Shape Gradient Methods for Shape Optimization of an Unsteady Multiscale Fluid–Structure Interaction Model

We consider numerical shape optimization of a fluid–structure interaction model. The constrained system involves multiscale coupling of a two-dimensional unsteady Navier–Stokes equation and a one-dimensional ordinary differential equation for fluid flows and structure, respectively. We derive shape gradients for both objective functionals of least-squares type and energy dissipation. The state and adjoint state equations are numerically solved on the time-dependent domains using the Arbitrary-Lagrangian–Eulerian method. Numerical results are presented to illustrate effectiveness of algorithms.