Numerical evaluation of various levels of singular integrals, arising in BEM and its application in hydrofoil analysis Hassan Ghassemi * , Ahmad Reza Kohansal Department of Marine Technology, Amirkabir University of Technology, Hafez Ave., 15875-4413 Tehran, Iran article info Keywords: Boundary element Singular integrals Gaussian quadrature Transformation method Foil model Potential and pressure distribution Induced velocity field abstract The sound implementation of the boundary element method (BEM) is highly dependent on an accurate numerical integration of singular integrals. In this paper, a set of various types of singular domain integrals with three-dimensional boundary element discretization is evaluated based on a transformation integration technique. In the BEM, the integration domain (body surface) needs to be discretized into small elements. For each element, the integral I(x p , x) is calculated on the domain dS. Several types of integrals IB a and IC a are numerically and analytically computed and compared with the relative error. The method is extended to evaluate singular integrals which arise in the solution of the three-dimensional Laplace’s equation. An example of the elliptic hydrofoil is performed to study the physical accuracy. The results obtained using both numerical and analytical methods are shown in good agreement with the experimental data. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction Various orders of singular integrals develop in the basic boundary integral equations, associated with the use of funda- mental solutions of the governing equations using Green’s identity in hydrodynamic of the lifting problems; like hydrofoils and propellers. The evaluation of these singular integrals accurately is an essential task in solving these problems using the BEM. Much effort has been made to remove the singularities resulting in the weakly, strongly, hyper- and super-singular integral kernels. Several examples may be given by the integrals in elasticity and potential flow having integrands containing 1 r a , r a @ @n 1 r  , LogðrÞ r a and LogðrÞr a @ @n 1 r  , where a is the exponential value. Generally, the name we associate with every type of singularity is dependent on the order of r (the distance between the field point and source point, that is, r = jx p  x q j) in the denominator of the integrand of a singular integral. In three-dimen- sional problems, the weakly, strongly, hyper- and super-singularity will refer to the singular integrands of order one, two, three and higher than three, respectively [1–3]. Several procedures have been proposed to perform various levels of singular integrals when the collocation point ap- proaches to the integration boundary. One of these procedures is the transformation method. The essential approach of this method is based on some algebraic and numerical operations. Using this scheme, successive simple addition and subtraction in local polar coordinates is employed to arrive at a regular integral [4]. A numerical treatment for solving the integral equa- tion of the second kind with Cauchy kernel was presented by Abdou and Nasr [5]. The singular term has been removed and the solution in the Legendre polynomial form has been used to obtain a system of linear algebraic equation. Jun et al. [6] devised a procedure based on a simple partition of the closest elements to the source point followed by Gaussian quadrature. The transition between the two methods was chosen according to the distance of the source point from the subdivision. 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.03.019 * Corresponding author. E-mail addresses: gasemi@aut.ac.ir, hmaaa2002@yahoo.com (H. Ghassemi). Applied Mathematics and Computation 213 (2009) 277–289 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc