Shape Derivative of Drag Functional

Jan Sokolowski (1) , Pavel Plotnikov (2)

1 IECN - Institut Élie Cartan de Nancy
2 Lavryentyev Institute of Hydrodynamics

In the paper compressible, stationary Navier-Stokes (N-S) equations are considered. The model is well-posed, there exist weak solutions in bounded domains, subject to inhomogeneous boundary conditions. The shape sensitivity analysis is performed for N-S boundary value problems, in the framework of small perturbations of the so-called {\it approximate solutions}. The approximate solutions are determined from Stokes problem and the small perturbations are given by solutions to the full nonlinear model. Such solutions are unique. The differentiability of the specific solutions with respect to the coefficients of differential operators implies the shape differentiability of the drag functional. The shape gradient of the drag functional is derived in the classical and useful for computations form, an appropriate adjoint state is introduced to this end. The shape derivatives of solutions to the Navier-Stokes equations are given by smooth functions, however the shape differentiability can be shown only in a weak norm. The proposed method of shape sensitivity analysis is general. The differentiability of solutions to the Navier-Stokes equations with respect to the data leads to the first order necessary conditions for a broad class of optimization problems. The boundary shape gradient as well as the boundary value problems for the shape derivatives of solutions to state equations and the adjoint state equations are obtained in the form of singular limits of volume integrals. This method of shape sensitivity analysis seems to be new and appropriate for nonlinear problems. It is an important observation for the numerical methods of shape optimization in fluid mechanics.

Mots clés: Compressible Navier-Stokes equations; drag minimization; shape derivative; necessary optimality conditions

Domaines: Equations aux dérivées partielles [math.AP]

Jan Sokolowski, Pavel Plotnikov. Shape Derivative of Drag Functional. 2009. ⟨hal-00382657⟩