Homogeneização assintótica com transformada de Laplace na modelagem de meios microperiódicos

Abstract

This work starts our study on the combination of complete transforms with homogenization in an integrative methodology aiming at simplifying the resolution of differential equations in partial derivatives with rapidly oscillating coefficients, and problems of boundary and / or initial values. Specifically, the following mathematical tools were studied: Laplace transform, the asymptotic homogenization method (MHA), and the method multilayer advection-diffusion (ADMM). First, we present a brief introduction on the Laplace transform method, containing: its definition, principal properties and analytical inversion. Furthermore, an example is developed in which we solve a problem of initial and boundary values for the heat equation using the direct and inverse Laplace transforms, and the solution obtained is validated by solving the same problem by separating variables. In addition, we studied the fixed Talbot algorithm for the numerical inversion of the Laplace transform. From this algorithm we analyze its precision and its convergence, for which test functions were employed and evaluating the absolute maximum error for several values of the relevant parameters. Next, we present in detail the development of MHA in the case of a problem of boundary values for the nonhomogeneous elliptic equation with periodic coefficient that describes a thermal field stationary. In addition, the results of the combination of Laplace transform with the MHA to formally solve a problem of initial and boundary values for the nonhomogeneous parabolic equation with periodic coefficient. Specifically, the equation resulting from applying the transform of Laplace is an elliptic equation similar to that presented previously and containing an advective term and, therefore, the application of MHA is almost immediate. Therefore, the development of discretization for one layer, two layers and n-layers is presented in the ADMM, and its application is illustrated by means of a numerical example. Finally, the pure application of the ADMM and the combination with the MHA is compared to solve a problem of initial and boundary values for the nonhomogeneous heat equation with periodic coefficients. An example is resolved in detail, including the use of the Fixed Talbot algorithm, and the accuracy and computational cost of both approaches are checked.