Abstract
The topological derivative method is used to solve a pollution sources reconstruction problem governed by a steady-state convection-diffusion equation. To be more precise, we are dealing with a shape optimization problem which consists of reconstruction of a set of pollution sources in a fluid medium by measuring the concentration of the pollutants within some subregion of the reference domain. The shape functional measuring the misfit between the known data and solution of the state equation is minimized with respect to a set of ball-shaped geometrical subdomains representing the pollution sources. The necessary conditions for optimality are derived with the help of the topological derivative method which consists in expanding the shape functional asymptotically and then truncate it up to the second order term. The resulting expression is trivially minimized with respect to the parameters under consideration which leads to a noniterative second-order reconstruction algorithm. Two different cases are considered. Firstly, when the velocity of the leakages is given and we reconstruct the support of the unknown sources, including their locations and sizes. In the second case, we consider the size of the pollution sources to be known and find out the mean velocity of the leakages and their locations. Numerical examples are presented showing the capability of the proposed algorithm in reconstructing multiple pollution sources in both cases.
Highlights
The topic addressed in this article belongs to a class of inverse problems where the main objective is to identify the location and size of pollution sources
Applied Mathematics & Optimization (2021) 84:1493–1525 the rate of impurity emission from the pollution sources using the measurements of the concentration of the pollutants in some monitoring region is an interesting issue to investigate
We assume that the velocity field V is given and we reconstruct the topology of the pollution sources by recovering their locations ξ and sizes α
Summary
The topic addressed in this article belongs to a class of inverse problems where the main objective is to identify the location and size of pollution sources. In our simplified mathematical setting, we are dealing with an inverse reconstruction problem whose forward counterpart is governed by a steady-state convection-diffusion equation In this case, since the pollutant concentration is known in a small subregion of the fluid domain, the inverse problem can be written in the form of an over-determined boundary value problem. The topological derivative is defined as the first term (correction) of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturbations, such as holes, inclusions, defects, source-terms and cracks This relatively new concept has applications in many different fields such as shape and topology optimization, inverse problems, image processing, multi-scale material design and mechanical modeling including damage and fracture evolution phenomena.
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