Numerical strategy for unbiased homogenization of random materials

Abstract : This paper presents a numerical strategy that allows to lower the costs associated to the prediction of the value of homogenized tensors in elliptic problems. This is done by solving a coupled problem, in which the complex microstructure is confined to a small region and surrounded by a tentative homogenized medium. The characteristics of this homogenized medium are updated using a self-consistent approach and are shown to converge to the actual solution. The main feature of the coupling strategy is that it really couples the random microstructure with the deterministic homogenized model, and not one (deterministic) realization of the random medium with a homogenized model. The advantages of doing so are twofold: (a) the influence of the boundary conditions is significantly mitigated, and (b) the ergodicity of the random medium can be used in full through appropriate definition of the coupling operator. Both of these advantages imply that the resulting coupled problem is less expensive to solve, for a given bias, than the computation of homogenized tensor using classical approaches. Examples of 1D and 2D problems with continuous properties, as well as a 2D matrix-inclusion problem, illustrate the effectiveness and potential of the method.
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Régis Cottereau. Numerical strategy for unbiased homogenization of random materials. International Journal for Numerical Methods in Engineering, Wiley, 2013, 95 (1), pp.71-90. ⟨10.1002/nme.4502⟩. ⟨hal-00837633⟩

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