J. ChungGraOuat e Research Assisia nt .

G. M. Hu|bertAssistanI Professo r .

ASSOC. Mem . ASM E.

Depan mem Mec hanica | Eng ineer in9 ,and App lied Mech anic s ,

The Univef Sity Mic higan ,Ann Arbor , M | 48 1% 2125

1 I n t r o d u ct i o nT h e n eed h as b een I ng r ecogn ized f o r st eF by -step t im e

i n tegr at i o n aIg r i t hn1S t possg s a lgo r i t hm i c d am p in g w henso lv i n g St ru ct u r al dyn am ics p ro b lem s. I n par t i cu la r , i t k d e-si r abIe t o hav e con t ro l lab le n um er ica l d issi pa t io n m the h i gherf r equency m od es sin ce, u si ng st an d ar d fi n i t e elem en t s t d is-cr et i ze t h e spa t i a l d o m ai n , t he sp ati al r eso l u t io n o f t h@e h igh -f r equen cy m od es t y p i@ l l y i s poo r - B y u sin g alg i t hm s w it hh i gh - fr eq u ency d issipa t i n , sp u r io u s hi gh - f r eq uen cy respo n sem ay be da m ped o u t . A lso , w hen so lv ing h igh ly n n li n ear p r o b -lem s, hi gh - f r eq u ency n u m er i caI d i ssi pa t io n has b een f o u n d t oim pr o ve th e co nv er gen ce o f i ter at i v e eq ua t io n so lv er s. H ow -ev er , t h e ad d i t io n o f h igh - f r eq u en cy d issi pat i o n sh o u ld no ti ncu r a loss o f accu r acy no r in t r o du ce ex cessiv e algo r i t hI11i cd am p i ng in t he im po rta n t l o w f r eq uen cy m o d es. F o r ex am p le ,t he N ew m ar k f am i ly o f al g r i t Inn s co n t ai n s m eth od s t h at po s- Ss high - f r equ en cy d issip at i o n (N ewm ar k , l 959) ; h o w ever ,t h@e m et hod s ar e o n ly f i r st - r d er accu r ate and X e t oo d is-sipal i ve in t he lo w - f r eq uency do m ai n . N um er ou s d i ssipa t iv eal go r i t hm s hav e been d ev elo ped t h at at tai n h igh - f r eq uen cyd i ssi pat i n w it h l in k lo w - f req u ency d am p ing w h i le m ai n t ai n -i ng seco n d-o rd er accu r acy ; e.g ., th e 8 m et hod o f W i ls n ( I 968) ,t he H H T -a m et hod o f H i lb er , H u ghg and T ay lo r ( l 977) , theW BZ -a m etho d f W oo d , B o ssak , and Z ien k i ew i cz ( I 98 I ) ,t he p m et hod o f Bazzi an d A nd er h eggen ( l 9 82) an d t he 8 l-

Cot ri bu ted by tht A pp li ed M echn ics Di vi si on o f T HE A l g l lC S@CI TYOF MI Ca m CAL EX6Ri ER S f@r prei t n !al in aI l he First h in t A SCE -EKI D -A S{ EA M D , SES k a i ng , Char lot t eg i lle, V A . June 6- 9, I993 .

Discussi o n OB lh i s ppef Sbw ld be dd ressed lo l he Technici l Ed itOf . F r@f ess r U w i s T . W heeler , Depg m i enI r M echi nici l Engi neer ing- U niv er sit y o fH Oll i m n - H uSi on - TX 77204 4 72 , nd wi lI k ccep ted un Ii l fur moIII lls af Ier pubUCi i i the pgper it sel f in l he A SM E JURN OF A PFLI P M ECl i ASICS.

ki l nl l i ch p1 r@g ived by the A SM E A pp l ied M echan ics Divi si on . Dec . 2, l 99 l ;f inal revi sion , A ug. n , l 9q2. A g od ate T echni cal E4 i IOr : C . F . Shi h .

