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It is well known that the solution of an exterior acoustic problem governed by the Helmholtz equation is violated at the eigenfrequencies of the associated interior problem when the boundary element method (BEM) based on the conventional boundary integral equation (CBIE) is applied without any special treatment to solve it. To tackle this problem, the Burton-Miller formulation using a linear combination of the CBIE and its normal derivative (NDBIE) emerges as an effective and efficient formula which is proved to yield a unique solution for all frequencies if the imaginary part of the coupling constant of the two equations is nonzero. The most difficult part in implementing the Burton-Miller formulation is that the NDBIE is a hypersingular type, and it is often regularized by using the fundamental solution of the Laplace's equation. But various regularization procedures in the literature give rise to integrals which are still difficult and/or extremely time consuming to evaluate in general. However, when constant triangular elements are used to discretize the boundary, all the strongly-singular and hypersingular integrals can be evaluated in finite-part sense explicitly without any difficulty, and the numerical computation becomes more efficient than any other singularity-subtraction technique. Therefore, in this paper, these singular integrals are evaluated rigorously for triangular constant element as finite parts of the divergent integrals by canceling out the divergent terms which appears in the limiting process explicitly. The correctness of the formulation is also demonstrated through some numerical test examples.

Explicit Evaluation of Hypersingular Boundary Integral Equation for 3-D Helmholtz Equation Discretized with Constant Triangular Element