Complas 2023

COMPLAS 2023

FFT-accelerated Discrete Green's Operator Preconditioned Finite Element Solver for Periodic Homogenization

Ladecký, Martin (Czech technical university in Prague)
Leute, Richard Josef (University of Freiburg)
Falsafi, Ali (École Polytechnique Fédérale de Lausanne)
Pultarová, Ivana (Czech technical university in Prague)
Pastewka, Lars (University of Freiburg)
Junge, Till (École Polytechnique Fédérale de Lausanne)
Zeman, Jan (Czech technical university in Prague)

In session: THa-1707 - - IS1707 - Computational Approaches for Heterogeneous Materials

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We generalize and provide a linear algebra perspective on finite element homogenization schemes, pioneered by Schneider et al. [1] and Leuschner and Fritzen [2]. The efficiency of the scheme is based on a discrete Green's operator preconditioned, well-scaled reformulation allowing for the use of the conjugate gradient or other Krylov subspace solvers. The discretized Green’s operator of a periodic homogeneous reference problem in a generic regular mesh has a block-diagonal structure in the Fourier space, which permits its efficient inversion using fast Fourier transform (FFT) techniques. This implies that the computational scheme scales as O(n log(n)), like FFT, rendering it equivalent to spectral solvers in terms of computational efficiency. However, in contrast to spectral solvers, the proposed scheme works with FE shape functions with local supports and does not exhibit the Fourier ringing phenomenon. We also discuss the equivalence between our displacement-based scheme and the strain-based homogenization technique with finite element projection [3]. [1] Schneider M., Merkert D., Kabel M. FFT-based homogenization for microstructures discretized by linear hexahedral elements. International Journal for Numerical Methods in Engineering, 109 (2017) 1461–1489. [2] Leuschner M., Fritzen F. Fourier-Accelerated Nodal Solvers (FANS) for homogenization problems. Computational Mechanics, 62 (2018) 359–392. [3] Leute, J.R., Ladecký, M., Falsafi, A., Pultarová, I., Zeman, J., Junge, T., and Pastewka, L. Elimination of ringing artifacts by finite-element projection in FFT-based homogenization. Journal of Computational Physics, 453 (2022) 110931. [4] Ladecký, M., Leute, J.R., Falsafi, A., Pultarová, I., Pastewka, L., Junge, T., and Zeman, J. An optimal preconditioned FFT-accelerated finite element solver for homogenization. Applied Mathematics and Computation, 446 (2023) 127835.