Abstract
Quadrature rules for evaluating singular integrals that typically occur in the boundary element method (BEM) for two-dimensional and axisymmetric three-dimensional problems are considered. This paper focuses on the numerical integration of the functions on the standard domain [-1, 1], with a logarithmic singularity at the centre. The substitution x = tp, where p (≥ 3) is an odd integer is given particular attention, as this returns a regular integral and the domain unchanged. Gauss-Legendre quadrature rules are applied to the transformed integrals for a number of values of p. It is shown that a high value for p typically gives more accurate results.
Highlights
In this paper the problem of determining an efficient quadrature rule for and integral of the form ∫−1 f ( x)dx, (1)in which f ( x) is a continuously differentiable function, except for having weak logarithmic singularity at its mid-point, is considered
The handling of singular integrals is a continual area of enquiry in the implementation of the boundary element method
The issue of singular integration in the boundary element method is considered in one of the simplest forms; with a logarithmic singularity ( ~ ln r ) for 2D problems, or when the integral is resolved onto the generator in axisymmetric 3D
Summary
In which f ( x) is a continuously differentiable function, except for having weak logarithmic singularity at its mid-point (when x = 0 ), is considered. A weak logarithmic singularity is one in which the behaviour near the singularity is of the ( ) form f ( x) = O log x. The governing partial differential equation is reformulated as a boundary integral equation, and the latter is solved in order to determine unknown boundary properties. By applying a suitable integral equation method, such as collocation, the boundary integral equation is resolved into a linear system of equations, with each row of the matrices representing a geometrical line integral of each panel with respect to a particular boundary observation (e.g. collocation) point
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