This paper presents a new robust and efficient time integration algorithm suitable for various complex nonlinear structural dynamicfinite element problems. Based on the idea of composition, the three-point backward difference formula and a generalized centraldifference formula are combined to constitute the implicit algorithm.Theoretical analysis indicates that the composite algorithm isa single-solver algorithmwith satisfactory accuracy, unconditional stability, and second-order convergence rate. Moreover, withoutany additional parameters, the composite algorithm maintains a symmetric effective stiffness matrix and the computational costis the same as that of the trapezoidal rule. And more merits of the proposed algorithm are revealed through several representativefinite element examples by comparing with analytical solutions or solutions provided by other numerical techniques. Results showthat not only the linear stiff problem but also the nonlinear problems involving nonlinearities of geometry, contact, and materialcan be solved efficiently and successfully by this composite algorithm. Thus the prospect of its implementation in existing finiteelement codes can be foreseen.
1. Introduction
Modeling of engineering problems leads in many cases toordinary and partial differential equations which are often ofnonlinear nature. A powerful tool to solve these differentialequations is the finite element method which was developed50 years ago andhas achievedmany important breakthroughsincluding various solution algorithms [1–4]. And it is believedthat the use of finite elementmethodwill continue to increasein the future because it provides a more rapid and lessexpensive way to evaluate design concepts and design details[5, 6].
Dynamic transient analysis is necessary for many engi-neering problems, such as earthquake analysis or the vibra-tion of structures. In particular when structures possessnonlinear features, the step-by-step time integration schememay be the only tool available for dynamic transient analysis.The widely used step-by-step integration algorithms can begrouped into two main categories: explicit and implicit [7].The explicit algorithm is appropriate to the simulation ofwavepropagation problems where high frequency responses are
dominated, and themost crucial difficulty in explicitmethodsis the numerical instability which may be time consumingbecause a very small time step should be utilized [8, 9];thus giant computers and parallel computing technologies areusually necessary. In the view of the fact that high frequencyresponses which are even totally spurious and are caused byspatial discretization are not the dominated response in long-term structural dynamic analysis, implicit algorithms whichallow for larger time steps turn out to be the more efficientstrategies [10].
Many researchers have been dedicated to the develop-ment of effective unconditionally stable implicit algorithms[11], and the research efforts can be classified, in essence,into three categories [12]. The first category introducesnumerical dissipation by using various parameters to satisfythe energy criterion. These well-known methods, includingthe generalized-𝛼method [13], Wilson-𝜃method [14], HHT-𝛼 [15], and WBZ-𝛼 [16], have one thing in common: oneor more parameters must be chosen before the solutionprocedure. Obviously personal judgement and problem spec-ifications will affect the accuracy and analysis will not be
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 907023, 11 pageshttp://dx.doi.org/10.1155/2015/907023
2 Mathematical Problems in Engineering
automatic [17, 18]. In particular, for nonlinear problems,it is difficult for an analyst to identify the appropriateparameters. In practical engineering applications, integrationschemes with no parameter are more popular [10, 19]. In thesecond category, energy conservation criterion is enforcedby employing Lagrange multipliers. It will definitely increasethe unknowns and result in nonsymmetric tangent stiffnessmatrices, thus leading to costly computation [20]. Despiteall these, failure in Newton-Raphson iteration of equilibriumhad been found in some cases [21]. And the last categoryobtains conservation algorithmically, such as the energy-momentum method proposed by Simo and Tarnow [22].However, the energy-momentum method lacks generalitysince different schemes have to be developed for each dif-ferent system of equations to be solved [23] and it must bemodified to damp out the spurious high frequencies at theprice of accuracy loss [24, 25]. Besides, the tangential stiffnessmatrices of all the generalized energy-momentum methodsare unsymmetric [12].
Based on the above summary here and more develop-ments described elsewhere [26, 27], we can see that a timeintegration algorithm suitable for various complex nonlineardynamic structural systems is lacking, which should inheritthe desirable properties including unconditional stability,at least second-order accuracy, ease of use (no additionalparameter), ease of implementation, and high computationalefficiency. Some delightful latest achievements are indicatingthat the conception of composition can be the outlet [28–30]. Generally speaking, there are two kinds of compositionregimes.
The first regime is assembling different independentintegration schemes in several substeps into a compositealgorithm in one whole time step, whose solution dependson both these independent schemes and the partition of thesesubsteps. High accuracy of the composite algorithm relies onthe equilibrium equation satisfaction at each moment of thesubsteps. The most representative one is the Bathe method,which combines the trapezoidal rule with a three-pointbackward differencemethod in two equal substeps [10, 28, 31].Unfortunately, the computational cost of the Bathe algorithmis about twice as expensive as the trapezoidal rule due tomore iterations in the substeps [28]. Moreover, generalizingthe Bathe scheme to more substeps receives algorithms withless period elongation but amplitude decay and computa-tional cost are increasing [28]. On the other hand, differentsegmentations of the substeps require additional parametersto be introduced [32].
The second regime is the composition of different differ-ence formulae in one whole time step to inherit their advan-tages [30, 33, 34]. For example, Liu et al. have proposed anefficient time integration algorithm by compositing the two-point and three-point backward Euler formulae [30], whichpresents good stability for nonlinear structural dynamicproblems, with no parameter and the same computationalefficiency as the trapezoidal rule. However, the accuracy ofthis algorithm is less than satisfactory for its high numericaldamping in the intermediate-frequency range.
