer, Netherlands, Dordrecht · doi:10.1007/978-94-011-0059-5_20
[3] Moulinec, H.; Suquet, P., A numerical method for computing the overall response of nonlinear composites with complex microstructure, Computer Methods in Applied Mechanics and Engineering, 157, 1, 69-94 (1998) · Zbl 0954.74079 · doi:10.1016/S0045-7825(97)00218-1
[4] Wicht D, Schneider M, Böhlke T (2020) On Quasi-Newton methods in fast Fourier transform-based micromechanics, International Journal for Numerical Methods in Engineering 121 (8) (2020) 1665-1694. doi:10.1002/nme.6283, doi:10.1002/nme.6283. doi:10.1002/nme.6283
[5] Vondřejc, J.; Zeman, J.; Marek, I.; Lirkov, I.; Margenov, S.; Waśniewski, J., Analysis of a fast fourier transform based method for modeling of heterogeneous materials, Large-Scale Scientific Computing, 515-522 (2012), Berlin Heidelberg, Berlin, Heidelberg: Springer, Berlin Heidelberg, Berlin, Heidelberg · Zbl 1354.74235 · doi:10.1007/978-3-642-29843-1_58
[6] F. Willot, B. Abdallah, Y.-P. Pellegrini, Fourier-based schemes with modified green operator for computing the electrical response of heterogeneous media with accurate local fields, International Journal for Numerical Methods in Engineering 98 (7) (2014) 518-533. doi:10.1002/nme.4641, doi:10.1002/nme.4641. doi:10.1002/nme.4641 · Zbl 1352.80013
[7] Willot F (2015) Fourier-based schemes for computing the mechanical response of composites with accurate local fields. Comptes Rendus Mécanique 343(3):232-245 doi:10.1016/j.crme.2014.12.005. http://www.sciencedirect.com/science/article/pii/S1631072114002149
[8] Mishra N, Vondřejc J, Zeman J (2016) A comparative study on low-memory iterative solvers for fft-based homogenization of periodic media. Journal of Computational Physics 321:151-168 doi:10.1016/j.jcp.2016.05.041. http://www.sciencedirect.com/science/article/pii/S0021999116301863 · Zbl 1349.94072
[9] To, V-T; Monchiet, V.; To, Q., An FFT method for the computation of thermal diffusivity of porous periodic media, Acta Mechanica, 228, 9, 3019-3037 (2017) · Zbl 1381.74239 · doi:10.1007/s00707-017-1885-5
[10] Eisenlohr, P.; Diehl, M.; Lebensohn, R.; Roters, F., A spectral method solution to crystal elasto-viscoplasticity at finite strains, International Journal of Plasticity, 46, 37-53 (2013) · doi:10.1016/j.ijplas.2012.09.012
[11] Zeman, J.; Vondřejc, J.; Novák, J.; Marek, I., Accelerating a fft-based solver for numerical homogenization of periodic media by conjugate gradients, Journal of Computational Physics, 229, 21, 8065-8071 (2010) · Zbl 1197.65191 · doi:10.1016/j.jcp.2010.07.010
[12] Brisard S, Dormieux L (2010) Fft-based methods for the mechanics of composites: A general variational framework. Computational Materials Science 49(3):663-671 doi:10.1016/j.commatsci.2010.06.009. http://www.sciencedirect.com/science/article/pii/S0927025610003563
[13] Milton G (2002) The Theory of Composites. Cambridge University Press, · Zbl 0993.74002
[14] Nemat-Nasser, S.; Taya, M., On effective moduli of an elastic body containing periodocally distributed voids, Quarterly of Applied Mathematics, 39, 1, 43-59 (1981) · Zbl 0532.73009 · doi:10.1090/qam/99626
[15] Nemat-Nasser, S.; Iwakuma, T.; Hejazi, M., On composites with periodic structure, Mechanics of Materials, 1, 3, 239-267 (1982) · doi:10.1016/0167-6636(82)90017-5
[16] Iwakuma, T.; Nemat-Nasser, S., Composites with periodic microstructure, Computers & Structures, 16, 1, 13-19 (1983) · Zbl 0498.73116 · doi:10.1016/0045-7949(83)90142-6
