On the singularities of Green's formula and its normal derivative,with an application to surface‐wave–body interaction problems

The paper presents the non‐singular forms of Green's formula and its normal derivative of exterior problems for three‐dimensional Laplace's equation. The main advantage of these modified 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 flux 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 efficacy of the modified formulations is examined by a sphere, in an infinite domain, subject to Neumann and Dirichlet conditions, respectively. The modified 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 ‘irregular’ frequencies is performed by linearly combining the standard integral equation with its normal derivative. Computations are presented of the added‐mass and damping coefficients and wave exciting forces on a floating hemisphere. Comparing the numerical results with that by other approaches demonstrates the effectiveness of the method. Copyright © 2000 John Wiley & Sons, Ltd.     

A direct approach for boundary integral equations with high‐order singularities

Boundary integral equations with extremely singular (i.e., more than hypersingular) kernels would be useful in several fields of applied mechanics, particularly when second‐ and third‐order derivatives of the primary variable are required. However, their definition and numerical treatment pose several problems. In this paper, it is shown how to obtain these boundary integral equations with still unnamed singularities and, moreover, how to efficiently and reliably compute all the singular integrals. This is done by extending in full generality the so‐called direct approach. Only for definiteness, the method is presented for the analysis of the deflection of thin elastic plates. Numerical results concerning integrals with singularities up to order r⁻⁴ are presented to validate the proposed algorithm. Copyright © 2000 John Wiley & Sons, Ltd.     

Evaluation of 2‐D Green's boundary formula and its normal derivative using Legendre polynomials,with an application to acoustic scattering problems

This paper presents the non‐singular forms, in a global sense, of two‐dimensional Green's boundary formula and its normal derivative. The main advantage of the modified formulations is that they are amenable to solution by directly applying standard quadrature formulas over the entire integration domain; that is, the proposed element‐free method requires only nodal data. The approach includes expressing the unknown function as a truncated Fourier–Legendre series, together with transforming the integration interval [a, b] to [‐1,1] ; the series coefficients are thus to be determined. The hypersingular integral, interpreted in the Hadamard finite‐part sense, and some weakly singular integrals can be evaluated analytically; the remaining integrals are regular with the limiting values of the integrands defined explicitly when a source point coincides with a field point. The effectiveness of the modified formulations is examined by an elliptic cylinder subject to prescribed boundary conditions. The regularization is further applied to acoustic scattering problems. The well‐known Burton–Miller method, using a linear combination of the surface Helmholtz integral equation and its normal derivative, is adopted to overcome the non‐uniqueness problem. A general non‐singular form of the composite equation is derived. Comparisons with analytical solutions for acoustically soft and hard circular cylinders are made. Copyright © 2001 John Wiley & Sons, Ltd.     

Regularized meshless method for antiplane shear problems with multiple inclusions

In this paper, we employ the regularized meshless method to solve antiplane shear problems with multiple inclusions. The solution is represented by a distribution of double‐layer potentials. The singularities of kernels are regularized by using a subtracting and adding‐back technique. Therefore, the troublesome singularity in the method of fundamental solutions (MFS) is avoided and the diagonal terms of influence matrices are determined. An inclusion problem is decomposed into two parts: one is the exterior problem for a matrix with holes subjected to remote shear, the other is the interior problem for each inclusion. The two boundary densities, essential and natural data, along the interface between the inclusion and matrix satisfy the continuity and equilibrium conditions. A linear algebraic system is obtained by matching boundary conditions and interface conditions. Finally, numerical results demonstrate the accuracy of the present solution. Good agreements are obtained and compare well with analytical solutions and Gong's results. Copyright © 2007 John Wiley & Sons, Ltd.