Paper No . 93-A Pk, .ZO.

Journa| of App|ied Mec hanics

A Time l ntegrat ion Algorit hm torSt ruct ural Dynamics With|n1proved Numer ical Dissipat ion :The Generalized- Met hodA new f amily qf t ime integration algor ithms k p r@ented f or soIuing st ructuraldy namics p r oblems. The new method, denoted as the generalized-a method , p os-S@Ses numer ical d iS lPat ion that can he controlled by the user - I n pa iculari i t i sshown that the generaIized-a method ach ieve5highf i ei1uency d issipation while min-im izing unwanted Iow-j kequency d issipation- CompGr isons aM gi uen qf the gener -alized -a method with other numer icaIIy d isslPatiue t ime integration methods; thesera Il lts highl ight the irnp r oued perf ormance qf the new aIgor ithm. 77Ie new algor i thmcan be easi ly imp lemented into p rograms that already include the N ewmark andf f i b er -H ughes-Tay Ior-@ t ime integration methods.

I988b) (also Hoff a .method of Hoff and PahI (I988&( I989)).

T h e m at r i x eq u ati o n o f l i n ear st r u ct u r al dy nam ics i sRI X + C V + K X = F ( l )

w h ere M , C , and K ar e t he m ass, dam p i ng , and st i f f ness m a-t r i ces, r e3pect i v ely , F k t he v ect o r o f ap p l ied load s (a gi venf u n ct i o n o f t i m e) , X k t he v ec to r o f d i sp lacem en t u n k now n s,an d super po sed d o g i n d icate d i f f eren t iat i o n w i t h r espe t ot im e- T he ini t i al v al u e p r o b lem co nsi sts o f f i nd in g a f u nct i o nX = X ( t ) w h ich sat i sf i es ( 1) f o r a l l t e IO, t , tN 0 , and t h ei ni t ial co ndi t io n s

X (0 ) = d (2)X (0) - v (3)

w here d and v ar e gi ven v ecto r s o f i n it ia l d isp lacem en t s andv el oci t ies , r espect iv ely . Th e n um er i cal l y d i ssi p at i v e t im e in -t egr at ion al go r i t h m s Ii st ed abo ve have t h e f o ll w ing co m m o nto m : d ,,, an d e gi ven app r ox im at i ns t o X ( t) ,X ( t, ) , an d X ( t ) , r espect iv ely . E x p r essi n s f o r d R+ I an d Vn+ lar e sp eci f ied as l in ear co m b in at io ns o f d ,, , v ,,, a, an d a+ l A nad di I i on al eq uat io n k need ed t o det er m i ne g* l T hi s ad d i -t io na l eq uat i o n r ep resen ts a m d i n ed ver sio n o f E q . ( l ) . E achal go r i t h m i s def in ed b y t he spec i f i c f o rm o f t h e d isp lacem entan d vel o ci t y u pd a t e equ at i o n s an d th e m od i f ied b al an ce eq ua-t i n . A lgo r i t hm s havi ng t h i s f o rm m ay be classi f ied g o ne-Step , t hr ee-St age (o r t h ree- lev el ) t i m e in teg rat io n m et ho d s. T healgo ri t hm s ar e o ne-st ep m et hods because t h e so lu t io n at t im et l + l d epen ds o n l y o n t h e so lu t i n hist o ry at t i m e . T he th r ee-st age desig l ati o n re fer s t o t he so l u ti o n bein g d escr ibed by t heth r ee so lu t i o n v ect o r s: 4 , VR, and an

I n t h i s paper , w e p r esen t a new f am i ly f o n e-st ep , t h r ee-stage , n um er i ca l ly d i ssip a t iv e t i m e in tegrat io n algo r i t hm s. T h isf am i ly , w h ich w e cal l t he gen er al izt d @ m et hod , c m ai ns th eH H T -a and W B Z @ al g r i t h m s; in ad d it io n , w e id en t i f y a n ew

