timar

*, M

ssach

; acce

Transient analysis of nonlinear problems in structural and solid mechanics is mainly carried out using direct time

Transient analysis of nonlinear problems in solid andwhereas we assumeM and C to be constant (an assump-tion that can be removed) [1].

at the discrete time points Dt apart up to time t. Explicitintegration techniques use Eq. (1) at the time(s) forwhich the displacements are known, to obtain the solu-tion at time t + Dt. This computation is relatively inex-pensive to carry out for each time step since no

0045-7949/$ - see front matter 2005 Elsevier Ltd. All rights reserved.

* Corresponding author. Tel.: +1 617 253 6645; fax: +1 617253 2275.

E-mail address: kjb@mit.edu (K.J. Bathe).

Computers and Structures 83 (2structural mechanics requires the stable and accuratesolution of the equations

MU C _U FU; time Rtimeplus initial conditions 1

where M is the mass matrix, C is the damping matrix, Ris the vector of externally applied nodal loads, F is thevector of nodal forces equivalent to the element stresses,U is the vector of nodal displacements (including rota-tions), and a time derivative is denoted by an overdot.

A widely used approach to solve Eq. (1) is direct timeintegration in which the equilibrium relations are satis-ed at discrete time points Dt apart. The solution isstepped forward in time by assuming time variationsof displacements, velocities and accelerations withinthe time interval Dt. The assumptions used result intoa specic algorithm and directly aect the stability andaccuracy of the procedure.

Direct integration techniques can be either explicit orimplicit. Assume that the solutions have been obtainedintegration of the equations of motion. For reliable solutions, a stable and ecient integration algorithm is desirable.Methods that are unconditionally stable in linear analyses appear to be a natural choice for use in nonlinear analyses,but unfortunately may not remain stable for a given time step size in large deformation and long time range responsesolutions. A composite time integration scheme is proposed and tested in some example solutions against the trapezoi-dal rule and the Wilson h-method, and found to be eective where the trapezoidal rule fails to produce a stable solution.These example results are indicative of the merits of the composite scheme. 2005 Elsevier Ltd. All rights reserved.

Keywords: Direct time integration; Nonlinear dynamic analysis; Stability

1. Introduction We note that F depends on the displacements and time,On a composite implicitfor nonline

Klaus-Jurgen Bathe

Massachusetts Institute of Technology, 77 Ma

Received 6 June 2005

Abstractdoi:10.1016/j.compstruc.2005.08.001e integration proceduredynamics

irza M. Irfan Baig

usetts Avenue, Cambridge, MA 02139, USA

pted 12 August 2005

005) 25132524

www.elsevier.com/locate/compstruc

or after simplication,

2514 K.J. Bathe, M.M.I. Baig / Computers and Structures 83 (2005) 25132524solution of coupled linear equations is needed (assumingM to be a diagonal matrix, and C as well, if present). Awidely used explicit technique is the central dierencemethod which, however, is only conditionally stable;that is, the time step size that can be employed withoutlosing the stability of the algorithm must be smallerthan, or equal to, the critical time step. This restrictioncan result in a time step size that can be several ordersof magnitude smaller than the step size which shouldbe adequate to accurately resolve the response. In suchcases, the use of an implicit integration procedure canbe much more eective.

Implicit methods use Eq. (1) at a time for which thesolution is not known, to obtain the response at timet + Dt. The need for the solution of a coupled systemof equations makes implicit methods considerably moreexpensive, computationally, per time step. Hence uncon-ditionally stable implicit schemes are desirable since thenthe time step size is chosen to satisfy accuracy require-ments alone. The use of larger time steps means, ofcourse, that much fewer steps are used than with anexplicit, conditionally stable procedure.

Implicit integration schemes like the trapezoidal ruleand the Wilson h-method [2,3] are unconditionally stablein linear analyses, and are also employed in nonlinearanalyses. As pointed out long time ago, when using animplicit method in nonlinear analysis, it can be of greatimportance that equilibrium iterations be carried out ateach time step [1,4]. Unfortunately, even with NewtonRaphson iterations carried out to very tight convergencetolerances, a scheme that is unconditionally stable in lin-ear analysis may become unstable in a nonlinear solu-tion. In particular, the trapezoidal rule which is knownto be unconditionally stable in linear analysis, maybecome unstable in nonlinear analysis when a long timeresponse and very large deformations are considered. Ifpresent, the instability is clearly seen in that the displace-ments, velocities and accelerations become unrealisti-cally large.

