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.
Resumo: Inúmeras aplicações na ciência e na tecnologia têm mostrado a utilidade da transformada de Laplace como ferramenta que auxilia na resolução dos modelos matemáticos correspondentes. Contudo, em diversas situações práticas, surge a necessidade de abordagens numéricas na inversão desta transformada por causa da dificuldade em se calcular analiticamente a integral de Bromwich que a define. No entanto, não existe nenhum método universal para a inversão numérica que funcione satisfatoriamente em todas as situações. Neste trabalho, apresenta-se um estudo comparativo do desempenho de três algoritmos de inversão numérica da transformada de Laplace, a saber, o algoritmo Talbot
Abstract:Numerous applications in science and technology have shown the usefulness of the Laplace transform as a tool that assists in the resolution of the corresponding mathematical models. However, in several practical situations, the need for numerical approaches in the inversion of this transform arises because of the difficulty in analytically calculating the Bromwich integral that defines it. However, there is no universal method for numerical inversion that works satisfactorily in all situations. In this work, a comparative study of the performance of three numerical inversion algorithms of the Laplace transform, namely, the Abate and Valkó Fixed-Talbot algorithm, the Stehfest algorithm, and the Zakian algorithm, is presented.Keywords: Fixed-Talbot algorithm. Laplace transform. Numerical inversion. Stehfest algorithm. Zakian algorithm.
IntroduçãoAs equações diferenciais descrevem o modo como certas grandezas variam com o tempo e/ou no espaço, tal como a corrente em um circuito elétrico, a oscilação de uma membrana vibrante ou ao fluxo de calor através de um condutor isolado. Essas equações geralmente estão associadas às condições iniciais que descrevem o estado do sistema no instante de tempo inicial ou de maneira mais geral, às condições de contorno [1]. Essas equações são aplicadas em diversas áreas da ciência, sendo extensamente utilizada em Engenharia elétrica e Engenharia química.Dentre as técnicas mais utilizadas na resolução de equações diferenciais encontra-se a transformada de Laplace, que é uma ferramenta operacional usualmente empregada na resolução de problemas de valor inicial que permite, por exemplo, transformar equações diferenciais ordinárias em equações algébricas.
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.
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