On adaptivity and error criteria for meshfree methods

Belytschko, Ted; Liu, Wing Kam; Singer, Mike A.

doi:10.1016/s0922-5382(98)80012-3

Cited by 14 publications

(13 citation statements)

References 12 publications

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“…These estimators can be used as error indicators for adaptivity purposes. For adaptive approaches in meshless method, we refer the reader to [76] and [23].…”

Section: A-posteriori Error Estimation Adaptivity and Computation Omentioning

confidence: 99%

Survey of meshless and generalized finite element methods: A unified approach

Babuška¹,

Banerjee²,

Osborn³

2003

Acta Numerica 2003

125

221

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In the last few years meshless methods for numerically solving partial differential equations came into the focus of interest, especially in the engineering community. This class of methods was essentially stimulated by difficulties related to mesh generation. Mesh generation is delicate in many situations, e.g., when the domain has complicated geometry; when the mesh changes with time, as in crack propagation, and remeshing is required at each time step; when a Lagrangian formulation is employed, especially with non-linear PDE's. In addition, a need to have flexibility in the selection of approximating functions (e.g., the flexibility to use non-polynomial approximating functions), played a significant role in the development of meshless methods. There are many recent papers, and two books, on meshless methods; most of them are of engineering character, without any mathematical analysis.In this paper we address meshless methods and the closely related generalized finite element methods for solving linear elliptic equations, using variational principles. We give a unified mathematical theory with proofs, briefly address implementational aspects, present illustrative numerical examples, and provide a list of reference to the current literature.The aim of the paper is to provide a survey of a part of this new field, with emphasis on mathematics. We present proofs of essential theorems because we feel these proofs are essential for understanding the mathematical aspects of meshless methods, which has approximation theory as a major ingredient. As always, any new field is stimulated by and related to older ideas. This will be visible in our paper. AMS(MOS 1 Report Documentation PageForm Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

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“…These estimators can be used as error indicators for adaptivity purposes. For adaptive approaches in meshless method, we refer the reader to [76] and [23].…”

Section: A-posteriori Error Estimation Adaptivity and Computation Omentioning

confidence: 99%

Survey of meshless and generalized finite element methods: A unified approach

Babuška¹,

Banerjee²,

Osborn³

2003

Acta Numerica 2003

125

221

View full text Add to dashboard Cite

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“…In their work, the effectivity index is optimized if the domain of influence in the MLSA of the recovered stress is the smallest necessary to retain the regularity of AðxÞ. In [3] is presented a residue-based error estimator and considered a multi-resolution analysis. Another important contribution, presented in [8], is a residue based error estimator that was applied in their proposed h-p cloud method.…”

Section: Error Analysismentioning

confidence: 99%

Recovery based error estimation and adaptivity applied to a modified element-free Galerkin method

Rossi

Alves

2004

Computational Mechanics

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In this work, we propose an error estimator of the recovery type, which considers the equilibrium and boundary traction conditions, and an h-refinement procedure that is applied to the modified element-free Galerkin (MEFG) method. The approximate solution obtained by the MEFG method satisfies accurately the essential boundary condition. However, the approximate MEFG stress field presents some discontinuities on a neighborhood of the essential boundary condition and may present some spurious oscillations at regions of high stress gradients or discontinuities. Thus, the h-adaptive scheme is responsible for the reduction not only of the global error but also of the local errors associated with the discontinuities and oscillations of the approximate stress field. In order to validate the proposed procedures, we present some numerical solution for some simple problems and consider the analysis of a complex component.

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“…The importance of the smoothness of meshfee approximations to enhance the definition of error measures is also pointed out by [14], for the case of the Reproducing Kernel Particle Method, RKPM [27,28].…”

Section: Introductionmentioning

confidence: 98%

p-Adaptive C k generalized finite element method for arbitrary polygonal clouds

2007

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A p-Adaptive Generalized Finite Element Method (GFEM) based on a Partition of Unity (POU) of arbitrary smoothness degree is presented. The shape functions are built from the product of a Shepard POU and enrichment functions. Shepard functions have a smoothness degree directly related to the weighting functions adopted in their definition. Here the weighting functions are obtained from boolean R-functions which allow the construction of C k approximations, with k arbitrarily large, defined over a polygonal patch of elements, named cloud. The Element Residual Method is used to obtain error indicators by taking into account the typical nodal enrichment scheme of the method. This procedure is enhanced by using approximations with a high degree of smoothness as it eliminates the discontinuity of the stress field in the interior of each cloud. Adaptive analysis of plane elasticity problems are presented, and the performance of the technique is investigated.

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On adaptivity and error criteria for meshfree methods

Cited by 14 publications

References 12 publications

Survey of meshless and generalized finite element methods: A unified approach

Survey of meshless and generalized finite element methods: A unified approach

Recovery based error estimation and adaptivity applied to a modified element-free Galerkin method

p-Adaptive C k generalized finite element method for arbitrary polygonal clouds

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