A novel high-performance quadrature rule for BEM formulations

This paper describes a general approach to compute the boundary integral equations that appear when the boundary element method is applied for solving common engineering problems. The proposed procedure consists of a new quadrature rule to accurately evaluate singular and weakly singular integrals in the sense of the Cauchy Principal Value by an exclusively numerical procedure. This procedure is based on a system of equations that results from the finite part of known integrals, that include the shape functions used to approximate the field variables. The solution of this undetermined system of equations in the minimum norm sense provides the weights of the quadrature rule. A MATLAB script to compute the quadrature rule is included as supplementary material of this work. This approach is implemented in a boundary element method formulation based on the Bézier–Bernstein space as an approximation basis to represent both geometry and field variables for verification purposes. Specifically, heat transfer, elastostatic and elastodynamic problems are considered.