In this paper effective material properties of randomly distributed short fiber composites are calculated with a developed comprehensive tool for numerical homogenization. We focus on the influence of change in volume fraction and length/diameter aspect ratio of fibers. Two types of fiber alignments are considered: fiber orientations with arbitrary angles and parallel oriented fibers. The algorithm is based on a numerical homogenization technique using a unit cell model in connection with the finite element method. To generate the three-dimensional unit cell models with randomly distributed short cylindrical fibers, a modified random sequential adsorption algorithm is used, which we describe in detail. For verification of the algorithm and checking the influence of different parameters, unit cells with various fiber embeddings are created. Numerical results are also compared with those from analytical methods.
The Advection-Diffusion Multilayer Method (ADMM) emerged to address the solution of advection-diffusion equations with variable coefficients in the context of pollutant dispersion modeling. The ADMM is based on the piecewise-constant approximation of the variable coefficients and the application of the Laplace transform. Applications of ADMM in other areas are potentially relevant for modeling the behavior of heterogeneous media. However, if such heterogeneity is characterized by rapidly oscillating coefficients, the direct application of the ADMM would increase the computational effort needed, as it would require a very fine discretization of the domain. In order to overcome such a drawback, in this contribution, an alternative approach combining the ADMM with the Asymptotic Homogenization Method (AHM) is presented. The ADMM-AHM integrated approach is compared to the direct application of the ADMM in order to assess the accuracy of the estimations of the solution of the original problem, and the computational efficiency.
The asymptotic homogenization method is applied to obtain formal asymptotic solution and the homogenized solution of a Dirichlet boundary-value problem for an elliptic equation with rapidly os- cillating coefficients. The proximity of the formal asymptotic solution and the homogenized solution to the exact solution is proved, which provides the mathematical justification of the homogenization pro- cess. Preservation of the symmetry and positive-definiteness of the effective coefficient in the homogenized problem is also proved. An example is presented in order to illustrate the theoretical results.
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