Differential quadrature time element method for structural dynamics

Acta Mech. Sin. (2012) 28(3):782–792DOI 10.1007/s10409-012-0081-z

RESEARCH PAPER

Differential quadrature time element method for structural dynamics

Yu-Feng Xing · Jing Guo

Received: 27 July 2011 / Revised: 20 January 2012 / Accepted: 27 February 2012©The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2012

Abstract An accurate and efficient differential quadraturetime element method (DQTEM) is proposed for solving ordi-nary differential equations (ODEs), the numerical dissipationand dispersion of DQTEM is much smaller than that of thedirect integration method of single/multi steps. Two methodsof imposing initial conditions are given, which avoids thetediousness when derivative initial conditions are imposed,and the numerical comparisons indicate that the first method,in which the analog equations of initial displacements andvelocities are used to directly replace the differential quadra-ture (DQ) analog equations of ODEs at the first and the lastsampling points, respectively, is much more accurate thanthe second method, in which the DQ analog equations ofinitial conditions are used to directly replace the DQ analogequations of ODEs at the first two sampling points. On thecontrary to the conventional step-by-step direct integrationschemes, the solutions at all sampling points can be obtainedsimultaneously by DQTEM, and generally, one differentialquadrature time element may be enough for the whole timedomain. Extensive numerical comparisons validate the effi-ciency and accuracy of the proposed method.

Keywords Differential quadrature rule · Direct integrationmethod · Time element · Phase error · Artificial damping

1 Introduction

The discretization of the spatial domain of structural dy-namic problems governed by partial differential equations

The project was supported by the National Natural Science Foun-dation of China (11172028, 10772014).

Y.-F. Xing (�) · J. GuoThe Solid Mechanics Research Center,Beihang University, 100191 Beijing, Chinae-mail: [email protected]

results in a system of ordinary differential equations (ODEs)with time as the independent variable. Various methods areavailable for numerical integration of ODEs. Most com-monly used one, based on the finite difference concept, isthe direct integration methods (DIM) [1] which is based onthe following two ideas. First, the dynamic equations aresought at discrete time or sampling points within the intervalof solution. The second is that a variation of displacements,velocities, and accelerations within each time interval Δt isassumed. And the variation form within each time intervaldetermines the accuracy, stability, and efficiency.

The commonly known drawbacks of DIM such as theclassical Runge–Kutta (RK) methods of different stages andorders [2, 3], the Wilson θ method [4, 5], the Newmarkmethod [6, 7] and so on, are numerical dissipation and dis-persion or artificial damping and phase errors, althoughthe symplectic integration methods [8] have the energy-conserving property.

Alternatively, the ODEs can be discretized using timefinite element methods (TFEM), which are based on theHamilton’s variational principles ([9, 10] for example) orHamilton’s law of varying action ([11, 12] for example) orGurtin’s variational principle ([13] for example) and theweighted residual methods ([14, 15] for example). The fea-tures of TFEM include: (1) In principle, desired accuracy ofarbitrary order can be achieved by taking high-order poly-nomials as primary functions; (2) The geometric property ofthe time domain is simple, there is no need to handle compli-cated boundary shapes; (3) It is suitable for nonlinear prob-lems and leads to more feasibilities; (4) The formulations ofTFEM are generally low order.

People generally constructed TFEM based on the varia-tional principles and the weighted residual methods. In a dif-ferent way, the ODEs can also be solved using the differen-tial quadrature (DQ) method [16, 17] wherein the ODEs arediscretized by DQ rule. The differential quadrature method(DQM) [18] is a simple and highly efficient numerical tech-nique for initial-value problems, and could yield accurate re-sults using a considerably small number of sampling points.

Differential quadrature time element method for structural dynamics 783

The literature on DQM up to 1996 may be found in a sur-vey paper [19]. And there is an increasing interest in apply-ing DQM to solve dynamic problems [16, 20–23]. Since thegoverning ODEs for structural dynamic problems are of thesecond order with two initial conditions, the application ofDQM is not as straight-forward as the first order ODEs [22].The method of imposing initial conditions is an importantissue.

The imposition of the initial/boundary conditions inDQM can be very tricky [19]. Several methods have beenused to deal with initial conditions, such as expressingthe initial conditions as differential quadrature (DQ) analogequations at the sampling grid points [16, 24, 25], modify-ing DQ rule [16, 26], modifying the trial functions [27, 28],or modifying the weighting coefficient matrices [21]. Butthe second and the last have been shown to be identical byRefs. [16, 22]. Hitherto, there are problems in feasibility,generality and simplicity in imposing the initial conditions.Especially no paper investigated the accuracy of differentmethods in imposing initial conditions.

The advantage of the Fung’s method [16, 22] is the ac-curacy of state variables at the end of time interval (which isnot a sampling point) can reach the order of 2n using onlyn + 1 sampling points, additionally, unconditional stable andnon-dissipative algorithm can be constructed. However, thedisadvantages of the Fung’s method [16] are also evident,such as the accuracy at the sampling points is of the orderof n; the powers of polynomials of displacement and ve-locity are the same, hence the differential relation betweendisplacement and velocity is violated, the initial velocity isnot exactly satisfied in general; In addition, the variables atthe end of time interval must be interpolated by the knownvariables at sampling points. In a similar way to Ref. [16],the beam problem subjected to a moving load was studied inRef. [17], but the imposing methods of initial conditions andthe analysis of stability were not involved.