JU N E 1gg3, Vo l - 6O I 371

algo r i t hm t ha t ach iev es an o p t im al co m b in at i o n f h i gh - f r e-q uency an d lo w - f r eq u ency d issipa t io n . T h at is , f o r a gi v enval ue o f h igh - f r eq u en cy d issip at io n , t h i s o p thn al al go r i thmm in i m i zes t he lo w - f req uency d issi pat io n . F ro m a p ract i ca lv i ew po in t , t he gener al i7x d -@m et ho d is p ar t i cu lar ly co n ven ien tsin ce th e al go r it hm ic par am eter s ar e def ined in t erm s o f t h ed g i r d am o u n t o f h igh - f r eq u en cy d i ssip at io n . T h e d esign o ft h e gener al i zed -@ m eth od is d iscussed and co m p ar iso n s w i t ho t her n um er ical l y d i ssi pa t i ve a lgo ri t h m s ar e p ro v id ed t ha tdem o n st rate t he im pr o ved per f rm an ce o f t he new t im e i I1-t egr at i o n m et hod -

2 T he General ized-a A lgori thmT he basic f o r m o f t h e gen er al ia d -a m et ho d is g iv en by

d l - da + Am + A F ( G - a) , + A R+ l)

Vn* I == v ,, + A t ( ( l - y )an + 7i , + l )

l l a n + l + C v, + l - af + I n + I - af = F ( 1, + l - gIf )

do = d

Vo = vao = M - l (F ( - C v - K d )

w h er ed + l - - f ( l - @ d + l + @/ R

Vl * l - ( I - a A Vn + l + ,,

. l * l - I,,, = ( I - - a , ,) R+ l + a ,,. ,

t , + l - - f I I ( l - a j + I + a j i ,

i n w hi ch n f { 0 , l , . . . ,N - , N i s th e n um ber o f t i m e st ep s an dA t i s t he t i m e step - T h e d i sp lacem en t an d v elo ci t y u pd a t e E q s .(4 ) and (5) ar e id en t ical t o t ho se o f t he N ew m ar k al go r i t hm .T he st ru ct u r e o f t h ese u pd ate eq uat io n s w as o b ta i ned by re-st r i ct i ng t he su m o f t he coef f i ci en t s o f t hei r acceler at i o n t erm sto eq ual t h e cef f i ci en t o f t h e accelerat io n t ern1 i n a T ay lo rser i es ex p an sio n o f d ( tR+ l ) and v ( t ,, + l) abo u t t ,,. Si m p le nu -m er i a l ex per im en t s have sh o w n m at t h is u pd at e eq ua t io ng m u re resu l t s i n a m o no to n e in cr ease per peI io d m t he pea kd isp l acem en t and vel oc i t y er r o r s. T he m od i f i ed b al an ce eq u a-t i o n , (6 ) , i s ef f ec t i vely a c m b in a t i on o f th e H H T -a an d W B Z -a bal an ce eq u at i o ns. T he cru ci al task k t o d et erm i n e r ela-t i nshi p s betw een the al go r i t h m ic pa r am eter s, @f , a , ,, a , an dy . W it h ap p r o p d ate ex p r essio ns f o r 7 an d B , i f a = 0 , th eaIgo r i t h m red uces to th e H H T -a m et h od ; @f = O pr d u ces t h eW B Z @ m et h d ; @,,, = @j = O gi v es r i se t o t h e N ew m ar k f am i ly .W e Sh an sh ow th at Ot her ch o ices o f t h e al go r i t hm ic p ar am eter sen gender t i m e i n teg r at i o n al go r i t hm s w i t h bet t er n u m er i cald i ssip at i o n ch ar acter i st i @.