In this paper, we devised a new efficient time integrationalgorithm based on the second composition regime. The
three-point backward difference formula was retained dueto its good performance on stiff problems [30], and at thesame time, in order to achieve satisfactory accuracy, a gen-eralized central difference formula was derived to calculatethe acceleration term. It will be displayed that, maintainingall those above mentioned desired properties for nonlinearstructural analysis, this new proposed composite method hasgained significant improved accuracy comparing to the back-ward Euler algorithm. In the following, the composite timeintegration method is introduced in Section 2, the propertiessuch as accuracy and stability are evaluated theoretically, andalso a linear stiff system will be investigated deeply with thehelp of the new composite algorithm. In Section 3, to illustrateits potential and to assess its accuracy and efficiency onnonlinear structural dynamic analysis, three finite elementexamples, involving nonlinearities on geometry, contact, andmaterial, respectively, are discussed. Finally, some concludingremarks are summarized in Section 4.
2. The Composite Time Integration Algorithm
In this section the detailed derivation process of the com-posite scheme is first presented, and then the accuracy,stability, and convergence rate of the scheme are evaluatedtheoretically. In addition, a typical stiff linear system is anal-yzed to probe valuable insights into the properties of thescheme.
2.1. The Basic Equations of the Integration Algorithm. Fornonlinear transient structural dynamic problems, the govern-ing equations of equilibrium discretized by finite elementstake the common second-order ordinary differential form as
Mu + f𝑑 (u) + f𝑠 (u) = p (1)
with the initial conditions
0u = u, 0u = u, (2)
whereM is themassmatrix,u, u, and u are the nodal displace-ment, velocity, and acceleration, respectively, f𝑑 and f𝑠 are thenodal damping force and the elastic force corresponding tothe element internal stresses, p is the externally applied load,and u and u are the given initial displacement and velocity,respectively. In linear cases, f𝑑 = Cu and f𝑠 = Ku, where Cand K are the damping matrix and stiffness matrix.
The step-by-step time integration solves (1) and (2) ata set of discrete time points. Assume that the solutions arecompletely known up to time 𝑡, and then the solution at time𝑡 + Δ𝑡 is to be calculated. The equilibrium equation at time𝑡 + Δ𝑡 can be rewritten as
M𝑡+Δ𝑡u =𝑡+Δ𝑡p− 𝑡+Δ𝑡f𝑑 −
𝑡+Δ𝑡f𝑠 . (3)
For the good performance on stiff problems, the second-order accuracy three-point Euler backward difference for-mula (4) is reserved here to combine with the generalized
Mathematical Problems in Engineering 3
central difference formula (5), whose derivation will be givenbriefly below:
𝑡+Δ𝑡u =
1
2Δ𝑡
(3𝑡+Δ𝑡u− 4 𝑡u+ 𝑡−Δ𝑡u) , (4)
𝑡+Δ𝑡u =
𝑡+Δ𝑡u− 2 𝑡u+ 𝑡−Δ𝑡uΔ𝑡2
+
𝑡+Δ𝑡u− 2 𝑡u+ 𝑡−Δ𝑡uΔ𝑡
. (5)
The acceleration 𝑡+Δ𝑡u can be obtained by the classicalforward Euler integration formula:
𝑡+Δ𝑡u =𝑡u+Δ𝑡 𝑡 ...u . (6)
To obtain an algorithmwith higher accuracy, substitutionof the common central difference formulae of the accelerationand its first-order derivative, as shown in (7), into (6) yieldsthe generalized central difference formula (5). Consider
𝑡u =
𝑡+Δ𝑡u− 2 𝑡u+ 𝑡−Δ𝑡uΔ𝑡2
,𝑡 ...u =
𝑡+Δ𝑡u− 2 𝑡u+ 𝑡−Δ𝑡uΔ𝑡2
.
(7)
Since the nodal damping forces and the internal forcesdepend nonlinearly on the current values, the linearizationof (3) yields the following matrices:
𝑡+Δ𝑡C =
𝜕𝑡+Δ𝑡f𝑑
𝜕𝑡+Δ𝑡u
,𝑡+Δ𝑡K =
𝜕𝑡+Δ𝑡f𝑠
𝜕𝑡+Δ𝑡u
. (8)
Substituting (4) and (5) into linearized equation (3) andemploying the displacement increment
𝑡+Δ𝑡u(𝑘) = 𝑡+Δ𝑡u(𝑘−1) + Δu(𝑘) (9)
yield the following Newton-Raphson iteration:𝑡+Δ𝑡
K(𝑘−1)Δu(𝑘) = 𝑡+Δ𝑡p(𝑘−1), (10)
where𝑡+Δ𝑡
K(𝑘−1) = 5
2Δ𝑡2M +
3
2Δ𝑡
𝑡+Δ𝑡C(𝑘−1) + 𝑡+Δ𝑡K(𝑘−1), (11)
𝑡+Δ𝑡p(𝑘−1) = MΔ𝑡2(4𝑡u− 3
2
𝑡−Δ𝑡u+Δ𝑡 (2 𝑡u− 𝑡−Δ𝑡u)
−
5
2
𝑡+Δ𝑡u(𝑘−1))
+ (𝑡+Δ𝑡p − 𝑡+Δ𝑡f(𝑘−1)
𝑑−𝑡+Δ𝑡f(𝑘−1)𝑠
)
(12)
are the effective stiffness matrix and effective load vector,respectively, with iterationnumber 𝑘 = 1, 2, 3, . . .. Clearly, theeffective stiffness matrix is symmetric as long as the dampingmatrix possesses symmetry.
