An FFT-based fast gradient method for elastic and inelastic unit cell homogenization problems. (English) Zbl 1439.74505
Summary: Building upon the previously established equivalence of the basic scheme of Moulinec-Suquet’s FFT-based computational homogenization method with a gradient descent method, this work concerns the impact of the fast gradient method of Nesterov in the context of computational homogenization. Nesterov’s method leads to a significant speed up compared to the basic scheme for linear problems with moderate contrast, and compares favorably to the (Newton-)conjugate gradient (CG) method for problems in digital rock physics and (small strain) elastoplasticity. We present an efficient implementation requiring twice the storage of the basic scheme, but only half of the storage of the CG method.
MSC:
| 74S99 | Numerical and other methods in solid mechanics |
| 65T50 | Numerical methods for discrete and fast Fourier transforms |
| 74Q05 | Homogenization in equilibrium problems of solid mechanics |
References:
| [1] | Moulinec, H.; Suquet, P., A fast numerical method for computing the linear and nonlinear mechanical properties of composites, C. R. Acad. Sci. Ser. II, 318, 11, 1417-1423 (1994) · Zbl 0799.73077 |
| [2] | Moulinec, H.; Suquet, P., A numerical method for computing the overall response of nonlinear composites with complex microstructure, Comput. Methods Appl. Mech. Engrg., 157, 69-94 (1998) · Zbl 0954.74079 |
| [3] | Zeman, J.; Vondřejc, J.; Novak, J.; Marek, I., Accelerating a FFT-based solver for numerical homogenization of periodic media by conjugate gradients, J. Comput. Phys., 229, 21, 8065-8071 (2010) · Zbl 1197.65191 |
| [4] | Bonnet, G., Effective properties of elastic periodic composite media with fibers, J. Mech. Phys. Solids, 55, 5, 881-899 (2007) · Zbl 1170.74042 |
| [5] | Vondřejc, J., Improved guaranteed computable bounds on homogenized properties of periodic media by Fourier-Galerkin method with exact integration, Internat. J. Numer. Methods Engrg., 107, 13, 1106-1135 (2016) · Zbl 1352.74269 |
| [6] | Brisard, S.; Dormieux, L., FFT-based methods for the mechanics of composites: A general variational framework, Comput. Mater. Sci., 49, 3, 663-671 (2010) |
| [7] | Brisard, S.; Dormieux, L., 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. Engrg., 217-220, 197-212 (2012) · Zbl 1253.74101 |
| [8] | Schneider, M.; Ospald, F.; Kabel, M., Computational homogenization of elasticity on a staggered grid, Internat. J. Numer. Methods Engrg., 105, 9, 693-720 (2016) · Zbl 07868636 |
| [9] | Willot, F., Fourier-based schemes for computing the mechanical response of composites with accurate local fields, C. R. Mech., 343, 3, 232-245 (2015) |
| [10] | Schneider, M.; Merkert, D.; Kabel, M., FFT-based homogenization for microstructures discretized by linear hexahedral elements, Internat. J. Numer. Methods Engrg., 1-29 (2016), in press, Available Online |
| [11] | Eyre, D. J.; Milton, G. W., A fast numerical scheme for computing the response of composites using grid refinement, Eur. Phys. J. Appl. Phys., 6, 1, 41-47 (1999) |
| [12] | Richardson, L. F., The approximate arithmetical solution by finite differences of physical problems involving differential equations with an application to the stresses in a masonry dam, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., 210, 307-357 (1911) · JFM 42.0873.02 |
| [13] | Mishra, N.; Vondřejc, J.; Zeman, J., A comparative study on low-memory iterative solvers for FFT-based homogenization of periodic media, J. Comput. Phys., 321, 151-168 (2016) · Zbl 1349.94072 |
| [14] | Michel, J. C.; Moulinec, H.; Suquet, P., A computational scheme for linear and non-linear composites with arbitrary phase contrast, Internat. J. Numer. Methods Engrg., 52, 139-160 (2001) |
| [15] | Monchiet, V.; Bonnet, G., A polarization-based FFT iterative scheme for computing the effective properties of elastic composites with arbitrary contrast, Internat. J. Numer. Methods Engrg., 89, 11, 1419-1436 (2012) · Zbl 1242.74197 |
| [16] | Moulinec, H.; Silva, F., Comparison of three accelerated FFT-based schemes for computing the mechanical response of composite materials, Internat. J. Numer. Methods Engrg., 97, 13, 960-985 (2014) · Zbl 1352.74223 |
| [17] | Hestenes, M. R.; Stiefel, E., Methods of conjugate gradients for solving linear systems, J. Res. Natl. Bur. Stand., 49, 6, 409-436 (1952) · Zbl 0048.09901 |
| [18] | Winther, R., Some superlinear convergence results for the conjugate gradient method, SIAM J. Numer. Anal., 17, 1, 14-17 (1980) · Zbl 0447.65021 |
| [19] | 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, 430-439 (2013) |
| [20] | Kabel, M.; Böhlke, T.; Schneider, M., Efficient fixed point and Newton-Krylov solvers for FFT-based homogenization of elasticity at large deformations, Comput. Mech., 54, 6, 1497-1514 (2014) · Zbl 1309.74013 |
