A New Error Upper Bound Formula for Gaussian Integration in Boundary Integral Equations

This paper proposes a new error upper bound formula for the Gaussian integration of the near-singular integral using the Boundary Element Method. First, this study found through numerical tests that the maximum relative error of the Gaussian integration has a downward concave shape but an approximately linear relationship with the relative distance, which is defined as the ratio of the distance from the source point to the element over the element length in a semi-logarithmic plot. Thus, the error upper bound can be defined as a line that closely approaches the computed error data points from the upper side. This line can be obtained by connecting two specified data points that are located outside, but very close to, the considered error range. Further research indicates that one parameter of the fitted line has a linear relationship with the number of Gaussian integration points and singularity orders and the other parameter can be treated as a constant, which together make the proposed Gaussian integration error upper bound formula widely applicable. Compared to the Lachat and Watson criterion, the proposed formula requires fewer integration points when the source point is very close to the element and thus serves to improve computational efficiency. The proposed formula also avoids calculation failure that can occur when using the Davies and Bu criterion. The numerical example results show that the proposed error upper bound formula can evaluate the integration accuracy well and improve computational efficiency when using an adaptive Gaussian integration method.