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.