REITERATED HOMOGENIZATION OF DEGENERATE NONLINEAR ELLIPTIC EQUATIONS
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The current-driven homogenization theory combined with FEM is studied in this paper. The extracted effective parameters are used to compute the scattering parameters of homogenization medium. The computed results are compared with related periodic structure. Two results agree with very well.
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The mechanical response of a heterogeneous medium results from the interactions of mechanisms spanning several length scales. The computational homogenization method captures direct influence of underlying constituents and morphology with a numerically efficient framework. This study reviews the performance of first order computational homogenization technique with a flat punch indentation problem. Results obtained are benchmarked against those using direct numerical simulations (DNS) with full microstructural details. It is shown that the computational homogenization method is able to capture structural response adequately, even for constituent materials with nonlinear behavior. However, the first order computational homogenization method becomes problematic when localized macroscopic deformation occurs. In this context, some re- cent trends addressing the issues are discussed.
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In this paper we study homogenization for a class of monotone systems of first-order time-dependent periodic Hamilton-Jacobi equations. We characterize the Hamiltonians of the limit problem by appropriate cell problems. Hence we show the uniform convergence of the solution of the oscillating systems to the bounded uniformly continuous solution of the homogenized system.
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In this paper, we study regularity of solutions of elliptic systems in divergence form with directional homogenization. Here directional homogenization means that the coefficients of equations are rapidly oscillating only in some directions. We will investigate the different regularity of solutions on directions with homogenization and without homogenization. Actually, we obtain uniform interior $W^{1, p}$ estimates in all directions and uniform interior $C^{1, γ}$ estimates in the directions without homogenization.
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A mixed homogenization method is developed by combining the general homogenization method with 3D-elastic sublaminate method. For a laminate it is a three- dimensional problem to determine the stiffness by the general homogenization method, while the 3D-elastic sublaminate method meets difficult in dealing with damaged laminate stiffness. The mixed homogenization method reduces the problem to two-dimensional scale for determining laminate stiffnesses with a potential advantage in calculating damaged stiffnesses. Comparison of the new method with general 3D-homogenization method shows excellent accuracy.
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