# Fast Boundary Element solvers in the frequency domain to simulate coupled acoustic-elastic problems in the time domain

3D rapid transient acoustic problems are known to be difficult to solve numerically when dealing with large geometries, because numerical methods based on geometry discretisation, such as the BEM or the FEM, often require to solve a linear system (from the spacial discretisation) for each time step. In a first part, I will present a numerical method to efficiently deal with 3D rapid transient acoustic problems set in large exterior domains. Using the Z-transform and the convolution quadrature method (CQM), a straightforward way to reframe the problem to the solving of a large amount of frequency-domain BEMs is derived. Then, taking advantage of a well-designed high-frequency approximation (HFA), the number of frequency-domain BEMs to be solved is drastically reduced, with little loss of accuracy.

In a second part, I will discuss how to consider the coupled, i.e. Fluid Structure Interaction, problem. A first approach consists in iteratively solving the BEM-FEM coupling by alternating Neumann solutions in each domain. Unfortunately this simple approach fails. We can show that the transient BEM-FEM coupling based on Neumann-Neumann iterations is problematic since energy estimates indicate that each iteration degrades the regularity of boundary traces (unlike in the elliptic case). To avoid this issue, an iterative algorithm based on Robin boundary conditions for the coupled elastodynamic/acoustic problem will be presented and proved to converge. The efficiency of these numerical methods will be demonstrated in the context of the simulation of underwater explosions