-
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: [email protected] (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.
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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