A Regularized Algorithm for the Calculation of Second Derivative Values in Potential BEM

摘要

In the past decade, experiments have shown that materials display strong size effects when the characteristic length scale is on the order of microns. The conventional elasto-plasticity theory, however, can not predict this size dependence because its constitutive model possesses no internal length scale. So far, a number of models considering this length scale have been proposed, see e.g. [1], and numerical simulations have been made to this phenomenon through the finite element method (FEM) [2] and the boundary element method (BEM) [3]. In those numerical analyses, strain gradients were generally calculated in an indirect way, i.e. interpolating from the obtained strains or interpolating twice from the displacements. This procedure obviously induces significant numerical errors. Therefore, the aim of the present work is to develop a new algorithm to improve the numerical accuracy of the strain gradients. For the first step, this paper presents our new algorithm in the context of potential BEM to show how the potential second derivatives can be calculated in an agreeable accuracy.The boundary integral equations (BIEs) of potential second derivatives are of third order singularities; here the strong singularity is referred to as the first order singularity and the hyper-singularity as the second order one. Obviously the direct calculation of the third order singular integrals is rather difficult, which has been seldom met in the BEM analysis so far. The idea of the present paper is to use a special indirect algorithm which is based on a regularized BIE formulation of the potential second derivatives, following the work of the present author and his coworkers [4]. In this algorithm, the singularities in the integrals are fully regularized, thus only a Gaussian quadrature is needed and the most common used C~0 continuous element can be used. Numerical examples show the accuracy and efficiency of the present algorithm.