On boundary conditions in the element-free Galerkin method

Mukherjee, Yu Xie; Mukherjee, Subrata

doi:10.1007/s004660050175

Cited by 156 publications

(73 citation statements)

References 11 publications

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“…Recently, the Lagrange multiplier technique was applied in the context of meshless methods without any theoretical analysis in [24], [57], [60], [67].…”

Section: The Lagrange Multiplier Methodsmentioning

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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“…Recently, the Lagrange multiplier technique was applied in the context of meshless methods without any theoretical analysis in [24], [57], [60], [67].…”

Section: The Lagrange Multiplier Methodsmentioning

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 this research, a quartic spline test function was used as shown in Eq. (4). Note that if the subdomain of a node overlaps the global domain, the test function will not force the area term in Eq.…”

Section: Formulation Of the Governing Equationsmentioning

confidence: 99%

“…[3]; however, in this research, a method called direct interpolation is adopted as introduced in ref. [4]. Instead of enforcing a weak solution to a local governing equation for a boundary node where the boundary value is known, the boundary value is determined by MLS interpolation.…”

Section: Boundary Conditionsmentioning

confidence: 99%

Application of the Meshless Local Petrov-Galerkin (MLPG) method to Rayleigh-Taylor instability

Susi¹,

Smith²

2012

AIP Conference Proceedings

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“…These include the use of Lagrange multipliers [2], use of traction as Lagrange multipliers [16] and the use of a layer of ÿnite elements along the boundary where essential conditions are prescribed [17]. However, this issue has been successfully resolved by Mukherjee and Mukherjee [18]. The strategy proposed in their paper for alleviating the above problem involves a new definition of the discrete norm used for the construction of the MLS interpolants.…”

Section: Introductionmentioning

confidence: 97%

The boundary node method for three-dimensional linear elasticity

Chati

Mukherjee

1999

Int. J. Numer. Meth. Engng.

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SUMMARYThe Boundary Node Method (BNM) is developed in this paper for solving three-dimensional problems in linear elasticity. The BNM represents a coupling between Boundary Integral Equations (BIE) and Moving Least-Squares (MLS) interpolants. The main idea is to retain the dimensionality advantage of the former and the meshless attribute of the later. This results in decoupling of the 'mesh' and the interpolation procedure. For problems in linear elasticity, free rigid-body modes in traction prescribed problems are typically eliminated by suitably restraining the body. However, an alternative approach developed recently for the Boundary Element Method (BEM) is extended in this work to the BNM. This approach is based on ideas from linear algebra to complete the rank of the singular sti ness matrix. Also, the BNM has been extended in the present work to solve problems with material discontinuities and a new procedure has been developed for obtaining displacements and stresses accurately at internal points close to the boundary of a body.

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On boundary conditions in the element-free Galerkin method

Cited by 156 publications

References 11 publications

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

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

Application of the Meshless Local Petrov-Galerkin (MLPG) method to Rayleigh-Taylor instability

The boundary node method for three-dimensional linear elasticity

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