Effects of the smoothness of partitions of unity on the quality of representation of singular enrichments for GFEM/XFEM stress approximations around brittle cracks

Torres, Diego Amadeu F.; Barcellos, Clóvis Sperb de; Mendonça, Paulo de Tarso R.

doi:10.1016/j.cma.2014.08.030

Cited by 13 publications

(8 citation statements)

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“…Firstly, because the singular enrichment functions themselves demand care. Additionally, the derivatives of the smooth PoU functions show an oscillatory behavior inside the clouds (Torres et al, 2015;Mendonça et al, 2011;Mendonça et al, 2013). Herein, the symmetrical Wandzura's triangular rule (Wandzura and Xiao, 2003) with 175 points was applied for the elements far from the crack segment and its tip.…”

Section: Numerical Integrationmentioning

confidence: 99%

“…Recently, the smooth version of the GFEM (Duarte et al, 2006) has shown to be very appropriate to model sharp gradients around crack tips in linear elastic fracture mechanics (LEFM) (Torres et al, 2015). Continuous stress fields around the singularity favor the estimation of crack severity parameters (Torres et al, 2019).…”

Section: Introductionmentioning

confidence: 99%

“…Such a strategy is called the topological enrichment and favors the emergence of convergence difficulties for the conventional C 0 -G/XFEM, as it is well-known since the works of Laborde et al (2005) and Béchet et al (2005). The topological strategy is still appealing as it demands fewer degrees of freedom (dof), besides negatively impacting less intensely the conditioning if compared to the geometrical enrichment (Gupta et al, 2013;Torres et al, 2015).…”

Section: Introductionmentioning

confidence: 99%

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On the topological enrichment for crack modeling via the generalized/extended FEM: a novel discussion considering smooth partitions of unity

Torres

2021

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Purpose It has been usual to prefer an enrichment pattern independent of the mesh when applying singular functions in the generalized/eXtended finite element method (G/XFEM). This choice, when modeling crack tip singularities through extrinsic enrichment, has been understood as the only way to surpass the typical poor convergence rate obtained with the finite element method (FEM), on uniform or quasi-uniform meshes conforming to the crack. Herein, the topological enrichment pattern is revisited in the light of a higher-order continuity obtained with a smooth partition of unity (PoU). Aiming to verify the smoothness' impacts on the blending phenomenon, a series of numerical experiments is conceived to compare the two GFEM versions: the conventional one, based on piecewise continuous PoU's, and another which considers PoU's with high-regularity. Design/methodology/approach The stress approximations right at the crack tip vicinity are qualified by focusing on crack severity parameters. For this purpose, the material forces method originated from the configurational mechanics is used. Some attempts to improve solution using different polynomial enrichment schemes, besides the singular one, are discussed aiming to verify the transition/blending effects. A classical two-dimensional problem of the linear elastic fracture mechanics (LEFM) is solved, considering the pure Mode I and the mixed-mode loading. Findings The results reveal that, in the presence of smooth PoU's, the topological enrichment can still be considered as a suitable strategy for extrinsic enrichment. First, because such an enrichment pattern still can treat the crack independently of the mesh and deliver some advantage in terms of convergence rates, under certain conditions, when compared to the conventional FEM. Second, because the topological pattern demands fewer degrees of freedom and impacts conditioning less than the geometrical strategy. Originality/value Several outputs are presented, considering estimations for the J–integral and the angle of probable crack advance, this last computed from two different strategies to monitoring blending/transition effects, besides some comments about conditioning. Both h- and p-behaviors are displayed to allow a discussion from different points of view concerning the topological enrichment in smooth GFEM.