Due to (4) and (5) being three time points, the compositealgorithm requires a starting step algorithm, which is recom-mended as follows:
Δ𝑡u =0u+Δ𝑡 0u+ Δ𝑡
2
10
(0u+ 4 Δ𝑡u) , (13)
Δ𝑡u =
3
2Δ𝑡
(Δ𝑡u− 0u) − 1
2
0u+ Δ𝑡
4
0u . (14)
Equation (13) results from the Newmark algorithm with𝛽 = 2/5. Using this particular value of parameter 𝛽 leads tothe same effective stiffness matrix of the starting step as thesubsequent time steps, but the effective load of (12) should bereplaced with
Δ𝑡p(𝑘−1) = MΔ𝑡2(0u+Δ𝑡0u + Δ𝑡
2
10
0u− 5
2
Δ𝑡u(𝑘−1))
+ (Δ𝑡p − Δ𝑡f(𝑘−1)
𝑑−Δ𝑡f(𝑘−1)𝑠
)
for 𝑡 = 0.
(15)
Obviously, the new composite time integration algorithmis of second-order accuracy with no additional artificialparameter and only one set of nonlinear equations withsymmetric effective stiffnessmatrix needs to be solved at eachtime step; hence the computational effort is the same as thatof the trapezoidal rule.
2.2. Stability Analysis. In this subsection, the stability ofthe proposed algorithm for linear problems is performedtheoretically while the stability performance for nonlinearcases will be investigated numerically in Section 3.
Now, we consider the following single degree-of-freedomsystem:
�� + 2𝜉𝜔�� + 𝜔2𝑢 = 0, (16)
where 𝜔 is the free vibration natural frequency and 𝜉 is thedamping ratio.
Considering (4) and (5) at any three successive timepoints results in
X𝑡+Δ𝑡 = AX𝑡, (17)
where X𝑡 = [𝑡𝑢,𝑡��/𝜔,𝑡−Δ𝑡
𝑢,𝑡−Δ𝑡
��/𝜔]𝑇 and A is the transfer
matrix expressed as
A =
[
[
[
[
[
[
[
[
[
[
[
8 (1 + 𝜉𝜔Δ𝑡)
𝐵
4𝜔Δ𝑡
𝐵
−3 − 2𝜉𝜔Δ𝑡
𝐵
−2𝜔Δ𝑡
𝐵
2 − 4𝜔2Δ𝑡2
𝐵𝜔Δ𝑡
6
𝐵
−2 + 𝜔2Δ𝑡2
𝐵𝜔Δ𝑡
−3
𝐵
1 0 0 0
0 1 0 0
]
]
]
]
]
]
]
]
]
]
]
, (18)
where 𝐵 = 5 + 6𝜉𝜔Δ𝑡 + 2𝜔2Δ𝑡2.
Figure 1 shows the spectral radii of the transfer matricesof the proposed method without damping as well as thoseof the trapezoidal rule [35], the generalized-𝛼 method [13],the Bathe method [31], and backward Euler method [30]. Itcan be seen that all of these mentioned methods except thetrapezoidal rule perform diminishing spectral radii with theincrease of Δ𝑡/𝑇 ratio value. The proposed method presentsthe spectral radius of 0.994 at the ratio Δ𝑡/𝑇 = 0.1, withalmost no amplitude decrease. As the Δ𝑡/𝑇 ratio increases,the spectral radius of the proposed method rapidly reducesto zero, indicating desired numerical damping in the high
4 Mathematical Problems in Engineering
0.0
0.3
0.6
0.9
1.2
10310−3 10210−2 10110010−1
Δt/T
Generalized-𝛼(𝜌∞ = 0)
Bathe method
Backward Euler methodNew composite method
Trapezoidal rule
Spec
tral
radi
us𝜌
Figure 1: Spectral radii of various methods with no damping.
0
2
4
6
8
10
12
Perio
d el
onga
tion
(%)
Δt/T
Generalized-𝛼(𝜌∞ = 0)
Bathe method
Backward Euler methodNew composite method
Trapezoidal rule
0.00 0.02 0.04 0.06 0.08 0.10
Figure 2: Period elongation of the new composite method compar-ing with other methods.
frequency range, which is an expected property for practicalengineering analysis. In particular, the spectral radius ofthe proposed method is smaller than that of the Bathemethod but larger than that of the backward Euler methodas well as the generalized-𝛼method.Therefore, the proposedcomposite method is unconditionally stable in linear case.
2.3. Accuracy Analysis. To evaluate the accuracy of the pro-posed composite method, the following spring-mass systemwithout damping is considered:
�� + 𝜔2𝑢 = 0. (19)
The initial conditions are 0𝑢 = 1, 0�� = 0, and 0�� = −𝜔2.
