  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.
