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Cracks and defects are inevitable during the long-term use of structures. This study focuses on determining the stress intensity factors of multi-cracked structures by using a new fast multipole dual boundary element method. Numerical examples show that the results of the present method agree well with analytic solutions. When the crack distribution changes, the most unfavorable conditions also change. The shape of the defect has an effect on the stress intensity factors of nearby cracks. Among triangular, rectangular, hexagonal, and circular defects, when the area of the defect is identical, the triangular pore is more likely to induce crack propagation, while the circular pore is more secure. The above results can be used as a reference for structural design and optimization.

Since only the boundary of the analyzed domain needs to be discretized, the boundary element method (BEM) inherently has the advantages of solving crack problems. In this paper, a micromechanics BEM scheme is applied to determine the effective elastic moduli of three-dimensional (3-D) solids containing a large number of parallel or randomly oriented microcracks. The 3-D analyses accelerated by the fast multipole method were carried out to investigate the relations between the effective elastic moduli and the microcrack density parameter. Numerical examples show that the results agree well with the available analytical solution and known micromechanics models. From the numerical examples, we can see that the FM-DBEM inherits the virtue of high accuracy from BIE besides dimension reduction. It makes the method be a promising approach for analyzing elastic materials with numerous microcracks of various shapes.

A fast multipole method (FMM) based on complex Taylor series expansions is applied to the dual boundary element method (DBEM) for large-scale crack analysis in linear elastic fracture mechanics. Combining multipole expansions with local expansions, both the computational complexity and memory requirement are reduced to O(N), where N is the number of DOF. An incremental crack-extension analysis based on the maximum principal stress criterion and the Paris law is used to simulate the fatigue growth of numerous cracks in a 2D solid. Some examples are presented to validate the numerical scheme.

The fast multipole method (FMM) is applied to the dual boundary element method (DBEM) for the analysis of finite solids with large numbers of microcracks. The application of FMM significantly enhances the run-time and memory storage efficiency. Combining multipole expansions with local expansions, computational complexity and memory requirement are both reduced to O(N), where N is the number of DOFs (degrees of freedom). This numerical scheme is used to compute the effective in-plane bulk modulus of 2D solids with thousands of randomly distributed microcracks. The results prove that the IDD method, the differential method, and the method proposed by Feng and Yu can give proper estimates. The effect of microcrack non-uniform distribution is evaluated, and the numerical results show that non-uniform distribution of microcracks increases the effective in-plane bulk modulus of the whole microcracked solid.