Burton-Miller method, a complex linear combination of conventional boundary element method (CBIE) and hypersinglar boundary element method (HBIE), is widely used to deal with exterior acoustic problems. The difficult in implementing Burton-Miller method is computing strongly singular integrals (2D problems). Although, many weakly singular/regularization methods have been presented to evaluate these integrals, these methods are still difficult or extremely time consuming. In this paper, analytical integration of strongly singular boundary integral equations discretized with constant element for 2D Helmholtz problems is presented. All singular and strongly singular integrals are analytically evaluated in finite part sense as constant elements are applied to discretize boundary. Contour integral is used for singular and strongly integrals. Validity of formulas is demonstrated with numerical examples.

An isogeometric analysis based on a kind of local bivariate B-spline function and Burton-Miller method is introduced in boundary element method (BEM) for 3D Helmholtz problems.The method avoid efficiently computation of singular and nearly singular integrals aroud singular point of parametric surface.Numerical examples show that the algorithm has good performance in accuracy, efficiency and convergence.The algorithm performs isogeometric BEM (IBEM) with traditional global B-spline shape functions on CPU time and accuracy.The method has potential in engineering applications.