Article

# A DIFFERENTIAL QUADRATURE FINITE ELEMENT METHOD

(Impact Factor: 1.62). 04/2012; 02(01). DOI: 10.1142/S1758825110000470

ABSTRACT

This paper studies the differential quadrature finite element method (DQFEM) systematically, as a combination of differential quadrature method (DQM) and standard finite element method (FEM), and formulates one- to three-dimensional (1-D to 3-D) element matrices of DQFEM. It is shown that the mass matrices of C0 finite element in DQFEM are diagonal, which can reduce the computational cost for dynamic problems. The Lagrange polynomials are used as the trial functions for both C0 and C1 differential quadrature finite elements (DQFE) with regular and/or irregular shapes, this unifies the selection of trial functions of FEM. The DQFE matrices are simply computed by algebraic operations of the given weighting coefficient matrices of the differential quadrature (DQ) rules and Gauss-Lobatto quadrature rules, which greatly simplifies the constructions of higher order finite elements. The inter-element compatibility requirements for problems with C1 continuity are implemented through modifying the nodal parameters using DQ rules. The reformulated DQ rules for curvilinear quadrilateral domain and its implementation are also presented due to the requirements of application. Numerical comparison studies of 2-D and 3-D static and dynamic problems demonstrate the high accuracy and rapid convergence of the DQFEM.

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• "(6). It can be seen from [1] that most results therein agree with the results of the differential quadrature finite element method (DQFEM) [4] [5] or the results of Liew [6] obtained by the p-Ritz method for about 2–3 digits. The corrected results of this work agree with the DQFEM results or the p-Ritz results for almost all digits used for comparisons. "
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ABSTRACT: This paper presents correct results for a paper by Liu and Xing (2011) [1] that had errors, greatly simplifies its formulations, overcomes its numerical stability problem, and provides some new results. In their early paper, Liu and Xing obtained exact closed-form solutions for free vibrations of orthotropic rectangular Mindlin plates by using the separation of variables method. The equations presented therein are correct, but the numerical values of the frequencies listed have some errors because the shear rigidities of the three-ply laminates were not correctly computed. In addition, the exact characteristic equations of the early paper can also be greatly simplified and the compact equations are presented herein. The characteristic equations in the early paper may have computational stability problem when plates become very thin. This paper overcome such computational difficulty. Some new exact frequency parameters are also included in this work.
Composite Structures 12/2014; 118(1):316–321. DOI:10.1016/j.compstruct.2014.07.051 · 3.32 Impact Factor
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• "Using a mapping technique, it is transformed into a set of regular Cartesian parent elements. Thus, the external flux boundary conditions must be written following the outward unit normal vector n as reported in [Xing, Liu, and Liu (2010) "
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ABSTRACT: In the present paper the Generalized Differential Quadrature Finite El-ement Method (GDQFEM) is applied to deal with the static analysis of plane state structures with generic through the thickness material discontinuities and holes of various shapes. The GDQFEM numerical technique is an extension of the Gener-alized Differential Quadrature (GDQ) method and is based on the idea of conven-tional integral quadrature. In particular, the GDQFEM results in terms of stresses and displacements for classical and advanced plane stress problems with discon-tinuities are compared to the ones by the Cell Method (CM) and Finite Element Method (FEM). The multi-domain technique is implemented in a MATLAB code for solving irregular domains with holes and defects. In order to demonstrate the accuracy of the proposed methodology, several numerical examples of stress and displacement distributions are graphically shown and discussed.
Computer Modeling in Engineering and Sciences 12/2013; 94(4):331-369. DOI:10.3970/cmes.2013.094.331 · 1.03 Impact Factor
• ##### Article: Variational finite-difference methods in linear and nonlinear problems of the deformation of metallic and composite shells (review)
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ABSTRACT: Variational finite-difference methods of solving linear and nonlinear problems for thin and nonthin shells (plates) made of homogeneous isotropic (metallic) and orthotropic (composite) materials are analyzed and their classification principles and structure are discussed. Scalar and vector variational finite-difference methods that implement the Kirchhoff–Love hypotheses analytically or algorithmically using Lagrange multipliers are outlined. The Timoshenko hypotheses are implemented in a traditional way, i.e., analytically. The stress–strain state of metallic and composite shells of complex geometry is analyzed numerically. The numerical results are presented in the form of graphs and tables and used to assess the efficiency of using the variational finite-difference methods to solve linear and nonlinear problems of the statics of shells (plates)
International Applied Mechanics 11/2012; 48(6). DOI:10.1007/s10778-012-0544-8