THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume10 , Number 3/2009 , pp.000-000 ON THE SHAPE RECONSTRUCTION OF 3D STOKES FLOWS Dinel POPA 1 , Eliza MUNTEANU , Ligia MUNTEANU , Veturia CHIROIU 3 2 3 2 3 1 2 3 ( , , ) u uu u 1 University of Piteşti, dept of Applied Mechanics, Târgul din Vale 1, 110040 Piteşti National Institute for Aerospace Research "Elie Carafoli", Bd. Iuliu Maniu 220, 061126 Bucharest Institute of Solid Mechanics, Romanian Academy, Ctin Mille 15, 010141 Bucharest Corresponding author: Veturia CHIROIU, E-mail: veturiachiroiu@yahoo.com This paper deals with the ill-posed nonlinear problem of the shape reconstruction of the Stokes fluid flow. Shape parameters are estimated with a genetic algorithm inverse method by reducing the errors (objective function) between estimated and observed velocity-pressure data. Recommendation concerning the proposed technique is deduced with regard to the algorithm performance Key Words: Shape reconstruction, Stokes flow, Genetic algorithm. 1. INTRODUCTION There are few studies in the literature considering the shape reconstruction of an immersed obstacle using the genetic algorithms. A theoretical foundation of the shape reconstruction of 3D Stokes flows is given by Yan and Ma [1] by establishing the differentiability of the initial boundary value problem with respect to the interior boundary curve in the sense of a domain derivative. These authors solved the problem by a regularized Newton method. They established the domain derivative of the Stokes equations in a multiple bounded domain, and derived an efficient numerical approach for the solution of the 2D realizations of such problem. In [2], Yan and Ma solved a shape reconstruction problem for heat conduction with mixed condition, and, in [3], the same authors derived the expressions of domain derivative for the steady Navier–Stokes equations Chapko, Kress and Yoon [4], [5] consider the inverse boundary problem for the time-dependent heat equation in the case of perfectly conducting and insulating inclusions. Hettlich [6] and Kirsch [7] solved the inverse obstacle scattering problem for sound soft and sound hard obstacles. Other publications on closely related topics are revealed by Matsumoto and Kawahara [8], Newman III et al. [9], Katamine et al. [10], Bernad et al. [11], Carabineanu [12], Dumitrescu, Cardoş and Alexandrescu [13]. The problem addressed by this paper is the shape reconstruction of 3D flows governed by Stokes equations from pressure-velocity data by using a genetic algorithm GA. Genetic algorithms are a class of optimization algorithms that mimic genetic recombination and natural selection (Goldberg [14], Giuclea et al.[15], Preda [16]). To our knowledge, the shape reconstruction of the Stokes fluid flow has not yet been solved by using a genetic algorithm. In our case, the GA is based on the modeling of the unknown boundary as a n-ellipsoid with only 10 parameters (Bonnet [17], Chiroiu, Munteanu and Nicolescu [18]). This paper deals with the shape reconstruction of the Stokes fluid flow. Shape parameters are estimated with a genetic algorithm inverse method. 2. FORMULATION OF THE PROBLEM = and the pressure The aim of the problem is to find the velocity of the fluid p , defined in , 2 1 Ω 1 2 Ω 2 3 1 2 \ Ω=Ω Ω Ω⊂ , with Ω and two simply connected bounded domains of class C in R . The