Much research eort has been directed to improvethe stability of integration schemes for nonlinear dy-namic analysis of solids and structures. Kuhl and Cris-eld [5] have presented a survey of algorithms thathave been formulated with this aim. The basic idea fol-lowed in the research is to satisfy, either algorithmicallyor by constraint equations, conservation of momentaand energy, see [57] and the references therein. How-ever, high frequency modes which are inaccurately re-solved with the time step used may then deterioratethe overall solution accuracy. Also, these methods mayresult in non-symmetric tangent stiness matrices andthe solution of a scalar variable either at the integrationpoints or over each element in an averaged sense. Hence,these integration schemes are computationally costly.

In this paper we focus on the formulation and study

of a single step (but two sub-steps) composite integra-tcDtU tU t _UcDt t U tcDt U cDt2

24

Solving for tcDt U and tcDt _U from the above equations

tcDt U tcDtU tU t _UcDt 4c2Dt2

t U 5

tcDt _U tcDtU tU 2cDt

t _U 6tion procedure. First we present the basic scheme andthen we solve various example problems. The calculatedresults are compared with those obtained using the trap-ezoidal rule and the Wilson h-method. The compositeprocedure is attractive since it only operates on the usualglobal vectors, only uses the usual symmetric matrices,shows good stability characteristics and is of second-order accuracy.

2. The composite time integration procedure

In general, time integration algorithms formulatedusing backward dierence expressions display somenumerical damping and we might use this propertyto stabilize a time integration scheme. The Houboltmethod is such an example which uses a four-pointbackward dierence approximation [1].

A composite, single step, second-order accurate inte-gration scheme for solving rst-order equations arisingin the simulation of silicon devices and circuits was pre-sented by Bank et al. [8]. This composite scheme isavailable in the ADINA program for uid ow struc-tural interaction problems. The rst-order uid owequations and second-order structural equations aresolved fully coupled in time using this procedure[9,10]. Some experience with the algorithm in the solu-tion of structural mechanics problems has been pre-sented in Ref. [11]. In this section we briey presentthe formulation of the algorithm, and in Section 5 wegive the solutions of some test problems for the evalu-ation of the scheme. For details on the notation used,see [1].

Assume that the solution is completely known up totime t, and the solution at time t + Dt is to be computed.Let t + cDt be an instant in time between times t andt + Dt, i.e., c 2 (0,1). Then using the trapezoidal ruleover the time interval cDt, we have the followingassumptions on velocity and displacement:

tcDt _U t _Ut U tcDt U

2cDt 2

and

tcDtU tUt _U tcDt _U

2cDt 3

rule), and at time t + Dt (using the three-point backwardmethod), respectively. The solution for t+DtU and thecalculation of the velocities and accelerations from thebackward dierence approximations in Eqs. (13) and(14) gives the complete response at time t + Dt. In ourstudies below we use c = 0.5.

3. Generalization of the composite scheme

The idea of using sub-steps in a given time step can ofcourse be generalized. For example, for n sub-steps, thetrapezoidal rule can be applied (n 1) times, and thenthe solution can be obtained at the end of the time stepby an (n + 1)-point backward dierence scheme. Ifn = 3, the sub-steps being equal in size, solutions attimes t + Dt/3 and t + 2Dt/3 can be obtained by two suc-cessive applications of the trapezoidal rule. The solutionat time t + Dt is then obtained by using the Houboltmethod based on the solutions at times t, t + Dt/3 and

0 0.02 0.04 0.06 0.080.08 0.1 0.12 0.14 0.16 0.18

0

2

4

6

8

10

12

14

Perc

enta

ge p

erio

d el

onga

tion

Trapezoidal ruleTwo substep composite scheme, =0.5 Wilson method, =1.4 Three substep composite scheme

