zbMATH — the first resource for mathematics

Efficient fixed point and Newton-Krylov solvers for FFT-based homogenization of elasticity at large deformations. (English) Zbl 1309.74013
Summary: In recent years the FFT-based homogenization method of Moulinec and Suquet has been established as a fast, accurate and robust tool for obtaining effective properties in linear elasticity and conductivity problems. In this work we discuss FFT-based homogenization for elastic problems at large deformations, with a focus on the following improvements. Firstly, we exhibit the fixed point method introduced by Moulinec and Suquet for small deformations as a gradient descent method. Secondly, we propose a Newton-Krylov method for large deformations. We give an example for which this methods needs approximately 20 times less iterations than Newton’s method using linear fixed point solvers and roughly 100 times less iterations than the nonlinear fixed point method. However, the Newton-Krylov method requires 4 times more storage than the nonlinear fixed point scheme. Exploiting the special structure we introduce a memory-efficient version with 40 % memory saving. Thirdly, we give an analytical solution for the micromechanical solution field of a two-phase isotropic St.Venant-Kirchhoff laminate. We use this solution for comparison and validation, but it is of independent interest. As an example for a microstructure relevant in engineering we discuss finally the application of the FFT-based method to glass fiber reinforced polymer structures.

74B20 Nonlinear elasticity
45G10 Other nonlinear integral equations
65T50 Numerical methods for discrete and fast Fourier transforms
Full Text: DOI
[1] Advani SG, Tucker III, CL (1987) The use of tensors to describe and predict fiber orientation in short fiber composites. J Rheol 31(8), 751-784, doi:10.1122/1.549945. URL http://link.aip.org/link/?JOR/31/751/1
[2] Agoras, M; Castañeda, PP, Multi-scale homogenization of semi-crystalline polymers, Phil Mag, 92, 925-958, (2012)
[3] Axelsson, O; Kaporin, IE, On the sublinear and superlinear rate of convergence of conjugate gradient methods, Numer Algorithm, 25, 1-22, (2000) · Zbl 0972.65024
[4] Bertram, A; Böhlke, T; Šilhavý, M, On the rank 1 convexity of stored energy functions of physically linear stress-strain relations, J Elast, 86, 235-243, (2007) · Zbl 1124.74004
[5] Boyd JP (1989) Chebyshev and Fourier spectral methods. Springer-Verlag, Berlin · Zbl 1242.74197
[6] Braess D (2007) Finite elements: theory. Fast solvers and applications in solid mechanics. Cambridge University Press, Cambridge · Zbl 1118.65117
[7] Brighi B, Bousselsal M (1995) On the rank-one-convexity domain of the Saint Venant-Kirchhoff stored energy function. Rendiconti del Seminario Matematico della Università di Padova 94, 25-45. URL http://eudml.org/doc/108375 · Zbl 0846.73010
[8] Brisard S, Dormieux L (2010) FFT-based methods for the mechanics of composites: A general variational framework. Comput Mater Sci 49(3), 663-671. doi:10.1016/j.commatsci.2010.06.009. URL: http://www.sciencedirect.com/science/article/pii/S0927025610003563 · Zbl 1054.74708
[9] Brisard S, Dormieux L (2012) Combining Galerkin approximation techniques with the principle of Hashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites. Comput Methods Appl Mech Eng 217-220(0):197-212. doi:10.1016/j.cma.2012.01.003. URL http://www.sciencedirect.com/science/article/pii/S0045782512000059 · Zbl 1253.74101
[10] Castañeda, PP, Exact second-order estimates for the effective mechanical properties of nonlinear composite materials, J Mech Phys Solids, 44, 827-862, (1996) · Zbl 1054.74708
[11] Castañeda, PP; Suquet, P, Nonlinear composites, Adv Appl Mech, 34, 171-302, (1998) · Zbl 0889.73049
[12] Ciarlet PG (1988) Mathematical elasticity: three-dimensional elasticity, vol I. Elsevier, Amsterdam · Zbl 0648.73014
