A Meshless Method Based on Least-Squares Approach for Steady- And Unsteady-State Heat Conduction Problems

Liu, Y.; Zhang, Xiong; Lu, Ming

doi:10.1080/10407790590901648

Cited by 71 publications

(26 citation statements)

References 22 publications

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“…It can be seen that the proposed method is an extension of the DLSM method proposed by Afshar and Arzani [16] and Firoozjaee and Afshar [17] for the solution of elliptic problems. The method also has some in common with those methods used by Liu et al [11] and Wang et al [15] to solve problems of elliptic nature.…”

Section: Collocated Discrete Least Squares Meshless (Cdlsm) Methodsmentioning

confidence: 93%

“…In this method, a set of over determined system of equations, in which the number of equations is greater than the unknowns, was constructed and the least squares technique was used to solve it. Liu et al [11] proposed a meshless weighted least squares (MWLS) method to solve steady and unsteady heat conduction problems. A Hermite-type collocation method was also suggested by Liu et al [12] to consider both governing equations and boundary conditions on Neumann boundaries.…”

Section: Introductionmentioning

confidence: 99%

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Collocated discrete least squares meshless (CDLSM) method for the solution of transient and steady‐state hyperbolic problems

Afshar

Lashckarbolok

Shobeyri

2008

Numerical Methods in Fluids

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SUMMARYA collocated discrete least squares meshless method for the solution of the transient and steady-state hyperbolic problems is presented in this paper. The method is based on minimizing the sum of the squared residuals of the governing differential equation at some points chosen in the problem domain as collocation points. The collocation points are generally different from nodal points, which are used to discretize the problem domain. A moving least squares method is employed to construct the shape functions at nodal points. The coefficient matrix is symmetric and positive definite even for non-symmetric hyperbolic differential equations and can be solved efficiently with iterative methods. The proposed method is a truly meshless method and does not require numerical integration. Advantages of the collocation points are shown to be threefold: First, the collocation points are shown to be responsible for stabilizing the method in particular when problems with shocked solution are attempted. Second, the collocation points are also shown to improve the accuracy of the solution even for problems with smooth solutions. Third, the collocation points are shown to contribute to the efficiency of the method when solving steady-state problems via faster convergence of the resulting algorithm. The ability of the method and in particular the effect of collocation points are tested against a series of one-dimensional transient and steady-state benchmark examples from the literature and the results are presented. A sensitivity analysis is also carried out to investigate the effect of the base polynomials on the accuracy and convergence characteristics of the method in solving steady-state problems. The results show the ability of the proposed method to accurately solve difficult hyperbolic problems considered. The method is also shown to be particularly stable for problems with shocked solution due to the inherent stabilizing mechanism of the method.

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Section: Collocated Discrete Least Squares Meshless (Cdlsm) Methodsmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Collocated discrete least squares meshless (CDLSM) method for the solution of transient and steady‐state hyperbolic problems

Afshar

Lashckarbolok

Shobeyri

2008

Numerical Methods in Fluids

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“…Singh and his colleagues [6][7][8][9] used the EFG method to solve a number of heat conduction problems. Liu et al [10] used meshless weighted least-squares (MWLS) method to solve steady-and unsteadystate heat conduction problems. Tan et al [11] applied leastsquares collocation meshless method to solve coupled radiative and heat conduction problems.…”

Section: Introductionmentioning

confidence: 99%

A new local meshless method for steady-state heat conduction in heterogeneous materials

Ahmadi

Sheikhy

Aghdam

et al. 2010

Engineering Analysis with Boundary Elements

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“…Arefmanesh et al [30] and Batra et al [31] put forward a meshless method for solving heat conduction problem based on MLPG. Liu et al [32,33] use the meshless least-squares method for solving the steady-state heat conduction problem. Sladek et al [34,35] use the meshless local boundary integral equation method (LBIE) for heat conduction analysis.…”

mentioning

confidence: 99%

The reproducing kernel particle method for two-dimensional unsteady heat conduction problems

Cheng

Liew

2009

Comput Mech

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The numerical solution to two-dimensional unsteady heat conduction problem is obtained using the reproducing kernel particle method (RKPM). A variational method is employed to furnish the discrete equations, and the essential boundary conditions are enforced by the penalty method. Convergence analysis and error estimation are discussed. Compared with the numerical methods based on mesh, the RKPM needs only the scattered nodes instead of meshing the domain of the problem. The effectiveness of the RKPM for two-dimensional unsteady heat conduction problems is examined by two numerical examples.

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A Meshless Method Based on Least-Squares Approach for Steady- And Unsteady-State Heat Conduction Problems

Cited by 71 publications

References 22 publications

Collocated discrete least squares meshless (CDLSM) method for the solution of transient and steady‐state hyperbolic problems

Collocated discrete least squares meshless (CDLSM) method for the solution of transient and steady‐state hyperbolic problems

A new local meshless method for steady-state heat conduction in heterogeneous materials

The reproducing kernel particle method for two-dimensional unsteady heat conduction problems

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