Energy-based comparison between the Fourier-Galerkin method and the finite element method. (English) Zbl 1433.65314
Summary: The Fourier-Galerkin method (in short FFTH) has gained popularity in numerical homogenisation because it can treat problems with a huge number of degrees of freedom. Because the method incorporates the fast Fourier transform (FFT) in the linear solver, it is believed to provide an improvement in computational and memory requirements compared to the conventional finite element method (FEM). Here, we systematically compare these two methods using the energetic norm of local fields, which has the clear physical interpretation as being the error in the homogenised properties. This enables the comparison of memory and computational requirements at the same level of approximation accuracy. We show that the methods’ effectiveness relies on the smoothness (regularity) of the solution and thus on the material coefficients. Thanks to its approximation properties, FEM outperforms FFTH for problems with jumps in material coefficients, while ambivalent results are observed for the case that the material coefficients vary continuously in space. FFTH profits from a good conditioning of the linear system, independent of the number of degrees of freedom, but generally needs more degrees of freedom to reach the same approximation accuracy. More studies are needed for other FFT-based schemes, non-linear problems, and dual problems (which require special treatment in FEM but not in FFTH).
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
energy-based comparison; finite element method; Fourier-Galerkin method; fast Fourier transform; numerical homogenisation; computational effectivenessReferences:
| [1] | de Geus, T. W.J.; van Duuren, J. E.P.; Peerlings, R. H.J.; Geers, M. G.D., Fracture initiation in multi-phase materials: A statistical characterization of microstructural damage sites, Mater. Sci. Eng. A, 673, 551-556 (2016) |
| [2] | de Geus, T. W.J.; Peerlings, R. H.J.; Geers, M. G.D., Microstructural modeling of ductile fracture initiation in multi-phase materials, Eng. Fract. Mech., 147, 318-330 (2015) |
| [3] | de Geus, T. W.J.; Popović, M.; Ji, W.; Rosso, A.; Wyart, M., How collective asperity detachments nucleate slip at frictional interfaces, Proc. Nat. Acad. Sci. (2019) |
| [4] | Němeček, J.; Králík, V.; Vondřejc, J., A two-scale micromechanical model for aluminium foam based on results from nanoindentation, Comput. Struct., 128, 136-145 (2013) |
| [5] | Němeček, J.; Králík, V.; Vondřejc, J., Micromechanical analysis of heterogeneous structural materials, Cem. Concr. Compos., 36, 1, 85-92 (2013) |
| [6] | Boyd, J. P., Chebyshev and Fourier Spectral Methods (2001), Courier Corporation: Courier Corporation New York · Zbl 0994.65128 |
| [7] | Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Thomas, Z. A., Spectral Methods: Fundamentals in Single Domains, Scientific Computation (2006), Springer: Springer Berlin, Heidelberg · Zbl 1093.76002 |
| [8] | J. Saranen, G. Vainikko, Periodic Integral and Pseudodifferential Equations with Numerical Approximation, in: Springer Monographs Mathematics, Berlin, Heidelberg, 2002. · Zbl 0991.65125 |
| [9] | Cain, A.; Ferziger, J.; Reynolds, W., Discrete orthogonal function expansions for non-uniform grids using the fast fourier transform, J. Comput. Phys., 56, 2, 272-286 (1984) · Zbl 0557.65097 |
| [10] | Cai, W.; Gottlieb, D.; Shu, C.-W., Essentially nonoscillatory spectral fourier methods for shock wave calculations, Math. Comp., 52, 186, 389 (1989) · Zbl 0666.65067 |
| [11] | Luciano, R.; Barbero, E., Formulas for the stiffness of composites with periodic microstructure, Int. J. Solids Struct., 31, 21, 2933-2944 (1994) · Zbl 0943.74513 |
