An improved moving least-squares Ritz method for two-dimensional elasticity problems

Zhang, Luwen; Liew, K.M.

doi:10.1016/j.amc.2014.07.001

Cited by 55 publications

(27 citation statements)

References 38 publications

(35 reference statements)

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“…An improved complex variable MLS Ritz (ICVMLS-Ritz) method is proposed in [74] for obtaining numerical solution of the 2D nonlinear Schrödinger equation. Authors of [75] proposed an IMLS-Ritz method with its element free framework developed for studying 2D elasticity problems.…”

Section: A Brief Review For Element-free Galerkin (Efg)mentioning

confidence: 99%

The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate

Dehghan

Abbaszadeh

Mohebbi

2015

Journal of Computational and Applied Mathematics

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Please cite this article as: M. Dehghan, M. Abbaszadeh, A. Mohebbi, The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin-Bona-Mahony-Burgers and regularized long-wave equations on non-rectangular domains with error estimate, Journal of Computational and Applied Mathematics (2015), http://dx. AbstractIn this paper a numerical technique is proposed for solving the nonlinear generalized Benjamin-Bona-Mahony-Burgers and regularized long-wave equations. Firstly, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the interpolating element-free Galerkin approach to approximate the spatial derivatives. The element-free Galerkin method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the element-free Galerkin method test and trial functions are moving least squares approximation shape functions. Also, in the element-free Galekin method, we do not use any triangular, quadrangular or other type of meshes. It is a global method while finite elements method is a local one. The element free Galerkin method is not a truly meshless method and for integration employs a background mesh. We prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method. We show that convergence order of the time discrete scheme is O(τ ). Since the shape functions of moving least squares approximation don't have Kronecker delta property, we can not implement the essential boundary condition, directly. Thus, the improved moving least squares shape functions that have the mentioned property are employed . An error estimate for the method proposed in the current paper is obtained. Also, the two-dimensional version of both equations on different complex geometries is solved. The aim of this paper is to show that the meshless method based on the weak form is also suitable for the treatment of the nonlinear partial differential equations and to obtain an error bound for the new method. Numerical examples confirm the efficiency of the proposed scheme. (M. Dehghan), a mohebbi@kashanu.ac.ir (A. Mohebbi), m.abbaszadeh@aut.ac.ir (M. Abbaszadeh).

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Section: A Brief Review For Element-free Galerkin (Efg)mentioning

confidence: 99%

The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate

Dehghan

Abbaszadeh

Mohebbi

2015

Journal of Computational and Applied Mathematics

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“…Meshless methods are efficient and accurate methods for solving the ordinary or partial differential equations [32][33][34][35][36][37][38][39][40][41][42]. As a meshless method, the Ritz method in space domain is implemented herein to deduce the governing equations from the expression of total energy function.…”

Section: Cnts Distributionmentioning

confidence: 99%

Low velocity impact response of functionally graded carbon nanotube reinforced composite beams in thermal environment

Jam

Kiani

2015

Composite Structures

111

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“…Because of the efficient and accurate of the meshless method for any geometry and boundary conditions, this method are vastly employed to solve the linear and nonlinear problem by researchers such as: an improved moving least-squares Ritz (IMLS-Ritz ) method for twodimensional elasticity problems [34], an improved complex variable moving least-squares Ritz (ICVMLS-Ritz) method to solve the two-dimensional nonlinear Schrodinger equations [35,36], the IMLS-Ritz method for elastodynamic cantilevered beam [37], the mesh free kpRitz method for static and dynamic of carbon nanotube reinforced FG cylindrical panels [38], the element-free approach for buckling analysis of FG-CNT reinforced composite thick skew plates [39], the IMLS approximation for vibration characteristic of moderately thick functionally graded carbon nanotube reinforced composite skew plates [40], and a review of meshless methods for laminated and functionally graded plates and shells [41]. 6 Numerical analysis of generalized regularized long wave equation using the element-free kpRitz method is performed by Guo et al [42].…”

Section: -Intruductionmentioning

confidence: 99%

Free vibration of viscoelastic double-bonded polymeric nanocomposite plates reinforced by FG-SWCNTs using MSGT, sinusoidal shear deformation theory and meshless method

Mohammadimehr

Navi

Arani

2015

Composite Structures

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An improved moving least-squares Ritz method for two-dimensional elasticity problems

Cited by 55 publications

References 38 publications

The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate

The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate

Low velocity impact response of functionally graded carbon nanotube reinforced composite beams in thermal environment

Free vibration of viscoelastic double-bonded polymeric nanocomposite plates reinforced by FG-SWCNTs using MSGT, sinusoidal shear deformation theory and meshless method

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