Linear algebraic perspective on FFT-accelerated finite element solvers for homogenisation 

Abstract

Originally introduced by Moulinec and Suquet [1], spectral solvers have been widely used in computational homogenisation due to their computational efficiency, low memory requirements and ease of implementation.

In this talk, we provide a linear algebra perspective on the fast Fourier transform (FFT)-accelerated finite element (FE) computational homogenisation scheme [2]. The computational complexity of the scheme is dominated by the FFT, making it equivalent to that of spectral solvers. However, unlike spectral solvers, the proposed scheme works with arbitrary FE shape functions with local supports which does not cause the Fourier ringing phenomenon.

In this iterative scheme, the periodic cell problem is preconditioned by a discrete Green's operator derived from a reference problem with constant data. All eigenvalues of the preconditioned matrix are bounded by the material data of the original and reference problems [3]. This explains why the number of steps required to solve the system by the preconditioned conjugate gradient (CG) method is (almost) independent of the grid size. In addition, guaranteed eigenvalue bounds allow us to use inexpensive and reliable estimates of the error in the homogenised properties in each iteration of the CG [4].

References:
[1] Moulinec, H. & Suquet, P. C R Acad Sci Paris 318, 1417–1423 (1994).

[2] Ladecký, M. et al. Appl Math Comput 446, 127835 (2023).
[3] Ladecký, M. & Pultarová, I. & Zeman, J. Appl. Math., 66, 21-42 (2021).
[4] Meurant, G. & Papež, J. & Tichý, P. Numer. Alg, 88, 1337-1359 (2021).

Le mardi 26 novembre 2024 à 11h00 / Amphithéâtre François Canac, LMA

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