In this context, this paper proposes an efficient and ac-curate time element method based on the DQ rule for solvingODEs of structural dynamics, called the differential quadra-ture time element method (DQTEM). The novelties of thepresent study are as follows:

(1) The numerical dissipation and dispersion are causedby lower order solution schemes including DIM and TFEM,and accumulated linearly step by step. And it is well knownthat the high order element is more efficient and accurate instandard FEM, this motivates us to develop high order timeelement method to solve numerical dissipation and disper-sion problems based on DQM.

(2) Give two methods of imposing initial conditions,and investigate their accuracy.

(3) Study the spectrum radius and phase error ofDQTEM.

(4) Compare the cost of DQTEM with the well-knownRK method and the widely used Newmark method.

An outline of the present paper is as follows. In Sect. 2,

the DQ rule is described briefly, the Chebyshev–Gauss–Lobatto sampling points are given and the DQM solution forODEs is derived. Section 3 presents the methods of impos-ing initial conditions, and their advantages compared withthe existent methods are highlighted. In Sect. 4, the stabilityand accuracy of DQTEM are studied. The present results arecompared with analytical solutions and those of RK methodand Newmark method in Sect. 5. Conclusions are drawn inSect. 6.

2 DQM solution of ordinary differential equation

2.1 The DQ rule

For clarity and completeness, DQM is briefly reviewed be-low, whose details can be found in Ref. [19] and a previouspaper [29] of the author.

The DQM idea is that the partial derivatives of a fieldvariable at the k-th discrete point in the computational do-main is approximated by a weighted linear sum of the fieldvariable along the line which passes through the k-th pointand is parallel to the coordinate direction of the derivatives.For example, the m-th derivatives of the field variable x(t) atpoint tk is approximated as

x(m)k =

N∑

j=1

A(m)k j x j, k = 1, 2, · · · ,N, (1)

where A(m)k j are the weighting coefficients associated with the

m-th order derivatives, and N is the number of samplingpoints. Note that Eq. (1) can be understood as the high orderdifference of the derivatives using variables. A key prob-lem to DQ rule is the computation of weighting coefficientswhich can be obtained directly from an explicit formula [30].The coefficients for the first order derivatives are given by

A(1)k j = l(1)

j (tk)

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

N∏

i=1,i�k, j

(tk − ti)

/ N∏

i=1,i� j

(t j − ti), k � j,

N∑

i=1,i�k

1(tk − ti)

, k = j,

(2)

where l j is the Lagrange polynomials. The weighting coef-ficients of higher order derivatives can be obtained using therecurrence relationship as

A(m)k j =

N∑

i=1

A(1)ki A(m−1)

i j . (3)

A crucial factor for improving the accuracy of theDQ solutions is how to space the sampling points. It hasbeen well shown [31] that the so-called Chebyshev–Gauss–Lobatto points, first used by Shu and Richards [32] and used

784 Y.-F. Xing, J. Guo

widely since then, are more accurate than the equally spaced,Legendre, and Chebyshev points in a variety of problems.The time sampling points are given by

tk = t0 + h(t − t0). (4)

For the Chebyshev–Gauss–Lobatto sampling points, h is

h =12

(1 − cos

k − 1N − 1

π). (5)

2.2 The DQM solution

In the present study we employ DQM to discretize the struc-tural dynamics ODEs as

M x + C x + Kx = f (t), (6)

where M = [Mi j]n×n, C = [Ci j]n×n and K = [Ki j]n×n are themass, damping and stiffness matrices, respectively; n is thedimension or the degree of freedom (DOF) of the system.The superposed dot denotes differentiation with respect totime. x(t) and f (t) are the displacement and the prescribedload vector, respectively, and have the forms

xT = [x1(t) x2(t) · · · xn(t)],

f T = [ f1(t) f2(t) · · · fn(t)].(7)

Equation (6) can be obtained through spatial discretizationmethod as finite element methods, finite difference methodand so on. Since each DOF is a function only of time, there-fore its derivative based on DQ rule can be given as

xi|t=tk = xik =

N∑

j=1

A(1)k j xi j, (8)

where xi is an element of displacement vector x(t), and xi j

is the value of xi at time sampling point t j. For the sake ofconvenience, let us define

xTi = [xi1 xi2 · · · xiN], (9)

f Ti = [ fi1 fi2 · · · fiN]. (10)

Then the velocity vector x i and acceleration vector x i can begiven, respectively, by using DQ rule, as

x i = Ax i, (11)

x i = Bx i, (12)

where the weighting coefficients matrices of order N ×N are

A = A (1), (13)

B = A (2). (14)

It is noteworthy that the weighting coefficients are dependentonly on the coordinates of time sampling points. Substitutionof Eqs. (11) and (12) into Eq. (6) leads to

MXBT + CX AT + KX = F , (15)

where

X T = [x1 x2 · · · xn],

FT = [ f 1 f 2 · · · f n],(16a)

BX T = [Bn1 Bn2 · · · Bnn],

AX T = [An1 An2 · · · Ann].(16b)

When there is no damping, Eq. (15) can be transformedto Sylvester equation whose solution can be found by themethod of Bartels and Stewart [33]. According to the theoryof matrix analysis, Eq. (15) can be transformed into anothereasy to solve form as

GZ = Q , (17)

where Z = cs(X ) and Q = cs(F) are the column expansionsof matrices X and F , respectively, that is

Z T = [x11 · · · xn1 x12 · · · xn2 · · · x1N · · · xnN],

QT = [ f11 · · · fn1 f12 · · · fn2 · · · f1N · · · fnN].

