Numerical treatment of acoustic problems with the hybrid boundary element method

Abstract

The symmetric hybrid boundary element method in the frequency and time domain is introduced for the computation of acoustic radiation and scattering in closed and infinite domains. The hybrid stress boundary element method in a frequency domain formulation is based on the dynamical Hellinger–Reissner potential and leads to a Hermitian, frequency-dependent stiffness equation. As compared to previous results published by the authors, new considerations concerning the interpretation of singular contributions in the stiffness matrix are communicated. On the other hand, the hybrid displacement boundary element method for time domain starts out from Hamilton's principle formulated with the velocity potential. The field variables in both formulations are separated into boundary variables, which are approximated by piecewise polynomial functions, and domain variables, which are approximated by a superposition of singular fundamental solutions, generated by Dirac distributions, and generalized loads, that are time dependent in the transient case. The domain is modified such that small spheres centered at the nodes are subtracted. Then the property of the Dirac distribution, now acting outside the domain, cancels the remaining domain integral in the hybrid principle and leads to a boundary integral formulation, incorporating singular integrals. In the time domain formulation, an analytical transformation is employed to transform the remaining domain integral into a boundary one. This approach results in a linear system of equations with a symmetric stiffness and mass matrix. Earlier 2D results are generalized in the present paper by a 3D implementation. Numerical results of transient pressure wave propagation in a closed domain are presented.