Optimal shape design for a time-dependent Brinkman flow using asymptotic analysis techniques

Article type: Research Article

Authors: Dhif, R.^a | Meftahi, H.^{b; *} | Rjaibi, B.^a

Affiliations: [a] Laboratory of Mathematical and Numerical Modelling in Engineering Sciences, University of Tunis El-Manar, Tunisia | [b] Institute of Computer Science Isikef/University of Jendouba, Tunisia

Correspondence: [*] Corresponding author. E-mail: houcine.meftahi@enit.utm.tn.

Abstract: In this paper, we consider the geometric inverse problem of recovering an obstacle ω immersed in a bounded fluid flow Ω governed by the time-dependent Brinkman model. We reformulate the inverse problem into an optimization problem using a least squares functional. We prove the existence of an optimal solution to the optimization problem. Then, we perform the asymptotic expansion of the cost function in a simple way using a penalty method. An important advantage of this method is that it avoids the truncation method used in the literature. To reconstruct the obstacle, we propose a fast algorithm based on the topological derivative. Finally, we present some numerical experiments in two- and three-dimensional cases showing the efficiency of the proposed method.

Keywords: Optimal shape design, inverse problem, time-dependent Brinkman equation, topological sensitivity analysis

DOI: 10.3233/ASY-221792

Journal: Asymptotic Analysis, vol. 132, no. 1-2, pp. 261-284, 2023