Moreover, the coefficients matrix of Eq. (17) is given by

G = B⊗M + A⊗C + E⊗K , (18)

where E is a N × N unit matrix, and ⊗ represents the Kro-necker product. Another form in terms of sub-matrix ofEq. (18) is

G jm = B jm M + A jmC + E jmK , (19)

where G jm of order n × n are the sub-matrices of G , j,m =1, 2, · · · ,N. Correspondingly, Q j and Z j of order n × 1 arethe sub-matrices of Q and Z , respectively, and

QTj = [ f1 j f2 j · · · fn j], (20)

Z Tj = [x1 j x2 j · · · xn j]. (21)

The above solution method is called the differential quadra-ture time element method (DQTEM).

Remarks:

(1) Equation (17) includes the dynamic equilibrium equa-tions DQ analog equations of at any time samplingpoints.

(2) The requests for highly accurate and efficient solutionmethods may be more necessary for the dynamics prob-lems without damping, and in these cases, Eq. (15) canbe directly solved by the method of Bartels and Stew-art [33].

(3) The state variables at all sampling points are solved si-multaneously in DQTEM, which results in a great de-crease, even to the extent of negligibility, in the accu-mulation of numerical dissipation and phase errors if thesampling points are more enough, not like the lower or-der step-by-step DIM.

(4) For nonlinear dynamic system with less number of DOF,it is convenient to solve Eq. (17) directly. Using DQTEMin conjunction with iteration method.


(5) When numbers of DOF and the sampling points arelarge, the storage of matrix G may become a challengeto the capacity of computer, unless there is a skillful so-lution strategy.

3 The methods of imposing initial conditions

When the dynamic problems are solved, the imposition ofinitial conditions has the same importance as the imposi-tion of boundary conditions in the analysis of structural staticproblems and free vibrations. When DIM and mode super-position method are used to solve ODEs, the imposition ofinitial conditions is not complicated, but it becomes trickywhen DQM is used.

The methods of imposing initial conditions include ex-pressing the initial conditions as DQ analog equations at thesampling grid points [16, 24, 25], modifying the trial func-tions [27, 28], modifying DQ rule [16] and modifying theweighting coefficient matrices [21] as briefly reviewed inSect. 1. Our methods given below are based on but differ-ent from the method in Refs. [16, 24, 25], and convenient forpractical use compared with those available methods. In thefollowing, we first present two practical and simple methodsfor imposing initial conditions in DQTEM, then give a fewremarks.

The initial conditions corresponding to the first time sam-pling point (usually t = 0) are as follows

x(t)|t=0 = x0,

x(t)|t=0 = x0,(22)

where x0 and x0 are the initial displacement and velocityvectors, respectively. Using DQ rule, Eq. (22) can be rewrit-ten as

⎡⎢⎢⎢⎢⎢⎣1 0 0 · · · 0 0

A11 A12 A13 · · · A1(N−1) A1N

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

xi1

xi2

...

xi(N−1)

xiN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎣xi0

xi0

⎤⎥⎥⎥⎥⎥⎦ , i = 1, 2, · · · , n, (23)

which are the DQ analog equations of the initial conditions.

3.1 The first method of imposing initial conditions

In this method, the first n equations of G , corresponding tothe first sampling point, are replaced by the DQ analog equa-tions of initial displacement conditions in Eq. (23), the lastn equations of G , corresponding to the last or terminal sam-pling point, are replaced by the DQ analog equations of ini-

tial velocity conditions in Eq. (23), that is

G11 = I , G1m = 0, Q1 = x0, m = 2, 3, · · · ,N, (24)

GNm = A1mI , QN = x0, m = 1, 2, · · · ,N, (25)

where I is a n×n unit matrix. The DQTEM with this methodof imposing initial conditions is denoted by DQTEM1.

3.2 The second method of imposing initial conditions

Alternatively, the first n equations of G , corresponding tothe first sampling point, are also replaced by the DQ analogequations of initial displacement conditions, but the second nequations of G , corresponding to the second sampling point,are replaced by the DQ analog equations of initial velocityconditions, as

G11 = I , G1m = 0, Q1 = x0, m = 2, 3, · · · ,N, (26)

G2m = A1mI , Q2 = x0, m = 1, 2, · · · ,N. (27)

The DQTEM with this method of imposing initial conditionsis denoted by DQTEM2.

In order to demonstrate the differences between thepresent method and the method used in Refs. [16, 24, 25],consider a single DOF system of

x + 2ξωx + ω2x = f . (28)

Using DQTEM to solve Eq. (28) with initial condition (22),Eq. (28) transform to the forms of⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 · · · 0

G21 G22 · · · G2N

......

. . ....

G(N−1)1 G(N−1)2 · · · G(N−1)N

AN1 AN2 · · · ANN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x1

x2

...

xN−1

xN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x0

f2

...

fN−1

x0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

for DQTEM1 , (29)

and⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 · · · 0

AN1 AN2 · · · ANN

G31 G32 · · · G3N

......

