COMPUFER METHODS IN APPLIED MECHANICS AND ENGINEERING 76 (1989) 87-93 NOR H I -HOLLAND

EXTENDED COMPARISON OF THE HILBER-HUGHES-TAYLOR a-METHOD AND THE O~-METHOD

C. HOFF SEMM, Department of Civil Engineering, University of California at Berkeley, CA 94720, U.S.A.

T.J.R. HUGHES, G. HULBERT Stanford University, Department of Mechanical Engineering, Stanford, CA 94305, U.S.A.

P.J. PAHL Department of Civil Engineering, Technical University of Berlin, DIO00 Berlin 12,

Federal Republic of Germany

Received 19 October 1988 Revised manuscript received 30 January 1989

Two unconditionally stable implicit methods with controllable numerical dissipation are compared, namely the improved version of the Hilber-Hughes-Taylor a-method and the recently developed Ol- method. The accuracy, spectral pxoperties and implementational aspects in nonlinear problems are discussed. Both methods maintain second order accuracy in the presence of viscous physical damping and numerical dissipation. They are overshooting-free and their spurious roots vanish for small time steps. Applied to nonlinear problems, two alternative implementations can be employed. Both methods are similar in the first implementation, whereas differences occur in the ~econd.

1. Introduction

For structures subject to low frequency excitations, vibrations tynk:ally ensue in the lower part of the eigenfrequeney spectrum. Under these circumstances, in numerical simulations time steps are taken as large as possible to save cqmputer time, but small enough to maintain sufficient accuracy. Although higher frequencies of the system are poorly approximated, a step-by-step integration method which damps out the high frequency oscillations may simulate the real physical behavior better than a nondissipative method like the often used trapezoidal tale of Newmark t~].'" 1

Many attempts have been made to introduce numerical dampil~g in a step-by-step integra- tion method. Hoff and Pahl [2] compared some methods with the Ol-method, which was recently ~atroduced in [3]. The first version of the a-method, see [4], was used in [2] to compare accuracy and stability properties. Hughes and McVerry improved the a-method with respect to the viscous damping term, see [5, 6], In the improved version, the a-method is also second order accurate in the presence of viscous and numerical damping. Since this was not considered in [2], an extended comparison is made herein. Both methods can now be presented within the generalized algorithm ,designed by Hoff and Pahl [3]. The order of accuracy and spectral properties are considered and some implementational aspects for

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88 C. Hoff et al., Extended comparison of the Hilber-Hughes-Taylor a-method and O-method

nonlinear problems are discussed. (Analyses of the a-method which did not consider physical damping were presented in [7, 8]).

2. Presen ta t ion of the a - m e t h o d a n d O l - m e t h o d in a genera l ized a lgo r i t hm

With the improvements of Hughes and McVerry, see [5, 6], the a-method fits also into the generalized algorithm designed by Hoff and Pahl [3]. To demonstrate how the parameters must be adjusted, the generalized algorithm for linear systems is written with the acceleration increment as the primary unknown:

(Or ~lU + 02~/ Ate + O3/3 At2K) Aa

= Pn + O,(p~+~- p . ) - U a . - C[v. + 01Ata.]

-K[d , + Ol Atv~ + ½02 At2a.], (1)

an + 1 = an + Aa ,

v,+t = v , + A t a , + ~/ At Aa , (2)

d,+ 1 = d,, + A t o n + ½ At2a. + ~ A t 2 A a ,

where M is the mass matrix, C is the damping matrix, K is the stiffness matrix, a n is the acceleration, v,, is the velocity, d, is the displacement and p,, is the external load.

In contrast to the original algorithm in [3], the param,~ter Ot for the load is replaced by the parameter 0t to get a consistent order of aocuracy in the load terms of (1). In Table I the six free parameters are adjusted to obtain the Or-method of Hoff and Pahl [3] and the Hilber-Hughes-Taylor a-method in the improved version, see [5]. The generalized algorithm is able to incorporate both algorithms, in the sense of an identical amplification matrix, so that the short and the long time behavior is exactly retained, see [3] for elaboration.

