A gradient‐based adaptation procedure and its implementation in the element‐free Galerkin method

Luo, Yunhua; Häußler‐Combe, Ulrich

doi:10.1002/nme.615

Cited by 11 publications

(10 citation statements)

References 30 publications

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“…It is demonstrated in [25] that averaged gradient of the strain energy density is an effective error indicator. Therefore, in the proposed r -adaptation algorithm, gradients of the strain energy density will be used as weight function.…”

Section: Laplacian Smoothing Weighted By Gradients Of Strain Energy Dmentioning

confidence: 99%

“…The other situation is that there is an insufficient number of elements nodes in the initial mesh; then, the iteration has to be stopped when the mesh has reached its maximum limit in improving solution accuracy. Both circumstances can be dealt with by the so-called mesh intensity [25]. Mesh intensity is defined as the ratio of a gradient of strain energy density to mesh density,…”

Section: Laplacian Smoothing Weighted By Gradients Of Strain Energy Dmentioning

confidence: 99%

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r‐Adaptation algorithm guided by gradients of strain energy density

Luo

2010

Numer Methods Biomed Eng

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SUMMARYIn this paper, a r -adaptation algorithm is presented. The algorithm is based on weighted Laplacian smoothing. In the proposed algorithm, the gradients of strain energy density are used as weight functions; Laplacian smoothing is iterated until the maximum deviation or standard deviation in mesh intensity is smaller than a prescribed value. Numerical results show that the algorithm is sensitive and robust. The algorithm can be extended to other finite element formulations by replacing strain energy density with its corresponding counterpart.

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Section: Laplacian Smoothing Weighted By Gradients Of Strain Energy Dmentioning

confidence: 99%

Section: Laplacian Smoothing Weighted By Gradients Of Strain Energy Dmentioning

confidence: 99%

r‐Adaptation algorithm guided by gradients of strain energy density

Luo

2010

Numer Methods Biomed Eng

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“…In meshless integral-free methods, a cloud refinement procedure based on gradients can be important and suitable for implementation. Previous gradient based methods have been used by Stein and Rust [28] for finite elements and by Luo and Häussler-Combe [29] for the element-free Galerkin method. The basic idea of the gradient based methods is simple: a larger variation of the gradients needs a richer cloud of nodes.…”

Section: Adaptivity Strategymentioning

confidence: 99%

“…In this paper a procedure like the one used by Luo and Häussler-Combe [29] has been adapted to the GFDM, but with the following differences:…”

Section: Adaptivity Strategymentioning

confidence: 99%

An Approach to Refinement of Irregular Clouds of Points Using Generalized Finite Differences

Gavete¹,

Gavete²,

Ureña

et al. 2015

Mathematical Problems in Engineering

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This paper describes a fully automatic adaptive refinement procedure performed in conjunction with the generalized finite difference method (GFDM) for solving a second-order partial differential equation (PDE) frequently encountered in engineering practice. A quadtree structure is used to organize the clouds of points. The adaptive procedure is based on the existing differences between the gradients of close points. The relative error between two successive adaptation processes is taken as the criterion to stop the algorithm. Using this adaptive procedure, the high gradient regions are easily detected and refined. Some benchmark examples are presented showing the accuracy of the method.

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“…[75] describes an error estimate for the EFGM based on a Taylor series with a higher order derivative and a structured grid is used instead of a cloud of points, which makes the implementation very straightforward. Error estimation based on the gradient of strain energy density is proposed in [76] and an adaptive analysis for EFGM is proposed in [77] based on background cells, error being estimated based on two different integration orders and a refinement algorithm based on local Delaunay triangulation is also proposed. In [78], the approach from [70] is used for error estimation and adaptive analysis in crack propagation problems.…”

Section: Introductionmentioning

confidence: 99%

Finite deformation elasto-plastic modelling using an adaptive meshless method

Ullah

Augarde

2013

Computers & Structures

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'Finite deformation elasto-plastic modelling using an adaptive meshless method. ', Computers and structures., Further information on publisher's website:http://dx.doi.org/10. 1016/j.compstruc.2012.04.001 Publisher's copyright statement: NOTICE: this is the author's version of a work that was accepted for publication in Computers and structures. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reected in this document. Changes may have been made to this work since it was submitted for publication. A denitive version will be subsequently published in Computers and structures, 2012Computers and structures, , 10.1016Computers and structures, /j.compstruc.2012 Additional information: Use policyThe 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.Please consult the full DRO policy for further details. Problems including both material and geometric nonlinearities are of practical engineering importance in many applications. Efficient computational modelling of these problems with minimum computer resources remains challenging. Often these problems are modelled with the finite element method (FEM) with adaptive analysis, in which an error estimation procedure automatically determines regions for coarse and fine discretization. Meshless methods offer the attractive possibility of simpler adaptive procedures involving no remeshing, simply insertion or deletion of nodes. However, as meshless methods are computationally more expensive than the FEM, the use of the minimum possible number and proper distribution of nodes are important issues. In this study an adaptive meshless approach for nonlinear solid mechanics is developed based on the element free Galerkin method. An existing error estimation procedure for linear elasto-static problems, is extended here for nonlinear problems including finite deformation and elasto-plasticity, and a new adaptive procedure is described and demonstrated.

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A gradient‐based adaptation procedure and its implementation in the element‐free Galerkin method

Cited by 11 publications

References 30 publications

r‐Adaptation algorithm guided by gradients of strain energy density

r‐Adaptation algorithm guided by gradients of strain energy density

An Approach to Refinement of Irregular Clouds of Points Using Generalized Finite Differences

Finite deformation elasto-plastic modelling using an adaptive meshless method

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