An effective Chebyshev tau meshless domain decomposition method based on the integration–differentiation for solving fourth order equations

Shao, Wenting; Wu, Xun

doi:10.1016/j.apm.2014.10.048

Cited by 12 publications

(4 citation statements)

References 35 publications

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“…Special treatment of the boundary conditions is presented in this work. Indeed, to the limited extent that (4) does not hold for the boundary conditions, the boundary conditions will require application using a tau method 49‐51 . A simple equation for calculating the internal collocation points,

C

P

, in the

x

and

y

dimensions is provided in Reference 52 according to the following model:

\left\{\begin{array}{c}C={N}_x+1-\nu \\ {}P={N}_y+1-\nu \end{array}\right.,

where

\nu

denotes the boundary conditions number for each direction.…”

Section: Spectral Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Efficient resolution of incompressible Navier–Stokes equations using a robust high‐order pseudo‐spectral approach

Drissi,

Mesmoudi,

Mansouri

2023

Numerical Methods in Fluids

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An accurate numerical tool is presented in this work to investigate the stationary incompressible Navier–Stokes equations. The proposed approach is based on a pseudo‐spectral method for discretizing the differential equations and the asymptotic numerical method to convert nonlinear systems into linear algebraic equations. The coupling of the spectral method with the asymptotic numerical method is considered as an efficient algorithm to solve any nonlinear differential equations. Their efficiency and robustness are examined here on the flow fluid in different canal with different geometries. These computational efficiency and performance have been analysed via several numerical and benchmark examples of incompressible fluid flow in lid‐driven cavity and vortex shedding over L‐Shaped cavity and fluid flow around a square obstacle. The validation of the proposed approach is made by comparison between the obtained results and those calculated using a finite element method or Ansys commercial code. This validation asserts that the presented numerical tool can be promise for solving fluid flow problems with high accuracy.

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C

P

, in the

x

and

y

dimensions is provided in Reference 52 according to the following model:

\left\{\begin{array}{c}C={N}_x+1-\nu \\ {}P={N}_y+1-\nu \end{array}\right.,

where

\nu

denotes the boundary conditions number for each direction.…”

Section: Spectral Methodsmentioning

confidence: 99%

“…Indeed, to the limited extent that (4) does not hold for the boundary conditions, the boundary conditions will require application using a tau method. [49][50][51] A simple equation for calculating the internal collocation points, C, P, in the x and y dimensions is provided in Reference 52 according to the following model:…”

Section: Spectral Methodsmentioning

confidence: 99%

Efficient resolution of incompressible Navier–Stokes equations using a robust high‐order pseudo‐spectral approach

Drissi,

Mesmoudi,

Mansouri

2023

Numerical Methods in Fluids

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“…Besides, the spectral methods have enormous rates of convergence, they also have high level of reliability. The spectral method were divided into four classifications: collocation [5,6,26,27], tau [8,20,24,41], Galerkin [14,15,23] and Petrov-Galerkin [4,29] method. The main idea of the spectral methods is to express the solution of the problem as a finite sum of given basis of functions (orthogonal polynomials or combination of orthogonal polynomials) and then to choose the coefficients in order to minimize the difference between the exact and the numerical solutions.…”

Section: Introductionmentioning

confidence: 99%

Shifted Jacobi spectral collocation method with convergence analysis for solving integro-differential equations and system of integro-differential equations

Doha

Abdelkawy

Amin

et al. 2019

NAMC

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This article addresses the solution of multi-dimensional integro-differential equations (IDEs) by means of the spectral collocation method and taking the advantage of the properties of shifted Jacobi polynomials. The applicability and accuracy of the present technique have been examined by the given numerical examples in this paper. By means of these numerical examples, we ensure that the present technique is simple and very accurate. Furthermore, an error analysis is performed to verify the correctness and feasibility of the proposed method when solving IDE.

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“…Then, Shao and Wu 8 use the Chebyshev tau meshless method (CTMM), which was based on the highest derivative, that solved the fourth-order equations in irregularly shaped domains with complex boundary conditions. In 2015, Shao and Wu 9 combined the CTMMID with the domain decomposition method and applied it to solve the fourth-order problem on irregular domains. This circumvented the ill-conditioning problem.…”

Section: Introductionmentioning

confidence: 99%

Vibration analysis of the plate with the regular and irregular domain by using the Barycentric Lagrange interpolation

Yeh

2018

Journal of Low Frequency Noise, Vibration and Active Control

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This paper uses the Barycentric Lagrange interpolation method to explore the free vibration of a plate with the regular and irregular domain using the Chebyshev function, allowing us to consider multiple dimensions. From our results, it can be shown that the Barycentric Lagrange interpolation method can solve three-dimensional problems. In the analysis, we can see that the Barycentric Lagrange interpolation method can solve the dynamic motion of the plate with regular domain, and the error of the simulation can be reduced to 0.15%. The effect of the geometric node number on the simulated error of the natural frequency of the plate is very profound. The Barycentric Lagrange interpolation method and the extrapolation difference method can solve the natural frequency of the plate with irregular domain. The error of the simulation on the natural frequency can be reduced to 1.084%. This allows us to understand the vibration of the plate with the regular and irregular domain under various boundary conditions quickly.

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An effective Chebyshev tau meshless domain decomposition method based on the integration–differentiation for solving fourth order equations

Cited by 12 publications

References 35 publications

Efficient resolution of incompressible Navier–Stokes equations using a robust high‐order pseudo‐spectral approach

Efficient resolution of incompressible Navier–Stokes equations using a robust high‐order pseudo‐spectral approach

Shifted Jacobi spectral collocation method with convergence analysis for solving integro-differential equations and system of integro-differential equations

Vibration analysis of the plate with the regular and irregular domain by using the Barycentric Lagrange interpolation

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