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International Journal of Applied Mechanics

Vol. 2, No. 1 (2010) 207–227

c ? Imperial College Press

DOI: 10.1142/S1758825110000470

A DIFFERENTIAL QUADRATURE FINITE

ELEMENT METHOD

YUFENG XING∗, BO LIU and GUANG LIU

The Solid Mechanics Research Center

Beijing University of Aeronautics and Astronautics

Beijing 100191, China

xingyf@buaa.edu.cn

Received 25 June 2009

Accepted 12 October 2009

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 C0finite 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 C0and C1differential

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 alge-

braic operations of the given weighting coefficient matrices of the differential quadrature

(DQ) rules and Gauss-Lobatto quadrature rules, which greatly simplifies the construc-

tions of higher order finite elements. The inter-element compatibility requirements for

problems with C1continuity 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.

Keywords: Differential quadrature method; finite element method; free vibration;

bending.

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 refine-

ment, 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 con-

vergence 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

∗Corresponding author.

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208Y. Xing, B. Liu & G. Liu

[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 C0and/or C1conti-

nuity 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 C0continuity is pos-

sible and seems to be simple for both p- and hp-versions [Shephard et al., 1997],

but the construction of shape functions satisfying the C1continuity 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 C0and/or C1continuity requirements

in present study.

Analytical calculation of derivatives in the h-version is possible and usually

straightforward; 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 devel-

oped 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 differ-

ence analogue of derivatives on the solution domain boundary and on an irregular

domain. Although the isoparametric mapping technique of the FEM was incorpo-

rated 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) gradu-

ally 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

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A Differential Quadrature Finite Element Method209

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 accurate in acquiring high approximation accuracy as compared to the conven-

tional 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 dimen-

sional 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 condi-

tions 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 implemen-

tations 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 C1problems, and the C1continuity requirements are

accomplished through modifying the nodal parameters using DQ rules, the nodal

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210Y. Xing, B. Liu & G. Liu

shapes functions as in standard FEM are not necessary. (3) The DQFE element

matrices are symmetric, well conditioned, and computed efficiently by simple alge-

braic 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 C1high order elements; the 1-D to 3-D DQFE stiffness and mass matrices and

load vectors for C0and C1problems are given explicitly, which are significant to

static and dynamic applications; it is shown that all C0DQFE 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 imple-

mentation 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 elastic-

ity problems, and the cubic 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.

2. The Reformulated Differential Quadrature Rule

The survey paper [Bert and Malik, 1996] has presented the details of DQM, only the

reformulated DQ rules for curvilinear quadrilateral domain and its implementations

are given below. DQM has been applied to irregular domains with the help of the

natural-to-Cartesian geometric mapping using the serendipity-family interpolation

functions [Bert and Malik, 1996; Xing and Liu, 2009] or the blending functions

which permit exact mapping [Malik and Bert, 2000].

The mapping using serendipity-family interpolation functions is applicable to

arbitrary domain. For an arbitrary quadrilateral domain as shown in Fig. 1, the

geometric mapping has the form

?x(ξ,η) =?Sk(ξ,η)xk

where xk, yk; (k = 1,2,...,Ns) are the coordinates of Ns boundary grid points

in the Cartesian x-y plane, Sk(ξ,η) the serendipity interpolations defined in the

natural ξ-η plane. Since the base function Skhas a unity value at the kth node and

zeros at the remaining (Ns−1) nodes, the domain mapped by Eq. (1) and the given

quadrilateral domain matches exactly at least at the nodal points.

y(ξ,η) =?Sk(ξ,η)yk

−1 ≤ξ, η ≤ 1(1)

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A Differential Quadrature Finite Element Method211

(a) (b)

Fig. 1.

in natural ξ-η plane.

(a) A curvilinear quadrilateral region in Cartesian x-y plane; (b) a square parent domain

Subsequently, we should express the derivatives of a function f(x,y) with respect

to x, y coordinates in terms of its derivatives in ξ-η coordinates. Using the chain

rule of differentiation results in

∂f

∂x=

1

|J|

?∂y

∂η

∂f

∂ξ−∂y

∂ξ

∂f

∂η

?

,

∂f

∂y=

1

|J|

?∂x

∂ξ

∂f

∂η−∂x

∂η

∂f

∂ξ

?

(2)

where the determinant |J| of the Jacobian J = ∂(x,y)/∂(ξ,η) is

|J| =∂x

∂ξ

∂y

∂η−∂y

∂ξ

∂x

∂η

(3)

Then the partial derivatives ∂f/∂x and ∂f/∂y at gird point xij = x(ξi,ηj),yij =

y(ξi,ηj) in the mapped curvilinear quadrilateral domain can be computed using DQ

rules, as

?∂f

?∂f

∂x

?

?

ij

=

1

|J|ij

??∂y

??∂x

∂η

?

?

ij

?

?N

n=1

M

?

?

m=1

A(1)

imfmj

?

?

−

?∂y

?∂x

∂ξ

?

?

ij

?N

n=1

?

m=1

?

M

?

B(1)

jnfin

??

??

(4)

∂y

ij

=

1

|J|ij

∂ξ

ij

B(1)

jnfin

−

∂η

ij

A(1)

imfmj

(5)

where M and N are the numbers of grid points in x (or ξ)-direction and y-(or η)

direction, respectively. A(r)

the rth-order and sth-order partial derivative of f with respect to ξ and η at the

discrete point ξiand ηj, respectively. Equations (4) and (5) define the DQ rules of

the first order partial derivatives with respect to the Cartesian x, y coordinates for

irregular domain. Certainly, these rules can also be written in a compact form using

ijand B(s)

ijare the weighting coefficients associated with