The boundary element method (BEM) is known to be an accurate and stable numerical method, especially useful in linear elastic fracture mechanic problems. In this case, the so called dual formulation is employed, where two different kinds of boundary integral equations are used to model the crack surfaces.

Special attention has to be taken at the crack tips, where the discontinuity enforced by the presence of the crack produces singular stress fields. For this reason, several approaches have been employed in order to model the displacements and stresses around the crack tip adequately. A simple approach is to use quarter-point elements, which displaces the position of the central node to include the proper displacement behaviour at the crack tip. Other approaches include the use of partition of unity in the elements containing the crack tips, however this particular approach can make the linear system of equations ill-conditioned.

More recently, the authors have developed an implicit enrichment, where only a couple of degrees of freedom are added to the problem for each crack tip, hence adding the enrichment to more elements does not increase the total number of degrees of freedom of the fracture problem. Furthermore, the stress intensity factors (SIF) are calculated as part of the displacement solution, thus avoiding the post-processing step to calculate them. This formulation has been extended for anisotropic materials.

However, when dealing with large meshes or problems with multiple cracks, the computational time used to solve the linear system of equations Ax = b increases drastically, becoming impracticable to be solved within a reasonable amount of time. The Adaptive Cross Approximation (ACA) is an acceleration technique, where admissible blocks obtained with hierarchical matrices are approximated by low-rank matrices.

In this work, we present for the first time a combined approach of both ACA and XBEM for fracture problems in 2D anisotropic materials.