A new method of numerical evaluation of singular integrals occuring in two-dimensional BIEM Roman Nahlik Technological and Automotive Institute, Technical University of Lddl, Department of Bielsko - Biala, Findera 32, 43-300 Bielsko - Biala, Poland Ryszard Bialecki Institut of Thermal Technology. Silesian Technical University, Konarskiego 22, 44-100 Gliwice, Poland (Received June 1982) A new quadrature rule for integrands having logarithmical singularities has been developed. This rule proved to be efficient especially in the context of the BIEM. Key words: mathematical model, BIEM, numerical integration, singular integrals Introduction The matrix formulation of BIEM can be expressed in a form:’ HU=CQ where : H and C are square matrices II and Q are vectors, discretized boundary potentials and fluxes, respectively. The diagonal terms of H and G are computed by the evalua- tion of singular integrals. The accuracy with which this is done has a considerable influence on the results, as the matrices H and G are normally strongly diagonal dominant. Another way of handling such integrals is to extract the singularity from the integrand. This is often accomplished by using weighted integration formulae, e.g. logarithmically weighted Gaussian quadrature.2 Finally, in some cases the integration can be performed analytically. One way of carrying out numerical integration of func- tions having singularities is to proceed as if there were no singularity at all, i.e. choosing quadrature rules which do not need the values of the integrand in its singularity points. This method is straightforward, however, it necessi- tates applying a quadrature rule which uses a large number of integration points thus, it is not numerically effective. The approximations we discuss here have the form:’ 1 I f(X) dx = 2 At .f(Xi) i=1 -1 (2) The weights At and the nodes Xi can be determined demand- ing that expression (2) is exact for all functionsf(x) of a form: fCx) = g Ck. rk(Xi) k=l (3) where: tk(x), k = 1 (1)M is a set of linear independent and integrable functions; ck, k = l(l)M are constants and M is the order of the quadrature. ln a general case, i.e. for any chosen set of tk(x), deter- mining the unknown nodes and weights in formula (2) is to be accomplished by solving a nonlinear set of coupled equa- tions. When solving such a set, especially for large N, numerical difficulties may appear. These difficulties can be We wish to make M as high as possible. When t&) = xk-r, the integration formula is the known Gaussian rule.3 In that case At and xi can be determined from a set of overcome by choosing arbitrary, e.g. equally spaced Xi and decoupled equations, exploiting the orthogonality of Legendre polynomials. This leads to quadratures of the highest possible order, M = 2N. 0307-904X/83/03169-04/$03.00 0 1983 Butterworth &Co. (Publishers) Ltd. Appl. Math. Modelling, 1983, Vol. 7, June 169