. . ....

GN1 GN2 · · · GNN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x1

x2

...

xN−1

xN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x0

x0

f3

...

fN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

for DQTEM2. (30)

When using the method in Refs. [16, 24, 25] to solve Eq. (28)with initial condition (22), one has to express x1 and xN byother variables from Eq. (23) as⎡⎢⎢⎢⎢⎢⎣

x1

xN

⎤⎥⎥⎥⎥⎥⎦ =⎡⎢⎢⎢⎢⎢⎣

1 0

A11 A1N

⎤⎥⎥⎥⎥⎥⎦−1 ⎧⎪⎪⎨⎪⎪⎩

⎡⎢⎢⎢⎢⎢⎣x0

x0

⎤⎥⎥⎥⎥⎥⎦ −⎡⎢⎢⎢⎢⎢⎣

0 · · · 0

A12 · · · A1(N−1)

⎤⎥⎥⎥⎥⎥⎦


×

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x2

...

xN−1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭. (31)

Then discarding the DQ analog equations of governing dif-ferential equation (28) at the first and the last samplingpoints, one has

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

B21 + 2ξωA21 B22 + 2ξωA22 · · · B2N + 2ξωA2N

B31 + 2ξωA31 B32 + 2ξωA32 · · · B3N + 2ξωA3N

......

. . ....

B(N−1)1 + 2ξωA(N−1)1 B(N−1)2 + 2ξωA(N−1)2 · · · B(N−1)N + 2ξωA(N−1)N

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x1

x2

...

xN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+ ω2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x2

x3

...

xN−1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

f2

f3

...

fN−1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (32)

Substituting Eq. (31) into Eq. (32), the variables x2, x3, · · · ,xN−1 can be solved, and xN is obtained from Eq. (31) finally.

Remarks:

(1) When the derivatives at sampling points are approxi-mated using DQ rule, the accuracy near the two end sam-pling points is lower than the accuracy at interior points,so DQTEM1 is more accurate than DQTEM2, which isvalidated by numerical results in Sect. 5.

(2) It can be seen that the method in Refs. [16, 24, 25] istedious in the imposition of initial conditions comparedwith the present method.

4 Stability and accuracy

Stability and accuracy should be taken into account when in-vestigating the effectiveness of an iterative or step-by-stepsolution method [1, 34]. Here the spectral approach is em-ployed as usual to examine the stability and accuracy ofDQTEM1 and DQTEM2 for the second-order ODEs.

4.1 Stability analysis

In stability analysis, it is convenient to work with the follow-ing single DOF system

x + (ωh)2x = 0, (33)

where ω denotes the angular frequency. A superposed dotdenotes differentiation with respect to dimensionless time τ,where τ = t/h, h and t represent the time element (or timestep size) and the real time, respectively. Since Eq. (33) is asingle DOF system, then x is given by

xT = [x1 x2 · · · xN], (34)

where xi (i = 1, 2, · · · ,N) denote the values of the displace-ments at time sampling points, and

G = B + (ωh)2E , (35)

Z = x , Q = 0. (36)

The stability analysis of DQTEMs is the same, hence onlyDQTEM1 is dealt with, see Eq. (29), then Eq. (33) is dis-cretized to be⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 · · · 0

G21 G22 G23 · · · G2N

......

.... . .

...

G(N−1)1 G(N−1)2 G(N−1)3 · · · G(N−1)N

A11 A12 A13 · · · A1N

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

×

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x1

x2

...

xN−1

xN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x0

0

...

0

x0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (37)

from which one can solve x as⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x1

x2

...

xN−1

xN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

S 11 S 12 · · · S 1(N−1) S 1N

S 21 S 22 · · · S 2(N−1) S 2N

......

. . ....

...

S (N−1)1 S (N−1)2 · · · S (N−1)(N−1) S (N−1)N

S N1 S N2 · · · S N(N−1) S NN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

×

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x0

0

...

0

x0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (38)

where S = G−1. Based on DQ rule in Eq. (11), the velocityof the N-th sampling point can be expressed as

xN = αx0 + βx0, (39)

where


α = AN1S 11 + AN2S 21 + · · · + ANNS N1, (40a)

β = AN1S 1N + AN2S 2N + · · · + ANNS NN . (40b)

From Eqs. (33) and (34), one can find the relations between(xN , xN) and (x0, x0) as⎡⎢⎢⎢⎢⎢⎣

xN

xN

⎤⎥⎥⎥⎥⎥⎦ = D

⎡⎢⎢⎢⎢⎢⎣x0

x0

⎤⎥⎥⎥⎥⎥⎦ , (41)

where the Jacobi matrix D has the form of

D =

⎡⎢⎢⎢⎢⎢⎣S N1 S NN

α β

⎤⎥⎥⎥⎥⎥⎦ . (42)

In fact we have the following expressions⎡⎢⎢⎢⎢⎢⎣

xN

xN

⎤⎥⎥⎥⎥⎥⎦ = DN

⎡⎢⎢⎢⎢⎢⎣xN−1

xN−1

⎤⎥⎥⎥⎥⎥⎦ ,⎡⎢⎢⎢⎢⎢⎣

xN−1

xN−1

⎤⎥⎥⎥⎥⎥⎦ = DN−1

⎡⎢⎢⎢⎢⎢⎣xN−2

xN−2

⎤⎥⎥⎥⎥⎥⎦ ,

· · · ,⎡⎢⎢⎢⎢⎢⎣

x1

x1

⎤⎥⎥⎥⎥⎥⎦ = D1

⎡⎢⎢⎢⎢⎢⎣x0

x0

⎤⎥⎥⎥⎥⎥⎦ .(43)

And the following relation can be readily verified

D = DN DN−1 · · · D1. (44)

But x1, x2, · · · , xN are simultaneously solved from Eq. (38)using (x0, x0), this is different from the classical single step-by-step method as Newmark method or RK method in whichthe results of a time step are the initial conditions of the nexttime step.

