Dual boundary element method for solving the two-dimensional Helmholtz equation for the damped wave equation.

Abstract

In this paper, the dual boundary integral formulation of the two-dimensional Helmholtz equation with complex wave number is derived. The presence of damping in the medium results in the Helmholtz equation incorporating complex wave numbers in mathematical models. To address the singular and hypersingular integrals, the addition theorem is used to expand the four kernel functions, originally expressed with complex variables in the dual formulation, into purely real-variable functions in a series form. Consequently, the singular and hypersingular integrals are successfully transformed into the summation of regular integrals in an infinite series through the proposed regularization technique. The regular integrals are then computed using the Gaussian quadrature rule. This paper examines the occurrence of eigenvalues in both interior and exterior Helmholtz problems to understand how damping influences resonances. To validate the proposed formulation, three cases with exact solutions are used as standard benchmarks to evaluate the convergence and accuracy of the developed dual boundary element method program. Finally, two more general cases with amoeba-shaped geometry, which lack an exact solution and pose challenges in obtaining a convergent solution due to their irregular shape, are considered to evaluate the applicability and effectiveness of the proposed formulation.