On the singularities of Greenʹs formula and its normal derivative, with an application to surface-wave-body interaction problems

The paper presents the non-singular forms of Greenʹs formula and its normal derivative of exterior problems for three-dimensional Laplaceʹs equation. The main advantage of these modi"ed formulations is that they are amenable to solution by directly using quadrature formulas. Thus, the conventional boundary element approximation, which locally regularizes the singularities in each element, is not required. The weak singularities are treated by both the Gauss #ux theorem and the property of the associated equipotential body. The hypersingularities are treated by further using the boundary formula for the associated interior problems. The e$cacy of the modi"ed formulations is examined by a sphere, in an in"nite domain, subject to Neumann and Dirichlet conditions, respectively. The modi"ed integral formulations are further applied to a practical problem, i.e. surface-wave}body interactions. Using the conventional boundary integral equation formulation is known to break down at certain discrete frequencies for such a problem. Removing the &irregularʹ frequencies is performed by linearly combining the standard integral equation with its normal derivative. Computations are presented of the added-mass and damping coe$cients and wave exciting forces on a #oating hemisphere. Comparing the numerical results with that by other approaches demonstrates the e!ectiveness of the method.