Meshless local Petrov-Galerkin (MLPG) method in combination with finite element and boundary element approaches

Liu, G. R.; Gu, Yuantong

doi:10.1007/s004660000203

Cited by 97 publications

(39 citation statements)

References 19 publications

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“…The unknown variable u in this LSWF is approximated by the nodal shape functions (10). To obtain the discrete equations from the LSWF (22), the RBF interpolation (9) is adopted to approximate the trial function u.…”

Section: Discretization and Numerical Implementationmentioning

confidence: 99%

Local Heaviside‐weighted LRPIM meshless method and its application to two‐dimensional potential flows

Saeedpanah

Jabbari

2008

Numerical Methods in Fluids

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SUMMARYIn this paper, the local radial point interpolation meshless method (LRPIM) is used for the analysis of twodimensional potential flows, based on a local-weighted residual method with the Heaviside step function as the weighting function over a local subdomain. Trial functions are constructed using radial basis functions. The present method is a truly meshless method based only on a number of randomly located nodes. Integration over the subdomains requires only a simple integration cell to obtain the solution. No element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions. The novelty of the paper is the use of a local Heaviside weight function in the LRPIM, which does not need local domain integration and integrations only on the boundary of the local domains are needed. Effects of the sizes of local subdomain and interpolation domain on the performance of the present method are investigated. The behavior of shape parameters of multiquadrics has been systematically studied. Two numerical tests in groundwater and fluid flows are presented and compared with closed-form solutions and finite element method. The results show that the use of a local Heaviside weight function in the LRPIM is highly accurate and possesses no numerical difficulties.

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Section: Discretization and Numerical Implementationmentioning

confidence: 99%

Local Heaviside‐weighted LRPIM meshless method and its application to two‐dimensional potential flows

Saeedpanah

Jabbari

2008

Numerical Methods in Fluids

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“…The MLPG method has been successfully used in analyses of solids Zhu, 1998, 2000;Atluri et al, 1999;Gu and Liu, 2001b, c). In addition, the MLPG method has also been combined with FEM and BEM (Liu and Gu, 2000b).…”

Section: Introductionmentioning

confidence: 99%

Comparisons of two meshfree local point interpolation methods for structural analyses

Liu

2002

Computational Mechanics

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As truly meshless methods, the local point interpolation method (LPIM) and the local radial point interpolation method (LR-PIM), are based on the point interpolations and local weak forms integrated in a local domain of very simple shape. LPIM and LR-PIM are examined and compared with each other. They are also compared with the established FEM and the meshless local Petrov-Galerkin (MLPG) method. The numerical implementations of these two methods are discussed in detail. Parameters that influence the performance of them are detailedly studied. The convergence and efficiency of them are thoroughly investigated. LPIM and LR-PIM formulations are developed for structural analyses of 2-D elastodynamic problems and 1-D Timoshenko beam problems in the first time. It is found that LPIM and LR-PIM are very easy to implement, and very efficient obtaining numerical solutions to problems of computational mechanics.

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“…Hegen [21] developed another coupled EFG/FEM technique based on the modified variational principle. Gu and Liu developed a group of coupled methods including EFG/BEM and MLPG/FEM/BEM [17][18][19]27,28] based on the FE interface element technique or the modified variational principle. In these coupled methods, the MM is used in a sub-domain where the MM is required to obtain high accuracy, and FEM or BEM is employed in other sub-domains where FEM or BEM is required to improve the computational efficiency.…”

Section: Introductionmentioning

confidence: 99%

A bridging transition technique for the combination of meshfree method with finite element method in 2D solids and structures

Yarlagadda

2009

Comput Mech

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For certain continuum problems, it is desirable and beneficial to combine two different methods together in order to exploit their advantages while evading their disadvantages. In this paper, a bridging transition algorithm is developed for the combination of the meshfree method (MM) with the finite element method (FEM). In this coupled method, the MM is used in the sub-domain where the MM is required to obtain high accuracy, and the FEM is employed in other sub-domains where FEM is required to improve the computational efficiency. The MM domain and the FEM domain are connected by a transition (bridging) region. A modified variational formulation and the Lagrange multiplier method are used to ensure the compatibility of displacements and their gradients. To improve the computational efficiency and reduce the meshing cost in the transition region, regularly distributed transition particles, which are independent of either the meshfree nodes or the FE nodes, can be inserted into the transition region. The newly developed coupled method is applied to the stress analysis of 2D solids and structures in order to investigate its' performance and study parameters. Numerical results show that the present coupled method is convergent, accurate and stable. The coupled method has a promising potential for practical applications, because it can take advantages of both the MM and FEM when overcome their shortcomings.

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Meshless local Petrov-Galerkin (MLPG) method in combination with finite element and boundary element approaches

Cited by 97 publications

References 19 publications

Local Heaviside‐weighted LRPIM meshless method and its application to two‐dimensional potential flows

Local Heaviside‐weighted LRPIM meshless method and its application to two‐dimensional potential flows

Comparisons of two meshfree local point interpolation methods for structural analyses

A bridging transition technique for the combination of meshfree method with finite element method in 2D solids and structures

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