3 A n al y si s o f t h e G en er a l i R d - a A lg o l-i i h II1

F o r p u rp ses o f an aly si s, i t i s ad v an tage u s t o r ed u ce t heco u p led eq u at io n s o f m o t io n an d t he al go r i t h m ic eq uad o ns t oa ser ies o f u n co u p led sing le-d egr ee-o f - f reed o m sy stem s - T h isis accom p lished by i nv k in g t h e p r o pen y o f eigen v ect o r o r -th o go nal i t y ; see , e.g . , (H u gh g , l 987) . T he si ng le-d egr ee-o f -f r eed o m p r o b lem is gi v en b y

i + 2* U + Zli = j ; ( 14)

an d u ( = d , i (0 ) = U. T o st ud y t he accu r acy a nd stab i l i t y p rOFer t ies o f a 1 alg ri t hm , it su f fi ces t o c I1si der f = O f o r al l t e m ,tN] .T h e gener a l i zed -a m et hod , ap p l i ed t o ( I 4 ) w i t h .f = 0 , m ay bew r i t t en in t h e co m pa fo r m

x + l = A X ,,, n e l O, 1, . . . ,N - I l ( l 5 )

372 I v l . 60 , J UN E 1993

in which X = I d AI u, , At2an l T; A is cal led t he ampli f icati onII1a I I lX .

The a uracy of an algor ithm may be determi ned f rom itslocal t ru ncation erro r , T, def ined by

T= A t - 2S ( - I )iA iU( t,,+ l - i ) ( I 6)i - o

whereA o= l , A l is the t race f A , A 2 is t he sum o f the pr incipalm inors o f A and A 1 is the determi nant of A . A n al gori t hm iskth order a ii rate i rOvided that T= O (A H ) . T he geR r alized-a method is second-o rder accurate, provided

7 = i - a m+ W ( I7)

Thestabi Iity , numeri cal dissipat in and numer ical disper sionof an algori thm depend On the eigenvalues o f i ts ampli f i-cation matrU . The spect ral radius, p , of an algori thm isdef i I1edby

p = max( | k l | , |k2| , | 3| ) ( I 8)where L i s the hh eigenval ue o f A . O f par ticu lar interest ism e var iat ion i n p as a funct ion of n = At . A n algor ithm isuncondi t ionaIly stable for linear pr oblem s i f p I for al lQeIO, ) (st r ict ly less than one for repeated real r ts) . Thegeneralized-@ met hod k uncondit ional ly stable, provi ded

(4 )

( 5 )

( a

n )( 8 )

(9 )l l Ia , , a r s - - a = - + - (a - a .m l 2 4 z J ( I 9)

( l

( l I )

( I

( I 3 )

T h e spect r al r ad iu s a lso i s a m easu re o f nu m er i cal d issi pa -t io n ; a sm al ler spect r al r ad iu s v al u e co r rg po n d s t o gr a tern um er i caI d issip at io n . A l go ri t hm i c d am p ing m t he hi gh f r e-q uen cy r eg im e i s d g i r ed ; i t sho u ld be con t r o l lab le and sh u ldbe achieved w i t ho u t ind u ci ng excessi ve d i ssi p at i o n i n t he i m -p o r t an t lo w f r eq uen cy r egi o n . I n t er m s o f i@ spect r al r adi us ,an al go r i t hm w i t h t hese desi r ab le d i ssip at i o n p r o pert i es hassp ect r al I ad i l l Sva l u e c lo se to un i t y i I1 t h e lo w - f r eq u en cy d o m ainan d t he v alu e sm o o t h ly d ecr eases as Q in cr eases. T h is sm o o th -ness p r o per ty im p o ses sever al r est r i ct io ns o n t he v al u es o f t h eam p l i fi ca ti o n m at r i x ei gen val u es.

