Elastic crack growth in finite elements with minimal remeshing

Belytschko, Ted; Black, T. Howard

doi:10.1002/(sici)1097-0207(19990620)45:5<601::aid-nme598>3.0.co;2-s

Cited by 4,085 publications

(2,201 citation statements)

References 15 publications

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“…Among the noteworthy SGEMs are the s-version of the finite element method [19,20,21,22] with application to strong [23,24] and weak [25,26,27,28] discontinuities, various multigrid-like scale bridging methods [29,30,31,32], the Extended Finite Element Method (XFEM) [33,34,35] and the Generalized Finite Element Method (GFEM) [36,37] both based on the Partition of Unity (PU) framework [38,39] and the Discontinuous Galerkin (DG) [40,41] method. Multiscale methods based on the concurrent resolution of multiple scales are often called as embedded, concurrent, integrated or hand-shaking multiscale methods.…”

Section: Introductionmentioning

confidence: 99%

“…In XFEM, which use local Partition of Unity [45], the enrichment functions are used to describe spatial features, such as asymptotic crack fields [33] or local flow fields [46]; in addition they provide means to model arbitrary discontinuities. For this discontinuous enrichment functions, the problem of linear dependency does not arise, but the issue of ensuring the integration errors to be significantly smaller than the approximation errors requires special attention [35].…”

Section: Introductionmentioning

confidence: 99%

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Multiscale enrichment based on partition of unity

Fish¹,

Yuan²

2004

Numerical Meth Engineering

139

110

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A new Multiscale Enrichment method based on the Partition of Unity (MEPU) method is presented. It is a synthesis of mathematical homogenization theory and the Partition of Unity method. Its primary objective is to extend the range of applicability of mathematical homogenization theory to problems where scale separation may not be possible. MEPU is perfectly suited for enriching the coarse scale continuum descriptions (PDEs) with fine scale features and the quasi-continuum formulations with relevant atomistic data. Numerical results show that it provides a considerable improvement over classical mathematical homogenization theory and quasi-continuum formulations.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Multiscale enrichment based on partition of unity

Fish¹,

Yuan²

2004

Numerical Meth Engineering

139

110

View full text Add to dashboard Cite

show abstract

“…The resolution of compressible fracture mechanics problems has been extensively studied in the context of the X-FEM for both 2D [39,18,40,32,24] and 3D fracture mechanics [21,22]. The most common enrichment strategy consists in using the asymptotic displacement field as an enrichment for the displacement finite element approximation.…”

Section: Incompressible Fracture Mechanicsmentioning

confidence: 99%

Stability of incompressible formulations enriched with X-FEM

Legrain

Moës

Huerta

2008

Computer Methods in Applied Mechanics and Engineering

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The treatment of (near-)incompressibility is a major concern for applications involving rubber-like materials, or when important plastic flows occurs as in forming processes. The use of mixed finite element methods is known to prevent the locking of the finite element approximation in the incompressible limit. However, it also introduces a critical condition for the stability of the formulation, called the infsup or LBB condition. Recently, the finite element method has evolved with the introduction of the partition of unity. The eXtended Finite Element Method (X-FEM) uses the partition of unity to remove the need to mesh physical surfaces or to remesh them as they evolve. The enrichment of the displacement field makes it possible to treat surfaces of discontinuity inside finite elements. In this paper, some strategies are proposed for the enrichment of mixed finite element approximations in the incompressible setting. The case of holes, material interfaces and cracks are considered. Numerical examples show that for well chosen enrichment strategies, the finite element convergence rate is preserved and the inf-sup condition is passed.

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“…The method makes use of the partition-of-unity property of finite element shape functions [1,36,37]. A collection of functions i , associated with nodes i, form a partition of unity if…”

Section: Exact Numerical Representation Of Discontinuitiesmentioning

confidence: 99%

Cohesive‐zone models, higher‐order continuum theories and reliability methods for computational failure analysis

Borst

Gutiérrez

Wells

et al. 2004

Numerical Meth Engineering

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SUMMARYA concise overview is given of various numerical methods that can be used to analyse localization and failure in engineering materials. The importance of the cohesive-zone approach is emphasized and various ways to incorporate the cohesive-zone methodology in discretization methods are discussed. Numerical representations of cohesive-zone models suffer from a certain mesh bias. For discrete representations this is caused by the initial mesh design, while for smeared representations it is rooted in the ill-posedness of the rate boundary value problem that arises upon the introduction of decohesion. A proper representation of the discrete character of cohesive-zone formulations which avoids any mesh bias can be obtained elegantly when exploiting the partition-of-unity property of finite element shape functions. The effectiveness of the approach is demonstrated for some examples at different scales. Moreover, examples are shown how this concept can be used to obtain a proper transition from a plastifying or damaging continuum to a shear band with gross sliding or to a fully open crack (true discontinuum). When adhering to a continuum description of failure, higher-order continuum models must be used. Meshless methods are ideally suited to assess the importance of the higher-order gradient terms, as will be shown. Finally, regularized strain-softening models are used in finite element reliability analyses to quantify the probability of the emergence of various possible failure modes.

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Elastic crack growth in finite elements with minimal remeshing

Cited by 4,085 publications

References 15 publications

Multiscale enrichment based on partition of unity

Multiscale enrichment based on partition of unity

Stability of incompressible formulations enriched with X-FEM

Cohesive‐zone models, higher‐order continuum theories and reliability methods for computational failure analysis

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