IMPROVED EIGENFUNCTION EXPANSIONS IN DIFFUSION PROBLEMS WITH MULTISCALE SPACE VARIABLE COEFFICIENTS

Renato Machado Cotta¹; Carolina Palma Naveira Cotta¹; Diego Campos Knupp²

¹ COPPE/UFRJ; ² UERJ

doi:10.20906/CPS/COB-2015-2714

Resumo

The Generalized Integral Transform Technique (GITT) is described for solving diffusion problems defined in physical domains with subregions of multiple materials and/or spatial scales. Through a single domain reformulation strategy that treats the multiregion problem as a single domain with space variable coefficients, the subregions transitions are incorporated into one single eigenvalue problem for the proposed eigenfunction expansions. Therefore, the eigenfunctions spatial behaviors are responsible for recovering the local information in each subregion, thus avoiding cumbersome domain decomposition schemes, while the transformed ordinary differential equations system handles the temporal behavior of the solution. The GITT itself is employed in the solution of the eigenvalue problem with space variable coefficients, by adopting a simpler auxiliary eigenvalue problem for the eigenfunction representation in each space coordinate, resulting in a matrix eigenvalue problem that is readily solved. An integral balance approach to improve the convergence behavior of the eigenfunction expansions, while offering explicit accounting for the space variable coefficients, is then here proposed. An example is provided that deals with heat conduction in heterogeneous media, such as in functionally graded materials, with multiscale variation of thermophysical properties across the domain. The convergence characteristics of the integral transforms solution is more closely examined and critically compared against fully converged results from the available exact solution.

Palavras-chave: Heterogeneous media; Multiscale coefficients; Integral transforms; Hybrid methods; Functionally graded material; Sturm-Liouville problem