Adjoint shape design sensitivity analysis of fluid–solid interactions using concurrent mesh velocity in ALE formulation

A coupled variational equation for fluid–solid interaction (FSI) problems is derived using a steady state Navier–Stokes equation for incompressible flows, an equilibrium equation for geometrically nonlinear solids, a traction continuity condition at interfaces, and a pseudo-equilibrium equation for mesh velocity. The moving boundary in arbitrary Lagrangian–Eulerian (ALE) formulation is included in the variational equations by the mesh velocity obtained from a displacement-loaded pseudo-structural problem at a concurrent configuration, which eventually facilitates to derive shape design sensitivity. A continuum-based adjoint shape sensitivity is derived under ALE formulation, which turns out to be very accurate and efficient due to the utilization of converged tangent and the linearity of both adjoint and sensitivity equations. Through numerical examples, the obtained sensitivity is verified in terms of accuracy and efficiency compared with finite difference sensitivity and further applied to the shape optimization problem of finding a stiff structure while satisfying a volume constraint.