Improving conditioning of 3D XFEM for fracture mechanics

Abstract

Since the early development stages of the eXtended Finite Element Method (XFEM), problems involving singularities were treated via use of enrichment in a fixed area around the singularity in order to ensure optimal convergence. It may further be demonstrated that within the domain of 3D fracture mechanics, geometrical enrichment may eliminate mesh dependence and significantly improve the quality of the computed Stress Intensity Factors (SIFs). Nonetheless, the use of geometrical enrichment often inflicts significant deterioration in the conditioning of the associated stiffness matrices, and consequently several remedies have been proposed in existing literature for improving conditioning. Such remedies include the use of special preconditioners [1, 2], degree of freedom gathering [3], which has also been extended to 3D [4], and the use of alternative enrichment functions [5]. All available approaches however, require either the introduction of a preconditioning step or modification of fundamental components of the method, therefore sacrificing part of the simplicity of XFEM, and rendering their application to existing XFEM codes problematic. In the present work we introduce a simple procedure, which significantly improves the conditioning of the resulting stiffness matrices, while imposing only minor modifications to the initial method. Also due to its generality, this procedure can be applied to different types of enrichment functions. The proposed scheme is tested in a number of 3D fracture mechanics problems, where it is further contrasted to benchmarks derived from existing literature. REFERENCES [1] B´echet, E and Minnebo, H and Mo¨es, N and Burgardt, B Improved implementation and robustness study of the X-FEM for stress analysis around cracks.Int. J. Numer. Meth. Engng. 64:1033–1056 (2005). [2] Menk, A and Bordas, SPA A robust preconditioning technique for the extended finite element method. Int. J. Numer. Meth. Engng. 85:1609–1632 (2011). [3] Laborde, P and Pommier, J and Renard, Y and Sala¨un, M High-order extended finite element method for cracked domains. Int. J. Numer. Meth. Engng. 64:354–381 (2005). [4] Agathos, K and Chatzi, E and Bordas, SPA and Talaslidis, D A well-conditioned and optimally convergent XFEM for 3D linear elastic fracture. Int. J. Numer. Meth. Engng. 105:643–677 (2015). [5] Chevaugeon, N and Mo¨es, N and Minnebo, H Improved crack tip enrichment functions and integration for crack modeling using the extended finite element method. Journal for Multiscale Computational Engineering. 11:597-631 (2013)