2

K.J. Bathe, M.M.I. Baig / Computers and Structures 83 (2005) 25132524 2515The equilibrium equation (1) at time t + cDt is

M tcDt U C tcDt _U tcDtR tcDtF 7Substituting for tcDt U and tcDt _U in the above equation,and linearizing, the following expression is obtained (see[1]),

tcDtKi1 M 4c2Dt2

C 2cDt

DUi

tcDtR tcDtFi1

M 4c2Dt2

tcDtUi1 tU 4cDt

t _U t U

C 2cDt

tcDtUi1 tU t _U

8

with t+cDtU(i) = t+cDtU(i1) + DU(i), and t+cDtK(i1) beingthe consistent tangent stiness matrix at the congura-tion corresponding to the displacement t+cDtU(i1). Oncethe displacements have been computed, the velocitiesand accelerations are obtained from the relations givenabove.

Let the derivative of a function f at time t + Dt bewritten in terms of the function values at times t,t + cDt and t + Dt as [12]

tDt _f c1tf c2tcDtf c3tDtf 9where

c1 1 cDtc 10

c2 11 ccDt 11

c3 2 c1 cDt 12

Evaluating velocities in terms of displacements andaccelerations in terms of velocities, we have

tDt _U c1tU c2tcDtU c3 tDtU 13tDt U c1t _U c2tcDt _U c3 tDt _U 14Eq. (1) at time t + Dt is

M tDt U C tDt _U tDtR tDtF 15and substituting the above expressions and proceedingas for Eq. (8), we obtain

tDtKi1 c3c3M c3CDUi

tDtR tDtFi1 Mc1 t _U c2tcDt _U c3c1 tU c3c2tcDtU c3c3tDtUi1 Cc1 tU c2tcDtU c3tDtUi1 16

Note that NewtonRaphson iterations are performed inEqs. (8) and (16) with i = 1,2,3, . . . and appropriatetight convergence tolerances [1], in order to establish dy-

namic equilibrium at time t + cDt (using the trapezoidal0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

0

t /T(b)

Fig. 1. Percentage period elongation and amplitude decay forthe trapezoidal rule, the Wilson h-method, the two sub-stepcomposite scheme (c = 0.5), and the three sub-step compositet / T

4

6

8

10

12

14

16

18

Perc

enta

ge a

mpl

itude

dec

ay

Two substep composite scheme, =0.5 Wilson method, =1.4 Three substep composite scheme

(a)scheme. (Here, Dt is of course always the total time step size.)

t + 2Dt/3, see [1]. We call this scheme the three sub-stepcomposite method.

4. Accuracy of analysis

The scheme presented in Section 2 is in linear analysisunconditionally stable and second-order accurate be-cause so are the trapezoidal rule and the three-pointbackward dierence method [12]. Following the ap-proach in Refs. [1,3], we can evaluate the percentage per-iod elongation and percentage amplitude decay. Theevaluations are carried out for a simple spring mass sys-tem without any physical damping, and with unit initialdisplacement and zero initial velocity, as functions of Dt/T, where Dt is always the complete time step size and T isthe natural period of the spring mass system. The curvesobtained are given in Fig. 1, along with the curves forthe trapezoidal rule and the Wilson h-method. The com-posite scheme is seen to perform well when compared tothe other methods.

The gures also show the curves calculated for thethree sub-step composite method dened in Section 3.This procedure shows considerably more amplitude

In this section we present numerical results for three

posite algorithm (with c = 0.5) and the trapezoidal rule.Since the composite scheme retains stability due tothe numerical damping introduced by the three-pointbackward dierence method, we also test the Wilsonh-method (which is known to introduce numerical

1.0

2516 K.J. Bathe, M.M.I. Baig / Computers and Structures 83 (2005) 251325240 5.0 10.0time(s)test problems, obtained using both the proposed com-

E = 200x109N/m2

thickness = 0.01mplane stress

= 8000kg/m3 = 0.30

p = f(time)N/m2

0.05 m

0.05 m

f(time)

y

z

Adecay than the two sub-step composite scheme.

