The Boundary Node Method for Potential Problems

Mukherjee, Yu Xie; Mukherjee, Subrata

doi:10.1002/(sici)1097-0207(19970315)40:5<797::aid-nme89>3.0.co;2-#

Cited by 320 publications

(62 citation statements)

References 12 publications

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“…Examples of these methods are the diffuse element method (DEM) (Nayroles et al, 1992), the element free Galerkin (EFG) method (Belytschko et al, 1994a), the reproducing kernel particle method (RKPM) (Liu et al, 1995), the point interpolation methods Wang and GR Liu, 2000), the meshless local Petrov-Galerkin method (MLPG) (Atluri 1998a), the boundary node method (BNM) (Mukherjee and Mukherjee, 1997), the boundary point interpolation method (BPIM) (Gu and GR Liu, 2001e,2002a, the meshfree weak-strong (MWS) form method (GR Liu and Gu, 2002d;, etc. These methods do not require a mesh at least for the field variable interpolations.…”

Section: Categories Of Meshfree Methods †mentioning

confidence: 99%

An Introduction to Meshfree Methods and Their Programming

Liu¹,

Gu²

2005

135

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All Rights Reserved © 2005 SpringerNo part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.Printed in the Netherlands.

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Section: Categories Of Meshfree Methods †mentioning

confidence: 99%

An Introduction to Meshfree Methods and Their Programming

Liu¹,

Gu²

2005

135

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“…In meshless methods, the number of nodes corresponding to each cell is arbitrary. Nonetheless, in order to carry out accurate integration by Gaussian quadrature, it is recommended to use a small number of nodes per cell [8,17,18]. Therefore, we assume that the following assumption is satisfied.…”

Section: Construction Of the Integration Background Cells F/jmentioning

confidence: 99%

“…LI AND J.L. ZHU the boundary node method (BNM) [8], the boundary cloud method [9], the hybrid boundary node method [10], the boundary point interpolation method [11], the boundary element-free method [12] and the Galerkin boundary node method (GBNM) [13]. In contrast with the domain type methods, they are superior in treating problems dealing with infinite or semi-infinite domains.…”

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confidence: 99%

Galerkin Boundary Node Method for Exterior Neumann Problems

Zhu¹

2011

JCM

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In this paper, we present a meshless Galerkin scheme of boundary integral equations (BIEs), known as the Galerkin boundary node method (GBNM), for two-dimensional exterior Neumann problems that combines the moving least-squares (MLS) approximations and a variational formulation of BIEs. In this approach, boundary conditions can be implemented directly despite the MLS approximations lack the delta function property. Besides, the GBNM keeps the symmetry and positive definiteness of the variational problems. A rigorous error analysis and convergence study of the method is presented in Sobolev spaces. Numerical examples are also given to illustrate the capability of the method.Mathematics subject classification: 65N12, 65N30, 65N38.i Key words: Meshless, Galerkin boundary node method. Boundary integral equations. Moving least-squares. Error estimate. / 244 X.L. LI AND J.L. ZHU the boundary node method (BNM) [8], the boundary cloud method [9], the hybrid boundary node method [10], the boundary point interpolation method [11], the boundary element-free method [12] and the Galerkin boundary node method (GBNM) [13]. In contrast with the domain type methods, they are superior in treating problems dealing with infinite or semi-infinite domains. However, most boundary type meshless methods found in the literature lack a rich mathematical foundation to justify their use.The BNM is formulated using the moving least-squares (MLS) approximations [1,14] and the technique of BIEs. This method exploits the dimensionality of BIEs and the meshless attribute of the MLS. Nevertheless, since the MLS approximations lack the delta function property, the BNM cannot accurately satisfy boundary conditions. The strategy employed in the BNM [8] involves a new definition of the discrete norm used for the construction of the MLS approximations, which doubles the number of system equations.The GBNM is a boundary type meshless Galerkin method which combines the MLS scheme and a variational formulation of BIEs. Compared with the BNM, boundary conditions in the GBNM can be satisfied directly and system matrices are symmetric. The main difiFerence between the GBNM and the traditional Galerkin BEM is the way in which the shape function is formulated. In the GBNM, the boundary variables are approximated by the MLS technique that only use the boundary nodes, but in the Galerkin BEM, interpolants of the boundary variables are related to the geometry of the elements. The GBNM has been used for Dirichlet problems of Laplace equation [13] and biharmonic equation [15], and for Stokes fiow [16]. In this paper, the GBNM is further developed for solving 2-D exterior Neumann problems.As in many other meshless methods such as the EFGM and the BNM, background cells are used in the GBNM for numerical integration over the boundary. Cells are used for integration only, and have no restriction on shape or compatibility. The topology of cells can be much simpler than that of elements in the BEM or the FEM, since cells can be divided into smaller ones without affecting...

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“…In order to resolve the above demerits, the boundary node method (BNM) [3], which is one of meshless methods, was proposed. Recently, the BNM has been reformulated without using integration cells and its performance has been investigated numerically [4], [5].…”

Section: Introductionmentioning

confidence: 99%

Performance Improvement of Extended Boundary Node Method

et al. 2015

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The extended boundary node method (X-BNM) with the radial point interpolation method (RPIM) shape function is proposed and its performance is numerically investigated by comparing with the dual reciprocity boundary element method (DRM). The results of computations show that the accuracy of the X-BNM is almost equal to that of the DRM. In addition, the speed of the X-BNM with the RPIM shape function is almost equal to that of the DRM. Therefore, not only the DRM but also the X-BNM with the RPIM shape function is a powerful method for solving the Poisson problem.Index Terms-Boundary value problems, Green's function methods, integral equations, least squares approximation.

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The Boundary Node Method for Potential Problems

Cited by 320 publications

References 12 publications

An Introduction to Meshfree Methods and Their Programming

An Introduction to Meshfree Methods and Their Programming

Galerkin Boundary Node Method for Exterior Neumann Problems

Performance Improvement of Extended Boundary Node Method

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