Fourier-based schemes with modified Green operator for computing the electrical response of heterogeneous media with accurate local fields

Abstract

Nota Bene: The present document constitutes a 'postprint' version of the published paper, in which a few errors in the proofs (on the present pages 2, 5, 15 and 17) have been corrected and an incomplete reference on p. 22 has been completed. These minor corrections are marked out in red. Results are unchanged. AbstractA modified Green operator is proposed as an improvement of Fourier-based numerical schemes commonly used for computing the electrical or thermal response of heterogeneous media. Contrary to other methods, the number of iterations necessary to achieve convergence tends to a finite value when the contrast of properties between the phases becomes infinite. Furthermore, it is shown that the method produces much more accurate local fields inside highly conducting and quasi-insulating phases, as well as in the vicinity of phase boundaries. These good properties stem from the discretization of Green's function, which is consistent with the pixel grid while retaining the local nature of the operator that acts on the polarization field. Finally, a fast implementation of the 'direct scheme' of Moulinec et al. (1994) that allows for parsimonious memory use is proposed.

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

“…Recently, Monchiet and Bonnet [340] proposed a polarization-based FFT iterative scheme to determine the overall properties of multiphase composites with arbitrary phase contrast, see also Refs. [341][342][343][344]. The use of a discrete element formulation to resolve the RVE problem has been addressed in Refs.…”

Deformation patterning in finite-strain crystal plasticity by spectral homogenization with application to magnesium

“…The standard spectral scheme described above, in the presence of discontinuities in τ(X) and F(X), exhibits numerical ringing artifacts associated with the Gibbs phenomenon for truncated Fourier series (Gibbs, 1898(Gibbs, , 1899Hewitt and Hewitt, 1979). Inspired by Willot et al (2014), Berbenni et al (2014) and Lebensohn and Needleman (2016), we use discrete (modified) spectral differentiation with a finite difference-based scheme (Müller, 1996). The simplest of these modified schemes for a uniform grid with spacing ∆X, whose derivation was presented in detail in 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Vidyasagar et al (2017), is a discrete analog of the Lanczos-σ factor (Lanczos, 1956).…”

“…The above can be extended to rotated (Willot et al, 2014) and higher-order schemes (Vidyasagar et al, 2017) by choosing different finite-difference stencils.…”

“…First-order finite difference-based approximations composed onto Fourier transforms have been shown to result in significantly mitigating oscillatory artifacts (Müller, 1996;Brown et al, 2002;Berbenni et al, 2014;Brisard and Dormieux, 2010;Lebensohn and Needleman, 2016;Schneider et al, 2016), while maintaining consistency with the original governing equations with h-refinement. Willot et al (2014) showed that rotated first-order schemes show a marked reduction in oscillatory artifacts. Similarly, the influence of higher-order finite-difference schemes on mitigating the ringing artifacts, ensuring convergence and smoothing out the fields, was surveyed by Vidyasagar et al (2017) and showed promising results for nonconvex pattern prediction (in linearized kinematics applied to ferroelectric domain patterns).…”

Numerical Simulation of Hindered Diffusion inγ-Alumina Catalyst Supports

“…Homogenization of diffusion process may be performed by different methods: [53,54]. This method is particularly well suited if the porous space is digitalized.…”