[13] Eisenlohr P., Diehl, M., Lebensohn, R., Roters, F.: A spectral method solution to crystal elasto-viscoplasticity at finite strains. Int J Plast 46(0), 37-53 (2013). doi:10.1016/j.ijplas.2012.09.012. URL http://www.sciencedirect.com/science/article/pii/S0749641912001428
[14] Eyre, DJ; Milton, GW, A fast numerical scheme for computing the response of composites using grid refinement, Eur Phys J, 6, 41-47, (1999)
[15] Feyel, F., Chaboche, J.L.: FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre sic/ti composite materials. Comput Methods Appl Mech Eng, 183(3-4), 309-330 (2000). doi:10.1016/S0045-7825(99)00224-8. URL http://www.sciencedirect.com/science/article/pii/S0045782599002248 · Zbl 0993.74062
[16] Francfort, G, Homogenization and linear thermoelasticity, SIAM J Math Anal, 14, 696-708, (1983) · Zbl 0525.73002
[17] Gélébart, L., Mondon-Cancel, R.: Non-linear extension of FFT-based methods accelerated by conjugate gradients to evaluate the mechanical behavior of composite materials. Comput Mater Sci 77(0), 430-439 (2013). doi:10.1016/j.commatsci.2013.04.046. URL http://www.sciencedirect.com/science/article/pii/S0927025613002188
[18] Hestenes, MR; Stiefel, E, Methods of conjugate gradients for solving linear systems, J Res Nat Bureau Stand, 49, 409-436, (1952) · Zbl 0048.09901
[19] Hill, R, On constitutive macro-variables for heterogeneous solids at finite strain, Proc R Soc Lond A, 326, 131-147, (1972) · Zbl 0229.73004
[20] Johnson SG, Frigo M (2007) A modified split-radix FFT with fewer arithmetic operations. Signal Process IEEE Trans on 55(1):111-119 · Zbl 1390.65168
[21] Kocks UF, Tome CN, Wenk HR (1998) Texture and anisotropy: preferred orientations in polycrystals and their effect on materials properties. Cambridge University Press, Cambridge · Zbl 0916.73001
[22] Krawietz A (1986) Materialtheorie. Springer-Verlag, Berlin · Zbl 0593.73001
[23] Kröner E (1971) Statistical continuum mechanics. Springer, Wien
[24] Kröner, E.: Bounds for effective elastic moduli of disordered materials. J Mech Phys Solids 25(2), 137-155 (1977). doi:10.1016/0022-5096(77)90009-6. URL http://www.sciencedirect.com/science/article/pii/0022509677900096
[25] Lahellec, N., Michel, J.C., Moulinec, H., Suquet, P.: Analysis of inhomogeneous materials at large strains using fast Fourier transforms. In: C. Miehe (ed.) IUTAM Symposium on computational mechanics of solid materials at large strains, Solid mechanics and its applications, vol. 108, pp. 247-258. Springer, Netherlands (2003). doi:10.1007/978-94-017-0297-3_22. URL http://dx.doi.org/10.1007/978-94-017-0297-3_22 · Zbl 1039.74040
[26] Michel, JC; Moulinec, H; Suquet, P, A computational scheme for linear and non-linear composites with arbitrary phase contrast, Int J Numer Methods Eng, 52, 139-160, (2001)
[27] Milton GW (2002) The theory of composites. Cambridge University Press, Cambridge · Zbl 0993.74002
[28] Monchiet, V; Bonnet, G, A polarization-based FFT iterative scheme for computing the effective properties of elastic composites with arbitrary contrast, Int J Numer Methods Eng, 89, 1419-1436, (2012) · Zbl 1242.74197
[29] Monchiet, V., Bonnet, G.: Numerical homogenization of nonlinear composites with a polarization-based FFT iterative scheme. Comput Mater Sci 79(0), 276-283 (2013). doi:10.1016/j.commatsci.2013.04.035. URL http://www.sciencedirect.com/science/article/pii/S0927025613002073
[30] Moulinec, H; Suquet, P, A fast numerical method for computing the linear and nonlinear mechanical properties of composites, Comptes Rendus de l’Académie des Sciences. Série II, Mécanique, Physique, Chimie, Astronomie, 318, 1417-1423, (1994) · Zbl 0799.73077
[31] Moulinec, H; Suquet, P, A numerical method for computing the overall response of nonlinear composites with complex microstructure, Comput Methods Appl Mech Eng, 157, 69-94, (1998) · Zbl 0954.74079
