We present a high-order discontinuous Galerkin Arbitrary Lagrangian Eulerian formulation for the Navier–Stokes equations coupled to various structure models including pitching airfoils, non-linear membranes, and neo-Hookean solids. We use standard continuous Galerkin discretizations of the solids, which are coupled to the fluid using standard techniques. Using implicit–explicit Runge–Kutta time integrators with explicit predictors for the coupling variables, we effectively decouple the system into two implicit problems—one for the fluid and one for the structure. This allows the use of efficient parallel domain specific solvers while maintaining a high formal order of accuracy in time. The talk is based on joint work with Per-Olof Persson (UC Berkeley).