The analytical solution of (19) is 𝑢(𝑡) = cos(𝜔𝑡). Figures 2and 3 show the period elongation and amplitude decay of theproposed composite scheme, respectively. For comparison,
0
2
4
6
8
10
Am
plitu
de d
ecay
(%)
Δt/T
Generalized-𝛼(𝜌∞ = 0)
Bathe method
Backward Euler methodNew composite method
Trapezoidal rule
0.00 0.02 0.04 0.06 0.08 0.10
Figure 3: Amplitude decay of the new composite method compar-ing with other methods.
the corresponding results of the trapezoidal rule [35], thegeneralized-𝛼 method [13], the Bathe method [31], andbackwardEulermethod [30] are also presented. Clearly, whenthe ratio Δ𝑡/𝑇 is no more than 0.01, all methods presentsatisfactory accuracy. With the increase of the ratio Δ𝑡/𝑇, thenew compositemethodmaintains high accuracy,much betterthan the backward Euler difference method under the sameefficiency, just a little worse than the Bathe method whichcosts more calculation effort.
2.4. Convergence Rate Analysis. In order to evaluate theconvergence rate, we consider the following linear resonanceproblem as discussed in [36]:
�� + 𝜔2𝑢 = sin (𝜔𝑡) (20)
with the initial conditions 0𝑢 = 1, 0�� = −𝜔2.
The exact solutions of this system for displacement,velocity, and acceleration, respectively, are
𝑢 (𝑡) =
2𝜔 + 1
2𝜔2
sin (𝜔𝑡) + 2𝜔 − 𝑡
2𝜔
cos (𝜔𝑡) ,
�� (𝑡) = cos (𝜔𝑡) − 2𝜔 − 𝑡
2
sin (𝜔𝑡) ,
�� (𝑡) =
1 − 2𝜔
2
sin (𝜔𝑡) + 𝜔 (𝑡 − 2𝜔)
2
cos (𝜔𝑡) .
(21)
The circular frequency 𝜔 is assigned to 2𝜋 rad/s corre-sponding to the period 𝑇 = 1 s. Figure 4 shows the relativeerror curves of displacement, velocity, and acceleration at thetime of 1.0 s. Clearly, this new composite method is second-order and convergent in terms of the displacement, velocity,and acceleration.
2.5. Linear Stiff System Demonstration. In this subsection, alinear spring-mass model analyzed by Bathe and Noh [31],
Mathematical Problems in Engineering 5Re
lativ
e err
or
12
10−310−4
10−4
10−6
10−8
10−2
10−2
10−1
Δt/T
100
DisplacementVelocity
Acceleration
Figure 4: Convergence rate of the new composite method.
m0 = 0 m1 = 1 m2 = 1k1 = 107 k2 = 1
u2u1u0 = sin 𝜔pt
Figure 5: Stiff linear spring-mass system.
which seems simple but can represent complex practicalstructural system, is investigated to test the proposed com-posite algorithm.
The governing equations of the stiff linear system, withparameters shown in Figure 5, are
[
1 0
0 1] {
��1
��2
} + [107+ 1 −1
−1 1
]{
𝑢1
𝑢2
} = {
107 sin𝜔𝑝𝑡0
} . (22)
It can be seen that the two springs have quite differentstiffness. The left stiff spring represents rigid connections inthe complex models, while the right flexible one representsthe flexible parts. Indeed, the high stiffness parts used inpractice usually have little physical meaning other than toprovide constraints, whose response would naturally notbe included in the mode superposition solution. In orderto verify the high frequency damping property of the newproposed composite method, the initial conditions 0𝑢1 = 0,0��1 = 2, 0��2 = 0, and 𝜔𝑝 = 2 rad/s are supposed, and thetime step used is 0.1 s. Hence we have Δ𝑡/𝑇1 = 50.0, Δ𝑡/𝑇2 =0.0159, and Δ𝑡/𝑇𝑝 = 0.0318, where 𝑇1 = 0.002 s, 𝑇2 =
6.283 s are the two distinct natural periods of the system and𝑇𝑝 = 3.1416 s is the period of the prescribed displacementboundary condition.
Comparing to the analytical solutions, the displacement,velocity, and acceleration responses shown in Figure 6demonstrate that the proposed composite method performsvery well on this stiff problem without adjustment of anyparameter. However, with the same computational cost, thetrapezoidal rule cannot even achieve stable result becauseof lacking appropriate numerical damping [31]; as shown in
Figure 7, the acceleration response of node 1 displays distinctinstability. As in practical finite element simulations, notonly the rigid connection but also the spatial finite elementdiscretization can bring in spurious high frequencies; thusa favorable algorithm should have appropriate numericaldamping in the high frequency range [19].
3. Nonlinear Dynamic FiniteElement Examples
In practical engineering, nonlinear behavior is more com-mon. According to the different sources of nonlinearities,they can be divided into three categories: geometrical non-linearity, contact nonlinearity, and material nonlinearity. Inthis section, three typical finite element examples involvingeach nonlinearity are solved to prove the precision andeffectiveness of the proposed composite time integrationalgorithm.
3.1. Geometrical Nonlinear Example. The stiff single pendu-lummodel, which involves large displacements and rotations,is a classical geometrical nonlinear example to demonstratethe ability of a new time integration algorithm in solvingnonlinear problems, and the failure of the trapezoidal rule inthis case has been illustrated by many scholars [10, 30, 37].
Figure 8 shows the geometry, material, boundary, andinitial conditions of the single pendulum model, whoseanalytical rotational period is 2.4777 s and the total energy is149 J.We solve this nonlinear problemusing geometrical non-linear finite elements with the time step of 0.01 s for as long as2500 s (more than 1000 cycles). The solutions of the first 15seconds and the last 15 seconds are shown in Figure 9. Thetotal amplitude decay of the whole 1000 cycles is only about0.0036% and the total period elongation is less than 0.0928%.We also ran this problem using a much larger time step of0.5 s, and still a stable solution but of coursewith less accuracywas achieved, which evidently proved the robust stabilityof the new composite scheme for geometrical nonlinearproblems.