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Section: Numerical Integrationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the topological enrichment for crack modeling via the generalized/extended FEM: a novel discussion considering smooth partitions of unity

Torres

2021

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“…This mapping is conceived so that the high gradients of the solution are taken into account by concentrating integration points in the region corresponding to the singularity. A variety of mapping methods were explored through years and it can be mentioned as recurrent strategies the Duffy transformation (Duffy, 1982), its modifications (Cano and Moreno, 2015; Lv et al , 2019) and the variations of the polar integration method (Laborde et al , 2005; Béchet et al , 2005; Park et al , 2009; Torres et al , 2015).…”

Section: Introductionmentioning

confidence: 99%

On the numerical integration in generalized/extended finite element method analysis for crack propagation problems

Campos

Barros

Penna

2020

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Purpose The purpose of this paper is to evaluate some numerical integration strategies used in generalized (G)/extended finite element method (XFEM) to solve linear elastic fracture mechanics problems. A range of parameters are here analyzed, evidencing how the numerical integration error and the computational efficiency are improved when particularities from these examples are properly considered. Design/methodology/approach Numerical integration strategies were implemented in an existing computational environment that provides a finite element method and G/XFEM tools. The main parameters of the analysis are considered and the performance using such strategies is compared with standard integration results. Findings Known numerical integration strategies suitable for fracture mechanics analysis are studied and implemented. Results from different crack configurations are presented and discussed, highlighting the necessity of alternative integration techniques for problems with singularities and/or discontinuities. Originality/value This study presents a variety of fracture mechanics examples solved by G/XFEM in which the use of standard numerical integration with Gauss quadratures results in loss of precision. It is discussed the behaviour of subdivision of elements and mapping of integration points strategies for a range of meshes and cracks geometries, also featuring distorted elements and how they affect strain energy and stress intensity factors evaluation for both strategies.

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“…They concluded that XFEM is an effective tool for modeling stable crack growth in homogeneous and bi-materials as it does not require the conformal meshing. The piecewise polynomial partition of unity (PoU) functions used in the conventional instance of the method may be not the most appropriate for some kinds of problems, as in case of singular enrichments (Torres et al, 2014). In this context, the C k -GFEM, which is quite similar to C 0 -GFEM, presents the high regularity of the approximation as an attractive feature.…”

Section: Introductionmentioning

confidence: 99%

Comparative Analysis of Ck- and C0-GFEM Applied to Two-dimensional Problems of Confined Plasticity

Freitas

Torres

Mendonça

et al. 2015

Lat. Am. j. solids struct.

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For many practical applications in engineering, a complex structure shows linear elastic behavior over almost all its extension, but exhibits confined plasticity contained in some small critical regions, e.g. stress concentrations in fillets and sharp internal corners. The behavior of C 0 -and C k -GFEM is investigated in this class of problems.The first goal of this study is to verify the actual formulation of the C k -GFEM for two-dimensional elastoplasticity, as a modification of the C 0 -GFEM formulation. The C k -GFEM is based on a set of basis functions with C k continuity over the domain. The approximation functions are constructed from a C k continuous partition of unity, over which polynomial enrichment functions (or any special function) can be applied, in the same fashion as in the usual C 0 -GFEM. In this way, the finite element approximations show continuous responses for both displacements and stresses across inter-element interfaces. An investigation is performed to assess the behavior of higherregularity partitions of unity against conventional C 0 counterparts. The irreversible response and hardening effects of the material is represented by the rate independent J 2 plasticity theory with linear isotropic hardening of material and von Mises yield criteria, being considered only monotonic loading and the kinematics of small displacements and small deformations. The focus herein is to enlighten any possible advantage of smoothness in the presence of plastification phenomena, seeking for improvements in capturing the evolution of the process zone.

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Effects of the smoothness of partitions of unity on the quality of representation of singular enrichments for GFEM/XFEM stress approximations around brittle cracks

Cited by 13 publications

References 77 publications

On the topological enrichment for crack modeling via the generalized/extended FEM: a novel discussion considering smooth partitions of unity

On the topological enrichment for crack modeling via the generalized/extended FEM: a novel discussion considering smooth partitions of unity

On the numerical integration in generalized/extended finite element method analysis for crack propagation problems

Comparative Analysis of Ck- and C0-GFEM Applied to Two-dimensional Problems of Confined Plasticity

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