A coupling procedure for modeling acoustic problems using finite elements and boundary elements

Finite element (FEM) and boundary element (BEM) methods have been used for a long time for the numerical simulation of acoustic problems. The development presented in this paper deals with a general procedure for coupling acoustic finite elements with acoustic boundary elements in order to solve efficiently acoustic problems involving non homogeneous fluids. Emphasis is made on problems where finite elements are used for a confined (bounded) fluid while boundary elements are selected for an external (unbounded) fluid. The discrete sets of equations related to the two models are first summarized. An elimination procedure is then proposed for solving the coupled problem. This procedure enables to take care of the sparseness of the finite element matrix and leads to solve an updated boundary element system. The solution sequence allows for the treatment of multiple load cases (velocity boundary conditions) in order to generate mutual impedance coefficients. 1. PROBLEM'S STATEMENT The problem to be solved involves (at least) two fluid regions with different acoustic properties (Figure 1). The first fluid region (Q1) is assumed to be bounded. His acoustic properties are pl (density) and cl (speed of sound). The second (usually external) fluid with properties p~ and CE occupies the region (QE) and shares a common interface (TEc) with the first region. On this interface, acoustic pressures and velocities have to respect the continuity constraints: p(x) = p(x*) vx E r, (1 ) vn(X-) = vn(X+) VX E rEC (2) where Xand X+ are points along boundary TEC inside QI and QE respectively. Additionally pressure, normal velocity and/or normal admittance boundary conditions can be constrained along the boundaries TI and TEU. The acoustic response inside region Ql and/or QE has to be evaluated in some frequency range taking into account radiation into the external region. Several unit velocity boundary conditions have to be considered in order to generate transfer functions related to sub-surfaces located on TI. 2. FEM AND BEM MODELS The solution within the bounded region Ql is obtained using the acoustic finite element method [I] with a conventional pressure formulation. The discretization process allows to relate the nodal Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jp4:1994568 C5-332 JOURNAL DE PHYSIQUE IV pressure vector P to the fixed velocity vector (along a part of TI) and the unknown velocities VA (along TEC): where the matrices K and M are the usual stiffness and mass matrices while the matrix An contains admittance values related to some part of TI. r~~ Figure 1 : Basic geometry of the fluid domain. /\ Within the external unbounded region SLE, the discrete solution can be evaluated from a direct boundary element model [2] leading to a set of relations between nodal pressures PE and nodal normal velocities V: along the closed boundary TE: [A](PE) = [B](v!) (4) where A and B are complex, frequency dependent, fully populated, non-symmetric matrices. 3. COUPLING STRATEGY External Region " E