A displacement formulation for coupled elastoacoustic problems that preserves flow irrotationality

Abstract

Two-way fluid–structure interaction (FSI) problems, in the sense that a flow induces the motion of a solid, which in turn modifies the flow boundary conditions, have been approached with very different strategies, the most common of which is probably the finite element method (FEM). In the case of elastoacoustics, the flow consists of an acoustic field interacting with a vibrating structure. When the problem is discretized with the FEM, an algebraic block matrix system is obtained and the coupling between the acoustic field and the structure takes place through a coupling matrix with off-diagonal terms. Usually the structure is characterized by its displacement field, while for the acoustics several options are available, ranging from pressure to acoustic displacement or velocity/displacement acoustic potentials. Depending on the formulation, symmetric or asymmetric systems are obtained and different types of numerical stability problems have to be faced. In this work, a monolithic strategy based on the Rayleigh–Ritz method is proposed. The displacement is used as the primary variable for both the structure and the acoustic field and is expanded in terms of Gaussians as basis functions. This provides an algebraic block matrix system for the global uncoupled problem. However, instead of resorting to a coupling matrix, the essential continuity conditions at the acoustic-structure interface are imposed by the nullspace method (NSM). That is, the solution of the uncoupled system is expanded in terms of a basis of the nullspace generated by the essential conditions of the problem, including the displacement continuity constraints at the interface, thus giving the solution of the coupled problem. As for natural conditions, they are imposed in a weak sense. For ease of explanation, a one-dimensional (1D) case is first introduced, followed by the coupling of a 2D acoustic cavity with a beam and a 3D one with a plate. The proposed method is validated with FEM simulations on fine meshes and the advantage of using Gaussian basis functions over trigonometric ones is also demonstrated.