5. Numerical examplesFig. 2. The rotating plate problem.

Fig. 3. The rotating plate problem; results using the trapezoidal

rule; Dt = 0.02 s.

damping as well, see Fig. 1) in the solution of the prob-lems, with h = 1.4. The test problems involve large dis-placements and rotations and the solutions illustratethe instabilities encountered using the trapezoidal rule.We use linear elastic constitutive relations, thereforethe nonlinearities in these problems are only due to largedeformations.

5.1. Rotating plate

A plate in plane stress conditions, modeled with four-node elements, is subjected to the loading shown inFig. 2. The load is applied normal to the plate boundaryfor 10 s to give the plate a reasonable angular velocity

K.J. Bathe, M.M.I. Baig / Computers and Structures 83 (2005) 25132524 2517Fig. 4. The rotating plate problem; results using the composite

scheme; Dt = 0.4 s.Fig. 5. The rotating plate problem; results using the Wilson

h-method; Dt = 0.02 s.

and is then taken o to have a conservative system fromthat instant onwards.

The problem is rst solved using the trapezoidal rulewith Dt = 0.02 s. The velocity and acceleration in the z-direction of point A on the plate are plotted along withthe angular momentum in Fig. 3. The response is mainly

in the rigid body rotational mode. The period of rigidbody rotation is about 12.5 s and therefore the time stepchosen should be suciently small to capture theresponse very accurately. However, after about threerevolutions of the plate, numerical errors start to accu-mulate signicantly, resulting eventually into very largeaccelerations. Consequently the angular momentum isnot conserved, and a point is reached at which the solu-tion cannot proceed any further.

The same problem is next solved using the proposedcomposite formula with Dt = 0.4 s (that is, 20 times thetime step used with the trapezoidal rule but of courseabout twice the computational eort per time step).Fig. 4 shows that the quality of response remains excel-lent. In fact, there is a negligible decay in the angularmomentum of the plate. This decay is less than 0.06%per revolution for the time step chosen. This solutionillustrates the superior and more robust performanceof the composite procedure in this long time durationproblem. It is also of interest to test the performanceof the Wilson h-method. Using Dt = 0.02 s, an accuratesolution is also obtained, see Fig. 5. However, the use ofa time step size Dt = 0.1 s resulted in a non-positive def-inite eective stiness matrix after only a few time steps,

2518 K.J. Bathe, M.M.I. Baig / Computers and Structures 83 (2005) 25132524Fig. 6. The rotating plate problem; results using the three sub-

step composite scheme; Dt = 0.4 s.probably because the solution at the discrete time t + Dtdoes not satisfy the dynamic equilibrium accurately (theNewtonRaphson iterations are used to satisfy dynamicequilibrium at time t + hDt, see [1,2]), and yet this solu-tion is used for the start of the next time step solution.

For comparison purposes, we also present the solu-tion obtained using the three sub-step composite scheme

0.01 m

0.5 m

60o

O

y

z

E = 200x109 N/m2 = 0.3 = 8000 kg/m3

Plane stress

g = 9.81m/s2

thickness = 0.01 mFig. 7. The compound pendulum in its initial conguration.

of Section 3 with Dt = 0.4 s, see Fig. 6. The integrationremains stable but has considerably more numericaldamping (and of course, for a given step size the compu-tational eort is larger than when using the two sub-stepcomposite algorithm).

5.2. Compound pendulum

Fig. 7 shows the compound pendulum considered.The bar is initially at rest and released to swing underthe action of the constant gravitational eld with a per-iod of about 1.25 s.

K.J. Bathe, M.M.I. Baig / Computers and Structures 83 (2005) 25132524 2519Fig. 8. The compound pendulum; results using the trapezoidal

rule; Dt = 0.005 s.Fig. 9. The compound pendulum; results using the composite

scheme; Dt = 0.01 s.

Forty four-node elements are used to model the pen-dulum, with 20 elements along the length and 2 in thethickness direction.

