A generalized finite element method for the simulation of three-dimensional dynamic crack propagation

Duarte, C. Armando; Hamzeh, Osama N.; Liszka, Tad J.; Tworzydlo, W. W.

doi:10.1016/s0045-7825(00)00233-4

Cited by 345 publications

(233 citation statements)

References 32 publications

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“…Especially the flexibility of these methods with respect to the employed shape functions lead to this development. The concept of so-called extrinsic enrichment for instance is employed in many MM [5,19,21], the extended finite element method (XFEM) [4,6,17], and the generalized finite element method (GFEM) [10][11][12]. Here, the fundamental assumption is the availability of a partition of unity (PU) in the discretization space V ⊂ H k (Ω).…”

Section: Introductionmentioning

confidence: 99%

Stable enrichment and local preconditioning in the particle-partition of unity method

Schweitzer

2010

Numer. Math.

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Summary. This paper is concerned with the stability and approximation properties of enriched meshfree methods. In particular we focus on the particle-partition of unity method (PPUM) yet the presented results hold for any partition of unity based enrichment scheme.The goal of our enrichment scheme is to recover the optimal convergence rate of the uniform h-version independent of the regularity of the solution. Hence, we employ enrichment not only for modeling purposes but rather to improve the approximation properties of the numerical scheme. To this end we enrich our PPUM function space in an enrichment zone hierarchically near the singularities of the solution. This initial enrichment however can lead to a severe ill-conditioning and can compromise the stability of the discretization. To overcome the ill-conditioning of the enriched shape functions we present an appropriate local preconditioner which yields a stable and optimally conditioned basis independent of the employed initial enrichment. The construction of this preconditioner is of linear complexity with respect to the number of discretization points.We obtain optimal error bounds for an enriched PPUM discretization with local preconditioning that are independent of the regularity of the solution globally and within the employed enrichment zone we attain a kind of super-convergence.The results of our numerical experiments clearly show that our enriched PPUM with local preconditioning recovers the optimal convergence rate of O(h p ) of the uniform h-version globally. For the considered model problems from linear elastic fracture mechanics we obtain an improved convergence rate of O(h p+δ ) with δ ≥ 1 2 for p = 1. The condition number of the stiffness matrix is independent of the employed enrichment zone.

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Section: Introductionmentioning

confidence: 99%

Stable enrichment and local preconditioning in the particle-partition of unity method

Schweitzer

2010

Numer. Math.

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“…In the literature, numerical techniques such as the extended finite element method (X-FEM) [17,18], generalized finite element method [23], or the element partition method [24] are all particular instances of the partition of unity method. In the X-FEM, the emphasis has been on modeling discontinuities (such as cracks) with minimal enrichment.…”

Section: Extended Finite Element Methodsmentioning

confidence: 99%

Finite element-based model for crack propagation in polycrystalline materials

Sukumar

Srolovitz

2004

Mat. apl. comput.

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Abstract.In this paper, we use an extended form of the finite element method to study failure in polycrystalline microstructures. Quasi-static crack propagation is conducted using the extended finite element method (X-FEM) and microstructures are simulated using a kinetic Monte Carlo Potts algorithm. In the X-FEM, the framework of partition of unity is used to enrich the classical finite element approximation with a discontinuous function and the two-dimensional asymptotic crack-tip fields. This enables the domain to be modeled by finite elements without explicitly meshing the crack surfaces, and hence crack growth simulations can be carried out without the need for remeshing. First, the convergence of the method for crack problems is studied and its rate of convergence is established. Microstructural calculations are carried out on a regular lattice and a constrained Delaunay triangulation algorithm is used to mesh the microstructure. Fracture properties of the grain boundaries are assumed to be distinct from that of the grain interior, and the maximum energy release rate criterion is invoked to study the competition between intergranular and transgranular modes of crack growth.Mathematical subject classification: 74R10.

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“…a meshing of the crack surface e.g. by triangle meshes, enable a simple consideration of given crack increments [8,9]. One has to determine the length and orientation of the increments at selected nodes on the crack front and then simply update the mesh.…”

Section: Introductionmentioning

confidence: 99%

A new description of cracks in the XFEM

Fries

2011

Proc Appl Math and Mech

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This work outlines an approach that combines the advantages of explicit and implicit descriptions of the crack geometry in the context of the extended finite element method (XFEM). The XFEM is typically combined with an implicit description of the crack which is realized by the level-set method. This facilitates the definition of the enrichments in the XFEM. However, the update of the crack geometry during the propagation of a crack is simpler by means of explicit crack descriptions where the crack surface is meshed.

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A generalized finite element method for the simulation of three-dimensional dynamic crack propagation

Cited by 345 publications

References 32 publications

Stable enrichment and local preconditioning in the particle-partition of unity method

Stable enrichment and local preconditioning in the particle-partition of unity method

Finite element-based model for crack propagation in polycrystalline materials

A new description of cracks in the XFEM

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