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

Agathos, Konstantinos; Bordas, Stéphane P.A.; Chatzi, Eleni

doi:10.1016/j.cma.2018.08.007

Cited by 78 publications

(38 citation statements)

References 53 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Many ideas have been proposed to improve the conditioning of GFEM/XFEM. We refer to References , and and the references therein. A stable GFEM (SGFEM) idea was introduced in Reference based on a simple modification of the enrichments as follows:

\sum_{i \in I_{h}} a_{i} ϕ_{i} + \sum_{i \in I_{E N R}} b_{i} ϕ_{i} (F - ℐ_{h} F),

where

ℐ_{h} F

is the FE interpolant of F , namely,

ℐ_{h} F (x, y) = \sum_{j \in I_{h}} v (x_{j}, y_{j}) ϕ_{j} (x, y) .

…”

Section: Conventional Xfem Sgfem and Dof‐gathering Xfemmentioning

confidence: 99%

“…The GFEM/XFEM have been successfully applied to the crack problems, and a lot of significant developments have been made in recent decades (see References and for review and recent progress). The GFEM/XFEM for the crack problems enrich the standard FEM with singular functions at nodes around the crack tip O and with the Heaviside functions at nodes in a neighborhood of the crack line Γ O .…”

Section: Introductionmentioning

confidence: 99%

“…The bad conditioning may lead to disastrous round‐off errors in elimination methods or slow convergence of iterative schemes to solve the underlying linear system. Various interesting ideas have been proposed to improve the conditioning of GFEM/XFEM, such as locally adapting positions of either nodes or interface curves to balance the ratios of the two volumes created by the intersections of the curves with the elements, preconditioning the stiffness matrices or orthogonalization, and correcting the enrichments by interpolant …”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stable generalized finite element methods for elasticity crack problems

Cui

2020

Numerical Meth Engineering

View full text Add to dashboard Cite

Generalized or eXtended finite element methods (GFEM/XFEM) for crack problems have been studied extensively. The GFEM/XFEM are called extrinsic if additional functions are enriched at every node in certain domains, while they are called degree of freedom (DOF)-gathering if the singular enriched functions are gathered using cutoff functions. The DOF-gathering GFEM/XFEM save the additional DOFs compared with the extrinsic approach. Both extrinsic and DOF-gathering GFEM/XFEM suffer from difficulties of stabilities in a sense that their scaled condition numbers (SCN) of stiffness matrices could be much larger than those of the standard FEM. A GFEM/XFEM is referred to as stable GFEM (SGFEM) if it reaches optimal convergence orders, and its SCN is of same order as that of FEM. An extrinsic SGFEM was established in Zhang et al for the Poisson crack problems. Objective of this article is to propose the SGFEM for elasticity crack problems; both extrinsic and DOF-gathering schemes areaddressed. The main idea is to modify the enriched functions by subtracting their FE interpolants, which was developed by Babuška and Banerjee. To remove local almost linear dependence introduced by multifold enrichments at one node, we propose a local principal component analysis technique to identify and analyze "contributions" of multifold enrichments at one node. Numerical studies demonstrate that the proposed SGFEM and DOF-gathering SGFEM are of optimal convergence and have the SCNs of same order as in the FEM.

show abstract

\sum_{i \in I_{h}} a_{i} ϕ_{i} + \sum_{i \in I_{E N R}} b_{i} ϕ_{i} (F - ℐ_{h} F),

where

ℐ_{h} F

is the FE interpolant of F , namely,

ℐ_{h} F (x, y) = \sum_{j \in I_{h}} v (x_{j}, y_{j}) ϕ_{j} (x, y) .

…”

Section: Conventional Xfem Sgfem and Dof‐gathering Xfemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Stable generalized finite element methods for elasticity crack problems

Cui

2020

Numerical Meth Engineering

View full text Add to dashboard Cite

show abstract

“…With respect to tip enrichment, ill-conditioning can arise when geometrical enrichment is used due to the fact that away from the singularity the tip enrichment functions tend to become linearly dependent both with respect to the FE part of the approximation and each other [101,129]. To overcome this issue several alternatives have been proposed such as altering the partition of unity basis used to pre-multiply the tip enrichment functions [105,121,130], preconditioners [106,125], stabilization [127] and enrichment function orthogonalization [101,129]. Moreover, vector enrichment functions have been shown to lead to improved conditioning [114], and if further combined to the stable GFEM [111,112] they can lead to optimal growth rates of the scaled condition number.…”

Section: Ill-conditioningmentioning

confidence: 99%

Discrete and Phase Field Methods for Linear Elastic Fracture Mechanics: A Comparative Study and State-of-the-Art Review

et al. 2019

Self Cite

View full text Add to dashboard Cite

Three alternative approaches, namely the extended/generalized finite element method (XFEM/GFEM), the scaled boundary finite element method (SBFEM) and phase field methods, are surveyed and compared in the context of linear elastic fracture mechanics (LEFM). The purpose of the study is to provide a critical literature review, emphasizing on the mathematical, conceptual and implementation particularities that lead to the specific advantages and disadvantages of each method, as well as to offer numerical examples that help illustrate these features.

show abstract

“…However, the use of these enrichment functions brings extra unknowns to the problem, as both thermal and mechanical fields require a set of enrichment terms to model crack discontinuities in temperature and displacement respectively. This increase in the number of unknowns and can lead to numerical difficulties such as ill-conditioned linear systems requiring solutions [27], where possible solutions are obtained either using quasi-orthogonalization of enrichment functions [28,29] or decomposing the system stiffness matrix to remove the diagonal zeros [30]. Alternatively, meshless methods can model crack discontinuities by disabling the influence between nodes on the two sides of the crack, which can avoid the use of enrichment functions, an advantage over element-based methods.…”

Section: Accepted Manuscriptmentioning

confidence: 99%

Thermoelastic fracture modelling in 2D by an adaptive cracking particle method without enrichment functions

Augarde

2019

International Journal of Mechanical Sciences

View full text Add to dashboard Cite

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.

show abstract

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

Cited by 78 publications

References 53 publications

Stable generalized finite element methods for elasticity crack problems

Stable generalized finite element methods for elasticity crack problems

Discrete and Phase Field Methods for Linear Elastic Fracture Mechanics: A Comparative Study and State-of-the-Art Review

Thermoelastic fracture modelling in 2D by an adaptive cracking particle method without enrichment functions

Contact Info

Product

Resources

About