An element-free IMLS-Ritz method for numerical solution of three-dimensional wave equations

Zhang, Luwen; Huang, Dong; Liew, K.M.

doi:10.1016/j.cma.2015.08.018

Cited by 57 publications

(10 citation statements)

References 35 publications

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“…Ritz method as a powerful tool is used extensively in the solution of linear and nonlinear problems arising in various fields of structural mechanics, see e.g. [41][42][43][44]. In the present research also, the Ritz method is used to discretize the governing system of equations.…”

Section: Solution Methodsmentioning

confidence: 98%

Nonlinear free vibration of temperature-dependent sandwich beams with carbon nanotube-reinforced face sheets

Mirzaei

Kiani

2016

Acta Mech

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In this research, large amplitude free vibrations of a sandwich beam with stiff core and carbon nanotube (CNT)-reinforced face sheets are analysed. The distribution of CNTs across the thickness of the face sheets may be uniform or functionally graded. The equivalent single-layer theory of Timoshenko is used to construct the Hamiltonian of the beam under the von Kármán type of geometrical nonlinearity assumptions. A uniform temperature field through the beam is also included in the formulation. The Ritz method with polynomial basis functions is used to discrete the equations of motion and establish the matrix representation of the governing equations. A nonlinear eigenvalue problem is obtained and solved using a standard continuation procedure. After validating the developed solution method and formulation, parametric studies are conducted to examine the influences of thermal environment, core thickness-to-face sheet thickness ratio, boundary conditions, amplitude of vibrations, CNTs volume fraction and their distribution pattern. It is concluded that an increase in the volume fraction of CNTs results in higher fundamental frequency and decreases the nonlinear-to-linear frequency ratio.

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Section: Solution Methodsmentioning

confidence: 98%

Nonlinear free vibration of temperature-dependent sandwich beams with carbon nanotube-reinforced face sheets

Mirzaei

Kiani

2016

Acta Mech

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“…As one of the most widely known energy-based methods, the Ritz method is used in the present research. The effectiveness and efficiency of various types of Ritz methods has been the subject of many studies [36–39]. In this study, the approximation of the displacement field is carried out using the Ritz method whose shape functions are written in terms of the Chebyshev polynomials.…”

Section: Modelingmentioning

confidence: 99%

Free vibration of functionally graded carbon-nanotube-reinforced composite plates with cutout

Mirzaei¹,

Kiani²

2016

Beilstein J. Nanotechnol.

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SummaryDuring the past five years, it has been shown that carbon nanotubes act as an exceptional reinforcement for composites. For this reason, a large number of investigations have been devoted to analysis of fundamental, structural behavior of solid structures made of carbon-nanotube-reinforced composites (CNTRC). The present research, as an extension of the available works on the vibration analysis of CNTRC structures, examines the free vibration characteristics of plates containing a cutout that are reinforced with uniform or nonuniform distribution of carbon nanotubes. The first-order shear deformation plate theory is used to estimate the kinematics of the plate. The solution method is based on the Ritz method with Chebyshev basis polynomials. Such a solution method is suitable for arbitrary in-plane and out-of-plane boundary conditions of the plate. It is shown that through a functionally graded distribution of carbon nanotubes across the thickness of the plate, the fundamental frequency of a rectangular plate with or without a cutout may be enhanced. Furthermore, the frequencies are highly dependent on the volume fraction of carbon nanotubes and may be increased upon using more carbon nanotubes as reinforcement.

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“…The EFG and IEFG ideas and also their improvements have been employed for solving several problems such as 2D elastoplasticity problems, 2D potential problems, 2D and 3D Stokes flow problems, nonlinear p‐Laplacian model, 2D large deformation problems, heat conduction problems, incompressible Navier‐Stokes equation, 2D Schrodinger equation, 2D linear elastodynamics, 3D wave equations, study the MLS and IMLS analytically, 2D elasticity problems, temperature field problems, 2D solute transport problems, 2D time‐fractional diffusion‐wave model, and so on. Also, an improved IIMLS method based on the nonsingular weight function is developed in Reference .…”

Section: Introductionmentioning

confidence: 99%

A proper orthogonal decomposition variational multiscale meshless interpolating element‐free Galerkin method for incompressible magnetohydrodynamics flow

Abbaszadeh

Dehghan

Navon

2020

Numerical Methods in Fluids

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Summary In the recent decade, the meshless methods have been handled for solving most of PDEs due to easiness of the meshless methods. One of the popular meshless methods is the element‐free Galerkin (EFG) method that was first proposed for solving some problems in the solid mechanics. The test and trial functions of the EFG are based on the special basis. Recently, some modifications have been developed to improve the EFG method. One of these improvements is the variational multiscale EFG procedure. In the current article, the shape functions of interpolation moving least squares approximation have been applied to the variational multiscale EFG technique for solving the incompressible magnetohydrodynamics flow. In order to reduce the elapsed CPU time of simulation, we employ a reduced‐order model based on the proper orthogonal decomposition technique. The current combination can be referred to as the reduced‐order variational multiscale EFG technique. To illustrate the reduction in CPU time used as well as the efficiency of the proposed method, we applied it for the two‐dimensional cases.

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An element-free IMLS-Ritz method for numerical solution of three-dimensional wave equations

Cited by 57 publications

References 35 publications

Nonlinear free vibration of temperature-dependent sandwich beams with carbon nanotube-reinforced face sheets

Nonlinear free vibration of temperature-dependent sandwich beams with carbon nanotube-reinforced face sheets

Free vibration of functionally graded carbon-nanotube-reinforced composite plates with cutout

A proper orthogonal decomposition variational multiscale meshless interpolating element‐free Galerkin method for incompressible magnetohydrodynamics flow

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