DQTEM would be stable if spectral radius ρ(D) ≤ 1.The curves of ρ(D) with respect to the number N of the sam-ple points in one time element are plotted in Figs. 1 and 2 forDQTEM1 and DQTEM2, respectively. It follows that ρ(D)of DQTEM1 is larger than or equal 1 for different value of N,and ρ(D) of DQTEM2 can be smaller than, equal or largerthan 1. It is apparent from Fig. 1 that when N became larger,the upper limit ωh of the first stable interval is larger, andthere are several intervals of ωh with ρ(D) = 1 except forN = 3, and similarly from Fig. 2 that when N became larger,the upper limitωh of the first stable interval is also larger, and

the cases for N = 3 and N = 4 are unconditionally stable, seeTables 1 and 2 whose elements are obtained using time stepsize h = 0.01 and (ρ−1) < 10−6, and from which one can seethat the usage of Chebyshev–Gauss–Lobatto points can in-crease the accuracy and efficiency. The elements of columnswith head “No.” in Tables 1 and 2 represent the number ofstable intervals with respect to different sampling point N.

Fig. 1 Spectral radii for different values of N for DQTEM1

Fig. 2 Spectral radii for different values of N for DQTEM2

Table 1 The stable intervals of ωh with ρ(D) = 1 for DQTEM1

N Interval 1 Interval 2 Interval 3 Interval 4 No. Stability

3 0.00 < ωh ≤ 2.82 - - - 1 Conditionally

4 0.00 < ωh ≤ 2.95 3.27 ≤ ωh ≤ 5.65 - - 2 Conditionally

5 0.00 < ωh ≤ 3.09 3.14 ≤ ωh ≤ 5.65 6.93 ≤ ωh ≤ 8.84 - 3 Conditionally

10 0.00 < ωh ≤ 9.41 9.43 ≤ ωh ≤ 12.53 12.69 ≤ ωh ≤ 15.65 16.16 ≤ ωh ≤ 17.81 6 Conditionally





Table 2 The stable intervals of ωh with ρ(D) ≤ 1 for DQTEM2

N Interval 1 Interval 2 Interval 3 Interval 4 No. Stability

3 - - - - - Unconditionally

4 - - - - - Unconditionally

5 0.00 < ωh ≤ 0.11 ωh ≥ 3.13 - - 2 Conditionally





Although there are several stable intervals forDQTEM1 and DQTEM2 when N > 4, only the first inter-val is of practice for multi DOF dynamic system.

4.2 Accuracy analysis

Accuracy refers to the difference between the numerical so-lution and the exact solution when the numerical solutionprocess is stable. If the eigenvalues of D remain complex,as

λ1,2 = a ± ib =√

a2 + b2eiθ, (45)

θ = arctan(b/a), (46)

where i =√−1 and b � 0, θ is the phase of DQTEM for

one time step or element. The single-step phase error is ameasure of numerical dispersion, and usually defined as

Δθ = ωh − θ = ωh − arctan(b/a). (47)

The time step h of DQTEM is usually much larger than thatof the commonly used DIM as RK method, especially whenN is large. When ωh is larger than π, the single-step phaseerror of DQTEM can not be directly calculated by Eq. (47),and should be calculated by

Δθ = mod(ωh/π)π − |arctan(b/a)| ,0 < mod(ωh/π) ≤ 1/2,

(48)

Δθ = mod(ωh/π)π − (π − |arctan(b/a)|),mod(ωh/π) > 1/2,

(49)

where mod(ωh/π) represents the remainder of ωh/π. Notethat Eqs. (48) and (49) are suitable for all intervals in Ta-bles 1 and 2, and Δθ are virtually negligible when N is large,see Table 3 which shows that for each N, all upper limitsωh of the first interval of DQTEM1 are larger than thoseof DQTEM2. If one uses the same ωh in both schemes,the phase errors of DQTEM1 are much smaller than that ofDQTEM2, and its reason has been given in Sect. 3.

The above discussion shows that DQTEM is an effec-tive and accurate algorithm, the larger the time element, the

more superior the DQTEM with Chebyshev–Gauss–Lobattosampling points.

5 Numerical analysis

This section aims at demonstrating the high accuracy and ef-ficiency of DQTEM.

