A differential quadrature finite element method

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, which 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.

1. Introduction

The finite element method (FEM) is a powerful tool for the numerical solution of a wide range of engineering problems. In conventional FEM, the low order schemes are generally used and the accuracy is improved through mesh refinement, this approach is viewed as the h-version FEM. The p-version FEM employs a fixed mesh and convergence is sought by increasing the degrees of element. The hybrid h-p version FEM effectively marries the previous two concepts, whose convergence is sought by simultaneously refining the mesh and increasing the element degrees [Bardell, 1996]. The theory and computational advantages of adaptive p- and hp-versions for solving problems of mathematical physics have been well documented [Babuska et al., 1981; Oden and Demkowicz, 1991; Shephard et al., 1997]. Many studies have focused on the development of optimal p- and hp-adaptive strategies and their efficient implementations [Campion et al., 1996; Demkowicz et al., 1989; Zhong and He, 1998]. Issues associated with element-matrix construction can be summarized as(1) Efficient construction of the shape functions satisfying the C0 and/or C1 continuity requirements.(2) Efficient and effective evaluations of element matrices and vectors.(3) Accounting for geometric approximations of elements that often cover large portions of the domain. 

The efficient construction of shape functions satisfying the C0 continuity is possible and seems to be simple for both p- and hp-versions [Shephard et al., 1997], but the construction of shape functions satisfying the C1 continuity is difficult for displacement-based finite element formulation [Duan et al., 1999; Rong and Lu, 2003]. The geometry mapping for the p- and hp-version can be achieved through both the serendipity family interpolations and the blending function method [Campion and Jarvis, 1996], thus we focus on the first two issues for efficiently constructing FEM formulation satisfying the C0 and/or C1 continuity requirements in present study.

 Analytical calculation of derivatives in the h-version is possible and usually traightforward; nevertheless, explicit differentiation is extremely complicated or even impossible in the p- and hp-versions. As a result, numerical differentiation has to be used, but which increases the computational cost [Campion and Jarvis, 1996]. An alternative method of deriving the FEM matrices is to combine the finite difference analogue of derivatives with numerical integral methods to discretize the energy functional. This idea was originated by Houbolt [1958], and further developed by Griffin and Varga [1963], Bushnell [1973], and Brush and Almroth [1975]. As the approach is based on the minimum potential energy principle, it was called the finite difference energy method (FDEM). Bushnell [1973] reported that FDEM tended to exhibit superior performance normally and required less computational time to form the global matrices than the finite element models. However, during the further applications of the FDEM [Atkatsh et al., 1980; Satyamurthy et al., 1980; Singh and Dey, 1990], it was found that it is difficult to calculate the finite difference analogue of derivatives on the solution domain boundary and on an irregular domain. Although the isoparametric mapping technique of the FEM was incorporated into FDEM to cope with irregular geometry [Barve and Dey, 1990; Fielding et al., 1997], the lack of geometric flexibility of the conventional finite difference approximation holds back the further development of the FDEM. Consequently, it has lain virtually dormant thus far.

During the last three decades, the differential quadrature method (DQM) gradually emerges as an efficient and accurate numerical method, and has made noticeable success over the last two decades [Bellman and Casti, 1971; Bert et al., 1988; Bert and Malik, 1996; Shu, 2000]. The essence of DQM is to approximate the partial derivatives of a field variable at a discrete point by a weighted linear sum of the field variable along the line that passes through that point. Although it is analogous to the finite difference method (FDM), it is more flexible in selection of nodes, and more powerful in acquiring high approximation accuracy as compared to the conventional FDM. The late significant development of the DQM has motivated an interest in the combination of the DQM with a variational formulation. Striz et al. [1995] took an initiative and developed the hybrid quadrature element method (QEM) for two-dimensional plane stress and plate bending problems, and plate free vibration problems [Striz et al., 1997]. The hybrid QEM essentially consists of a collocation method in conjunction with a Galerkin finite element technique to combine the high accuracy of DQM with the generality of FEM. This results in superior accuracy with fewer degrees of freedom than conventional FEM and FDM. However, the hybrid QEM needs shape functions, and has been implemented for rectangular thin plates only.

Chen and New [1999] used the DQ technique to discretize the derivatives of variable functions existing in the integral statements for variational methods, the Galerkin method, and so on, in deriving the finite element formulation, the discretizations of the static 3-D linear elasticity problem and the buckling problem of a plate by using the principle of minimum potential energy were illustrated. This method is named as the differential quadrature finite element method (DQFEM). Later, Haghighi et al. [2008] developed the coupled DQ-FE methods for two dimensional transient heat transfer analysis of functionally graded material. Nevertheless, shape functions are needed in both methods.

Zhong and Yu [2009] presented the weak form QEM for static plane elasticity problems by discretizing the energy functional using the DQ rules and the Gauss-Lobatto integral rules, whereas each sub-domain in the discretization of solution domain was called a quadrature element. This weak form QEM differs fundamentally with that of [Striz et al., 1995; Striz et al., 1997], and the strong form QEM of [Striz et al., 1994; Zhong and He, 1998]. The weak form QEM is similar with the Ritz–Rayleigh method as well as the p-version while it exhibits distinct features of high order approximation and flexible geometric modeling capability.

Xing and Liu [2009] presented a differential quadrature finite element method (DQFEM) which was motivated by the complexity of imposing boundary conditions in DQM and the unsymmetrical element matrices in DQEM, the name is the same as that of [Chen and New, 1999], but the starting points and implementations are different. Compared with [Zhong and Yu, 2009] and [Chen and New, 1999], DQFEM [Xing and Liu, 2009] has the following novelties: (1) DQ rules are reformulated, and in conjunction with the Gauss-Lobatto integral rule are used to discretize the energy functional to derive the finite element formulation of thin plate for both regular and irregular domains. (2) The Lagrange interpolation functions are used as trial functions for C1 problems, and the C1 continuity requirements are accomplished through modifying the nodal parameters using DQ rules, the nodal shapes functions as in standard FEM are not necessary. (3) The DQFE element matrices are symmetric, well conditioned, and computed efficiently by simple algebraic operations of the known weighting coefficient matrices of the reformulated DQ rules and Gauss-Lobatto integral rule.

In this paper, the differential quadrature finite element method is studied systematically, and the following novel works are included: DQFEM is viewed as a general method of formulating finite elements from lower order to higher order, the difficulty of formulating higher order finite elements are alleviated, especially for C1 high order elements; the 1-D to 3-D DQFE stiffness and mass matrices and load vectors for C0 and C1 problems are given explicitly, which are significant to static and dynamic applications; it is shown that all C0 DQFE mass matrices are diagonal, but they are obtained by using non-orthogonal polynomials and different from the conventional diagonal lumped mass matrices; the reformulated DQ rules for curvilinear quadrilateral domain and its implementation are also presented to improve its application; furthermore, the free vibration analyses of 2-D and 3-D plates with continuous and discontinuous boundaries and bending analyses of thin and Mindlin plates with arbitrary shapes are carried out.

The outline of this paper is as follows. The reformulation of DQM and its implementation are presented in Sec. 2. In Sec. 3, the DQFE stiffness and mass matrices and load vectors are given explicitly for rod, beam, plate, 2-D and 3-D elasticity problems, and the third order Euler beam element matrices of DQFEM are compared with that of FEM. In Sec. 4, the numerical results are compared with some available results. Finally the conclusions are outlined.