This problem was also solved using the Bathe methodunder the same calculation conditions and convergenceprecisions. Results showed that at the time of 150 s, the energydecay of the Bathe method was 0.0248%, much larger thanthat of the composite method (0.0005%). And on average,the Bathe method needed 6 iterations while the compositemethod only needed 4 iterations in one time step to reachthe convergence precisions. In addition, [30, 37] have provedthat the trapezoidal rule cannot achieve stable results forthis nonlinear problem. Thus the new composite method isof high accuracy and efficiency for geometrical nonlinearproblems.
3.2. Contact Nonlinear Example. Another particular difficultnonlinear behavior is the contact between two or morebodies, which can be considered stemming from boundarynonlinearity. In this subsection, a benchmark test aboutcenter-to-center collision without friction between two iden-tical elastic bars [38, 39] using Lagrange multiplier to impose
6 Mathematical Problems in Engineering
0 2 4 6 8 10
0.0
0.6
1.2
1.8D
ispla
cem
ent (
m)
Time (s)
−0.6
−1.2
(a) Displacement responses
0 2 4 6 8 10Time (s)
0
1
2
3
Velo
city
(m/s
)
−2
−1
(b) Velocity responses
0
2
4
6
−4
−2Acce
lera
tion
(m/s2)
0 2 4 6 8 10Time (s)
Numerical result of node 1 Analytical result of node 1Numerical result of node 2 Analytical result of node 2
(c) Acceleration responses
Figure 6: Responses of the stiff spring system using the new composite method: (a) displacement; (b) velocity; (c) acceleration.
05
1015202530
Resp
onse
s
0 2 4 6 8 10Time (s)
−5
−10
−15
−20
−25
Acceleration of node 1Velocity of node 1Displacement of node 1
Figure 7: Responses of node 1 in stiff spring system using thetrapezoidal rule.
z
x
t = 2.2 s
t = 0
Δt = 0.1 s
l0 = 3.0443m
𝜌0A0 = 6.57kg/m
EA0 =
u0 = 7.72m/s
u0 = 19.6m/s2
1010 N
Figure 8: Stiff single pendulum with boundary and initial condi-tions.
contact constraints is performed with time step of 0.008 s.Each bar is 100 meters long and discretized by 50 equal eight-node 3D finite elements with the same size of 1m × 1m × 2m.
Mathematical Problems in Engineering 7
02468
Disp
lace
men
t (m
)
−4
−2
0 5 10 2485 2490 2495 2500
Time (s)
(a) Displacement responses
0
5
10
Velo
city
(m/s
)
−5
−100 5 10 2485 2490 2495 2500
Time (s)
(b) Velocity responses
0
10
20
30
−10
−20
x componentz component
Acce
lera
tion
(m/s2)
0 5 10 2485 2490 2495 2500
Time (s)
(c) Acceleration responses
Figure 9: Responses of the stiff single pendulum: (a) displacement; (b) velocity; (c) acceleration.
Gap = 0.1m
Each bar length: 100mCross-section area:
Young’s modulus: 106 PaDensity: 100 kg/m3
Poisson’s ratio: 0� = 0
� = 1.0m/s
1m × 1m
Figure 10: Dynamic contact model of two identical elastic bars.
The detailed material parameters and initial conditions areshown in Figure 10.
The longitudinal velocity responses of the contacting pairnodes of each bar are illustrated in Figure 11, which clearlyshows that impact occurs at the time of 0.1 s, and since thenthe two contacting pair nodes move at the same velocity of0.5m/s. And the contacting pair nodes keepmoving togetheruntil the stress waves caused by the impact propagate back tothe contacting ends at the time of 2.1 s. After that, the two barsseparate and accomplish the exchange of their initial veloci-ties. One distinct thing is that steep fluctuations with an inter-val of 2.0 s appear on the velocity responses after separation;the reason is that the stress waves are propagating withoutloss back and forth at the elastic wave velocity of 100m/s.As shown in Figures 12 and 13, there is some little oscillationin the contact pressure and the total energy, which are
0 1 2 3 4 5 6
0.0
0.5
1.0
1.5
Velo
city
(m/s
)
Time (s)
−0.5
Right barLeft bar
Figure 11: Longitudinal velocities of the contacting nodes.
attributed to the fact that the impenetrability constraintsare only imposed on displacements in our procedure forsimplicity, which should also be imposed on the rates ofdisplacement for better accuracy [40–42]. As a whole, thecomposite method displays robust stability for dynamiccontact nonlinear problems.
3.3. Material Nonlinear Example. Material nonlinearity,characterized by a nonlinear response function between
8 Mathematical Problems in Engineering
0
2
Con
tact
pre
ssur
e (kP
a)
−2
−4
−6
−80 1 2 3 4 5 6
Time (s)
Analytical solutionSimulation results
Figure 12: Contact pressure of the contacting nodes.
