A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics

Atluri, Satya N.; Zhu, Tengteng

doi:10.1007/s004660050346

Cited by 1,932 publications

(933 citation statements)

References 8 publications

(4 reference statements)

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“…Equidistant, i.e. circular domains around each node are applied in typical MLPG implementations [16], with a radius at specific fraction of the local inter-node distance (usually around 60%). This approach, though straightforwardly applicable, either does not cover all the area of the domain or it is leading to significant overlapping in the domain integration.…”

Section: T H ðXþ ¼ P T ðXþaðxþ ð1þmentioning

confidence: 99%

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Transient thermal conduction with variable conductivity using the Meshless Local Petrov–Galerkin method

Karagiannakis

Bourantas

Kalarakis

et al. 2016

Applied Mathematics and Computation

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a b s t r a c tA numerical solution of the transient heat conduction problem with spatiotemporally variable conductivity in 2D space is obtained using the Meshless Local Petrov-Galerkin (MLPG) method. The approximation of the field variables is performed using Moving Least Squares (MLS) interpolation. The accuracy and the efficiency of the MLPG schemes are investigated through variation of (i) the domain resolution, (ii) the order of the basis functions, (iii) the shape of the integration site around each node, (iv) the conductivity range, and (v) the volumetric heat capacity range. Steady-state boundary conditions of the essential type are assumed. The results are compared with those calculated by a typical Finite Element method. Specific rectangular-type integration sites are introduced during both steady-state and transient MLPG integration, in order to provide complete surface coverage of the domain without overlapping, and the accuracy of the method is demonstrated in all cases studied. Computational efficiency is also investigated with this MLPG method and found to be slower than FE methods during construction stage, but it clearly surpasses that of FEM approaches during the solution stage on a wide parameter range.

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Section: T H ðXþ ¼ P T ðXþaðxþ ð1þmentioning

confidence: 99%

“…This truly meshless formulation based on the recently developed [16] local symmetric weak form with the Local PetrovGalerkin approach is proposed here to solve transient non-linear heat conduction problems. The essential boundary conditions in the present formulation were imposed by 1st order MLS description.…”

Section: Introductionmentioning

confidence: 99%

Transient thermal conduction with variable conductivity using the Meshless Local Petrov–Galerkin method

Karagiannakis

Bourantas

Kalarakis

et al. 2016

Applied Mathematics and Computation

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“…As an important example of such methods, we mention the Meshless Local Petrov-Galerkin (MLPG) method introduced by S.N. Atluri and his colleagues [1,2,3]. It is a truly meshless method in weak form which is based on local subdomains, rather than a single global domain.…”

Section: Introductionmentioning

confidence: 99%

Direct Meshless Local Petrov–Galerkin (DMLPG) method: A generalized MLS approximation

Mirzaei

Schaback

2013

Applied Numerical Mathematics

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The Meshless Local Petrov-Galerkin (MLPG) method is one of the popular meshless methods that has been used very successfully to solve several types of boundary value problems since the late nineties. In this paper, using a generalized moving least squares (GMLS) approximation, a new direct MLPG technique, called DMLPG, is presented. Following the principle of meshless methods to express everything "entirely in terms of nodes", the generalized MLS recovers test functionals directly from values at nodes, without any detour via shape functions. This leads to a cheaper and even more accurate scheme. In particular, the complete absence of shape functions allows numerical integrations in the weak forms of the problem to be done over low-degree polynomials instead of complicated shape functions. Hence, the standard MLS shape function subroutines are not called at all. Numerical examples illustrate the superiority of the new technique over the classical MLPG. On the theoretical side, this paper discusses stability and convergence for the new discretizations that replace those of the standard MLPG. However, it does not treat stability, convergence, or error estimation for the MLPG as a whole. This should be taken from the literature on MLPG.

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“…Among different kinds of meshfree methods proposed so far, see [18][19][20][21][22][23][24][25][26], Element free Galerkin (EFG) [21] and Meshless local Petrov-Galerkin (MLPG) [25] have gained the much attention and both used Moving least square (MLS) approximation as the shape function construction. More recently, Liu and Gu [26] introduced a point interpolation method which uses Radial basis (RB) function to construct the shape function.…”

Section: Introductionmentioning

confidence: 99%

Meshfree modeling and homogenization of 3D orthogonal woven composites

Wen

Aliabadi

2011

Composites Science and Technology

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In this paper, evaluation of 3D orthogonal woven fabric composite elastic moduli is achieved by applying Meshfree methods on the micromechanical model of the woven composites. A new, realistic and smooth fabric unit cell model of 3D orthogonal woven composite is presented. As an alternative to Finite Element Method, Meshfree Methods show a notable advantage, which is the simplicity in meshing while modelling the matrix and different yarns. Radial basis function and moving kriging interpolation are used for the shape function constructions. The Galerkin method is employed in formulating the discretized system equations. AbstractIn this paper, evaluation of 3D orthogonal woven fabric composite elastic moduli is achieved by applying Meshfree methods on the micromechanical model of the woven composites. A new, realistic and smooth fabric unit cell model of 3D orthogonal woven composite is presented. As an alternative to Finite Element Method, Meshfree Methods show a notable advantage, which is the simplicity in meshing while modelling the matrix and different yarns. Radial basis function and moving kriging interpolation are used for the shape function constructions. The Galerkin method is employed in formulating the discretized system equations. The numerical results are compared with the Finite Element and the experimental results.

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A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics

Cited by 1,932 publications

References 8 publications

Transient thermal conduction with variable conductivity using the Meshless Local Petrov–Galerkin method

Transient thermal conduction with variable conductivity using the Meshless Local Petrov–Galerkin method

Direct Meshless Local Petrov–Galerkin (DMLPG) method: A generalized MLS approximation

Meshfree modeling and homogenization of 3D orthogonal woven composites

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