Shape optimization of Navier–Stokes flows by a two-grid method

We consider the energy dissipation minimization constrained by steady Navier–Stokes flows. The nonlinearity of the Navier–Stokes equation causes its numerical solver computationally expensive and thus leads to inefficiency of shape optimization algorithms. We propose a new efficient shape gradient algorithm of distributed type by using a two-grid solver for the Navier–Stokes equation. Asymptotically optimal convergence rate is shown for mixed finite-element approximation of the two-grid distributed shape gradient. Moreover, a Uzawa iterative scheme for Stokes-type systems is adopted in the two-grid solver and a new scalar-type shape gradient flow is proposed to improve significantly the computational efficiency especially for 3D. Numerical experiments are presented to verify theoretical analysis and show effectiveness of two-grid shape gradient optimization algorithms.