Second Order Topological Derivative for the Inverse Problem in Diffusion with Retention
Jairo Rocha de Faria1; Ana Paula Pintado Wyse1; Antônio José Boness Santos1; Luiz Bevilacqua2; Flavio Pietrobon Costa3
1 anawyse@ci.ufpb.br; 2 Instituto Alberto Coimbra, COPPE, Universidade Federal do Rio de Janeiro; 3 Departamento de Ciências Exatas e Tecnológicas -- Universidade Estadual de Santa Cruz
doi:10.20906/CPS/CILAMCE2015-0411
Resumo
Several relevant phenomena studied in scientific computing, like population spreading with partial hold up of the population to guarantee territorial domain, chemical reactions inducing adsorption processes and multiphase flow thorough porous media, Brownian motion in the speckle light field, for instance, are modeled by diffusion with retention. A new and general formulation of this problem was presented recently by Bevilacqua et. al., where the resulting governing equation is a fourth order differential partial equation. The new parameters associated to this higher order term need to be estimated, which leads to an inverse problem. In this work we shall consider the retention phenomena as a singular perturbation of the classical diffusion and rewrite the inverse problem as an optimization problem, taking into account the usual shape functional given by squared residues between experimental data and predicted values. Then, we derive the first and second order topological asymptotic expansion of this functional. Numerical results obtained using first and second topological derivative are presented and discussed.
Palavras-chave: Topological Derivative; Inverse Problem; Diffusion-Retention