| [21] | Polyak, B., Gradient methods for minimizing functionals, Zh. Vychisl. Mat. Mat. Fiz., 643-653 (1963), (in Russian) · Zbl 0196.47701 |
| [22] | Qin, Z.; Goldfarb, D., Structured sparsity via alternating direction methods, J. Mach. Learn. Res., 13, 1, 1435-1468 (2012) · Zbl 1303.68108 |
| [23] | Beck, A.; Teboulle, M., A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2, 183-202 (2009) · Zbl 1175.94009 |
| [24] | Becker, S.; Bobin, J.; Candès, E. J., NESTA: A fast and accurate first-order method for sparse recovery, SIAM J. Imaging Sci., 4, 1, 1-39 (2011) · Zbl 1209.90265 |
| [25] | Sutskever, I.; Martens, J.; Dahl, G.; Hinton, G., On the importance of initialization and momentum in deep learning, (Proceedings of the 30th International Conference on Machine Learning (2013)), 1139-1147 |
| [26] | Nesterov, Y., A method for solving the convex programming problem with convergence rate \(O(1 / k^2)\), Dokl. Akad. Nauk SSSR, 269, 3, 543-547 (1983) |
| [27] | Nesterov, Y., Smooth minimization of non-smooth functions, Math. Program., 103, 127-152 (2005) · Zbl 1079.90102 |
| [28] | Su, W.; Boyd, S.; Candes, E., A differential equation for modeling Nesterov’s accelerated gradient method: theory and insights, (Ghahramani, Z.; Welling, M.; Cortes, C.; Lawrence, N.; Weinberger, K., Advances in Neural Information Processing Systems, vol. 27 (2014), Curran Associates, Inc.), 2510-2518 |
| [29] | Attouch, H.; Peypouquet, J.; Redont, P., A dynamical approach to an intertial forward-backward algorithm for convex minimization, SIAM J. Optim., 24, 1, 232-256 (2014) · Zbl 1295.90044 |
| [30] | Riesz, F., Sur une espéce de géométrique analytique des systémes de fonctions sommables, C. R. Acad. Sci. Paris, 144, 1409-1411 (1907) · JFM 38.0422.03 |
| [31] | Palais, R., Morse theory on Hilbert manifolds, Topology, 2, 299-340 (1963) · Zbl 0122.10702 |
| [32] | Polyak, B., Some methods of speeding up the convergence of iteration methods, USSR Comput. Math. Math. Phys., 27, 5, 1-17 (1964) · Zbl 0147.35301 |
| [33] | Alvarez, F., On the minimizing property of a second order dissipative system in Hilbert spaces, SIAM J. Control Optim., 38, 4, 1102-1119 (2000) · Zbl 0954.34053 |
| [34] | Attouch, H.; Goudou, X.; Redont, P., The heavy ball with friction method. The continuous dynamical system, global exploration of the local minima of a real-valued function, Commun. Contemp. Math., 2, 1, 1-34 (2000) · Zbl 0983.37016 |
| [35] | Attouch, H.; Maingé, P.; Redont, P., A dynamical approach to an inertial forward-backward algorithm for convex minimization, SIAM J. Optim., 24, 1, 232-256 (2014) · Zbl 1295.90044 |
| [36] | X. Meng, H. Chen, Accelerating Nesterov’s method for strongly convex functions with Lipschitz gradient, 2011, arXiv preprint, arXiv:1109.6058; X. Meng, H. Chen, Accelerating Nesterov’s method for strongly convex functions with Lipschitz gradient, 2011, arXiv preprint, arXiv:1109.6058 |
| [37] | Chambolle, A.; Dossal, C., On the convergence of the iterates of the “fast iterative shrinkage/thresholding algorithm”, SIAM J. Imaging Sci., 4, 1, 1-39 (2011) |
| [38] | Nečas, J., Direct Methods in the Theory of Elliptic Equations (2012), Springer: Springer Berlin · Zbl 1246.35005 |
| [39] | Cooley, J. W.; Tukey, J. W., An algorithm for the machine calculation of complex Fourier series, Math. Comp., 19, 297-301 (1965) · Zbl 0127.09002 |
| [40] | Finite elements for elastic materials and homogenization (FeelMath), Fraunhofer ITWM, http://www.itwm.fraunhofer.de/en/departments/flow-and-material-simulation/mechanics-of-materials/feelmath.html; Finite elements for elastic materials and homogenization (FeelMath), Fraunhofer ITWM, http://www.itwm.fraunhofer.de/en/departments/flow-and-material-simulation/mechanics-of-materials/feelmath.html |
| [41] | GeoDict Math2Market GmbH, http://www.geodict.de; GeoDict Math2Market GmbH, http://www.geodict.de |
| [42] | Frigo, M.; Johnson, S., FFTW: an adaptive software architecture for the FFT, (Acoustics, Speech and Signal Processing. Proceedings of the 1998 IEEE International Conference on, vol. 3 (1998)), 1381-1384 |
| [43] | Andrä, H.; Combaret, N.; Dvorkin, J.; Glatt, E.; Han, J.; Kabel, M.; Keehm, Y.; Krzikalla, F.; Lee, M.; Madonna, C.; Marsh, M.; Mukerji, T.; Saenger, E.; Sain, R.; Saxena, N.; Ricker, S.; Wiegmann, A.; Zhan, X., Digital rock physics benchmarks - part II: Computing effective properties, Comput. Geosci., 50, 33-43 (2013) |
| [44] | Fliegener, S.; Luke, M.; Gumbsch, P., 3D microstructure modeling of long fiber reinforced thermoplastics, Compos. Sci. Technol., 104, 136-145 (2014) |
| [45] | Dembo, R. S.; Eisenstat, S. C.; Steihauf, T., Inexact newton methods, SIAM J. Numer. Anal., 19, 2, 400-408 (1982) · Zbl 0478.65030 |
| [46] | Schneider, M., The sequential addition and migration method to generate representative volume elements for the homogenization of short fiber reinforced plastics, Comput. Mech. (2016), in press |
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