A numerical spectral approach for solving elasto-static field dislocation and g-disclination mechanics

Numerical implementation of non-local polycrystal plasticity using fast Fourier transforms

“…(1999), in which derivatives are calculated by means of corresponding central finite difference expressions, to which discrete Fourier transforms are taken and the shift theorem is applied (see Berbenni et al (2014) for details). In the case of the approximation given by Eq.…”

“…As pointed out by Berbenni et al (2014) in the context of another application of the FFT-based approach to calculate stress fields associated to the presence of disclinations in elastic media, also requiring the evaluation of higher order derivatives of incompatible distortion fields, the use of the DFT expressions (e.g. Eqs.…”

“…In these EVP-FFT implementations, a local formulation of crystal plasticity was adopted, rendering their predictions size-independent. While the potential for an efficient non-local plasticity FFT-based implementation has been recognized, overcoming the numerical challenges associated with the required computation and use in the constitutive relation of higher order derivatives of the plastic distortion field was only possible considering the alternative use of discrete Fourier transforms 5 (DFT) (Müller, 1998;Dreyer et al, 1999;Willot and Pellegrini, 2008;Willot, 2015) in lieu of the original Moulinec-Suquet continuous Fourier transform (CFT)-based algorithm, in particular drawing upon recent work by Berbenni et al (2014) devoted to the calculation of stress fields associated to the presence of disclinations in elastic media using the FFT-based approach.…”

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).…”

“…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.…”