The problem is rst solved using the trapezoidal rulewith a time step Dt = 0.005 s. This time step size should

be small enough to capture the evolution of responseaccurately. Fig. 8 shows the calculated velocity andacceleration in the z-direction at the tip of the bar, alongwith the kinetic energy of the system. The trapezoidalrule performs well for a certain length of time after

2520 K.J. Bathe, M.M.I. Baig / Computers and Structures 83 (2005) 25132524Fig. 10. The compound pendulum; results using the composite

scheme; Dt = 0.02 s.Fig. 11. The compound pendulum; results using the composite

scheme; Dt = 0.04 s.

which the predicted velocity and acceleration responsedeteriorates noticeably, eventually resulting in very largevelocity and acceleration.

The problem is next solved using the compositescheme with Dt = 0.01 s, which requires about the same

solution eort as using the trapezoidal rule withDt = 0.005 s. The composite scheme performs well, giv-ing a good velocity and acceleration response, as seenin Fig. 9 which also shows the evolution of kinetic en-ergy of the bar. Although the response for only the rst

K.J. Bathe, M.M.I. Baig / Computers and Structures 83 (2005) 25132524 2521Fig. 12. The compound pendulum; results using the Wilson

h-method; Dt = 0.005 s.Fig. 13. The compound pendulum; results using the Wilson

h-method; Dt = 0.02 s.

backward dierence approximation is used. There are no

N/m2

0.02

odeled

2522 K.J. Bathe, M.M.I. Baig / Computers and Structures 83 (2005) 2513252420 s is shown, the problem was actually run for a totaltime of 150 s, and the solution was observed to stay sta-ble and accurate. Also, in our experience, the algorithmremains stable if a larger time step is used, introducing,however, greater numerical damping resulting in re-duced accuracy of the solution. This loss of accuracydue to the increase in numerical damping is illustratedin Fig. 10, which shows the results obtained using thecomposite scheme with Dt = 0.02 s, and more so inFig. 11 which shows the results obtained using the com-posite scheme with Dt = 0.04 s.

Next we solve the problem using the Wilson h-method with Dt = 0.005 s. Fig. 12 shows the solutionobtained which is very accurate. Fig. 13 shows the solu-tion calculated using the Wilson h-method withDt = 0.02 s and this gure shows that the solution accu-racy is similar to when using the composite scheme withDt = 0.04 s. Since the composite scheme uses two solu-

500 f(time)

0.4 m

E=70 x 109 N/m2

plane strain

=2700 kg/m3=0.33

f(time)

0

1.0

Fig. 14. Cantilever beam mtions per time step, the solution eort is about the samein these two cases. However, the use of a larger timestep, e.g., Dt = 0.03 s, with the Wilson h-method resultedin a non-positive denite eective stiness matrix afteronly a few steps (an instability we also encountered inthe previous example).

5.3. Cantilever beam

The response solved for in the two previous test prob-lems involved large rigid body motions over long timeintervals. Here we consider the cantilever beam shownin Fig. 14, modeled with a 400 1 mesh of nine-node ele-ments and subjected to pressure loading. The beam issupported to prevent rigid body motion but undergoeslarge displacements.

Figs. 1517 show the calculated response of the beamat its tip using the trapezoidal rule, the compositescheme and the Wilson h-method. As in the solutionspecial calculations to start the time integration. Thecomposite scheme is available in the ADINA program,in particular for the solution of uid ow structuralinteraction problems, where rst- and second-order sys-tem equations are fully coupled.of the previous problems considered, the solution pro-vided by the trapezoidal rule is not acceptable, whereasthe composite scheme and the Wilson h-method performwell.

6. Conclusions

We focused on a composite single step direct timeintegration scheme. The procedure uses two sub-stepsper time step Dt: in the rst sub-step the usual trapezoi-dal rule is used and in the second sub-step a three-point

0.001 m

time (s)0.04

using nine-node elements.In this paper we presented the scheme for nonlineardynamic analysis of structures and demonstrated its per-formance relative to the trapezoidal rule and the Wilsonh-methodrepresentative of other widely used timeintegration methodsby solving three test problemsthat are useful to test time integration methods in non-linear dynamics. For a given time step size the compositescheme is about twice as expensive computationally asthe usual trapezoidal rule and the Wilson h-method,and hence we are primarily interested in the schemewhen the other techniques are not stable. The numericalexamples solved show the algorithm to remain stable forlarge time step sizes; but when the time step is too largethe numerical damping can be appreciable.

