Journal of Computational Acoustics

In this paper we present various numerical methods for solving time-dependent fluid-structure interaction problem in two or three dimensions that we claim to be efficient, robust and highly accurate. These methods, based on mixed variational formulations, are explicit and conservative and can be of arbitrary high order in space. Their accuracy will be illustrated via a comparison with analytical solutions in simple configuration.

This paper is dedicated to the optimal convergence properties of a domain decomposition method involving two-Lagrange multipliers at the interface between the subdomains and additional augmented interface operators. Most methods for optimizing these augmented interface operators are based on the discretization of continuous approximations of the optimal transparent operators.^1–5 Such approach is strongly linked to the continuous equation, and to the discretization scheme. At the discrete level, the optimal transparent operator can be proved to be equal to the Schur complement of the outer subdomain. Our idea consists of approximating directly the Schur complement matrix with purely algebraic techniques involving local condensation of the subdomain degree of freedom on small patch defined on the interface between the subdomains. The main advantage of such approach is that it is much more easy to implement in existing code without any information on the geometry of the interface and of the finite element formulation used. Such technique leads to a so-called "black box" for the users. Convergence results and parallel efficiency of this new and original algebraic optimization technique of the interface operators are presented for acoustics applications.

We present an incomplete LU preconditioner for solving discretized Helmholtz problems. The preconditioner is based on an analytic factorization of the Helmholtz operator. This allows us to take the physical properties of the acoustics problem modeled by the Helmholtz equation into account in the preconditioner. We show how the parameters in the preconditioner can be chosen in order to make it effective. Numerical experiments show that the new preconditioner leads to convergent iterative methods even for large wave numbers, and it outperforms classical ILU preconditioners by a large margin.

This study is devoted to some numerical issues in the boundary integral solution of the scattering of an acoustic wave by an open surface. More precisely, it deals with the construction of a cheap analytical preconditioner to enhance the iterative solving of this kind of equation. Detailed attention is paid to bring out the reasons that make this construction much more difficult than for closed surfaces. This preconditioner is carefully tested and compared to two more usual ones for two and three dimensional problems. It is shown that this preconditioner provides a cheap and efficient tool making reliable the iterative solving. The discussion also precisely brings out the issues where further studies are still needed to improve its efficiency.

A dual-primal variant of the FETI-H domain decomposition method is designed for the fast, parallel, iterative solution of large-scale systems of complex equations arising from the discretization of acoustic scattering problems formulated in bounded computational domains. The convergence of this iterative solution method, named here FETI-DPH, is shown to scale with the problem size, the number of subdomains, and the wave number. Its solution time is also shown to scale with the problem size. CPU performance results obtained for the acoustic signature analysis in the mid-frequency regime of mockup submarines reveal that the proposed FETI-DPH solver is significantly faster than the previous generation FETI-H solution algorithm.

The non-overlapping Schwarz method with absorbing boundary conditions instead of the Dirichlet boundary conditions is an efficient variant of the overlapping Schwarz method for the Helmholtz equation. These absorbing boundary conditions defined on the interface between the subdomains are the key ingredients to obtain a fast convergence of the iterative Schwarz algorithm. In a one-way subdomains splitting, non-local optimal absorbing boundary conditions can be obtained and leads to the convergence of the Schwarz algorithm in a number of iterations equal to the number of subdomains minus one. This paper investigates different local approximations of these optimal absorbing boundary conditions for finite element computations in acoustics. Different approaches are presented both in the continuous and in the discrete analysis, including high-order optimized continuous absorbing boundary conditions, and discrete absorbing boundary conditions based on algebraic approximation. A wide range of new numerical experiments performed on unbounded acoustics problems demonstrate the comparative performance and the robustness of the proposed methods on general unstructured mesh partitioning.

We presented recently a new method to design a non-conforming space-time scheme for the wave equation.¹⁰ We introduce here a new concept of stability for domain decomposition, including perturbations on the boundaries of the numerical subdomains. We prove that our scheme is stable in that strong sense, and overall second order in time and space, for a constant velocity.