Interface problems with quadratic X-FEM: design of a stable multiplier space and error analysis

Ferté, Guilhem; Massin, Patrick; Moës, Nicolas

doi:10.1002/nme.4787

Cited by 17 publications

(21 citation statements)

References 47 publications

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“…The case of linear F (x) is slightly different, due to its link with the geometrical approximation of the level sets and more particularly of the interface established at the end of "Enrichment functions for weak discontinuities" section. Ferté et al [45,46] showed for strong discontinuities that the approximation of the level set impacts the expected theoretical convergence rates, in the case of quadratic elements: 2 is expected if a quadratic approximation of the level set is used (equivalently quadratic F (x)), while 1.5 is expected if a linear representation of the level set is used (equivalently linear F (x)). To illustrate this behavior, we also considered two models based on the enrichment designed for strong discontinuities, using Lagrange multipliers to enforce the continuity of the displacement through the interface: one with quadratic shape functions and a linear approximation of the level set and one with quadratic shape functions and a quadratic approximation of the level set.…”

Section: Circular Inclusion With Imposed Displacementmentioning

confidence: 99%

On the construction of approximation space to model discontinuities and cracks with linear and quadratic extended finite elements

Ndeffo

Massin

Moës

et al. 2017

Adv. Model. and Simul. in Eng. Sci.

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This paper presents a robust enrichment strategy to model weak and strong discontinuities as well as cracks for industrial applications. First, numerical issues encountered with popular extended finite element approximation spaces are pointed out. Then, the paper gives indications on how to circumvent those issues. The very originality of the paper relies on questioning the theoretical approximation spaces with respect to numerical results and to modify accordingly their design. The relationship between the new design and the previous designs is clearly established, in order to highlight the very small implementation cost of the modifications exposed here. Hence with minimal additional computational cost, gains in accuracy can be significant as shown later in the paper.

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Section: Circular Inclusion With Imposed Displacementmentioning

confidence: 99%

On the construction of approximation space to model discontinuities and cracks with linear and quadratic extended finite elements

Ndeffo

Massin

Moës

et al. 2017

Adv. Model. and Simul. in Eng. Sci.

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“…The choice of the V h and M h has to satisfy a uniform inf-sup condition [4] with respect to suitable norms. In the present study, as in [3,8], the following mesh-dependent L 2 norms are considered:…”

Section: Discretizationmentioning

confidence: 99%

“…As in [8,10], the Lagrangian multiplier components are defined on the parent nodes belonging to K h . They are based on the same linear shape functions N i .x/ that are used for the displacement field.…”

Section: Discretization Of the Lagrangian Multipliersmentioning

confidence: 99%

“…The authors report that a naive discretization of the Lagrangian multipliers violates the inf-sup condition, which determines the stability of the formulation. The problem can be fixed by defining a reduced Lagrangian multiplier space as proposed in [3,8,10,20]. In these works, only Heaviside functions are introduced to represent the jump across the implicit interface.…”

Section: Introductionmentioning

confidence: 99%

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Crack Lip Contact Modeling Based on Lagrangian Multipliers with X-FEM

Jin

Pierard

Wyart

et al. 2016

SEMA SIMAI Springer Series

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The eXtended Finite Element Method (X-FEM), developed intensively in the past 15 years has become a competitive tool for the solution of problems with evolving discontinuities and singularities. In the present study, we focus on the application of X-FEM on frictionless contact problems in the context of fracture mechanics. A promising approach in the literature counting for this problem consists in applying Lagrangian multipliers. :597-631, 2013). The influence of the tip enrichment functions on the stability of the formulation is addressed. We show evidences that the vector enrichment functions can improve the conditioning of the problem without jeopardizing the simulation accuracy in the presence of contact.

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“…More recently, new models such as TLS (Thick Level Set) approach, developed by [17], permit the description of initiation and propagation of defects in a unified framework [18]. However, the introduction of partial inter-facial cracks is not fully established yet, even if there is many promising works on this subject [19][20]. In addition, all these models and approaches to solve the problem may be too heavy to be contained in practical software.…”

Section: Introductionmentioning

confidence: 99%

A half-analytical elastic solution for 2D analysis of cracked pavements

Nasser

Chabot

2018

Advances in Engineering Software

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This paper presents a half-analytical elastic solution convenient for parametric studies of 2D cracked pavements. The pavement structure is reduced to three elastic and homogeneous equivalent layers resting on a soil. In a similar way than the Pasternak's modelling for concrete pavements, the soil is modelled by one layer, named shear layer, connected to Winkler's springs in order to ensure the transfer of shear stresses between the pavement structure and the springs. The whole four-layer system is modelled using a specific model developed for the analysis of delamination in composite materials. It reduces the problem by one dimension and gives access to regular interface stresses between layers at the edge of vertical cracks allowing the initial debonding analysis. In 2D plane strain conditions, a system of twelve-second order differential equations is written analytically. This system is solved numerically by the finite difference method (Newmark) computed in the free Scilab software. The calculus tool allows analysis of the impact of material characteristics changing, loads and locations of cracks in pavements on the distribution of mechanical fields. The approach with fracture mechanic concepts is well suited for practical use and for some subsequent numerical developments in 3D.

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Interface problems with quadratic X-FEM: design of a stable multiplier space and error analysis

Abstract: International audienc

Cited by 17 publications

References 47 publications

On the construction of approximation space to model discontinuities and cracks with linear and quadratic extended finite elements

On the construction of approximation space to model discontinuities and cracks with linear and quadratic extended finite elements

Crack Lip Contact Modeling Based on Lagrangian Multipliers with X-FEM

A half-analytical elastic solution for 2D analysis of cracked pavements

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