Enrichment and coupling of the finite element and meshless methods

Huerta, Antonio; Fernández‐Méndez, Sonia

doi:10.1002/1097-0207(20000820)48:11<1615::aid-nme883>3.0.co;2-s

Cited by 173 publications

(159 citation statements)

References 19 publications

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“…Some notable drawbacks of MLS based methods are that the basis functions are not strictly positive and, in addition, they do not posses the Kronecker-Delta property on the boundary of the domain. This requires additional efforts to impose essential boundary conditions [19]. Other meshless schemes where these problems are avoided have been later developed.…”

Section: Introductionmentioning

confidence: 99%

A stabilized formulation with maximum entropy meshfree approximants for viscoplastic flow simulation in metal forming

et al. 2014

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The finite element method is the reference technique in the simulation of metal forming and provides excellent results with both Eulerian and Lagrangian implementations. The latter approach is more natural and direct but the large deformations involved in such processes require remeshing-rezoning algorithms that increase the computational times and reduce the quality of the results. Meshfree methods can better handle large deformations and have shown encouraging results. However, viscoplastic flows are nearly incompressible, which poses a challenge to meshfree methods. In this paper we propose a simple model of viscoplasticity, where both the pressure and velocity fields are discretized with maximum entropy approximants. The inf-sup condition is circumvented with a numerically consistent stabilized formulation that involves the gradient of the pressure. The performance of the method is studied in some benchmark problems including metal forming and orthogonal cutting.

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Section: Introductionmentioning

confidence: 99%

A stabilized formulation with maximum entropy meshfree approximants for viscoplastic flow simulation in metal forming

et al. 2014

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“…Thus, some strategies have been developed to circumvent this issue [13,15]. Apart from mixing meshless methods and finite elements [16] another alternative to overcome the re-meshing issue in finite elements is to use the eXtended Finite Element Method (X-FEM) based on the partition of unity framework introduced by Babuška and Melenk [17]. Proper enrichment of the finite element basis makes it possible to model crack, material inclusions and holes with non-conforming meshes.…”

Section: Introductionmentioning

confidence: 96%

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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“…An important effort has been dedicated to the coupling of FE and meshless methods in the last years. Most of the literature is devoted to the coupling of FE with EFG or RKPM, see for instance [11][12][13][14][15][16]. However, the attention paid to the development of formulations linking FE with SPH methods is more modest [17,18].…”

Section: Introductionmentioning

confidence: 99%

Continuous blending of SPH with finite elements

Fernández‐Méndez

Bonet²,

Huerta

2005

Computers & Structures

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This paper proposes a methodology for the continuous blending of the finite element method and Smooth Particle Hydrodynamics. The coupled approximation with finite elements and particles, and the discretization of the boundary value problem with a coupled integration, are described. An integration correction is also proposed to stabilize the solution. Some numerical examples demonstrate the applicability of the method.

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Enrichment and coupling of the finite element and meshless methods

Cited by 173 publications

References 19 publications

A stabilized formulation with maximum entropy meshfree approximants for viscoplastic flow simulation in metal forming

A stabilized formulation with maximum entropy meshfree approximants for viscoplastic flow simulation in metal forming

Stability of incompressible formulations enriched with X-FEM

Continuous blending of SPH with finite elements

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