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A simplified method for upscaling composite materials with high contrast of the conductivity. (English) Zbl 1202.80025

The authors compute the effective conductivity coefficient for composite materials with high contrasts, such as fibrous insulating materials and metal foams. They start from the elliptic problem \(\nabla \cdot (K(x)\nabla u(x))=0\) in a domain \(\Omega \) of \(\mathbb{R}^{n}\), \(n=2,3\). The domain \( \Omega \) is the union of two subdomains \(\Omega _{M}\) and \(\Omega _{A}\) respectively filled in with two constituents having substantially different thermal conductivities \(K_{M}=1\) and \(K_{A}=\delta \) with \(\delta \ll 1\). Dirichlet boundary conditions \(u=g\) are imposed on the boundary \(\partial \Omega \). Here \(K(x)\) is the conductivity matrix which takes the values \( K_{M}\) and \(K_{A}\). The authors first recall the construction of the effective coefficient as \(\widetilde{K}e_{i}=\delta \left| \Omega _{A}\right| \left\langle \nabla u_{i}\right\rangle _{\Omega _{A}}/\left| \Omega \right| +\left| \Omega _{M}\right| \left\langle \nabla u_{i}\right\rangle _{\Omega _{M}}/\left| \Omega \right| \) where \(u_{i}\) is the solution of an auxiliary problem, \( \left\langle \cdot \right\rangle \) denotes the mean value of the corresponding quantity in the domain under consideration and \(\left| \cdot \right| \) denotes the volume of the corresponding domain. The authors first prove uniform estimates (with respect to \(\delta \)) on the solution of this elliptic problem assuming that \(g\in H^{1/2}\) and that the boundaries are Lipschitz. The next part of the paper presents a function \(v\) such that the mean value of \(K\nabla v\) builds an approximation of the mean value of \(K\nabla u\) in \(\Omega _{M}\) or in \(\Omega \setminus \widetilde{\Omega }_{M}\). The last part of the paper proposes an algorithm for the computation of the approximate conductivity coefficient and numerical values of this approximate conductivity coefficient.

MSC:

80M40 Homogenization for problems in thermodynamics and heat transfer
80M35 Asymptotic analysis for problems in thermodynamics and heat transfer
35J25 Boundary value problems for second-order elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data

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