| [12] | Vondřejc, J.; Zeman, J.; Marek, I., An FFT-based Galerkin method for homogenization of periodic media, Comput. Math. Appl., 68, 3, 156-173 (2014) · Zbl 1369.65151 |
| [13] | Moulinec, H.; Suquet, P., A fast numerical method for computing the linear and nonlinear mechanical properties of composites, C. R. Acad. Sci. Sér. II, 318, 11, 1417-1423 (1994) · Zbl 0799.73077 |
| [14] | Zeman, J.; Vondřejc, J.; Novák, 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 |
| [15] | Vondřejc, J.; Zeman, J.; Marek, I., Analysis of a fast fourier transform based method for modeling of heterogeneous materials, (Lirkov, I.; Margenov, S.; Waśniewski, J., Large-Scale Scientific Computing. Large-Scale Scientific Computing, Lecture Notes in Computer Science, vol. 7116 (2012), Springer: Springer Berlin, Heidelberg), 512-522 · Zbl 1354.74235 |
| [16] | Vondřejc, J.; Zeman, J.; Marek, I., Guaranteed upper-lower bounds on homogenized properties by FFT-based Galerkin method, Comput. Methods Appl. Mech. Engrg., 297, 258-291 (2015) · Zbl 1423.74806 |
| [17] | Schneider, M., Convergence of FFT-based homogenization for strongly heterogeneous media, Math. Methods Appl. Sci., 38, 13, 2761-2778 (2014) · Zbl 1328.65256 |
| [18] | 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 |
| [19] | Zeman, J.; de Geus, T. W.J.; Vondřejc, J.; Peerlings, R. H.J.; Geers, M. G.D., A finite element perspective on non-linear FFT-based micromechanical simulations, Internat. J. Numer. Methods Engrg., 111, 10, 903-926 (2017) · Zbl 07867079 |
| [20] | de Geus, T. W.J.; Vondřejc, J.; Zeman, J.; Peerlings, R. H.J.; Geers, M. G.D., Finite strain FFT-based non-linear solvers made simple, Comput. Methods Appl. Mech. Engrg., 318, 412-430 (2017) · Zbl 1439.74045 |
| [21] | Vondřejc, J., Improved guaranteed computable bounds on homogenized properties of periodic media by the fourier-Galerkin method with exact integration, Internat. J. Numer. Methods Engrg., 107, 13, 1106-1135 (2016) · Zbl 1352.74269 |
| [22] | Michel, J.-C.; Moulinec, H.; Suquet, P., Effective properties of composite materials with periodic microstructure: a computational approach, Comput. Methods Appl. Mech. Engrg., 172, 1-4, 109-143 (1999) · Zbl 0964.74054 |
| [23] | Prakash, A.; Lebensohn, R. A., Simulation of micromechanical behavior of polycrystals: finite elements versus fast fourier transforms, Modelling Simulation Mater. Sci. Eng., 17, 6, Article 064010 pp. (2009) |
| [24] | 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, Modelling Simulation Mater. Sci. Eng., 18, 8, 85005 (2010) |
| [25] | Dunant, C. F.; Bary, B.; Giorla, A. B.; Péniguel, C.; Sanahuja, J.; Toulemonde, C.; Tran, A.-B.; Willot, F.; Yvonnet, J., A critical comparison of several numerical methods for computing effective properties of highly heterogeneous materials, Adv. Eng. Softw., 58, 1-12 (2013) |
| [26] | Robert, C.; Mareau, C., A comparison between different numerical methods for the modeling of polycrystalline materials with an elastic-viscoplastic behavior, Comput. Mater. Sci., 103, 134-144 (2015) |
| [27] | Leclerc, W.; Ferguen, N.; Pélegris, C.; Haddad, H.; Bellenger, E.; Guessasma, M., A numerical investigation of effective thermoelastic properties of interconnected alumina/al composites using FFT and FE approaches, Mech. Mater., 92, 42-57 (2016) |
| [28] | Willot, F., Fourier-based schemes for computing the mechanical response of composites with accurate local fields, C. R. Méc., 343, 232-245 (2015) |
| [29] | 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 |
| [30] | Anglin, B. S.; Lebensohn, R. A.; Rollett, A. D., Validation of a numerical method based on fast fourier transforms for heterogeneous thermoelastic materials by comparison with analytical solutions, Comput. Mater. Sci., 87, 209-217 (2014) |
