Shape Optimization for Navier-Stokes Flow
  This paper is concerned with the optimal shape design of the newtonian viscous incompressible fluids driven by the stationary nonhomogeneous Navier-Stokes equations. We use three approaches to derive the structures of shape gradients for some given cost functionals. The first one is to use the Piola transformation and derive the state derivative and its associated adjoint state; the second one is to use the differentiability of a minimax formulation involving a Lagrangian functional with a function space parametrization technique; the last one is to employ the differentiability of a minimax formulation with a function space embedding technique. Finally we apply a gradient type algorithm to our problem and numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible. 
Zhiming Gao, Yichen Ma, Hongwei Zhuang