Asymptotic Study of a Singular Time-Dependent Brinkman Flow with Application

In this article we address the problem of locating point forces within a time-dependent singular Brinkman flow. The context of the study is framed as an approximation of cerebrospinal fluid (CSF) around the central nervous system, with the point forces representing a model for the blood-brain barrier. We approach the problem by reformulating the identification task as an optimization problem, employing a tracking shape functional. A notable challenge in this study arises from the irregularity in the solution of the partial differential equation (PDE), which complicates the exploration of sensitivity analysis. To overcome this issue, we employ a relaxation method and compute the topological derivative of the cost function. The topological derivative, commonly used in shape optimization problems, offers insights into how the cost function responds to small perturbations in the domain. To determine the optimal position of the point forces, we employ a one-shot algorithm based on the derived topological gradient. Finally, we present numerical results that showcase the efficiency of our method in addressing the identified problem.