Method of particular solutions using polynomial basis functions for the simulation of plate bending vibration problems

Lin, Ji; Chen, Ching-Shyang; Wang, Fajie; Dangal, Thir

doi:10.1016/j.apm.2017.05.012

Cited by 47 publications

(23 citation statements)

References 41 publications

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“…As one of them, the meshless methods including the wave-based method [14][15] , the boundary node method [16][17] , and the method of fundamental solutions [18][19] have been all developed for plate bending problems to a certain extent. The above methods seek the approximations with the expansion functions to satisfy the governing equations and then obtain the unknown weight parameters by requiring the approximations satisfying the given boundary conditions [20] . It is difficult to solve the problems with complex boundary conditions, such as Ref.…”

Section: Introductionmentioning

confidence: 99%

“…Among them, the radial basis functions (RBFs) are more successful to find the particular solutions for some PDEs [22][23] ; however, the RBFs cannot solve the optimization problem of shape parameters well [22,25] . Additionally, although the particular solutions based on Chebyshey polynomials can avoid such difficulties and simultaneously possess relatively high accuracy [26][27] , there is an obvious disadvantage in using Chebyshey polynomials as basis functions, i.e., the forcing term of the corresponding differential equation needs to be smoothly extendable to the exterior of the solution domain in the case of irregular domains [20,25] , and the selection of the corresponding collocation points at this time must be treated specially. Recently, a novel method to derive the particular solutions to a PDE by using the standard polynomial basis functions was presented [25] , where the collocation points could be selected arbitrarily inside the solution domain.…”

Section: Introductionmentioning

confidence: 99%

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Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory

Feng

Yan

Lin

et al. 2020

Appl. Math. Mech.-Engl. Ed.

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Based on the nonlocal theory and Mindlin plate theory, the governing equations (i.e., a system of partial differential equations (PDEs) for bending problem) of magnetoelectroelastic (MEE) nanoplates resting on the Pasternak elastic foundation are first derived by the variational principle. The polynomial particular solutions corresponding to the established model are then obtained and further employed as basis functions with the method of particular solutions (MPS) to solve the governing equations numerically. It is confirmed that for the present bending model, the new solution strategy possesses more general applicability and superior flexibility in the selection of collocation points. Finally, the effects of different boundary conditions, applied loads, and geometrical shapes on the bending properties of MEE nanoplates are evaluated by using the developed method. Some important conclusions are drawn, which should be helpful for the design and applications of electromagnetic nanoplate structures.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory

Feng

Yan

Lin

et al. 2020

Appl. Math. Mech.-Engl. Ed.

Self Cite

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“…Several numerical examples were conducted and compared with those from other approaches. Later, Lin et al [42] presented an improved polynomial expansion method for solving plate bending vibration problems. In order to alleviate the conditioning of the matrix, the multiple-scale method was also employed.…”

Section: Introductionmentioning

confidence: 99%

A Novel Meshfree Approach with a Radial Polynomial for Solving Nonhomogeneous Partial Differential Equations

Xiao

Liu

2020

Mathematics

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In this article, a novel radial–based meshfree approach for solving nonhomogeneous partial differential equations is proposed. Stemming from the radial basis function collocation method, the novel meshfree approach is formulated by incorporating the radial polynomial as the basis function. The solution of the nonhomogeneous partial differential equation is therefore approximated by the discretization of the governing equation using the radial polynomial basis function. To avoid the singularity, the minimum order of the radial polynomial basis function must be greater than two for the second order partial differential equations. Since the radial polynomial basis function is a non–singular series function, accurate numerical solutions may be obtained by increasing the terms of the radial polynomial. In addition, the shape parameter in the radial basis function collocation method is no longer required in the proposed method. Several numerical implementations, including homogeneous and nonhomogeneous Laplace and modified Helmholtz equations, are conducted. The results illustrate that the proposed approach may obtain highly accurate solutions with the use of higher order radial polynomial terms. Finally, compared with the radial basis function collocation method, the proposed approach may produce more accurate solutions than the other.

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“…For some RBFs, it is difficult to obtain the optimal shape parameters, such as the multiquadric radial basis functions (MQ-RBFs) [29] and the Gaussian radial basis functions [30,31], which is crucial to the accuracy of the numerical algorithms. What's worse, the coefficient matrix can be highly ill-conditioned for large scale problems [32]. In order to overcome these drawbacks, researchers come up with the localized approaches including the compactly supported RBFs, the localized RBFs collocation method, the local radial basis functions based differential quadrature collocation method [33,34], and the localized method of approximate particular solutions (LMAPS) [35].…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulation of Non-Linear Schrodinger Equations in Arbitrary Domain by the Localized Method of Approximate Particular Solution

Hong¹

2019

AAMM

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The aim of this paper is to propose a fast meshless numerical scheme for the simulation of non-linear Schrödinger equations. In the proposed scheme, the implicit-Euler scheme is used for the temporal discretization and the localized method of approximate particular solution (LMAPS) is utilized for the spatial discretization. The multiple-scale technique is introduced to obtain the shape parameters of the multiquadric radial basis function for 2D problems and the Gaussian radial basis function for 3D problems. Six numerical examples are carried out to verify the accuracy and efficiency of the proposed scheme. Compared with well-known techniques, numerical results illustrate that the proposed scheme is of merits being easy-to-program, high accuracy, and rapid convergence even for long-term problems. These results also indicate that the proposed scheme has great potential in large scale problems and real-world applications.

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Method of particular solutions using polynomial basis functions for the simulation of plate bending vibration problems

Cited by 47 publications

References 41 publications

Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory

Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory

A Novel Meshfree Approach with a Radial Polynomial for Solving Nonhomogeneous Partial Differential Equations

Numerical Simulation of Non-Linear Schrodinger Equations in Arbitrary Domain by the Localized Method of Approximate Particular Solution

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