[32] Moulinec, H., Suquet, P.: Comparison of FFT-based methods for computing the response of composites with highly contrasted mechanical properties. Physica B: Condens Matter 338(1-4), 58-60 (2003). doi:10.1016/S0921-4526(03)00459-9. URL http://www.sciencedirect.com/science/article/pii/S0921452603004599. Proceedings of the Sixth International Conference on Electrical Transport and Optical Properties of Inhomogeneous Media
[33] Mura T (1987) Micromechanics of defects in solids. Mechanics of elastic and inelastic solids, 2nd edn. Martinus Nijhoff Publishers, Dordrecht · Zbl 0229.73004
[34] Nemat-Nasser S (1993) Micromechanics: overall properties of heterogeneous materials, North-Holland series in applied mathematics and mechanics. Elsevier Science Publishers B.V, Amsterdam · Zbl 1124.74004
[35] Ortega, JM, The Newton-Kantorovich theorem, Am Math Mon, 75, 658-660, (1968) · Zbl 0183.43004
[36] Ortega JM, Rheinboldt W (1970) Iterative solution of nonlinear equations in several variables. Academic Press, New York · Zbl 0241.65046
[37] Paige, CC; Saunders, MA, Solution of sparse indefinite systems of linear equations, SIAM J Numer Anal, 12, 617-629, (1975) · Zbl 0319.65025
[38] Schladitz K, Peters S, Reinel-Bitzer D, Wiegmann A, Ohser J (2006) Design of acoustic trim based on geometric modeling and flow simulation for non-woven. Comput Mater Sci 38(1), 56-66. doi:10.1016/j.commatsci.2006.01.018. URL http://www.sciencedirect.com/science/article/pii/S092702560600019X
[39] Schneider M (2014) Convergence of FFT-based homogenization for strongly heterogeneous media. Mathematical methods in the applied sciences n/a(n/a), n/a-n/a. doi:10.1002/mma.3259. URL http://dx.doi.org/10.1002/mma.3259
[40] Šilhavý M (1997) The mechanics and thermodynamics of continuous media. Springer, New York · Zbl 0870.73004
[41] Smit RJM, Brekelmans WAM, Meijer HEH (1998) Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling. Comput Methods Appl Mech Eng 155(1-2):181-192. doi:10.1016/S0045-7825(97)00139-4. URL http://www.sciencedirect.com/science/article/pii/S0045782597001394
[42] Spahn J, Andrä H, Kabel M, Müller R (2014) A multiscale approach for modeling progressive damage of composite materials using fast Fourier transforms. Compu Methods Appl Mech Eng 268(0):871-883. doi:10.1016/j.cma.2013.10.017. URL http://www.sciencedirect.com/science/article/pii/S0045782513002697 · Zbl 1295.74006
[43] Truesdell C, Noll W (1965). The non-linear field theories of mechanics, encyclopedia of physics, vol. III. Springer URL http://books.google.de/books?id=dp84F_odrBQC · Zbl 0779.73004
[44] Vinogradov, V; Milton, GW, An accelerated FFT algorithm for thermoelastic and non-linear composites, Int J Numer Methods Eng, 76, 1678-1695, (2008) · Zbl 1195.74302
[45] Vondrejc J, Zeman J, Marek I (2011) Analysis of a fast Fourier transform based method for modeling of heterogeneous materials. In: Lirkov I, Margenov S, Wasniewski J (eds) LSSC Lecture Notes Computer Science. Springer, Berlin, pp 515-522. doi:10.1007/978-3-642-29843-1_58 · Zbl 1354.74235
[46] Vondřejc J (2013) FFT-based method for homogenization of periodic media: theory and applications. Ph.D. thesis, Department of Mechanics, Faculty of Civil Engineering, Czech Technical University, Czech Republic, Prague (2013).
[47] Zeller, R; Dederichs, PH, Elastic constants of polycrystals, Physica Status Solidi (b), 55, 831-842, (1973)
[48] Zeman J, Vondřejc J, Novák J, Marek I (2010) Accelerating a FFT-based solver for numerical homogenization of periodic media by conjugate gradients. J Comput Phys 229:8065-8071. doi:10.1016/j.jcp.2010.07.010. URL http://www.sciencedirect.com/science/article/pii/S0021999110003931 · Zbl 1197.65191
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.