Improvements of generalized finite difference method and comparison with other meshless method

Gavete, L.; Gavete, María Lucía; Benito, Juan José

doi:10.1016/s0307-904x(03)00091-x

Cited by 145 publications

(55 citation statements)

References 16 publications

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“…Orkisz [28] and Benito et al [29] considered h-adaptive versions of the GFD method, whereas Luo and Häussler-Combe [30] proposed a version of the GFD scheme based on the minimization of a global residue. Gavete et al [31] carried out a comparison of the GFD method with the other meshfree schemes, such as the element-free Galerkin method. The GFD approach has also been applied to incompressible viscous flows by Ding et al [22] and to compressible flows by Luo et al [32].…”

Section: The Weighted Svd-gfd Methodsmentioning

confidence: 99%

A singular‐value decomposition (SVD)‐based generalized finite difference (GFD) method for close‐interaction moving boundary flow problems

Ang

Yeo

Chew

et al. 2008

Numerical Meth Engineering

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SUMMARYIn this paper, we present a study on a singular-value decomposition (SVD)-based generalized finite difference (GFD) method and a nodal selection scheme for moving body/boundary flow problems formulated on a hybrid Cartesian cum meshfree grid system. The present study shows that the SVD-based method is more robust and accurate than the conventional least-squares-based GFD scheme. A nodal selection scheme is also introduced to overcome the problem of numerical instability associated with the clustering of computational nodes. Such nodal clustering occurs dynamically when moving bodies or boundaries approach within close proximity of each other, resulting in the overlap of their meshfree grids. The nodal scheme is applied to close-interaction flow problems as exemplified by the squeezing action of a circular cylinder through a very narrow slot and the close proximity bypass interaction of two oscillating circular cylinders.

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Section: The Weighted Svd-gfd Methodsmentioning

confidence: 99%

A singular‐value decomposition (SVD)‐based generalized finite difference (GFD) method for close‐interaction moving boundary flow problems

Ang

Yeo

Chew

et al. 2008

Numerical Meth Engineering

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“…The local quadrature rule was tested and found to be very robust, because there is no drawback of obtaining an ill-posed set of local nodes, as possibly encountered in the generalized finite difference method using scattered nodes [22]. Using whole derivative formulas, Liu and Li [23] constructed a mesh-free method, which is in fact equivalent to the LDQ rules, with the aid of linear interpolation, Equations (2) as follows:…”

Section: (K)mentioning

confidence: 99%

An interpolation‐based local differential quadrature method to solve partial differential equations using irregularly distributed nodes

Qin

2007

Commun. Numer. Meth. Engng.

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SUMMARYTo circumvent the constraint in application of the conventional differential quadrature (DQ) method that the solution domain has to be a regular region, an interpolation-based local differential quadrature (LDQ) method is proposed in this paper. Instead of using regular nodes placed on mesh lines in the DQ method (DQM), irregularly distributed nodes are employed in the LDQ method. That is, any spatial derivative at a nodal point is approximated by a linear weighted sum of the functional values of irregularly distributed nodes in the local physical domain. The feature of the new approach lies in the fact that the weighting coefficients are determined by the quadrature rule over the irregularly distributed local supporting nodes with the aid of nodal interpolation techniques developed in the paper. Because of this distinctive feature, the LDQ method can be consistently applied to linear and nonlinear problems and is really a mesh-free method without the limitation in the solution domain of the conventional DQM. The effectiveness and efficiency of the method are validated by two simple numerical examples by solving boundary-value problems of a linear and a nonlinear partial differential equation.

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“…Recently, Benito, Urena and Gavete gave a discussion about the influence of several factors in the GFD method, and a comparison between the GFD method and the EFGM in solving Laplace equation is also presented in [49][50][51]. The GFD method shows more accurate than the EFGM.…”

Section: Introductionmentioning

confidence: 99%

A Lagrangian Approach for Computational Acoustics with Meshfree Method

Zhang

Smith

Zhang

et al. 2017

Preprint

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Abstract:Although Eulerian approaches are standard in computational acoustics, they are less effective for certain classes of problems like bubble acoustics and combustion noise. A different approach for solving acoustic problems is to compute with individual particles following particle motion. In this paper, a Lagrangian approach to model sound propagation in moving fluid is presented and implemented numerically, using three meshfree methods to solve the Lagrangian acoustic perturbation equations (LAPE) in the time domain. The LAPE split the fluid dynamic equations into a set of hydrodynamic equations for the motion of fluid particles and perturbation equations for the acoustic quantities corresponding to each fluid particle. Then, three meshfree methods, the smoothed particle hydrodynamics (SPH) method, the corrective smoothed particle (CSP) method, and the generalized finite difference (GFD) method, are introduced to solve the LAPE and the linearized LAPE (LLAPE). The SPH and CSP methods are widely used meshfree methods, while the GFD method based on the Taylor series expansion can be easily extended to higher orders. Applications to modeling sound propagation in steady or unsteady fluids in motion are outlined, treating a number of different cases in one and two space dimensions. A comparison of the LAPE and the LLAPE using the three meshfree methods is also presented. The Lagrangian approach shows good agreement with exact solutions. The comparison indicates that the CSP and GFD method exhibit convergence in cases with different background flow. The GFD method is more accurate, while the CSP method can handle higher Courant numbers.

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Improvements of generalized finite difference method and comparison with other meshless method

Cited by 145 publications

References 16 publications

A singular‐value decomposition (SVD)‐based generalized finite difference (GFD) method for close‐interaction moving boundary flow problems

A singular‐value decomposition (SVD)‐based generalized finite difference (GFD) method for close‐interaction moving boundary flow problems

An interpolation‐based local differential quadrature method to solve partial differential equations using irregularly distributed nodes

A Lagrangian Approach for Computational Acoustics with Meshfree Method

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