Overview and applications of the reproducing Kernel Particle methods

Liu, W. K.; Chen, Y.; Jun, Sukky; Chen, J. S.; Belytschko, Ted; Pan, Chunhui; Uras, R. A.; Chang, C. T.

doi:10.1007/bf02736130

Cited by 273 publications

(134 citation statements)

References 54 publications

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“…There are well known references with excellent presentations of mesh-free methods, see for instance [8,9,10,11,12]. Here some basic notions will be recalled in order to introduce the notation and the approach employed in following sections.…”

Section: Preliminaries Of the Efg Methodsmentioning

confidence: 99%

Locking in the incompressible limit: pseudo-divergence-free element free Galerkin

Vidal

Villon

Huerta

2002

Revue Européenne des Éléments Finis

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SUMMARYLocking in finite elements has been a major concern since its early developments. It appears because poor numerical interpolation leads to an over-constrained system. This paper proposes a new formulation that asymptotically suppresses locking for the Element Free Galerkin (EFG) method in incompressible limit, i.e. the so-called volumetric locking. Originally it was claimed that EFG did not present volumetric locking. However, recently, performing a modal analysis, the senior author has shown that EFG presents volumetric locking. In fact, it is concluded that an increase of the dilation parameter attenuates, but never suppresses, the volumetric locking and that, as in standard finite elements, an increase in the order of reproducibility (interpolation degree) reduces the relative number of locking modes. Here an improved formulation of the Element Free Galerkin method is proposed in order to alleviate volumetric locking.Diffuse derivatives are defined in the thesis of the second author and their convergence to the derivatives of the exact solution, when the radius of the support goes to zero (for a fixed dilation parameter), it's proved. Therefore diffuse divergence converges to the exact divergence. Since the diffuse divergence-free condition can be imposed a priori, new interpolation functions are defined that asymptotically verify the incompressibility condition. Modal analysis and numerical results for classical benchmark tests in solids corroborate this issue.

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Section: Preliminaries Of the Efg Methodsmentioning

confidence: 99%

Locking in the incompressible limit: pseudo-divergence-free element free Galerkin

Vidal

Villon

Huerta

2002

Revue Européenne des Éléments Finis

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“…The approximation u will converge to the exact function as W (x, ρ) approaches the Dirac delta function δ(x). Moreover, the kernel function W can be easily designed so that the approximation u is exact for polynomials up to order m, that is u = u for any u polynomial of degree less or equal to m, see [20]. In that case, the approximation is said to have reproducibility of order m. Equivalently, some authors refer to this property as consistency of order m.…”

Section: Basic Equations Of Sphmentioning

confidence: 99%

“…In order to simplify the notation, let us consider approximations that can be written in the form (20) where the FE shape functions are as usual, N k := N h k and g k := ∇N h k for k ∈ N . The SPH shape functions are N k := W k for k ∈ P, as defined in Section 3.1 by (4), or by (7) for 0-order reproducibility.…”

Section: In Order To Define a Proper Integration The Particles In Thmentioning

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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“…Unlike FEA in which elements are connected by a topological mesh, meshfree particle methods were developed based on using a finite number of discrete particles to describe the state of a system [112][113][114][115][116]. Meshfree particle methods eliminate mesh constraints and demonstrate advantages in many applications including crack prop-agation simulations, large deformation analyses and fluid-structure interactions (FSI).…”

Section: Introductionmentioning

confidence: 99%

Multiscale modeling and simulation of rolling contact fatigue

Gharehbagh

Ali

2018

International Journal of Fatigue

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iii ABSTRACT In this thesis, a hierarchical multiscale method was developed to predict rolling contact fatigue lives of mechanical systems. In the proposed multiscale method, the molecular modeling and simulation of lubricant was conducted to investigate the friction between rolling contact surfaces. The calculated friction coefficient was passed to the continuum model of rolling contact components to predict fatigue lives.Molecular dynamics modeling and simulation of thin film lubrication and lubricated contact surfaces were carried out to investigate mechanisms of hydrodynamic lubrication at nano-scale first. Although various lubricant alkane chains were considered in the molecular model, the chain length of eight united molecules were mainly employed in this thesis. In addition, the effects of temperature and nano-particles (debris) on the friction forces were discussed. It was found that the existing of nano-particles (debris) could increase the friction force between contact surfaces with hydrodynamic lubrication.In the continuum model of the developed multiscale method, finite element analysis was employed to predict rolling contact fatigue life of rolling contact components, including bearing and gear-tooth. Specifically, the fatigue crack initiation of bearing was studied, and then the fatigue crack initiation and propagation in gear-tooth. In addition, the enhancement of gear-tooth fatigue life by using composite patches was discussed as well. It should be noted that the friction coefficient used in the continuum model was calculated in the molecular model. It is one-way message passing in iv the developed multiscale method.Another continuum method was studied and developed in this thesis to provide alternate methods for the continuum model in the proposed multiscale framework.Peridynamics method has advantages in modeling and simulation of discontinuities, including cracks, over the conventional finite element methods. The applications of Peridynamics in predicting fatigue crack initiation and propagation lives were discussed in this thesis. v PUBLIC ABSTRACTThis research aims to develop a hierarchical multiscale method to predict rolling contact fatigue lives of mechanical systems. In the proposed multiscale method, the molecular modeling and simulation of lubricant was conducted to investigate the friction between rolling contact surfaces. The calculated friction coefficient was passed to the continuum model of rolling contact components to predict fatigue lives.Molecular dynamics modeling and simulation of thin film lubrication and lubricated contact surfaces were carried out to investigate mechanisms of hydrodynamic lubrication at nano-scale first. In addition, the effects of temperature and nano-particles (debris) on the friction forces were discussed. It was found that the existing of nanoparticles (debris) could increase the friction force between contact surfaces with hydrodynamic lubrication.In the continuum model of the developed multiscale method, finite element analysis was employed to pred...

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Overview and applications of the reproducing Kernel Particle methods

Cited by 273 publications

References 54 publications

Locking in the incompressible limit: pseudo-divergence-free element free Galerkin

Locking in the incompressible limit: pseudo-divergence-free element free Galerkin

Continuous blending of SPH with finite elements

Multiscale modeling and simulation of rolling contact fatigue

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