0
1
2
3
4
5
6
Ener
gy (k
J)
0 1 2 3 4 5 6Time (s)
Kinetic energyStrain energy
Total energy
Figure 13: Energy of the contact bars.
stresses and strains or by a set of evolution equations, oftenoccurs in structural engineering. For example, in seismicdesign of buildings, the lead-rubber bearing (LRB for short)is an effective device designed with nonlinear constitutiverelationships to reduce the destructive transmission energyof the group motion to the superstructure. The LRB is madeof alternate layers of rubber and steel plates with one or morelead plugs inserted into the predrilled holes, and the lead coredeforms in shear providing both the initial rigidity and thebilinear response, which is crucial for seismic isolation [43].
In this example, a seven-story building isolated by theLRB is analyzed using the composite algorithm under EL-Centro, 1940 earthquake motion (North-South componentwith peak acceleration equal to 3.417m/s2). Rayleigh damp-ing consisting of mass matrix and stiffness matrix with coeffi-cients both equal to 0.01 is also considered. For simplicity, themass and the stiffness of each floor of the superstructure are
kept constant, with detailed parameters shown in Figure 14.And some additional assumptions are made: floors of eachstory of the superstructure are assumed to be rigid; theforce-deformation behavior of the LRB is considered bilineardefined by three parameters 𝑓0, 𝑘𝑦, and 𝑢0 (as shown inFigure 14(b)), which denote the yield strength, postyieldstiffness, and yield displacement, respectively.
Since it is difficult to obtain the analytical solution, resultsof the composite time integration algorithm are comparedand validated with that of the commercial finite elementsoftware Ansys (version of 12.1) using Newmark integrationmethod (𝛽 = 0.25, 𝛾 = 0.5). The absolute accelerationresponse of the top floor under time stepΔ𝑡 = 0.02 s is plottedin Figure 15 and the nonlinear force-displacement responseof the base floor is depicted in Figure 16. It can be seenthat the results of the composite algorithm are very close tothe Ansys results where Combin40 and Combin14 elements[44] are utilized to simulate the bilinear and the linearfloors, respectively. Meanwhile, we can see that, with the helpof the LRB, there is prominent reduction in the top flooracceleration of the building, and the fundamental frequencyof the superstructure is shifted away from the dominantfrequencies of the earthquake ground motion; thus theseismic safety of the building is guaranteed.
To verify the accuracy, another much smaller time step of0.002 s was also used in the solution procedure of both theAnsys Newmark method and the new composite algorithm.Focusing on the top acceleration response, the accuracy of thetime integration algorithm was quantificationally evaluatedusing the differences between the big time step (0.02 s) andthe small time step (0.002 s). For this material nonlineardynamic system, it is acceptable that result of small timestep is much closer to the true solution. And the 2-norm ofthe difference between the two methods both using smalltime step is about 0.7, indicating that the new compositeintegration and the Ansys Newmark integration are bothaccurate when time step is small enough. As shown inFigure 17, the difference between the solutions of the newcomposite algorithm using different two time steps, whose2-norm is about 1.4, is much smaller than that of the AnsysNewmark solutions, with 2-norm equal to 25.9. Obviouslythe composite algorithm did an excellent job for its highaccuracy. Moreover, considering its computational efficiencyand simple implementation, the composite algorithm hasgreat potential to be embedded into finite element codes fornonlinear structural dynamic engineering analysis.
4. Conclusions
In this paper, an excellent implicit time integration algo-rithm was proposed through composition of the three-point backward difference formula and a generalized centraldifference formula. Theoretical analysis and investigation ona simple frequently used stiff linear system which can repre-sent complicated practical structures have been carried outto give considerable insights into the composite time inte-gration algorithm. Besides, three finite element examplesinvolving nonlinearities on geometry, contact, and material
Mathematical Problems in Engineering 9
m7
m6
m5
m4
m3
m2
m1
mb
k7
k6
k5
k4
k3
k2
k1
kb
mb = 8 × 105 kg
mi = 5.4 × 105 kg
ki = 1.0 × 109 N/m
Base massBase isolatorFoundation
(i = 1, . . ., 7)
ub
ug
u7
(a) Seven-story building supported by the LRB
Force
Displacement
f0
u0
kyf0 = 4488.4 kNu0 = 0.05m
ky = 2892.784kPa
(b) Force-displacement behavior of the LRB
Figure 14: Model of seven-story building isolated by the LRB and the force-displacement behavior of the LRB.
0
4
8
12
New composite method (not isolated)New composite method (isolated)Ansys (isolated)
4035302520151050Time (s)
−4
−8
Abso
lute
acce
lera
tion
(m/s2)
Figure 15: Absolute acceleration of the top floor with and withoutthe LRB isolated.
0
2
4
6
Displacement (m)−0.10 −0.08 −0.06 −0.04 −0.02
−2
−4
−60.00 0.02 0.04 0.06 0.08 0.10
Forc
e (10
6N
)
Figure 16: Force-displacement response of the base isolation floor.
0
1
2
3
4035302520151050Time (s)
Small time step differenceAnsys differenceNew composite method difference
Acce
lera
tion
diffe
renc
e (m
/s2)
−2
−1
Figure 17: Top floor acceleration differences of different solutions.
were analyzed in detail to verify the excellent performanceon nonlinear problems.
Serving as a robust and efficient time integration methodvaluable for both linear and nonlinear dynamic engineeringanalysis, the attractive properties of the composite algorithmare emphasized herein: (1) it is of second-order accuracyand is unconditionally stable; (2) with appropriate numericaldamping in high frequency range, it is suitable for not onlylinear stiff problems but also nonlinear problems; (3) it issecond-order and convergent in terms of the displacement,velocity, and acceleration; (4) it only solves a symmetricmatrix equation once within each time step, resulting in thesame calculation efficiency as the trapezoidal rule but can dealwith the nonlinear problems where the trapezoidal rule fails;(5) without any additional parameters to be determined, it issimple and convenient to be implemented into existing finiteelement codes.
