A Stability Analysis of the Direct Interpolation Boundary Element Method applied to acoustic wave propagation problems using the Modal Superposition Technique

Abstract

The formulations of the boundary element method for dynamic problems, whether acoustic or elastodynamic, that use a more straightforward fundamental solution and consequently require the use of Radial basis functions to transform domain integrals into boundary integrals present instability when reduced time increments are used. This problem does not arise in numerical techniques like finite element and finite difference methods. Despite the higher quality of results presented by the Direct Interpolation Method (DIBEM), a new radial basis formulation, instability problems persist for very small time steps, demanding mesh refinement. Several factors were examined, such as the robustness of the inertia DIBEM matrix concerning conditioning and positivity, densification of the number of internal interpolation points inside, the type of radial basis function, the composition of the time marching scheme, and others. This work evaluates if the cause of this stability is linked to the modal participation of non-real frequencies, which arise in the high spectrum due to numerical inaccuracy when dealing with non-symmetric matrices.