3. Accuracy, overshooting and spectral properties

To get second urder accuracy in the presence of physical and numerical damping, the parameters must fulfill the following condition, see Table 2 in [3]:

Table 1 Parameters of the Hilber-Hughes-Taylor a-method and the O~-method within the generalized algorithm

method Ot = ~)2 = 03 -- ~ = q = ~ =

1 1 1 HHT a (I + a) (I + a) (I + a) ~ (I - a) 2 ~ - a (I + a)

1 3 1 Oi O1 1 1

C. Hoff et al., Extended comparison of the Hilber-Hughes-Taylor a-method and O-method 89

"P= 01(~- 1)+ ½. (3)

It can be easily verified that the parameter values listed in Table 1 for the Hi lber -Hughes- Taylor a -me thod and the @l-method satisfy (3). The a -method is therefore also second order accurate for all a # 0 in the presence of physical damping. In both methods the spurious root goes to zero for small time steps because

~ 1 ~ = 1 (4)

is fulfilled, which guarantees good convergence behavior. Furthermore, both methods are theoretically overshooting-free because the parameters in Table 1 satisfy the condition

03. (5)

Conditions (4) and (5) were established in [3]. Numerical examples confirm these prop- erties. The practically useful ranges of the parameters are - 0 . 3 <~ a ~< 0 and 0.95 ~< O 1 <~ 1. For a < 0 or O1 < 1, the methods exhibit numerical damping. For a = 0 or e 1 = 1, the numerical damping vanishes and the methods become identical to the trapezoidal rule of Newmark.

The spectral curves for the physically undamped case were already presented in [2]. These curves are nearly identical for both methods. Small differences occur in the physically damped case. For example, the spectral radius and the algorithmic damping ratio are shown in Figs. 1 and 2 for 10% physical damping and the parameters a = -0 .1 and O1 = 0.955. (Note that Fig. 5 in [2] contains some errors for At~T>-0.3.)

4. Implementatlonal aspects for nonlinear problems

The use of step-by-step integration methods is called for in the case of the nonlinear problem:

0.951.00 ~ ......... ~ ~ . . . .

trapezoidal rule

~. 0.90

0.85 ~ f 01 method a method ~t~//.v.:~..~.~=..~,~,~.~ . . . .

o.SC.ol . . . . . . . il . . . . . . . . i . . . . . . . io . . . . . . ioo . . . . . iooo

at/T Fig. 1. Spectral radius in the case of physical damping, ~ = 10%.

90 C. Hoff et al., Extended comparison of the Hilber-Hughes-Taylor a-method and O-method

0.12 ..................................

o.lo

0.08 ~,~ f 01 method

" " ' ~ : : : . . . . . . . . . . . . . . . . . . . . ._-2.-.

o.oo -/2Zo2

0.04 trspezoidal rule

0.02

, l

°o~ . . . . . . 11 ........ i . . . . . . . io l oo . . . . i'0oo

~t/T

Fig. 2. Algorithmic damping ratio ~ in the case of physical damping, ~ ffi 10%.

Mii(t) + f(u(t) , fL(t)) = p(t) ,

with the tangent matrices

Ou ' Of~

u(O) = Uo, u(o) = Uo, (6)

(7)

A concise implementation becomes a crucial aspect for any method and is therefore discussed shortly. In contrast to the linear case, the nonlinear problem offers the two possibilities of putting the parameters either inside or outside the nonlinear operator f(u, ~). In parabolic problems a similar option exists which, in the case of one step methods, results either in the midpoint rule or in the trapezoidal rule, see [5] and references therein for a detailed discussion. Regarding the generalized algorithm, we describe first the case when the parameters remain inside the nonlinear operator, in which (1) and (2) can be written in terms of a predictor multicorrector scheme.

1. Predict the state vectors:

il ~i)- d. + 01 Atv , + ½0 2 At2a. ,

A

0 "~= v. + 0 1 A t a . ,

d ( i ) = a n •

(8)

2. Satisfy equilibrium at the ~ point in a loop over multicorrector iterations:

C. Hoff et al., Extended comparison of the Hilber-Hughes-Taylor a-method and O-method 91

M * Aa (i) -- r (i) ,

(9)

re ' )= ~lP.+, + (1 - ~I)P. - M # O _ f ( ~ o ) #Co),

a<'+') = ~<') + ~ At'- Aa co ,

(10)

~(,+1) _ ~ ( , ) + ~I~ Aa(O.

3. Calculate the final state vectors at the end of the time step:

a.+, = a. + ~ (#~+~)- a . ) , eJtT1

1 v.+l = v . + A t a . + *1 ~.'~ At(riO+t) - a . ) , (11)

71t7 t

I d.+x = d . + atv. + ~ Ataa.

As a consequence of putting the parameters inside the nonlinear operator, the internal forces f (a ¢°, ~o)) must be evaluated at an intermediate point L All stresses, external forces and boundary conditions must be related to this point. Only one vector of internal forces must be kept during the iteration. In this implementation the Hilber-Hughes-Taylor a-method and the Or-method are almost identical so that their computational effort is the same. For a - 0 and O, - 1, the trapezoidal rule of Newmark ensues and the difference between the inter- mediate ~ point and the final point t.+t vanishes. In this case, the vectors a c~÷1~, oct+t), #~+ ~) are identical to the final state vectors d.+,, v.+ ~, a.+,. The trapezoidal rule is therefore the most concise method.

