Abstract: In this paper, we consider a geometric inverse source
problem for the heat equation with Dirichlet and Neumann boundary
data. We will reconstruct the exact form of the unknown source
term from additional boundary conditions. Our motivation is to
detect the location, the size and the shape of source support.
We present a one-shot algorithm based on the Kohn-Vogelius
formulation and the topological gradient method. The geometric
inverse source problem is formulated as a topology optimization
one. A topological sensitivity analysis is derived from a source
function. Then, we present a non-iterative numerical method for the
geometric reconstruction of the source term with unknown support
using a level curve of the topological gradient. Finally, we give
several examples to show the viability of our presented method.
Abstract: The aeration process via injectors is used to combat
the lack of oxygen in lakes due to eutrophication. A 3D numerical
simulation of the resulting flow using a simplified model is presented.
In order to generate the best dynamic in the fluid with respect to
the aeration purpose, the optimization of the injectors location is
considered. We propose to adapt to this problem the topological
sensitivity analysis method which gives the variation of a criterion
with respect to the creation of a small hole in the domain. The main
idea is to derive the topological sensitivity analysis of the physical
model with respect to the insertion of an injector in the fluid flow
domain. We propose in this work a topological optimization algorithm
based on the studied asymptotic expansion. Finally we present some
numerical results, showing the efficiency of our approach