5.1 Example 1

Consider the free vibration of a two DOF system for whichthe governing equilibrium equations are⎡⎢⎢⎢⎢⎢⎣

2 0

0 1

⎤⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎣

x1

x2

⎤⎥⎥⎥⎥⎥⎦ +⎡⎢⎢⎢⎢⎢⎣

6 −2

−2 4

⎤⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎣

x1

x2

⎤⎥⎥⎥⎥⎥⎦ =⎡⎢⎢⎢⎢⎢⎣

0

0

⎤⎥⎥⎥⎥⎥⎦ . (50)

The initial conditions are xT0 = [0 1], xT

0 = [1 0], theangular frequencies are ω1 =

√2 and ω2 =

√5. The relative

error used below is defined as

RelErr = abs(exact − present)/‖exact‖∞, (51)

5.1.1 Accuracy comparison of equally and unequally spacedsampling points

There are two typical methods to space sampling pointswhen DQM is used as shown above, one is the equallyspaced points, and the other is the Chebyshev–Gauss–Lobatto points. Let N = 41 (the first stable interval is0 < ωh ≤ 65.96 for DQTEM1), time element size h is 20 orthe time interval is 0–20. It can be seen from Figs. 3 and 4that the DQTEM1 with Chebyshev–Gauss–Lobatto points ismuch more accurate than that with the equally spaced points.

5.1.2 Accuracy comparison of two methods of imposing ini-tial conditions

Assume N = 100, consider time domain [0, 50]. It followsfrom Figs. 5 and 6 that DQTEM1 is much more accuratethan DQTEM2, their RelErrs are about 10−14 and 10−10, re-spectively, for this case.


Table 3 The comparison of ωh and single-step phase error of DQTEM1 and DQTEM2

NDQTEM1 DQTEM2

ωh1 h1/T Phase error for ωh1 Phase error for ωh2 ωh2 h2/T Phase error

5 3.09 0.49 −3.165 9×10−2 −2.096 0×10−9 0.11 0.02 −4.775 6×10−8

10 9.41 1.50 −4.402 1×10−3 7.053 9×10−9 2.02 0.32 5.245 0×10−7

15 15.70 2.50 1.275 3×10−3 1.677 4×10−9 5.87 0.93 5.644 6×10−8

20 25.12 4.00 −2.471 1×10−3 1.007 9×10−5 18.05 2.87 7.040 7×10−3

25 34.55 5.50 2.321 2×10−3 1.728 2×10−7 21.23 3.38 1.790 3×10−4

30 43.97 7.00 −3.316 6×10−3 4.371 8×10−9 24.73 3.94 5.423 0×10−6

35 53.40 8.50 2.335 7×10−3 6.810 7×10−10 30.01 4.78 3.059 7×10−7

40 62.82 10.00 −3.284 9×10−3 3.534 4×10−8 42.93 6.83 1.446 3×10−4

45 72.25 11.50 2.077 5×10−3 1.541 8×10−9 46.45 7.39 5.632 6×10−6

50 78.53 12.50 1.234 5×10−3 3.400 1×10−10 51.91 8.26 2.365 3×10−7

55 91.10 14.50 1.798 7×10−3 4.186 7×10−9 64.81 10.31 4.304 0×10−5

60 97.38 15.50 1.302 6×10−3 4.269 7×10−10 68.72 10.94 2.427 5×10−6

65 109.95 17.50 1.557 9×10−3 1.070 4×10−10 75.97 12.09 −1.223 9×10−6

70 116.23 18.50 1.292 7×10−3 5.698 5×10−10 86.87 13.83 9.603 0×10−6

75 128.80 20.50 1.357 8×10−3 1.721 7×10−10 91.77 14.61 6.834 8×10−7

80 135.08 21.50 1.236 0×10−3 6.876 3×10−10 105.47 16.79 3.035 5×10−5

85 153.91 24.50 −1.274 3×10−2 1.616 2×10−10 109.45 17.42 2.183 9×10−6

90 153.93 24.50 1.153 6×10−3 5.566 1×10−10 124.20 19.77 7.600 8×10−5

95 169.64 27.00 6.228 8×10−3 1.440 8×10−10 127.74 20.33 5.208 9×10−6

100 179.05 28.50 −1.023 5×10−2 8.570 6×10−11 133.24 21.21 2.135 0×10−7

Fig. 3 The displacement x1 and its RelErr for equally spaced points

Fig. 4 The displacement x1 and its RelErr for Chebyshev–Gauss–Lobbatto points


Fig. 5 The RelErr of x1 for DQTEM1 with the Chebyshev–Gauss–Lobbatto points

Fig. 6 The RelErr of x1 for DQTEM2 with the Chebyshev–Gauss–Lobbatto points

5.1.3 Efficiency comparison of RK method and DQTEM

Computational cost is a crucial issue for an algorithm. Letthe time domain be [0, 100]. In order to arrive at the sameRelErr of 10−10, DQTEM1 and DQTEM2 need at least 148and 162 sampling points, respectively. But h = 0.002 5 smust be used in RK method, that is, the number of time stepsequals to 40 000. Note that the RK method used here is theclassical one of stage four and order four with equal timestep. Table 4 gives the CPU comparison used in calculation.

5.2 Example 2

The free vibration of a fixed-free uniform rod is analyzedbelow. The Yang’s module E = 125 GPa, density ρ =8 980 kg/m3, diameter d = 0.1 m and length l = 1 m. The rodis meshed into 20 uniform constant-strain spatial elements.The initial velocity of the free-end is 1 m/s, all others arezero, so the problem is an analogous counterpart of impactproblem.

