Shape gradient of the dissipated energy functional in shape optimization for the viscous incompressible flow

This paper is concerned with a numerical simulation of shape optimization in a two-dimensional viscous incompressible flow governed by Navier–Stokes equations with mixed boundary conditions containing the pressure. The minimization problem of total dissipated energy was established in the fluid domain. We derive the structures of continuous shape gradient of the cost functional by using the differentiability of a minimax formulation involving a Lagrange functional with a function space parametrization technique. Finally a gradient type algorithm is effectively used for our problem.