FFT-based simulation using a reduced set of frequencies adapted to the underlying microstructure
1RWTH Aachen University, Institute of Applied Mechanics, 52074 Aachen, Germany.
2RWTH Aachen University, Material Mechanics, 52062 Aachen, Germany.
3Max-Planck-Institut für Eisenforschung GmbH, Microstructure Physics and Alloy Design,
40237 Düsseldorf, Germany.
Instead of the classical finite element (FE) based microstructure simulation, a Fast Fourier transform (FFT) based microstructure simulation, introduced by Moulinec and Suquet (1994, 1998), also enables the computation of highly resolved microstructural fields. In this context, the microscopic boundary value problem is captured by the Lippmann-Schwinger equation and solved by using Fast Fourier transforms (FFT) and fixed-point iterations. To decrease the computational effort of the fixed-point solver, Kochmann et al. (2019) introduced a model order reduction (MOR) technique based on solving the Lippmann-Schwinger equation in Fourier space with a reduced set of frequencies. Thereby, the accuracy of this MOR technique depends on the number of used frequencies and the choice of frequencies that are considered within the simulation. Instead of the earlier proposed fixed (Kochmann et al., 2019) or geometrically adapted (Gierden et al., 2021b) sampling patterns, we propose a sampling pattern which is updated after each load step based on the current strain. To show the precision of such a strain-based sampling pattern, an elasto-plastic two-phase composite microstructure is investigated.
Gierden, C., Waimann, J., Svendsen, B. & Reese, S. (2021). FFT-based simulation using a reduced set of frequencies adapted to the underlying microstructure. Computer Methods in Materials Science, 21(1), 51–58. https://doi.org/10.7494/cmms.2021.1.0742
Brisard, S., & Dormieux, L. (2010). FFT-based methods for the mechanics of composites: A general variational framework. Computational Materials Science, 49(3), 663–671. https://doi.org/10.1016/j.commatsci.2010.06.009.
Garcia-Cardona, C., Lebensohn, R., & Anghel, M. (2017). Parameter estimation in a thermoelastic composite problem via adjoint formulation and model reduction. International Journal for Numerical Methods in Engineering, 112(6), 578–600. https://doi.org/10.1002/nme.5530.
Gibbs, J.W. (1898). Fourier’s series. Nature, 59(1522), 200.
Gierden, C., Kochmann, J., Waimann, J., Kinner-Becker, T., Sölter, J., Svendsen, B., & Reese, S. (2021a). Efficient two-scale FE-FFT-based mechanical process simulation of elasto-viscoplastic polycrystals at finite strains. Computer Methods in Applied Mechanics and Engineering, 374, 113566. https://doi.org/10.1016/j.cma.2020.113566.
Gierden, C., Waimann, J., Svendsen, B., & Reese, S. (2021b). A geometrically adapted reduced set of frequencies for a FFTbased microstructure simulation. Cornell University, arXiv.org, arXiv:2103.10203.
Hashin, Z., & Shtrikman, S. (1962). On some variational principles in anisotropic and nonhomogeneous elasticity. Journal of the Mechanics and Physics of Solids, 10(4), 335–342. https://doi.org/10.1016/0022-5096(62)90004-2.
Kochmann, J., Wulfinghoff, S., Reese, S., Mianroodi, J.R., & Svendsen, B. (2016). Two-scale FE–FFT- and phase-field-based computational modeling of bulk microstructural evolution and macroscopic material behavior. Computer Methods in Applied Mechanics and Engineering, 305, 89–110. https://doi.org/10.1016/j.cma.2016.03.001.
Kochmann, J., Wulfinghoff, S., Ehle, L., Mayer, J., Svendsen, B., & Reese, S. (2018). Efficient and accurate two-scale FE-FFTbased prediction of the effective material behavior of elasto-viscoplastic polycrystals. Computational Mechanics, 61(6), 751–764. https://doi.org/10.1007/s00466-017-1476-2.
Kochmann, J., Manjunatha, K., Gierden, C., Wulfinghoff, S., Svendsen, B., & Reese, S. (2019). A simple and flexible model order reduction method for FFT-based homogenization problems using a sparse sampling technique. Computer Methods in Applied Mechanics and Engineering, 347, 622–638. https://doi.org/10.1016/j.cma.2018.11.032.
Kröner, E. (1959). Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Archive for Rational Mechanics and Analysis, 4(1), 273–334.
Liu, Z., Bessa, M.A., & Liu, W.K. (2016). Self-consistent clustering analysis: an efficient multi-scale scheme for inelastic heterogeneous materials. Computer Methods in Applied Mechanics and Engineering, 306, 319–341. https://doi.org/10.1016/j.cma.2016.04.004.
Moulinec, H., & Suquet, P. (1994). 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.
Moulinec, H. & Suquet, P. (1998). A numerical method for computing the overall response of nonlinear composites with complex microstructure. Computer Methods in Applied Mechanics and Engineering, 157(1–2), 69–94. https://doi.org/10.1016/S0045-7825(97)00218-1.
Smit, R.J.M., Brekelmans, W.A.M., & Meijer, H.E.H. (1998). Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling. Computer Methods in Applied Mechanics and Engineering, 155, 181–192. https://doi.org/10.1016/S0045-7825(97)00139-4.
Spahn, J., Andrä, H., Kabel, M., & Müller, R. (2014). A multiscale approach for modeling progressive damage of composite materials using fast Fourier transforms. Computer Methods in Applied Mechanics and Engineering, 268, 871–883. https://doi.org/10.1016/j.cma.2013.10.017.
Vondřejc, J., Liu, D., Ladecký, M., & Matthies, H.G. (2020). FFT-based homogenisation accelerated by low-rank tensor approximations. Computer Methods in Applied Mechanics and Engineering, 364, 112890. https://doi.org/10.1016/j.cma.2020.112890.
Waimann, J., Gierden, C., Schmidt, A., Svendsen, B., & Reese, S. (2021). Microstructure simulation using self‐consistent clustering analysis. PAMM. Proceedings in Aplied Mathematics and Mechanics, 20(1), e202000263. https://doi.org/10.1002/pamm.202000263.
Willis, J.R. (1981). Variational and related methods for the overall properties of composites. In C.-S. Yih (Ed.), Advances in applied mechanics (Vol. 21, pp. 1–78). Elsevier. https://doi.org/10.1016/S0065-2156(08)70330-2.
Willot, F. (2015). Fourier-based schemes for computing the mechanical response of composites with accurate local fields. Comptes Rendus Mécanique, 343(3), 232–245. https://doi.org/10.1016/j.crme.2014.12.005.
Wulfinghoff, S., Cavaliere, F., & Reese, S. (2018). Model order reduction of nonlinear homogenization problems using a Hashin–Shtrikman type finite element method. Computer Methods in Applied Mechanics and Engineering, 330, 149–179. https://doi.org/10.1016/j.cma.2017.10.019.
Zeman, J., Vondřejc, J., Novák, J., & Marek, I. (2010). Accelerating a FFT-based solver for numerical homogenization of periodic media by conjugate gradients. Journal of Computational Physics, 229(21), 8065–8071. https://doi.org/10.1016/j.jcp.2010.07.010.