[17] Nemat-Nasser, S.; Yu, N.; Hori, M., Solids with periodically distributed cracks, International Journal of Solids and Structures, 30, 15, 2071-2095 (1993) · Zbl 0782.73061 · doi:10.1016/0020-7683(93)90052-9
[18] Fotiu, PA; Nernat-Nasser, S., Overall properties of elastic-viscoplastic periodic composites, International Journal of Plasticity, 12, 2, 163-190 (1996) · Zbl 0858.73054 · doi:10.1016/S0749-6419(96)00002-2
[19] Chen Y (2004) Percolation and Homogenization Theories for Heterogeneous Materials, Ph.D. thesis at Massachusetts Institute of Technology
[20] Morawiec A (2004) Orientations and Rotations-Computations in Crystallographic Textures. Springer, · Zbl 1084.74002
[21] Torquato, S., Effective stiffness tensor of composite media-I. Exact series expansions, Journal of the Mechanics and Physics of Solids, 45, 9, 1421-1448 (1997) · Zbl 0974.74553 · doi:10.1016/S0022-5096(97)00019-7
[22] Vondřcjc J (2013) FFT-based method for homogenization of periodic media: Theory and applications, dissertation, Czech Technical University in Prague. https://pdfs.semanticscholar.org/5cf3/a69eedfb1e6ee260e30a02a28a899faaea46.pdf
[23] Stoer J, Bulirsch R (2007) Numerische Mathematik 1. Springer, Berlin · Zbl 0257.65001
[24] Saad, Y., Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics (2003) · Zbl 1031.65046 · doi:10.1137/1.9780898718003
[25] Stoer, J.; Bulirsch, R., Numerische Mathematik 2, Springer Verlag (2005) · Zbl 1070.65002 · doi:10.1007/978-3-540-45390-1
[26] Hackbusch, W., Iterative Solution of Large Sparse Systems of Equations, Springer International Publishing (2016) · Zbl 1347.65063 · doi:10.1007/978-3-319-28483-5
[27] Lee CE, Ozdaglar AE, Shah D Solving systems of linear equations: Locally and asynchronously, arXiv:abs/1411.2647
[28] Lee CE, Ozdaglar AE, Shah D Solving for a single component of the solution to a linear system, asynchronously
[29] Ozdaglar A, Shah D, Yu C Lee (2019) Asynchronous approximation of a single component of the solution to a linear system, IEEE Transactions on Network Science and Engineering, 1-12 doi:10.1109/TNSE.2019.2894990
[30] Milaszewicz, JP, Improving Jacobi and Gauss-Seidel iterations, Linear Algebra and its Applications, 93, 161-170 (1987) · Zbl 0628.65022 · doi:10.1016/S0024-3795(87)90321-1
[31] Gunawardena, AD; Jain, SK; Snyder, L., Modified iterative methods for consistent linear systems, Linear Algebra and its Applications, 154-156, 123-143 (1991) · Zbl 0731.65016 · doi:10.1016/0024-3795(91)90376-8
[32] Kotakemori, H.; Harada, K.; Morimoto, M.; Niki, H., A comparison theorem for the iterative method with the preconditioner (I+Smax), Journal of Computational and Applied Mathematics, 145, 2, 373-378 (2002) · Zbl 1003.65029 · doi:10.1016/S0377-0427(01)00588-X
[33] Urekew, TJ; Rencis, JJ, The importance of diagonal dominance in the iterative solution of equations generated from the boundary element method, International Journal for Numerical Methods in Engineering, 36, 20, 3509-3527 (1993) · Zbl 0800.73503 · doi:10.1002/nme.1620362007
[34] Duvigneau, F.; Liefold, S.; Höchstetter, M.; Verhey, JL; Gabbert, U., Analysis of simulated engine sounds using a psychoacoustic model, Journal of Sound and Vibration, 366, 544-555 (2016) · doi:10.1016/j.jsv.2015.11.034
[35] Andersen R, Borgs C, Chayes J, Hopcraft J, Mirrokni VS, Teng S-H (2007) Local computation of PageRank contributions, in: A. Bonato, F. R. K. Chung (Eds.), Algorithms and Models for the Web-Graph, Springer Berlin Heidelberg, pp. 150-165. doi:10.1007/978-3-540-77004-6 · Zbl 1136.68316
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.