A new semi-analytic algorithm of nearly singular integrals in high order boundary element analysis of 2D elasticity

The calculation of the nearly singular integrals on high order elements is difficult in boundary element method (BEM) at present. In this paper, a new semi-analytic algorithm is established to deal with the nearly strongly singular and hyper-singular integrals for high order elements in two dimensional (2D) BEM. By analyzing the geometric feature of high order elements by local coordinates, the relative distance from a source point to the element is defined. For the nearly singular integrals of the high order elements, the leading singular part of the integral kernel function is separated into the explicit formulation by a series of deduction. Then the nearly singular integrals on the high order elements close to the source point are transformed to both the non-singular part and singular part by the subtraction, where the former is computed by the numerical quadrature and the later is evaluated by the analytic algorithm. Consequently, the quadratic element with the new semi-analytic algorithm was applied to calculate the displacements and stresses very close to the boundary and thin-walled structures in the boundary element analysis of 2D elasticity. Three examples were given to demonstrate that the computed results of the quadratic element with the semi-analytic algorithm are more accurate than those of the linear element with the analytic algorithm for the nearly singular integrals. In fact, the boundary element analysis with the linear element has been greatly more advantageous compared with the finite element method in analyzing the thin bodies.