International Journal of Applied Mechanics (Impact Factor: 1.29). 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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    ABSTRACT: In this paper, an advanced version of the classic GDQ method, called the Generalized Differential Quadrature Finite Element Method (GDQFEM) is for-mulated to solve plate elastic problems with inclusions. The GDQFEM is com-pared with Cell Method (CM) and Finite Element Method (FEM). In particular, stress and strain results at fiber/matrix interface of dissimilar materials are pro-vided. The GDQFEM is based on the classic Generalized Differential Quadrature (GDQ) technique that is applied upon each sub-domain, or element, into which the problem domain is divided. When the physical domain is not regular, the map-ping technique is used to transform the fundamental system of equations and all the compatibility conditions. A differential problem defined on the regular master ele-ment in the computational domain is turned into an algebraic system. With respect to the very well-known Finite Element Method (FEM), the GDQFEM is based on a different approach: the direct derivative calculation is performed by using the GDQ rule. The imposition of the compatibility conditions between two boundaries are also used in the CM for solving contact problems. Since the GDQFEM is a higher-order tool connected with the resolution of the strong formulation of the system of equations, the compatibility conditions must be applied at each disconnection in order to capture the discontinuity between two boundaries, without losing accu-racy. A comparison between GDQFEM, CM and FEM is presented and very good agreement is observed.
    Computer Modeling in Engineering and Sciences 12/2013; 94(5):421-458. · 1.18 Impact Factor
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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. · 1.18 Impact Factor
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    ABSTRACT: This paper provides a new technique for solving free vibration problems of composite arbitrarily shaped membranes by using Generalized Differential Quadrature Finite Element Method (GDQFEM). The proposed technique, also known as Multi-Domain Differential Quadrature (MDQ), is an extension of the classic Generalized Differential Quadrature (GDQ) procedure. The multi-domain method can be directly applied to regular sub-domains of rectangular shape, as well as to elements of general shape when a coordinate transformation is considered. The mapping technique is used to transform both the governing differential equations and the compatibility conditions between two adjacent sub-domains into the regular master element in the parent space, called computational space. The numerical implementation of the global algebraic system obtained by the technique at issue is simple and straightforward. Computer investigations concerning a large number of membrane geometries have been carried out. GDQFEM results are compared with those presented in literature and a perfect agreement is observed. Membranes of complex geometry with a material inhomogeneity are also carefully examined. Numerical results referring to some new unpublished geometric shapes are reported to let comparisons with further research on this subject.
    International Journal of Mechanical Sciences 02/2014; 79(3):216-251. · 2.06 Impact Factor


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