Consider time domain [0, 3×10−3], which is separatedinto 10 equal DQTEM time elements with 50 Chebyshev–Gauss–Lobbatto sample points in each element, so the sizeof DQTEM element is 3×10−4. Note that the maximum fre-quency of the discretized rod isω20 = 2.578 9×105, based onwhich one can see that 3 × 10−4ω20 is within the first stableinterval (0.00 < ωh ≤ 78.53), see Table 1.

Since DQTEM1 is more accurate than DQTEM2, thusonly the solution of DQTEM1 for this dynamic problem iscompared with the exact solution obtained by the mode su-perposition method. DQTEM1 is also compared with the av-erage acceleration scheme (a member of Newmark algorithmfamily) since it is one of the most commonly used direct in-tegration method, this comparison can validate the value ofthe present method for engineering problem.

Table 4 The CPU time of three methods

RelErr/10−10 RelErr/10−4

h N CPU/s h N CPU/s

DQTEM1 - 148 0.030 2 - 127 0.018 7

DQTEM2 - 162 0.037 5 - 141 0.027 9

RK 0.002 5 100/h = 4 × 104 7.597 5 0.05 100/h = 2 000 0.287 3

Figure 7 shows an excellent agreement of the free-enddisplacement of DQTEM1 with the exact solution, but hereonly 10 time elements are used, if DQTEM1 is taken as astep-by-step direct integration method, a time element is atime step. In order to reach the same accuracy, Newmarkmethod needs 180 000 time steps and much more time thanDQTEM1, see Table 5. Figures 8 and 9 present the RelErr ofDQTEM1 and the RelErr of the Newmark method, it followsthat the RelErr of Newmark method increases with the in-

crease of number of time steps, while the RelErr of DQTEMnot.

It is obvious that DQTEM requires less computationalcost and are more effective than RK and Newmark methods,and some remarks are given as follows:

(1) When a small number of sampling points are used,that is to say, the time step size is larger, the solutions ofsuch conventional schemes as RK method would encountera sharp drop of accuracy for the long-term responses due to


Fig. 7 The displacement of free end of the rod

Fig. 8 The RelErr of xfree−end for DQTEM1 with the Chebyshev–Gauss–Lobbatto points

Fig. 9 The RelErr of xfree−end for Newmark method

Table 5 The CPU time of two methods (RelErr 10−5)

h Time steps CPU/s

DQTEM1 - 10 (50 points of an element) 0.226 0

Newmark 1.666 7×10−8 180 000 158.983 7

the artificial damping and the phase error accumulation whileDQTEM can hold high accuracy and efficiency. In otherwords, the artificial damping and the phase error accumu-lation of DQTEM are significantly smaller compared withthe conventional step-by-step methods.

(2) For the whole time domain interested, one DQ timeelement is usually enough if N is large enough, and the preci-sion can be significantly improved by increasing a few num-ber of the sample points.

(3) In the conventional step-by-step schemes, the re-sults of the present time step are the initial conditions of thenext time step, so all the initial conditions are approximateexcept for the first time step; and the DQTEM solutions forall time sampling points are calculated simultaneously usingthe same exact initial conditions when only one time elementis employed. Nevertheless, DQTEM can also serve as an ac-curate step-by-step method, as shown in Sect. 5.

(4) One can find from the above numerical results thatDQTEM can achieve more accurate solutions than otherstep-by-step schemes by using a considerably smaller num-ber of sample points. In some sense, DQTEM solutions canbe regarded as benchmarks to validate other lower order in-tegration methods.

(5) The same problem can be solved using the “ode45”(a RK solver) in MATLAB, by which one can obtain the re-sults with only a relative difference error of 10−3 at most.

6 Conclusions

In this paper, the ideas, advantages and disadvantages of thedirect integration methods and the time finite element meth-ods were reviewed. It was concluded that for structural dy-namic ODEs, the numerical simulation would not be satis-factory for somewhat long time history by using the availabledirect integration method of single/multi steps and TFEMbecause of their accumulated artificial damping and phaseerror.

A time element method called DQTEM was presentedfor solving structural dynamic ODEs on the basis of DQ rule,extensive numerical results shown that DQTEM possessesadvantages over TFEM and the direct integration methods ofsingle/multi steps which have been widely used nowadays.Particularly, DQTEM with the Chebyshev–Gauss–Lobbattosampling points and using the first method of imposing ini-tial conditions has much smaller artificial damping and phaseerror, implying that this method is practical and exhibits abetter capability of long-term numerical simulation.

References

1 Bathe, K.J., Wilson, E.L.: Numerical Methods in Finite Ele-ment Analysis. Prentice-Hall, Englewood Cliffs, New Jersey


(1976)2 Runge, C.: Uber die numerische Auflosung von Differential-

gleichungen. Mathematische Annalen 46, 167–178 (1895)3 Kutta, W.: Beitrag zur naherungsweisen Integration totaler Dif-

ferentialgleichungen. Z. Math. Phys 46, 435–453 (1901)4 Wilson, E.L., Farhoomand, I., Bathe, K.J.: Nonlinear dynamic

analysis of complex structures. Earthquake Engineering andStructural Dynamics 1, 242–252 (1973)

5 Bathe, K.J., Wilson, E.L.: Stability and accuracy analysis ofdirect integration methods. Earthquake Engineering and Struc-tural Dynamics 1, 283–291 (1973)

