TY - JOUR

T1 - A general algorithm for numerical integration of three-dimensional crack singularities in PU-based numerical methods

AU - Lv, Jia He

AU - Jiao, Yu Yong

AU - Rabczuk, Timon

AU - Zhuang, Xiaoying

AU - Feng, Xia Ting

AU - Tan, Fei

N1 - Funding information: The authors acknowledge the financial support from the National Natural Science Foundation of China (Nos. 51979253 , 41731284 , 41920104007 , 11672360 , 51621006 and 51879245 ).

PY - 2020/2/19

Y1 - 2020/2/19

N2 - With the development of PU-based numerical methods for crack problems, the evaluation of various orders of vertex/edge singularity has been one of the most critical issues, which restrains the computational efficiency of PU-based methods, especially for 3D crack problems. In this paper, based on the conventional Duffy transformation, a general algorithm for numerical integration of three-dimensional crack singularities is proposed for the vertex/edge singularity problems, which takes the integration cell shape into full consideration. Besides, the corresponding 3D conformal preconditioning strategy is constructed to fully eliminate the shape influence of tetrahedron elements. Extensive numerical examples, including ill-shaped integration cells and crack-front tetrahedron elements with parallel/nonparallel crack front, are given to validate the feasibility and accuracy of the proposed method. As a result, for each crack-front element, several hundreds of Gauss points are sufficient to achieve the precision of 10−6 for both kernels 1∕r and 1∕r, in sharp contrast with around ten thousands of Gauss points using the conventional Duffy transformation.

AB - With the development of PU-based numerical methods for crack problems, the evaluation of various orders of vertex/edge singularity has been one of the most critical issues, which restrains the computational efficiency of PU-based methods, especially for 3D crack problems. In this paper, based on the conventional Duffy transformation, a general algorithm for numerical integration of three-dimensional crack singularities is proposed for the vertex/edge singularity problems, which takes the integration cell shape into full consideration. Besides, the corresponding 3D conformal preconditioning strategy is constructed to fully eliminate the shape influence of tetrahedron elements. Extensive numerical examples, including ill-shaped integration cells and crack-front tetrahedron elements with parallel/nonparallel crack front, are given to validate the feasibility and accuracy of the proposed method. As a result, for each crack-front element, several hundreds of Gauss points are sufficient to achieve the precision of 10−6 for both kernels 1∕r and 1∕r, in sharp contrast with around ten thousands of Gauss points using the conventional Duffy transformation.

KW - Distance transformation

KW - Duffy transformation

KW - Numerical quadrature

KW - PU-based numerical methods

KW - Singular integrals

UR - http://www.scopus.com/inward/record.url?scp=85079558017&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2020.112908

DO - 10.1016/j.cma.2020.112908

M3 - Article

AN - SCOPUS:85079558017

VL - 363

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0045-7825

M1 - 112908

ER -