Abstract.A method is developed for the computation of the weights and nodes of a numerical quadrature which integrates functions containing singularities up to order 1/x 2 , without the requirement to know the coefficients of the singularities exactly. The work is motivated by the need to evaluate such integrals on boundary elements in potential problems and is a simplification of a previously published method, but with the advantage of handling singularities at the endpoints of the integral. The numerical performance of the method is demonstrated by application to an integral containing logarithmic, first, and second order singularities, characteristic of the problems encountered in integrating a Green's function in boundary element problems. It is found that the quadrature is accurate to 11-12 decimal places when computed in double precision. 1. Introduction. Boundary integral methods employing hypersingular integrals have become increasingly popular over recent decades with applications in potential problems such as acoustics [6] and fracture mechanics [13]. Hypersingular integrals arise naturally when it is required to compute the field quantities in such problems, for example, the potential and its gradients, and when specialized techniques for the avoidance of "interior resonance" are used, such as that of Burton and Miller [5]. When such integrals arise, they require special numerical treatment and cannot be evaluated using the standard tools of Gaussian quadrature. Instead, approaches tailored to the problem must be used, which can add considerable complexity to the code. This paper introduces a method for the design of numerical quadrature rules which evaluate hypersingular integrals without requiring a detailed analysis of the integrand. The method is based on that of Kolm and Rokhlin [12], who developed a procedure for the design of such rules, but is considerably simplified by the use of Brandão's approach to finite part integrals [4] and is extended to the case of integrals with endpoint singularities-essential if such a scheme is to be used with boundary elements.To fix ideas, we assume that we are dealing with a two-dimensional potential problem, such as the Laplace or Helmholtz equation:
A method is derived for the fast, exact prediction of acoustic fields around rotating sources by using a series expansion which generalizes a previously published method for a circular piston. The technique gives exact predictions for the field outside the sphere containing the rotor in a computational time two orders of magnitude less than that required for direct numerical evaluation of the acoustic integrals. Its use is demonstrated by application to two sample problems characteristic of real aircraft propellers.
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