The basis of meshless domain discretization: the meshless local Petrov?Galerkin (MLPG) method

Atluri, Satya N.; Shen, Shengping

doi:10.1007/s10444-004-1813-9

Cited by 105 publications

(91 citation statements)

References 54 publications

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“…Results for MLPG1 and DMLPG1 turn out to behave similarly. As we know, MLPG1 is more expensive than MLPG5 [1,2], but there is no significant difference between computational costs of DMLPG5 and DMLPG1. Therefore the differences between CPU times used for MLPG1 and DMLPG1 are absolutely larger.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…In MLPG categories, this is MLPG2 [1,2]. All functionals are local, and strong in the sense that they do not involve integration over test functions.…”

Section: Problems In Local Weak Formsmentioning

confidence: 99%

“…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%

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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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Section: Numerical Resultsmentioning

confidence: 99%

“…In MLPG categories, this is MLPG2 [1,2]. All functionals are local, and strong in the sense that they do not involve integration over test functions.…”

Section: Problems In Local Weak Formsmentioning

confidence: 99%

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

Mirzaei

Schaback

2013

Applied Numerical Mathematics

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“…[2]. This means the function interpolations do not pass through the nodal values, making boundary conditions difficult to impose.…”

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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“…Extensive developments have been made in several varieties of meshless methods and applied to many applications in science and engineering. These methods exist under different names, such as: the diffuse element method (DEM) [1], the hp-cloud method [2], Meshless Local PetrovGalerkin (MLPG) method [3,4,5,6,7,8,9,10,11,12], the meshless local boundary integral equation (LBIE) method [13,14], the partition of unity method (PUM) [15], the meshless collocation method based on radial basis functions (RBFs) [16], the smooth particle hydrodynamics (SPH) [17], the reproducing kernel particle method (RKPM) [18], the radial point interpolation method [19] and so on. Inverse problems arise in many heat transfer situations when experimental difficulties are encountered in measuring or producing the appropriate boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Numerical simulations of 1D inverse heat conduction problems using overdetermined RBF-MLPG method

Shirzadi¹

2013

Communications in Numerical Analysis

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This paper proposes a numerical method to deal with the one-dimensional inverse heat conduction problem (IHCP). The initial temperature, a condition on an accessible part of the boundary and an additional temperature measurements in time at an arbitrary location in the domain are known, and it is required to determine the temperature and the heat flux on the remaining part of the boundary. Due to the missing boundary condition, the solution of this problem does not depend continuously on the data and therefore its numerical solution requires special care especially when noise is present in the measured data. In the proposed method, the time variable is eliminated by using finite differences approximation. The method uses a weak formulation of the problem to enjoy the stability condition. To avoid the numerical integration on the whole domain, the weak form equations are constructed on local subdomains. The approximate solution is assumed to be a linear combination of Multi Quadric (MQ) radial basis function (RBF) constructed on nodal points in the domain and on the boundary. Since the problem is known to be ill-posed, Thikhonov regularization strategy is employed to solve effectively the discrete ill-posed resultant linear system.

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The basis of meshless domain discretization: the meshless local Petrov?Galerkin (MLPG) method

Cited by 105 publications

References 54 publications

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

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

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

Numerical simulations of 1D inverse heat conduction problems using overdetermined RBF-MLPG method

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