Improving the conditioning of XFEM/GFEM for fracture mechanics problems through enrichment quasi-orthogonalization

in Computer Methods in Applied Mechanics and Engineering (2018)

Partition of unity enrichment is known to significantly enhance the accuracy of the finite element method by allowing the incorporation of known characteristics of the solution in the approximation space ... [more ▼]

Partition of unity enrichment is known to significantly enhance the accuracy of the finite element method by allowing the incorporation of known characteristics of the solution in the approximation space. However, in several cases it can further cause conditioning problems for which a number of remedies have been proposed in the framework of the extended/generalized finite element method (XFEM/GFEM). Those solutions often involve significant modifications to the initial method and result in increased implementation complexity. In the present work, a simple procedure for the local near-orthogonalization of enrichment functions is introduced, which significantly improves the conditioning of the resulting system matrices, while requiring only minor modifications to the initial method. Although application to different types of enrichment functions is possible, the resulting scheme is specialized for the singular enrichment functions used in linear elastic fracture mechanics and tested through benchmark problems. [less ▲]

Extended Finite Element Method with Global Enrichment

Scientific Conference (2015, July)

A variant of the extended finite element method is presented which facilitates the use of enriched elements in a fixed volume around the crack front (geometrical enrichment) in 3D fracture problems. The ... [more ▼]

A variant of the extended finite element method is presented which facilitates the use of enriched elements in a fixed volume around the crack front (geometrical enrichment) in 3D fracture problems. The major problem associated with geometrical enrichment is that it significantly deteriorates the conditioning of the resulting system matrices, thus increasing solution times and in some cases making the systems unsolvable. For 2D problems this can be dealt with by employing degree of freedom gathering [1] which essentially inhibits spatial variation of enrichment function weights. However, for the general 3D problem such an approach is not possible since spatial variation of the enrichment function weights in the direction of the crack front is necessary in order to reproduce the variation of solution variables, such as the stress intensity factors, along the crack front. The proposed method solves the above problem by employing a superimposed mesh of special elements which serve as a means to provide variation of the enrichment function weights along the crack front while still not allowing variation in any other direction. The method is combined with special element partitioning algorithms [2] and numerical integration schemes [3] as well as techniques for the elimination of blending errors between the standard and enriched part of the approximation in order to further improve the accuracy of the produced results. Additionally, a novel benchmark problem is introduced which enables the computation of displacement and energy error norms as well as errors in the stress intensity factors for the general 3D case. Through this benchmark problem it is shown that the proposed method provides optimal convergence rates, improved accuracy and reduced computational cost compared to standard XFEM. [less ▲]

Stable 3D extended finite elements with higher order enrichment for accurate non planar fracture

in Computer Methods in Applied Mechanics and Engineering (2015)

We present an extended finite element method (XFEM) for 3D nonplanar linear elastic fracture. The new approach not only provides optimal convergence using geometrical enrichment but also enables to ... [more ▼]

We present an extended finite element method (XFEM) for 3D nonplanar linear elastic fracture. The new approach not only provides optimal convergence using geometrical enrichment but also enables to contain the increase in conditioning number characteristic of enriched finite element formulations: the number of iterations to convergence of the conjugate gradient solver scales similarly to and converges faster than the topologically-enriched version of the standard XFEM. This has two advantages: (1) the residual can be driven to zero to machine precision for at least 50% fewer iterations than the standard version of XFEM; (2) additional enrichment functions can be added without significant deterioration of the conditioning. Numerical examples also show that our new approach is up to 40% more accurate in terms of stress intensity factors, than the standard XFEM. [less ▲]