L.W. Zhang

Computation of vibration solution for functionally graded carbon nanotube-reinforced composite thick plates resting on elastic foundations using the element-free IMLS-Ritz method

This paper explores the element-free IMLS-Ritz method for computation of vibration solution of thick functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates resting on elastic foundations. The shear deformation effect is incorporated through the first-order shear deformation theory (FSDT). The cubic spline weight function and linear basis are utilized in the approximation. Regular node arrangements and cell background meshes are employed in the numerical integration. The penalty method is adopted to impose the essential boundary conditions. Numerical stability and applicability of the IMLS-Ritz method are examined through solving a few numerical example problems. The influence of Winkler modulus parameters on the vibration behavior of FG-CNTRC plates is studied. Besides, the effects of CNT volume fraction, CNT distribution, plate thickness-to-width ratio, plate aspect ratio on FG-CNTRC plates are investigated under different boundary conditions. The vibration frequencies and mode shapes of the FG-CNTRC plates on different Winkler foundations are presented.

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

We propose an improved moving least-squares Ritz (IMLS-Ritz) method with its element-free framework developed for studying two-dimensional elasticity problems. Using the IMLS approximation for the field variables, the discretized governing equations of the problem are derived via the Ritz procedure. In the IMLS, an orthogonal function system with a weight function is employed as the basis for construction of its displacement field. By using the element-free IMLS-Ritz method, solutions of the two-dimensional elasticity problems are obtained. The applicability of the element-free IMLS-Ritz method is illustrated through three selected example problems. The convergence characteristics of the method are examined by varying the number of nodes and geometric parameters of these examples. The accuracy of the method is validated by comparing the computed results with the EFG and exact solutions.

Modeling of biological population problems using the element-free kp-Ritz method

A degenerate parabolic equation arising in the spatial diffusion of biological population is analyzed using the mesh-free kp-Ritz method. The mesh-free kernel particle estimate is employed to approximate the 2D displacement field. A system of discrete equations is obtained through application of the Ritz minimization procedure to the energy expressions. To validate the accuracy of the results and stability of the present method, convergence studies were carried out based on influences of support size and number of nodes. Effectiveness of the mesh-free kp-Ritz method for biological population model is investigated by numerical examples in this paper. The present results were compared with results reported in extant literature and were found to be in good agreement with the literature.

Buckling of FG-CNT reinforced composite thick skew plates resting on Pasternak foundations based on an element-free approach

Buckling behavior of functionally graded carbon nanotube (FG-CNT) reinforced composite thick skew plates is studied. The element-free IMLS-Ritz method is used to obtain the buckling solutions to this problem. The first-order shear deformation theory (FSDT) is employed for formulation of the energy functional to incorporate the effects of transverse shear deformation and rotary inertia. Using the IMLS approximation in the field variables and minimizing the energy functional via the Ritz procedure, a discretized eigenvalue equation of the problem is derived. The buckling solution can be obtained through solving this eigenvalue problem. The numerical stability of the IMLS-Ritz method is validated through convergence studies. The accuracy of the IMLS-Ritz results is examined by comparing with the known solutions. Close agreement is found from the comparison study. Besides, parametric studies are conducted for various types of CNTs distributions, CNT ratios, aspect ratios, plate geometries and thickness-to-height ratios under different boundary conditions.

The improved complex variable element-free Galerkin method for two-dimensional Schrödinger equation

Abstract A numerical study of two-dimensional Schrodinger equation is carried out using the improved complex variable element-free Galerkin (ICVEFG) method. The ICVEFG method involves employment of the improved complex variable moving least-squares (ICVMLS) in the element-free Galerkin (EFG) procedure for numerical approximation. The ICVMLS is used to construct trial functions for the two-dimensional Schrodinger equation in the form of one-dimensional basis function that effectively reduces the number of unknown coefficients. In this study, the applicability of the ICVEFG method is examined through a number of numerical example problems. Convergence studies are carried out for these example problems by varying the number of nodes to ascertain convergent results are achieved as the number of nodes increases. The stability and accuracy of the ICVEFG method are validated by comparing the computed results with the exact solutions.

