Combining Galerkin approximation techniques with the principle of Hashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites

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FFT‐based homogenization for microstructures discretized by linear hexahedral elements (2017, methodology): “Regarding the second item on the list, for linear problems, both the Hashin–Shtrikman scheme of Brisard–Dormieux [25, 28] and the Fourier–Galerkin method of Bonnet [29], recently revitalized and extended by Vondřejc [30], produce bounds on the elastic moduli.”
Convergence of FFT‐based homogenization for strongly heterogeneous media (2015, methodology): “The convergence results in [33] and [59] both rely upon particular regularizations of the material coefficients.”
Fourier‐based schemes with modified Green operator for computing the electrical response of heterogeneous media with accurate local fields (2013, background): “This lea ds to a rule for computing (σ − σ0)−1P on voxels that lie on interfaces, and to a representation of t he Green operator as a slowly converging series for which approximat ions are available [21].”
Computational homogenization of elasticity on a staggered grid (2016, methodology): “Unfortunately, the Fourier series representation of Brisard–Dormieux’s 0-operator converges very slowly for three-dimensional problems, and the result depends on the choice of the reference material.”