If the parameters are put outside the nonlinear operator, we get for the a-method the following predictor multicorrector scheme:

1. Predict the state vectors:

d(l) .+l = d. + A t v . + ½ At2a. ,

t , ( t ) = vn + Ata n n + l (12)

a ( l ) _ a n n + l

2. Satisfy equilibrium in a loop over multicorrector iterations:

92 C. Hof f et al., Extended comparison o f the Hilber-Hughes-Taylor a-method and O-method

M * A a (i) = r (i) ,

M*=M+(1 +a) ( i -a )AtC+( l+a)~ (1 -a )2At2K,

_(i, (1 + a)f(d~i)+ " (i) ) + af(d~ v~) r (i) = (1 + a)p~+ 1 - a p . - Ma,,+l - 1, v~+l , ,

d( i+ l ) = d¢i) + ¼(1 - a ) 2 A t 2 Aa ~i) n + l - - n + l

v ( i + I ) . (i) + ( 1 __ a ) At Aa (i) n + l -~- g / n + l

a(i+ t) _(i) + Aa( i ) n + l --'- U n + l

03)

(14)

The state vectors of the converged solution in the multicorrector loop are defined as the = a~i+l) =,,(i+1) _o+1) In the a-method, the internal final state vectors, dn+ l "n+l , Vn+t "n+l , an+l = Un+l "

forces

f~ ,~ = ( l + a)f(d,,+~, v ~ + l ) - af(d, , , v,,) (15)

must be stored at the points (d~, v,,) and (d,,+l, v,,+l), whereas in the ~91-method

f,,+o =f(d~+,, v~+,) - [ f ( d . + (1 - 01) A t v . , v . + (1 - 01) Ata~) - f ( d ~ , v.)] (16)

would need additional evaluation at the argument (dn + ( 1 - 0 1 ) A t v ~, v~ + ( 1 - 0 1 ) A t a ~ ) , which can be considered as one more iteration step within the multicorrector. If the parameters remain outside the nonlinear operator, the a-method thus provides a more concise scheme than the Or-method. (See [9] for an alternative implementation of an implicit-explicit a-method.)

Since all accuracy and stability considerations are determined from the linearized problem, further investigations are necessary for arbitrary nonlinear systems. In this regard, dissipation of energy in the higher modes is only ensured if the internal forces f(u) are monotonic in u within the time interval, and the choice of the free parameters a and 191 with respect to the linearized system may not be optimal from the standpoint of accuracy for the" nonlinear system.

5. Conclusion

The two dissipative methods compared in this paper are both second order accurate in the presence of physical and numerical damping. They possess similar spectral properties and are both free of overshooting effects. In the implementation for nonlinear problems, the free parameters of the methods may be put inside or outside the nonlinear operator. The two methods can be implemented in the same way if the parameters remain inside the nonlinear operator. The a-method provides a more concise scheme than the Ol-method if the parameters are taken out of the nonlinear operator. Further investigations are necessary for nonlinear problems to determine the best treatment of the nonlinear operator and to define optimal values of the parameters a and O 1 with respect to accuracy and stability.

C. Hoff et al., Extended comparison of the Hilber-Hughes-Taylor a-method and O-method 93

References

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algorithms in structural dynamics, Comput. Mcihods Appl. Mech. Engrg. 67 (1988) 87-110. [3] C. Hoff and t~.J. Pahi, Development of an implicit method with numerical dissipation from a generalized single

step algorithm for structural dynamics, Comput. Methods Appl. Mech. Engrg. 67 (1988) 367-385. [4] H.M. Hilber, Analysis and Design of Numerical Integration Methods in Structural Dynamics, PB-264410,

Earthquake Engineering Research Center, University of California, Berkeley, 1976. [5] T.~.R. Hughes, Analysis of transient algorithms with particular reference to stability behavior, in: T.

Belytschko and T.J.R. Hughes, eds., Computational Methods for Transient Analysis (North-Holland, Am- sterdam, 1983) 67-155.

[6] T.J.R. Hughes, The Finite Element Method (Prentice-Hall, Englewood Cliffs, NJ, 1987). [7] H.M. Hilber, T.J.R. Hughes and R.L. Taylor, Improved numerical dissipation for time integration algorithms

in structural dynamics, Earthquake Engrg. Struct. Dyn. 5 (1977) 283-292. [8] H.M. Hilber and T.J.R. Hughes, Collocation, dissipation and 'overshoot' for time integration schemes in

structural dynamics, Earthquake Engrg. Struct. Dyn. 6 (1978) 99-118. [9] I. Miranda, R.M. Ferencz, and T.J.R. Hughes, An improved implicit-explicit time integration method for

structural dynamics, Earthquake Engrg. Struct. Dyn., to appear.