An element-free based solution for nonlinear Schrödinger equations using the ICVMLS-Ritz method

An improved complex variable moving least-squares Ritz (ICVMLS-Ritz) method is proposed for predicting numerical solution of the two-dimensional nonlinear Schrodinger equation. In this element-free solution procedure, the ICVMLS approximation is employed to reduce the number of unknown coefficients in the trial function. It follows by the Ritz procedure to derive the final algebraic equation system through discretizing the constructed energy formulation of the nonlinear Schrodinger equation. The central differencing scheme and Newtons algorithm are adopted to solve the nonlinear equation system. Numerical experiments are conducted on the final form of the governing equation system to demonstrate the accuracy and efficiency of the element-free ICVMLS-Ritz method by comparing the computed results with the available analytical solutions.

Numerical analysis of generalized regularized long wave equation using the element-free kp-Ritz method

Abstract A numerical analysis of the generalized regularized long wave (GRLW) equation by application of the element-free kp-Ritz method is performed. To approximate the displacement field, an element-free kernel particle estimation is employed and a system of nonlinear discrete equations is achieved by using the Ritz minimization procedure to the energy expressions. Validation of the present numerical solutions with the exact results has been conducted by numerical examples and evidenced a good agreement.

Transient analysis of single-layered graphene sheet using the kp-Ritz method and nonlocal elasticity theory

In this paper, an investigation on the transient analysis of single-layered graphene sheets (SLGSs) is performed using the element-free kp-Ritz method. The classical plate theory is used to describe the dynamic behavior of SLGSs. Nonlocal elasticity theory, in which nonlocal parameter is introduced, is incorporated to reflect the small effect. Newmarks method is employed to solve the discretized dynamic equations. Several numerical examples are presented to examine the effect of boundary conditions, aspect ratio, side length load distribution type and load variation type on the transient behavior of SLGSs. The present work can serve as the foundation for further investigation of the transient response of multi-layered graphene sheets.

Nonlinear local response of foam-filled sandwich plates with laminated faces under combined transverse and in-plane loads

Abstract This paper deals with the nonlinear local response of foam-filled sandwich plates with composite laminated faces subjected to transverse patch load combined with in-plane compressive edge loads. The analysis is carried out by modeling loaded faces as thin laminated plates resting on a two-parameter (Vlasov-type) elastic foundation, from which the shearing interaction between the loaded face and the supporting core-layer are included. An analytical–numerical approach is developed to determine the local bending or postbuckling responses of foam-filled sandwich plates with different sets of boundary conditions. Both local displacement and stress fields are examined. A detailed numerical investigation of the influence played by a number of laminate parameters (fiber orientation, geometric dimensions) and foundation parameters (elastic modular ratio and thickness ratio between face and core layers) as well as the load parameters is performed and pertinent conclusions are outlined.

Meshless modeling of geometrically nonlinear behavior of CNT-reinforced functionally graded composite laminated plates

A geometrically nonlinear analysis of carbon nanotube reinforced functionally graded (CNTR-FG) composite laminated plates is presented. Single-walled carbon nanotubes (SWCNTs) are selected as reinforcement and the effective material properties of CNTR-FG plates are assumed to be graded through the thickness direction in each layer. The two-dimensional displacement fields of the plates are approximated by a set of meshless kernel particle functions. For the purpose of eliminating shear locking, a stabilized conforming nodal integration scheme is employed to evaluate the system bending stiffness, and the membrane and shear terms are calculated by the direct nodal integration method. Parametric studies are conducted to investigate the effect of various types of CNT distribution, CNT volume fraction, plate aspect ratio and boundary conditions on the nonlinear responses of CNTR-FG laminated plates. Moreover, the effects of the number of layers and lamination angle are also investigated.