An ultra-weak method for acoustic fluid-solid interaction

We introduce the ultra-weak variational formulation (UWVF) for fluid-solid vibration problems. In particular, we consider the scattering of time-harmonic acoustic pressure waves from solid, elastic objects. The problem is modeled using a coupled system of the Helmholtz and Navier equations. The transmission conditions on the fluid-solid interface are represented in an impedance-type form after which we can employ the well known ultra-weak formulations for the Helmholtz and Navier equations. The UWVF approximation for both equations is computed using a superposition of propagating plane waves. A condition number based criterion is used to define the plane wave basis dimension for each element. As a model problem we investigate the scattering of sound from an infinite elastic cylinder immersed in a fluid. A comparison of the UWVF approximation with the analytical solution shows that the method provides a means for solving wave problems on relatively coarse meshes. However, particular care is needed when the method is used for problems at frequencies near the resonance frequencies of the fluid-solid system.