T o d iscu ss t hese r est r i ct i o n s, w e shal l o rder t h e ei gen v al i eso f A SO t hat f o r a c n v er gen t al go r i t h m t h er e ex ist s a p sk iv ev al u e , e .g . , Oc, su ch t hat i f O I l Qc, t hen t w o o f th e ei gen -v al ues ar e co m p l ex co nj ugate. T h ese t w o r o t s, l aI1d z, m et he p r in ci p al r om s w h il e is t he so -ca l led sp u r i o us r t .T yp i cal ly , i n t h e lo w - f r eq uency d om ai n , | k 3 l | k l ,2 | . T h u s,f o r t h e spect r al r ad i u s t o decr ease sm t h ly as Q i ncr eases,l 3| s | l .z l f o r al l I l e IO, ) . V i o la t i o n o f t h i s co n d it i o n i sm an i f ested i n th e m r1rg p on di ng spect ral r ad iu s p l t as a lSP

w her e, as n i nc reases, t h e spect r al r ad i u s al so in cr eases. Su 1an al go r it hm p o ssesses m o r e n u m er i cal d i ssipa t i o n f o r val ueso f g nea r t h e l Sp t h an i n t he h i gh - f r eq u em y reg io n an d t h usis o f less p r act i ca l i n t er est .

T I1e b ehav i o r o f t h e p r inc ipal r o o t s as a f un ct io n o f Q is al soim po rt an t . T o m ax i m ize h igh - f r eq u ency d issipat io n , th e p r i n -ci p a l r o s sh o u ld r em a in co m p lg co nj u gate as g incr eases .l f ro o t b i f u r ca t io n occu rs (bo t h p r i n ci p al roo t s b eco m e real ) ,t yp ica l ly n e r oo t h1Cr ea ses in m agn i I u de w i t h i ncr t asi ng Q;t h is r esu l t s m a i n cr ease i n t he sp ect r al r ad ius w i t h in cr easi n gn w h ich co rr espo nd s t o a d ecrease i n t he h igh - f r equ ency d is -si pat i o n . H igh - f req u ency d i ssi pa t io n i s m ax i m ized i f t he m in -ci pal r oo t s beco m e r ea l i n t he h igh - f r eq uency l im i t . T h at IS,i f t he p ri nd pa l r oo t s ar e gi v en b y k l j ( Q ) = A ( n ) iB ( Q ) ,w here A an d B ar e r eal n u m ber s an d i = J - I , t h en h lghf r eq u ency d i ssi pat i o n i s m i m i zed p r ov id ed l im g- - B ( 0 ) = 0 .F o r t he gen er a l i a d -a m et hod , t h is co nd i t io n k sa t isfi ed w hen

B = i ( I - @m+ aA Z (

T o su m m ar i ze t h e sm o o t hness d esi gn co nd i t io n s, w e r eq uu e

Transac t i ns { t he A SM E

r *-- @|As| Lr T-a i

Fig. 1 Class il icat iOn oI genera|ized-a met1d i!1 @ta - a, p@e

t he m agn i tud e o f t he sp u r io u s r oo t t o be less t han o r eq ua l t ot he m ag i t ud e o f t h e p ri nd paJ r oo t s f o r al l Q an d m e p r i nci p a lr oo ts t o hav e r eal val ues onl y m the hi 1- f req u en cy lim it -

P r v id ed ( I , ( I 9) , an d (20 ) h o ld , t h e gen er al i a d -g m et h od@ n be d g cr i bed i n t er m s o f t h e n vo r em ai ni ng f r ee par am et er s ,a m and aj ; th i s is dep ic ted in F i g . I . T he st ab i l i t y regi o n isdenot ed by t h e sh aded ar eas; t hese ar ea s ar e bo u nd ed b y t w oUI1es: ( I ) = - I , w h i ch co r r espo nds t o a , , @f i n ( I ; (2) T = - l , w h ich c rr espo nds t o af I / 2 i n ( l 9 ) ; t he @ su per -scr i p t on k den o tes t h e h igh - f req uen cy l i m i t o f t he r oo t .

