A serendipity triangular patch for evaluating weakly singular boundary integrals

Element subdivision is the most widely used method for the numerical evaluation of weakly singular integrals in three-dimensional boundary element analyses. In the traditional subdivision method, the sub-elements, which are called patches in this paper, are obtained by simply connecting the singular point with each vertex of the element. Patches with large angles at the source point may be produced and thus, a large number of Gaussian quadrature points are needed to achieve acceptable accuracy. In this paper, a serendipity triangular patch with four-node is presented to solve the problem. Case studies have been made to investigate the effect of the location of the middle node of the serendipity patch on accuracy, and an optimal location is determined. Moreover, theoretical analysis validating the optimal location is also given with a new form of polar coordinate transformation. Numerical examples are presented to compare the new patch with the conventional linear patch with respect to both accuracy and efficiency. In all cases, the results are encouraging.