# Desingularization of matrix equations employing hypersingular integrals in boundary element methods using double nodes

## TL;DR

This paper addresses singularities in boundary element method matrix equations caused by double nodes at corners, proposing hypersingular integral equations to improve accuracy and reduce computational cost.

## Contribution

It introduces a novel application of hypersingular integral equations (HBIE) to resolve singularities in boundary element methods with double nodes, comparing partial and full HBIE approaches.

## Key findings

- Partial-HBIE improves accuracy over full-HBIE.
- Both HBIE approaches resolve singularities effectively.
- Partial-HBIE has lower computational cost.

## Abstract

In boundary element methods, the method of using double nodes at corners is a useful approach to uniquely define the normal direction of boundary elements. However, matrix equations constructed by conventional boundary integral equations (CBIE) become singular under certain combinations of double node boundary conditions. In this paper, we analyze the singular conditions of the CBIE formulation for cases where the boundary conditions at the double node are imposed by combinations of Dirichlet, Neumann, Robin, and interface conditions. To address this singularity we propose the use of hypersingular integral equations (HBIE) for wave propagation problems that obey Helmholtz equation. To demonstrate the applicability of HBIE, we compare three types of simultaneous equations: (i) CBIE, (ii) partial-HBIE in which HBIE is only applied to the double nodes at corners while CBIE is applied to the other nodes, and (iii) full-HBIE in which HBIE is applied to all nodes. Based on our numerical results, we observe the following results. The singularity of the matrix equations for problems with any combination of boundary conditions can be resolved by both full-HBIE and partial-HBIE, and partial-HBIE exhibits better accuracy than full-HBIE. Furthermore, the computational cost of partial-HBIE is smaller than that of full-HBIE.