t p . . deno t e t h e u ser -speci fi ed valu e o f th e spe@r al r ad i u sin th e hi gh - f r eq uen cy li m i t (user -sp ed fi ed h i gh - f r eq u ency d is-si p at io I1) . Sin ce , b y d g ign , w e requ i r e t ha t | l s | k l J | f o ra l l Q, p . . = | | . T h e v al ue o f a n k o b t ain ed usi ng

t lf - t l - 1 r z g . ( 2 l )

i t Xf - a m + I

T hus , gi v en A . , w e g n der ln e a ,,, an d af i n t er m s f P@ F orex am p le, th e H H T - m et hod i gi ven b y

l - pg@R IgI O, @f g T: (Z )

w her e p . I 1, I / 2I . Sim i l ar ly , u sin g (2 I ) an d r e@ ll i n g tha t aj = 0 ,t he W B Z -a m et hd i s gi ven by

Pg - I@f = 0 , a M = - - - ( 2 3 )

p . . + I

w h er e p . e [ I ,Ol . T h ese ar e m or e co nven i en t exp rg Sio ns f o r t heH H T and W B Z -@ al go r i t h m i c par a 11et er s t han ex i st m t h el i ter atu re as t hey d i rect ly rela te t he p ar am t t S t o th e d esir edval ue o f hi gh - f r eq uen cy d issi pat io n . l V e have f o u nd t hat f o ra gi v en l evel o f h i gh - eq u Cy d issi pa t io n , i .e . , f o r f ix d p .. ,low - f r eq uen cy di ssip at i n i s m i n im ized w hen g k . U sin g(2 I ) and

k = - gL , (24)af - l

low - f req uency d i ssi p at io n k m i ni m i zed w hen a= (@R + I ) / 3 .I t i s m o re c nv en ien t to d escr i be t h is o p t i m al case b y def in i nga , , an d @f in tern1S o f p . ;

2p - lq m = T , af = i @5)

T h e gen er al i zed -a m et hd , w i t h p ar am et r i c v aI l ies gi v en i n( I 7) , (20 ) , an d (25) , i s an u ncOM i t i nal l y st ab le , seco nd -o rd eraccu ra te aIg ri t hm po ssessing aII Op t im a l c m b i na t io n o f h igh -f r eq uency and lo w -f r eq u ency d issip at i o n .

J urna| f A pp |ie6 Mec h an ic s

A t / Tc mprisn , specrl rad ii l numerica||y d issipal ive a|goFig . 2

ri thn1S

Rem arks.( l ) T he opt imal generalized-@ met hod perm i ts hi gh-f re-

quency dissipation to vary fr om the no di ssipation case (p. = l )to the so@lled asymptot ic annih ilati on case @@= 0) . I n m elatter @se, hi gh-f requency respo nse iS ni hi lated af ter onetime step - I t is advantageo us, f r om the user view point , to ableto speci fy the algori thmic par ameters in term s o f the high-f requency dissipat ion , Pg. , since the degree of hi gh-f requencydissipat in desired is usual ly a known quant ity.

(2) H of f an d Pahl ( l 988a) presented a generalized alg-rithm possessing six f ree p aram et e ; a mdi ned version ofthi s algori thm was developed by H of f et al . (1989) for con-sistent tr eatm ent o f applied loads- The general ized-@ met hodis a member of the modi fi ed H f f -Pah l fami ly (for li nearproblems) by set ting 8l = & = & = I - w , i = ( 1 - ,,) / (1 - ,and R aI1d y gi ven by (20) and ( I 7), rg peed vely . H owever , weemphasize that t he preci se def init ions of t he algori t hm ic pa-ramet ers for the opt imal case ar e uni que t the present work .

