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.

We introduce several improvements on key numerical techniques,especially for free surface identification and solid boundary handling method.They make K2_SPH simulating breaking waves successfully.It includes running-up,over-turning and breaking processes.Numerical test of violent tank sloshing shows that K2_SPH method is more accurate and reliable than traditional smoothed particle hydrodynamics(SPH) method for wave profile and pressure distribution.

Smoothed particle hydrodynamics is a Lagrangian meshless partilcle method.It gets superiority in simulation of free surface flow and large deformation problems. But low accuracy of kernel approximation becomes an obstacle for widely applications as particles are distributed disorderly or near a boundary.Adopting Taylor expansion and solving integral equation matrix.a second order kernel approximation method,namely K2_SPH is obtained and discussed.With improvement of kernel approximation,improved numedcal techniques are adopted.such as free surface boundary and solid boundary.They are very important for K2_SPH method.With comparison of standard SPH for nonlinear water wave simulation,K2_SPH improves obviously in free surface simulation and distribution of some variables in total particle system.