An Introduction to Meshfree Methods and Their Programming

Liu, G. R.; Gu, Yuantong

doi:10.1007/1-4020-3468-7

Cited by 135 publications

(17 citation statements)

References 96 publications

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“…Heat transfer problem is described by a nonlinear governing Eq. (14) in the fluid region Ω 1 with a linear governing Eq. (18) in the wall region Ω 2 subject to the linear boundary conditions (19)- (23).…”

Section: Solution Of Heat Transfer Problem Using the Modified Methods mentioning

confidence: 99%

“…Heat transfer problem is governed by the dimensionless Eqs. (14) in the fluid region Ω 1 and (18) in the wall region Ω 2 . The boundary conditions are expressed as follows…”

Section: The Energy Equationmentioning

confidence: 99%

“…In numerical analysis of boundary value problems meshless methods are more and more popular [14,15]. The method of fundamental solutions (MFS) is one of such a method.…”

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confidence: 99%

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A meshless procedure for analysis of fluid flow and heat transfer in an internally finned square duct

Grabski

2019

Heat Mass Transfer

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Application of the method of fundamental solutions in combination with the global radial basis function collocation method for analysis of fluid flow and heat transfer in an internally finned square duct is presented in the paper. Fluid flow problem is solved using the modified method of fundamental solutions. After that, the average fluid velocity and product of friction factor and Reynolds number can be determined. Heat transfer problem in the fluid is governed by a nonlinear equation with linear boundary conditions. The Picard iteration method is employed in the paper in order to transform the nonlinear problem into a sequence of inhomogeneous problems. At each iteration step, the general solution is obtained using the modified method of fundamental solutions and the particular solution is obtained using the global radial basis function collocation method. When the iteration process is stopped, the Nusselt number can be determined. Nomenclature aHalf of internal width of channel [m] bThickness of wall [m]

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Section: Solution Of Heat Transfer Problem Using the Modified Methods mentioning

confidence: 99%

“…Heat transfer problem is governed by the dimensionless Eqs. (14) in the fluid region Ω 1 and (18) in the wall region Ω 2 . The boundary conditions are expressed as follows…”

Section: The Energy Equationmentioning

confidence: 99%

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A meshless procedure for analysis of fluid flow and heat transfer in an internally finned square duct

Grabski

2019

Heat Mass Transfer

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“…EFGM has been applied to several engineering problems because of its good performance and accuracy (Singh 2006;Samimi and Pak 2012;Zhang et al 2017). Liu and Gu (2005) found that both the convergence rate and the accuracy of EFGM are higher than bilinear FEM due to higher interpolation accuracy achieved using more support nodes in the shape functions. Also, compared to the meshless RPIM the maximum bandwidth of the final global matrix in EFGM is usually smaller and thus, the computational cost for solving the final system of equation is lower (Liu and Gu 2005).…”

Section: Introductionmentioning

confidence: 98%

“…These methods discretize the computational domain and its boundaries as a set of scattered nodes which can be more conveniently added or removed without disturbing the existing nodal configuration. These methods discretize the governing partial differential equations into a set of algebraic equations involving the nodal values of the problem variables (Liu and Gu 2005). Meshless methods have shown remarkable application potential in various engineering problems including solid mechanics, fluid mechanics, geotechnics and heat conduction (Samimi and Pak 2012;Khoshghalb and Khalili 2010).…”

Section: Introductionmentioning

confidence: 99%

Simulation of Groundwater Flow in an Unconfined Sloping Aquifer Using the Element-Free Galerkin Method

Pathania

Bottacin‐Busolin

Rastogi

et al. 2019

Water Resour Manage

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Numerical simulation of groundwater flows in real aquifers has been commonly based on Finite Difference (FDM) and Finite Element Methods (FEM). These mesh-based methods require a computational grid, which can be costly to generate in case of complex geometries and can significantly increase the computational cost when adaptive remeshing is required. This work presents the Element-Free Galerkin Method (EFGM) for solution of the groundwater equation and demonstrates its application to a real unconfined sloping aquifer. EFGM uses the Moving Least Square (MLS) method for approximating the unknown head and computing the EFGM shape function, leading to increased accuracy, stability in function approximation and higher convergence rate than mesh-based methods. In the present study, EFGM is initially applied to one-and two-dimensional hypothetical problems, and validated with analytical solutions. Subsequently, EFGM is applied to the simulation of groundwater flow in the Blue Lake Aquifer (BLA) in North California, and the simulation results are compared with those obtained using MODFLOW. The results demonstrate the potential of EFGM for analyzing real aquifer problems.

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Error estimate and adaptive refinement for incompressible Navier‐Stokes equations using the discrete least squares meshless method

Firoozjaee

Afshar

2011

Numerical Methods in Fluids

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SUMMARY In this paper, an adaptive refinement strategy based on a node‐moving technique is proposed and used for the efficient solution of the steady‐state incompressible Navier–Stokes equations. The value of a least squares functional of the residual of the governing differential equation and its boundary conditions at nodal points is regarded as a measure of error and used to predict the areas of poor solutions. A node‐moving technique is then used to move the nodal points to the zones of higher numerical errors. The problem is then resolved on the refined distribution of nodes for higher accuracy. A spring analogy is used for the node‐moving methodology in which nodal points are connected to their neighbors by virtual springs. The stiffness of each spring is assumed to be proportional to the errors of its two end points and its initial length. The new positions of the nodal points are found such that the spring system attains its equilibrium state. Some numerical examples are used to illustrate the ability of the proposed scheme for the adaptive solution of the steady‐state incompressible Navier–Stokes equations. The results demonstrate a considerable improvement of the results with a reasonable computational effort by using the proposed adaptive strategy. Copyright © 2011 John Wiley & Sons, Ltd.

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An Introduction to Meshfree Methods and Their Programming

Cited by 135 publications

References 96 publications

A meshless procedure for analysis of fluid flow and heat transfer in an internally finned square duct

A meshless procedure for analysis of fluid flow and heat transfer in an internally finned square duct

Simulation of Groundwater Flow in an Unconfined Sloping Aquifer Using the Element-Free Galerkin Method

Error estimate and adaptive refinement for incompressible Navier‐Stokes equations using the discrete least squares meshless method

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