A Generation of Special Triangular Boundary Element Shape Functions for 3D Crack Problems

Abstract

This paper focuses on tackling the two drawbacks of the dual boundary element method (DBEM) when solving crack problems with a discontinuous triangular element: low accuracy of the calculation of integrals with singularity and crack front element must be utilized to model the square-root property of displacement. In order to calculate the integrals with higher order singularity, the triangular elements are segmented into several subregions which consist of subtriangles and subpolygons. The singular integrals in those subtriangles are handled by the singularity subtraction technique in the integration space and can be regularized and accurately calculated. For the nearly singular integrals in those subpolygons, the element subdivision technique is employed to improve the calculation accuracy. In addition, considering the location of the crack front in the element, special crack front elements are constructed based on a 6-node discontinuous triangular element, in which the displacement extrapolation method is introduced to obtain the stress intensity factors (SIFs) without consideration of orthogonalization of the crack front mesh. Several numerical results are investigated to fully verify the validation of the presented approach.

A Subdivision Transformation Method for Weakly Singular Boundary Integrals in Thin-structural Problem

“…The boundary element method (BEM) is a more accurate and effective method developed after finite element method. Different from the basic idea of the finite element method (FEM), BEM only divides the elements on the boundary of the domain and approximate the boundary conditions with the functions satisfying the governing equation [4][5][6][7]. Therefore, compared with the FEM [8,9], BEM has the advantages of dimensionality reduction, fewer elements and simpler data preparation, which is suitable for solving stress concentration, infinite domain and semi-infinite domain problems [10,11].…”

“…Especially for thin-structural problems, poorly shaped elements (such as elements with large angles or narrow lengths) will appear in the discrete geometric model of thin structure [12][13][14], which will seriously affect the accuracy of singular and near singular integrals. Many works have been demonstrated that the near singular integrals can be accurately evaluated by analytical and semi-analytical methods [15,16], Sinh and other nonlinear transformation methods [4,[17][18][19], etc. Therefore, accurate and effective calculation of weakly singular integral in BIE is the key to the implementation of boundary element analysis [20][21][22][23].…”