(3) For t he ptimal case- the spurious rot vaIue in t helow-f requency li mit (g= lima- o3) is nonzero - W hi le concernhas been g p ressed in the Ii ter ature regar ding nonzero valuesof 3, we have shown that the impact on al gor ithmic accuracyi neg bk (H ulk rt , I 99 1) .(4) T hepr oblem o f over shoot in displacements obtai nedby time integrati n algor it hm s was Studied by H ilber andH ughg ( I 978) . W e note that the generalized-@ met hod doesnot exhibit the overshot phenomenon -

4 C o m par iso n o f A lgo r i t hm sIn this sect im1, we compare algori thms that possess numer -

ia l dissipat ion , in par d la , the H H T - , W BZa , p, 8h andgeneralized-a met hods-

Spectral radii o f t he algo ri thms are p lotted in Fig. 2. Foral l co mpar iso ns, the algori t hmi c param eter s were chosen suchthat for each algori t hm , the spect ral r adius in the high-f re-quency limi t k 0.8; no physical dam ping is incl uded in themodeI problem . It k cla r t hat for t he gj ven value o f pg , t hegener al ized- met hod with opt imaI parameters given by (25)(hereaf ter refer red to as the generalized-a method ) has a sig-I1l f i@In ly better spectr al radi us t han the other algor i thms. I tslm proved per fo rmance can be seen more clear ly when cm-paring t he numerical dissipati on and di spersion o f the var iousalgori t hms.

M easures of numeri cal d issipati n and dispersion are pro-vi ded by t he al g i thm ic dam ping rat io and relative per ioden or - Provided the pnp rots are complex , these rotSmay be expressed as

JUN E 19g3 , v |. 6O l 373

- - -

- - -

l /

A t / TFig.3 A|9ri l l1mic damping ratis o! numerica||y dissipal ive algori thms

P - - - - -8 l - - -

V B ?r g - - - - -H H T -a - - -

G en g l i ed -a - - - - ap Oid a h ------

0.4 }

0.3 }

6.2 }

Fig . 4 Re ltiv e pe f i d en ,-s I t i m e in Wg rl io n a lg Ofi t h m s

l ,z = ex p ( A t ( - f ) (2b )w h er e and ar e t h e algo r i t hm ic d am p i n g r at io an d al g&ri t hm i c uency , r g pect i v ely . T h e R l ve per io d er r o r i sgi v en b y ( T - n / 7; w h er e T - 2 f / an d T = 2 f / .

F igu re 3 sho w s al go r i t hm ic d am p ing r at io s fo rn the d i ssipat i v eal go r i t h m s. F o r t h e sam e v al u e o f h igh - f r equ ency d i ssi pa t i o n ,i t can be seen t h at t he gen eral iR d - m et h o d h as su bst an t i al l ylg s lo w - f req uen cy d issi pat i o n . R elat i ve per i od err o rs ar e sh ow ni n F ig . 4 ; also i n clu d ed i s t he r el at i v e p er i o d er r o r o f t h et r apezo i da l r u le algo r i t h m . T h e t r apezo i da l r u l e p o ssessgs t hesm al ls t peI i od er r o r o f se nd - rd er accu r at e , u ncon di t i o n al l ystab le li near m u l t i st ep m et h od s and t h$ i t s per i od err o r m aybe u sed as a basi s t m par e t he per i od err o r s o f m e nu -m er i@ l l y d issi pat iv e m et h od s. F r o m F ig . 4 , i t can be seen t ha tt he per io d er r o r o f t he gen er al i zed -a m et ho d i s cl os@t t o t ha to f t he t r aperA id al ru le al go r i t h m .

A s a f in al co m par iso n , ( I 4) w as so lv ed usi n g d i f f er en t t i m eim eg at i o n algo r i t h m s w it h = 1 f = 0 , d = I aI1d u = I . T h edi sp lacem en t er r o r (d i f f er en ce bet w een t he ex a ct an d n u mer i ca ld isp lacem en t val ues) w as c m p u ted at tN = 0 .4 . F igu re 5 sh ow st h i s er ro r as a f u n ct i n o f t h e num ber o f t i m e steps , N , f o rvar i us algo r i t h m s- A l l al g r i t h m s sh ow n ach i ev e se n d -o r deraccu r acy ; t he p-m eth o d i s no t inc lud ed as i t i s on ly f i r st -o r deraccu r ate . D ah l q ll i st ( I 96 3) p r o v ed t h at t he t r apezoi d al n l le h ast he sm al l est er r r o f u n co n di t i o naJl y st ab le , seco nd -o rd er ac-cu ra t e m eth od s; f r o m F i g . 5 i t m ay be seen t ha t am o n gst t henu m er ical l y d issi p at i ve algo r i th m s i nvest i ga t ed , t he gen era l -i a d -a m et ho d h as t he sm al l est er r r .