6 Newmark, N.M.: A method of computation for structural dy-namics. ASCE Journal of the Engineering Mechanics Divisions85, 67–94 (1959)

7 Wood, W.L., Bossak, M., Zienkiewicz, O.C.: An alpha modifi-cation of Newmark’s method. International Journal for Numer-ical Methods in Engineering 15, 1562–1566 (1980)

8 Feng, K.: Difference scheme for Hamiltonian formalism andsymplectic geometry. Journal of Computational Mathematics4, 279–289 (1986)

9 Fried, I.: Finite-element analysis of time-dependent phenom-ena. AIAA Journal 7, 1170–1173 (1969)

10 Argyris, J.H., Scharpf, D.W.: Finite elements in time and space.Nuclear Engineering and Design 10, 456–464 (1969)

11 Riff, R., Baruch, M.: Time finite element discretization ofHamilton’s law of varying action. AIAA Journal 22, 1310–1318 (1984)

12 Simkins, T.E.: Finite elements for initial value problems in Dy-namics. AIAA Journal 10, 1357–1362 (1981)

13 Wilson, E.L., Nickell, R.E.: Application of the finite elementmethod to heat conduction analysis. Nuclear Engineering andDesign 4, 276–286 (1966)

14 Zienkiewicz, O.C.: A new look at the Newmark, Houbolt andother time stepping formulas. A weighted residual approach.Earthquake Engineering and Structural Dynamics 5, 413–418(1977)

15 Zienkiewicz, O.C., Wood, W.L., Taylor, R.L.: An alterna-tive single-step algorithm for dynamic problems. InternationalJournal for Numerical Methods in Engineering 8, 31–40 (1980)

16 Fung, T.C.: Solving initial value problems by differentialquadrature method—Part 2: second-and higher-order equa-tions. International Journal for Numerical Methods in Engi-neering 50, 1429–1454 (2001)

17 Eftekhari, S.A., Khani, M.: A coupled finite element-differ-ential quadrature element method and its accuracy for movingload problem. Applied Mathematical Modeling 34, 228–237(2010)

18 Bellman, R., Casti, J.: Differential quadrature and long term in-tegration. Journal of Mathematical Analysis and Applications34, 235–238 (1971)

19 Bert, C.W., Malik, M.: Differential quadrature method in com-putational mechanics: a review. Applied Mechanics Reviews

49, 1–28 (1996)20 Zong, Z.: A variable order approach to improve differential

quadrature accuracy in dynamic analysis. Journal of Sound andVibration 266, 307–323 (2003)

21 Tanaka, M., Chen, W.: Dual reciprocity BEM applied totransient elastodynamic problems with differential quadraturemethod in time. Computer Methods in Applied Mechanics andEngineering 190, 2331–2347 (2001)

22 Fung, T.C.: Stability and accuracy of differential quadraturemethod in solving dynamic problems. Computer Methods inApplied Mechanics and Engineering 191, 1311–1331 (2002)

23 Shu, C., Kha, W.: Numerical simulation of natural convectionin a square cavity by SIMPLE-generalized differential quadra-ture method. Computers and Fluids 31, 209–226 (2002)

24 Bert, C.W., Wang, X., Striz, A.G.: Differential quadrature forstatic and free vibration analyses of anisotropic plates. Interna-tional Journal of Solids and Structures 30, 1737–1744 (1993)

25 Shu, C., Du, H.: A generalized approach for implementing gen-eral boundary conditions in the GDQ free vibration analysis ofplates. International Journal of Solids and Structures 34, 837–846 (1997)

26 Fung, T.C.: Solving initial value problems by differentialquadrature method-Part 1: first-order equations. InternationalJournal for Numerical Methods in Engineering 50, 1411–1427(2001)

27 Wu, T.Y., Liu, G.R.: A differential quadrature as a numericalmethod to solve differential equations. Computational Mechan-ics 24, 197–205 (1999)

28 Wu, T.Y., Liu, G.R.: The generalized differential quadraturerule for initial-value differential equations. Journal of Soundand Vibration 233, 195–213 (2000)

29 Xing, Y.F., Liu, B.: High-accuracy differential quadrature fi-nite element method and its application to free vibrations ofthin plate with curvilinear domain. International Journal forNumerical Methods in Engineering 80, 1718–1742 (2009)

30 Quan, J.R., Chang, C.T.: New insights in solving distributedsystem equations by the quadrature methods — I. Analysis.Computers in Chemical Engineering 13, 779–788 (1989)

31 Bert, C.W., Malik, M.: The differential quadrature method forirregular domains and application to plate vibration. Interna-tional Journal of Mechanical Sciences 38, 589–606 (1996)

32 Shu, C., Richards, B.E.: Application of generalized differentialquadrature to solve two-dimensional incompressible Navier–Stokes equations. International Journal for Numerical Methodsin Fluids 15, 791–798 (1992)

33 Bartels, R.H., Stewart, G.W.: Solution of the matrix equationAX + XB = C. Communications of the ACM 15, 820–826(1972)

34 Hughes, T.J.R.: The Finite Element Method: Linear Static andDynamic Finite Element Analysis. Prentice-Hall, EnglewoodCliffs, NJ (1987)

Documents

Differential quadrature time element method for structural dynamics