Numerical homogenization of random media: the FFT method

Abstract : In recent years, Fourier-based methods, originally introduced by [Moulinec and Suquet, 1994], have become ubiquitous for computing numerically the properties of composite materials, with applications in domains ranging from linear elasticity [Willot et al., 2008], viscoplasticity [Lebensohn, 2001], crack propagation [Li et al., 2011], to thermal and electrical [Willot et al., 2013, Willot and Jeulin, 2011], but also optical properties [Azzimonti et al., 2013]. The success of the method resides in its ability to cope with arbitrarily complex and often very large microstructures, supplied as segmented images of real materials, e.g., multi-scale nano-composites [Jean et al., 2011], austenitic steel [Belkhabbaz et al., 2011], granular media [Willot et al., 2013] or polycrystals [Prakash and Lebensohn, 2009, Rollett et al., 2010, Lebensohn et al., 2005]. This technique allows maps of the local fields to be computed in realistic microstructures. Such fields are representative of the material behavior if the resolution is small enough, and if the system size is large enough, compare d with the typical length scale of the heterogeneities. Contrary to finite-element methods where matrix pre-conditioning often necessitates additional memory occupation, Fast-Fourier-Transform (FFT) methods are limited only by the amount of RAM or fast-access computer memory required to store the fields.
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François Willot. Numerical homogenization of random media: the FFT method. Colloque National Mécamat. Mécanique des matériaux biosourcés. De l'extraction au recyclage, conséquences sur les propriétés effectives., 2014, Aussois, France. ⟨hal-01940408⟩

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