Radial Basis Point Interpolation Collocation Method for 2-D Solid Problem

Liu, Xin; Liu, G. R.; Tai, Kang; Lam, K.Y.

doi:10.1142/9789812778611_0008

Cited by 19 publications

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“…There are two important parameters used in the present RFDM: the numbers of local supporting nodes and finite difference grid points. The first one has been well investigated [1,2,[19][20][21]. In this paper, the relations between the numbers of finite difference grid points and field nodes are discussed in details.…”

Section: Parameter Studymentioning

confidence: 99%

“…Radial point collocation method (RPCM) is also a meshfree strong-form method (see, e.g. References [19][20][21]) formulated using radial basis functions and nodes in local supporting domains. Like other strong-form methods, the RPCM suffers from problems of instability (e.g.…”

mentioning

confidence: 99%

“…Examples include the finite point method [23], Hermite-type collocation method (e.g. References [19,20]), fictitious point approach (e.g. Reference [21]), stabilized least-squares RPCM (LS-RPCM) [24] and meshfree weak-strong (MWS) form method [25,26].Based on the aforementioned existing work, this paper presents a radial point interpolation based finite difference method (RFDM) as an alternative meshfree strong-form method.…”

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confidence: 99%

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Radial point interpolation based finite difference method for mechanics problems

Liu

Zhang

et al. 2006

Numerical Meth Engineering

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SUMMARYA radial point interpolation based finite difference method (RFDM) is proposed in this paper. In this novel method, radial point interpolation using local irregular nodes is used together with the conventional finite difference procedure to achieve both the adaptivity to irregular domain and the stability in the solution that is often encountered in the collocation methods. A least-square technique is adopted, which leads to a system matrix with good properties such as symmetry and positive definiteness. Several numerical examples are presented to demonstrate the accuracy and stability of the RFDM for problems with complex shapes and regular and extremely irregular nodes. The results are examined in detail in comparison with other numerical approaches such as the radial point collocation method that uses local nodes, conventional finite difference and finite element methods.

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Section: Parameter Studymentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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Radial point interpolation based finite difference method for mechanics problems

Liu

Zhang

et al. 2006

Numerical Meth Engineering

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“…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. Xiaofei et al [13] carried out a sensitivity analysis on the MWLS parameters for solving the problems of a cantilever beam and an infinite plate with a central circular hole.…”

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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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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“…Radial point interpolation method (RPIM) was proposed by Liu and Wang [2], and has been improved and applied with Galerkin-based formulations [3][4]. In [8], an efficient radial basis point interpolation collocation-based formulation, namely radial point interpolation collocation method (RPICM), has been presented and applied to solve 2-D linear elastic problem. In RPICM, the collocation scheme, which is simple and efficient to solve partial differential equations, has been adopted without the need of numerical integrations.…”

Section: Introductionmentioning

confidence: 99%

Radial point interpolation collocation method (RPICM) for the solution of nonlinear poisson problems

Liu

Tai

et al. 2005

Comput Mech

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This paper applies radial point interpolation collocation method (RPICM) for solving nonlinear Poisson equations arising in computational chemistry and physics. Thin plate spline (TPS) Radial basis functions are used in the work. A series of test examples are numerically analysed using the present method, including 2D Liouville equation, Bratu problem and Poisson-Boltzmann equation, in order to test the accuracy and efficiency of the proposed schemes.Several aspects have been numerically investigated, namely the enforcement of additional polynomial terms; and the application of the Hermite-type interpolation which makes use of the normal gradient on Neumann boundary for the solution of PDEs with Neumann boundary conditions. Particular emphasis was on an efficient scheme, namely Hermite-type interpolation for dealing with Neumann boundary conditions. The numerical results demonstrate that a good accuracy can be obtained. The h-convergence rates are also studied for RPICM with coarse and fine discretization models.

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Radial Basis Point Interpolation Collocation Method for 2-D Solid Problem

Cited by 19 publications

References 0 publications

Radial point interpolation based finite difference method for mechanics problems

Radial point interpolation based finite difference method for mechanics problems

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

Radial point interpolation collocation method (RPICM) for the solution of nonlinear poisson problems

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