| [31] | Strang, G., Variational crimes in the finite element method, (Aziz, A. K., The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (1972), Academic Press: Academic Press New York, NY, USA), 689-710 · Zbl 0264.65068 |
| [32] | Efendiev, Y.; Hou, T. Y., Multiscale Finite Element Methods (2009), Springer New York: Springer New York New York, NY · Zbl 1163.65080 |
| [33] | Kanouté, P.; Boso, D. P.; Chaboche, J. L.; Schrefler, B. A., Multiscale methods for composites: A review, Arch. Comput. Methods Eng., 16, 1, 31-75 (2009) · Zbl 1170.74304 |
| [34] | Hughes, T. J.; Stewart, J. R., A space-time formulation for multiscale phenomena, J. Comput. Appl. Math., 74, 1-2, 217-229 (1996) · Zbl 0869.65061 |
| [35] | Efendiev, Y.; Galvis, J.; Hou, T. Y., Generalized multiscale finite element methods (gmsfem), J. Comput. Phys., 251, 116-135 (2013) · Zbl 1349.65617 |
| [36] | Brisard, S.; Dormieux, L., FFT-Based methods for the mechanics of composites: A general variational framework, Comput. Mater. Sci., 49, 3, 663-671 (2010) |
| [37] | 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. Eng., 217-220, 197-212 (2012) · Zbl 1253.74101 |
| [38] | Brisard, S., Reconstructing displacements from the solution to the periodic lippmann-schwinger equation discretized on a uniform grid, Internat. J. Numer. Methods Engrg., 109, 4, 459-486 (2017) · Zbl 07874364 |
| [39] | Willot, F.; Abdallah, B.; Pellegrini, Y.-P., Fourier-based schemes with modified green operator for computing the electrical response of heterogeneous media with accurate local fields, Internat. J. Numer. Methods Engrg., 98, 7, 518-533 (2014) · Zbl 1352.80013 |
| [40] | Schneider, M.; Merkert, D.; Kabel, M., FFT-Based homogenization for microstructures discretized by linear hexahedral elements, Internat. J. Numer. Methods Engrg., 109, 10, 1461-1489 (2017) · Zbl 1378.74056 |
| [41] | Vondřejc, J., FFT-Based Method for Homogenization of Periodic Media: Theory and Applications (2013), Czech Technical University in Prague, (Ph.D. thesis) |
| [42] | Kabel, M.; Merkert, D.; Schneider, M., Use of composite voxels in FFT-based homogenization, Comput. Methods Appl. Mech. Engrg., 294, 168-188 (2015) · Zbl 1423.74097 |
| [43] | Bensoussan, A.; Lions, J.-L.; Papanicolaou, G., Asymptotic Analysis for Periodic Structures (1978), North Holland: North Holland Amsterdam · Zbl 0411.60078 |
| [44] | Willis, J. R., The structure of overall constitutive relations for a class of nonlinear composites, IMA J. Appl. Math., 43, 3, 231-242 (1989) · Zbl 0695.73005 |
| [45] | Dvořák, J., Optimization of Composite Materials (1993), Charles University in Prague, (Ph.D. thesis) |
| [46] | Haslinger, J.; Dvořák, J., Optimum composite material design, RAIRO-Math. Model. Numer. Anal.-Model. Math. Anal. Numer., 29, 6, 657-686 (1995) · Zbl 0845.73049 |
| [47] | Ciarlet, P. G., Handbook of Numerical Analysis: Finite Element Methods (1991), Elsevier Science Publishers B.V. (North-Holland) · Zbl 0712.65091 |
| [48] | Alnæs, M. S.; Blechta, J.; Hake, J.; Johansson, A.; Kehlet, B.; Logg, A.; Richardson, C.; Ring, J.; Rognes, M. E.; Wells, G. N., The fenics project version 1.5, Arch. Numer. Softw., 3, 100, 9-23 (2015) |
| [49] | Ern, A.; Guermond, J.-l., Evaluation of the condition number in linear systems rising in finite element approximations, ESAIM Math. Model. Numer. Anal., 40, 1, 29-48 (2006) · Zbl 1149.65306 |
| [50] | Hlaváček, P.; Šmilauer, V.; Škvára, F.; Kopecký, L.; Šulc, R., Inorganic foams made from alkali-activated fly ash: Mechanical, chemical and physical properties, J. Eur. Ceram. Soc., 35, 2, 703-709 (2015) |
| [51] | Vondřejc, J., Double-grid quadrature with interpolation-projection (dogip) as a novel discretisation approach: An application to FEM on simplexes, Comput. Math. Appl., 78, 11, 3501-3513 (2019) · Zbl 1443.65373 |
| [52] | 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 |
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