10 Mathematical Problems in Engineering
It can be concluded that composition is a potentialidea for constructing robust and effective algorithms whichdeserves intensive studies in the future study of the finiteelement method fields, especially in nonlinear cases.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgments
This study was financially supported by the National NaturalScience Foundation of China (11372156), the Research Projectof State Key Laboratory of Hydroscience and Engineering ofTsinghua University (no. 2012-KY-4), and the National BasicResearch Program of China (2013CB035902).
References
[1] T. Belytschko, W. K. Liu, and B. Moran, Nonlinear FiniteElements for Continua and Structures , John Wiley & Sons,Chichester, UK, 2000.
[3] D. Soares Jr., “A new family of time marching procedures basedonGreen’s functionmatrices,”Computers and Structures, vol. 89,no. 1-2, pp. 266–276, 2011.
[4] X. D. Li and N.-E. Wiberg, “Structural dynamic analysis by atime-discontinuous Galerkin finite element method,” Interna-tional Journal for Numerical Methods in Engineering, vol. 39, no.12, pp. 2131–2152, 1996.
[5] K. J. Bathe, Finite Element Procedures, Prentice Hall, EnglewoodCliffs, NJ, USA, 1996.
[6] I. Romero, “An analysis of the stress formula for energy-momentum methods in nonlinear elastodynamics,” Computa-tional Mechanics, vol. 50, no. 5, pp. 603–610, 2012.
[7] M. A. Dokainish and K. Subbaraj, “A survey of direct time-integration methods in computational structural dynamics-I.Explicit methods,” Computers and Structures, vol. 32, no. 6, pp.1371–1386, 1989.
[8] H. Askes, D. C. D. Nguyen, and A. Tyas, “Increasing the criticaltime step: micro-inertia, inertia penalties and mass scaling,”Computational Mechanics, vol. 47, no. 6, pp. 657–667, 2011.
[9] A. Idesman and D. Pham, “Finite element modeling of linearelastodynamics problems with explicit time-integration meth-ods and linear elements with the reduced dispersion error,”Computer Methods in Applied Mechanics and Engineering, vol.271, pp. 86–108, 2014.
[10] K.-J. Bathe, “Conserving energy and momentum in nonlineardynamics: a simple implicit time integration scheme,” Comput-ers and Structures, vol. 85, no. 7-8, pp. 437–445, 2007.
[11] T. Belytschko and D. F. Schoeberle, “On the unconditionalstability of an implicit algorithm for nonlinear structural dyn-amics,” Journal of AppliedMechanics, vol. 42, no. 4, pp. 865–869,1975.
[12] D. Kuhl and M. A. Crisfield, “Energy-conserving and decayingalgorithms in non-linear structural dynamics,” InternationalJournal for Numerical Methods in Engineering, vol. 45, no. 5, pp.569–599, 1999.
[13] J. Chung and G. M. Hulbert, “A time integration algorithm forstructural dynamics with improved numerical dissipation: thegeneralized-𝛼 method,” ASME Journal of Applied Mechanics,vol. 60, no. 2, pp. 371–375, 1993.
[14] E. L.Wilson, “A computer program for the dynamic stress analy-sis of underground structures,” SESM Report 68-1, Departmentof Civil Engineering, University of California, Berkeley, Calif,USA, 1968.
[15] H. M. Hilber, T. J. R. Hughes, and R. L. Taylor, “Improvednumerical dissipation for time integration algorithms in struc-tural dynamics,” Earthquake Engineering & Structural Dynam-ics, vol. 5, no. 3, pp. 283–292, 1977.
[16] W. L. Wood, M. Bossak, and O. C. Zienkiewicz, “An alphamodification of Newmark’s method,” International Journal forNumericalMethods in Engineering, vol. 15, no. 10, pp. 1562–1566,1980.
[17] A. A. Gholampour and M. Ghassemieh, “New implicit methodfor analysis of problems in nonlinear structural dynamics,”International Journal for Engineering Modelling, vol. 23, no. 1–4,pp. 49–53, 2010.
[18] C.-C. Hung, W.-M. Yen, andW.-T. Lu, “An unconditionally sta-ble algorithm with the capability of restraining the influence ofactuator control errors in hybrid simulation,” Engineering Struc-tures, vol. 42, pp. 168–178, 2012.
[19] A. V. Idesman, M. Schmidt, and J. R. Foley, “Accurate finiteelement modeling of linear elastodynamics problems with thereduced dispersion error,”ComputationalMechanics, vol. 47, no.5, pp. 555–572, 2011.
[20] T. J. R. Hughes, T. K. Caughey, and W. K. Liu, “Finite-elementmethods for nonlinear elastodynamics which conserve energy,”Journal of AppliedMechanics, Transactions ASME, vol. 45, no. 2,pp. 366–370, 1978.
[21] D. Kuhl and E. Ramm, “Constraint energy momentum algo-rithm and its application to non-linear dynamics of shells,”Computer Methods in Applied Mechanics and Engineering, vol.136, no. 3-4, pp. 293–315, 1996.
