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Fourier-based schemes for computing the electrical and mechanical response of heterogeneous media with accurate local fields

François Willot CMM-Centre de Morphologie Mathématique Mathématiques et Systèmes, MINES Paristech

Fourier-based methods, originally introduced by Moulinec and Suquet, have become ubiquitous for computing numerically the properties of composite materials, with applications in domains ranging from linear elasticity, plasticity, viscoelasticity and crack propagation to thermal and electrical and also optical properties. 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, for example, multiscale nanocomposites, austenitic steel, granular media or polycrystals. This technique allows maps of the local fields to be computed in realistic microstructures. Recently, progress has been made in teh understanding of FFT methods. Making use of modified Green operators associated to finite-differences discretization, FFT schemes with much faster convergence rates have been derived. Such "discrete" schemes also produce more accurate local fields than the discretization commonly employed, in particular along interfaces.