When comparing the performance of the methods, thecomposite scheme can be signicantly more eective thanthe trapezoidal rule when large deformations over longtime ranges need be calculated. The Wilson h-methodalso provides quite eective solutions for the elastic

analysis problems considered herein. However, forinelastic and contact problems, integration methods thatestablish equilibrium at the actual physical times, anduse these equilibrium states to march forward arepreferable.

The basic approach used in the construction of thecomposite scheme is to use two second-order accurate(and in linear analysis, unconditionally stable) schemes,one of which introduces a small amount of numerical

K.J. Bathe, M.M.I. Baig / Computers and Structures 83 (2005) 25132524 2523Fig. 15. Cantilever beam; results using the trapezoidal rule;

Dt = 0.002 s.Fig. 16. Cantilever beam; results using the composite scheme;

Dt = 0.004 s.

nite element system. Of course, a small amount ofdamping can also be introduced in other ways, but the

2524 K.J. Bathe, M.M.I. Baig / Computers and Structures 83 (2005) 25132524damping. This approach is quite dierent from the ap-proach of constraining the energy and momenta of the

43651.[9] Bathe KJ. ADINA system. Encyclopaedia Math

1997;11:335. see also www.adina.com.

Fig. 17. Cantilever beam; results using the Wilson h-method;Dt = 0.002 s.[10] Bathe KJ, Zhang H. Finite element developments forgeneral uid ows with structural interactions. Int J NumMethods Eng 2004;60:21332.

[11] Baig MMI, Bathe KJ. On direct time integration in largedeformation dynamic analysis. In: Bathe KJ, editor.Computational uid and solid mechanics 2005. Proceed-ings of the third MIT conference on computational uidand solid mechanics, 2005, p. 10447.

[12] Collatz L. The numerical treatment of dierential equa-tions. third ed. New York: Springer-Verlag; 1966.basic aim then needs to be to preserve stability andsecond-order accuracy.

The composite scheme is attractive because only theusual symmetric stiness, mass and damping matricesare used, and no additional unknown variables (i.e., La-grange multipliers) need to be solved for. Indeed, theimplementation is as straightforward as the implementa-tion of the trapezoidal rule. Therefore the compositescheme is of value in analyses where the trapezoidal ruleand other techniques do not give suciently accuratesolutions.

References

[1] Bathe KJ. Finite element procedures. New York: PrenticeHall; 1996.

[2] Wilson EL, Farhoomand I, Bathe KJ. Nonlinear dynamicanalysis of complex structures. Int J Earthquake EngStruct Dyn 1973;1:24152.

[3] Bathe KJ, Wilson EL. Stability and accuracy analysis ofdirect integration methods. Int J Earthquake Eng StructDyn 1973;1:28391.

[4] Bathe KJ, Ramm E, Wilson EL. Finite element formula-tions for large deformation dynamic analysis. Int J NumMethods Eng 1975;9:35386.

[5] Kuhl D, Criseld MA. Energy-conserving and decayingalgorithms in non-linear structural dynamics. Int J NumMethods Eng 1999;45:56999.

[6] Simo JC, Tarnow N. The discrete energy-momentummethod. Conserving algorithms for nonlinear elastody-namics. Z Angew Math Phys 1992;43:75792.

[7] Laursen TA, Meng XN. A new solution procedure forapplication of energy-conserving algorithms to generalconstitutive models in nonlinear elastodynamics. ComputMethods Appl Mech Eng 2001;190:630922.

[8] Bank RE, Coughran Jr WM, Fichtner W, Grosse EH,Rose DJ, Smith RK. Transient simulations of silicondevices and circuits. IEEE Trans CAD 1985;CAD-4(4):

On a composite implicit time integration procedure for nonlinear dynamicsIntroductionThe composite time integration procedureGeneralization of the composite schemeAccuracy of analysisNumerical examplesRotating plateCompound pendulumCantilever beam

ConclusionsReferences