On a numerical approach for solving some geometrical shape optimization problems in fluid mechanics

Abstract

This paper is devoted to the numerical study of geometrical shape optimization problems in fluid mechanics, which consist in minimizing some criterion volume cost functionals on a family of admissible doubly connected domains, constrained by steady-state Stokes boundary value problems. We establish the existence of the shape derivative of the considered cost functionals, by means of Minkowski deformation, using the shape derivative formulas, recently established in Boulkhemair and Chakib (2014). This allows us to express the shape derivative by means of the support function and to avoid the tedious computations required when one use the gradient optimization process based on the classical shape derivative, involving the vector fields, notably when one opt for the finite element discretization. So, based on the established shape derivative formulas, we propose a shape optimization numerical process for solving these problems, using the gradient descent algorithm performed by the finite element discretization, for approximating the auxiliary boundary value Stokes problems. Finally, in order to show the validity and the effectiveness of the proposed approach, we present some numerical tests obtained by solving some shape optimization problems of minimizing different cost functionals on various configurations of domains, constrained by steady-state stokes boundary value problems with different boundary conditions. These numerical simulations include some comparison results showing that the proposed approach is more efficient than the gradient approach based on the classical shape derivative, in terms of the accuracy of the solution and central processing unit (CPU) time execution.