Details

Autor(en) / Beteiligte

• Ladecký, Martin
• Leute, Richard J.
• Falsafi, Ali
• Pultarová, Ivana
• Pastewka, Lars
• Junge, Till
• Zeman, Jan

Titel

An optimal preconditioned FFT-accelerated finite element solver for homogenization

Ist Teil von

• Applied mathematics and computation, 2023-06, Vol.446, p.127835, Article 127835

Ort / Verlag

Elsevier Inc

Erscheinungsjahr

2023

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• •We propose a specialized Green’s function preconditioned FE homogenization scheme that is optimized for regular grid discretizations.•Our formulation offers computational efficiency equivalent to spectral homogenization solvers.•Our formulation does not exhibit the ringing phenomena intrinsic to spectral methods.•All results are reproducible via an open-source project μ Spectre, available at https://muspectre.gitlab.io/muspectre. We generalize and provide a linear algebra-based perspective on a finite element (FE) homogenization scheme, pioneered by Schneider et al. (2017)[1] and Leuschner and Fritzen (2018)[2]. The efficiency of the scheme is based on a preconditioned, well-scaled reformulation allowing for the use of the conjugate gradient or similar iterative solvers. The geometrically-optimal preconditioner—a discretized Green’s function of a periodic homogeneous reference problem—has a block-diagonal structure in the Fourier space which permits its efficient inversion using fast Fourier transform (FFT) techniques for generic regular meshes. This implies that the scheme scales as O(nlog(n)), like FFT, rendering it equivalent to spectral solvers in terms of computational efficiency. However, in contrast to classical spectral solvers, the proposed scheme works with FE shape functions with local supports and does not exhibit the Fourier ringing phenomenon. We show that the scheme achieves a number of iterations that are almost independent of spatial discretization. The scheme also scales mildly with phase contrast. We also discuss the equivalence between our displacement-based scheme and the recently proposed strain-based homogenization technique with finite-element projection.