Solution of Helmholtz eigenvalue problems with non-regular domains using the direct interpolation technique

Abstract

This work aims to evaluate the performance of the Direct Interpolation Boundary Element Method solving Helmholtz problems that present no regular geometric shapes. Using the radial basis functions, the Direct Interpolation Method approximates the non-self-adjoint kernel of the domain integral equations, which appear in many partial differential equations of mathematics, physics, and engineering. Modeling the Helmholtz Equation, the Direct Interpolation Method approximates the inertia term by a sequence of radial basis functions. The technique has been used successfully in Poisson, Diffusive-advective, and Helmholtz problems, considering regular geometries and taking analytical solutions as a reference for performance evaluation. This paper evaluates the effects of the slenderness of the domain and the introduction of cuts on the boundary regarding the accuracy. Reference solutions are generated through Finite Element Method simulations using fine meshes.