[22] J. C. Simo and N. Tarnow, “The discrete energy-momentummethod. Conserving algorithms for nonlinear elastodynamics,”Journal of Applied Mathematics and Physics, vol. 43, no. 5, pp.757–792, 1992.
[23] K. Li and A. P. Darby, “A high precision direct integrationscheme for nonlinear dynamic systems,” Journal of Computa-tional and Nonlinear Dynamics, vol. 4, no. 4, Article ID 041008,pp. 1–10, 2009.
[24] F. Armero and E. Petocz, “A new class of conserving algorithmsfor dynamic contact problems,” in Proceedings of the 2nd ECCO-MAS Conference on Numerical Methods in Engineering, pp.861–867, Paris, France, September 1996.
[25] F. Armero and E. Petocz, “Formulation and analysis of conserv-ing algorithms for frictionless dynamic contact/impact prob-lems,”ComputerMethods inAppliedMechanics and Engineering,vol. 158, no. 3-4, pp. 269–300, 1998.
[26] K. K. Tamma, J. Har, X. Zhou, M. Shimada, and A. Hoitink, “Anoverview and recent advances in vector and scalar formalisms:space/time discretizations in computational dynamics—a uni-fied approach,” Archives of Computational Methods in Engineer-ing, vol. 18, no. 2, pp. 119–283, 2011.
[27] N. Mahjoubi, A. Gravouil, A. Combescure, and N. Greffet, “Amonolithic energy conservingmethod to couple heterogeneoustime integrators with incompatible time steps in structuraldynamics,” Computer Methods in Applied Mechanics and Engi-neering, vol. 200, no. 9–12, pp. 1069–1086, 2011.
Mathematical Problems in Engineering 11
[28] K.-J. Bathe and M. M. I. Baig, “On a composite implicit timeintegration procedure for nonlinear dynamics,” Computers andStructures, vol. 83, no. 31-32, pp. 2513–2524, 2005.
[29] M. Rezaiee-Pajand and S. R. Sarafrazi, “A mixed and multi-stephigher-order implicit time integration family,”Proceedings of theInstitution ofMechanical Engineers Part C: Journal ofMechanicalEngineering Science, vol. 224, no. 10, pp. 2097–2108, 2010.
[30] T. Y. Liu, C. B. Zhao, Q. B. Li, and L. H. Zhang, “An efficientbackward Euler time-integration method for nonlinear dyn-amic analysis of structures,” Computers and Structures, vol. 106-107, pp. 20–28, 2012.
[31] K.-J. Bathe and G. Noh, “Insight into an implicit time integra-tion scheme for structural dynamics,”Computers and Structures,vol. 98-99, pp. 1–6, 2012.
[32] W. T. Matias Silva and L. Mendes Bezerra, “Performance ofcomposite implicit time integration scheme for nonlinear dyn-amic analysis,”Mathematical Problems in Engineering, vol. 2008,Article ID 815029, 16 pages, 2008.
[33] M. Rezaiee-Pajand, S. R. Sarafrazi, and M. Hashemian,“Improving stability domains of the implicit higher order accu-racy method,” International Journal for Numerical Methods inEngineering, vol. 88, no. 9, pp. 880–896, 2011.
[34] T. C. Fung, “Complex-time-step Newmark methods with con-trollable numerical dissipation,” International Journal forNumerical Methods in Engineering, vol. 41, no. 1, pp. 65–93,1998.
[35] N. M. Newmark, “Amethod of computation for structural dyn-amics,” Journal of the Engineering Mechanics Division, vol. 85,no. 7, pp. 67–94, 1959.
[36] V. A. Leontyev, “Direct time integration algorithmwith control-lable numerical dissipation for structural dynamics: two-stepLambda method,” Applied Numerical Mathematics, vol. 60, no.3, pp. 277–292, 2010.
[37] P. B. Bornemann, U. Galvanetto, and M. A. Crisfield, “Someremarks on the numerical time integration of non-linear dyn-amical systems,” Journal of Sound and Vibration, vol. 252, no. 5,pp. 935–944, 2002.
[38] L. Shu, L. Jingbo, S. Jiaguang, and C. Yujian, “Amethod for ana-lyzing three-dimensional dynamic contact problems in visco-elastic media with kinetic and static friction,” Computers andStructures, vol. 81, no. 24-25, pp. 2383–2394, 2003.
[39] L. H. Zhang, T. Y. Liu, Q. B. Li, and T. Chen, “Modified dynamiccontact force method under reciprocating load,” EngineeringMechanics, vol. 31, no. 7, pp. 8–14, 2014 (Chinese).
[40] R. L. Taylor and P. Papadopoulos, “On a finite element methodfor dynamic contact/impact problems,” International Journal forNumericalMethods in Engineering, vol. 36, no. 12, pp. 2123–2140,1993.
[41] T. A. Laursen and G. R. Love, “Improved implicit integratorsfor transient impact problems—geometric admissibility withinthe conserving framework,” International Journal for NumericalMethods in Engineering, vol. 53, no. 2, pp. 245–274, 2002.
[42] H. Zolghadr Jahromi and B. A. Izzuddin, “Energy conservingalgorithms for dynamic contact analysis using Newmark meth-ods,” Computers and Structures, vol. 118, pp. 74–89, 2013.
[43] R. S. Jangid, “Optimum lead-rubber isolation bearings for near-fault motions,” Engineering Structures, vol. 29, no. 10, pp. 2503–2513, 2007.
