Application of self-detecting error technique in the boundary element method for solving exterior acoustic problems

Abstract

This paper proposes a self-detecting error technique to estimate discretization errors in the boundary element method (BEM) for solving exterior acoustic problems. The proposed technique not only offers the unique capability to quantify the error for each individual element but also provides an accurate estimation of total discretization error across various mesh sizes. This is achieved by utilizing the self-detecting error technique as an error estimator to enable adaptive mesh refinement, which locally adjusts element sizes (h-version). This adaptive mesh scheme achieves higher solution accuracy with fewer boundary elements compared to a uniform mesh distribution. Additionally, this technique is employed to identify fictitious frequencies that lead to a loss of uniqueness in the numerical solution, ensuring the reliability of the numerical solution. This paper presents a numerical investigation to understand how fictitious frequencies are influenced by the chosen approach, either the singular formulation or the hypersingular formulation of the BEM. The interior null field for the complementary domain becomes a nonzero field corresponding to the interior eigenmodes in case of the fictitious frequencies. To validate the proposed technique, a circular case with exact solutions is used as standard benchmark to evaluate the developed self-detecting error technique. Finally, two 2D acoustic problems, one involving radiation and the other involving scattering in a submarine-shaped domain, are presented to evaluate the applicability and effectiveness of the proposed technique.