374 l Vo |. 60 , J UN E 1993

a

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NFig. s c mParison oI disp|acemenl errors , l ime integ ral in algtF n 1l11s . N is th e nu m be r , t i m e si p s -

5 C OIl c l l l Si o n sA new f am i ly o f t i m e i n t egr a t i on m et h o d s has been p r esen ted

f o r so lv i ng st ru ct u r aI d y n am i @ p r b l eIn s . T h e new schem einchl d es b o t h t he H H T , a and W B Z -a m et h od s; in deed , t henew gen er al i zed -@ m et h od m ay be co n sid er ed as a syn t h @i s r gener al i zat io n o f t h ese tw o m et h od s . A n an al ysis o f t hegener ali zed -a m et hod w as per f o rm ed t o d ef ine app r o p r i ateval ues o f t he al go r i t h m i c pa r am et er s- Ex p ressi o n s w er e p r o -v ided t hat d ef i ne t h e al g r i t hm ic p aram a er s i n t er m s o f a user -speci f i ed v a l ue o f hi gh - f r eq u en cy di ssi pa t i o n . T h e r esu l t an ttun e in t egr at io n al go r i t h m i s u nco nd i t i on al l y St ab le , seco n d -o r der accu r ate an d po ssesses an o p t i m a l co m b in at io n f h igh -f req u ency an d lo w - f r eq u ency di ssip at i o n . T h at i s , f o r a d esi r edlevel o f h i gh - f r eq u m cy d issip at i o n , th e lo w - f r eq uency d i ssi -p at io n i s m i ni m i a d . T h e gener al i zed -a m et h o d w as sho w n t oh av e bet t er n u m er i cal d i ssip at i on char act er i st ics , an d sm al lerp er i o d and d i sp la cem en t err o r s t han o t her n um er ical l y d issi -pat i v e sch em e5. W e em p hasize t hat t he im p r ov ed per f o rm an ceo f t h e algo r i th m i s n t o b ta i ned by in cr easi ng th e co m p lex i t yo f t h e al go r i th m bey n d t h a t associ at ed w i t h t he H H T -a an dW B Z -a m et ho ds. T f 1i s i s i I I1p r t an t f r o m a p r act i cal v i ew po i n tsin ce t he gen er aIi a d -a m et hod can be di r ec t l y im p lem en tedw i th m i n i m al ad d i t i on al cod i n g in t o p r o gr am s t hat al r eadyi nclud e t he N ew m ar l H H T a o r 1V B Z -a m et hod s.

T he gen er al i zed -@ m et h o d has been p r esen t ed f o r st ru ct u raldyna m i @ . O f in t er est i s ext en d i n g t he m eth od t o syst em s o ff i r st - r der o rdi nar y di f f eren t i al eq u at i o n s- R esea r ch ef f o rt s areu n d er w ay i n t h is ar ea as w el l g d evelo p in g im pr o ved im p l i ci t -ex p l ici t t im e im eg a t i o n m et ho d s u si n g t he gen eral ized *m et hod as t h e par en t im p l i ci t algo r i thm -

A ck no w led gm eRtv e would li ke to acknow ledge suppon provided by a K orean

Government Overseas Sch lar ship for the f ir st author .

R e f e r e n c e sBaz . G ., Iid A nd- r heggen , E . , l 982, T he -Fm 1i ly o f A lgor i thm s for

T lme-Step l nteg l im wi Ih Im proved N uma ia l D issip- t in , Ear thq uake En-s ineer ing and Stm etral Dy aam ig , V o l . l O, p p . S37- 5% .

Dahlquist , G ., I963. A Sp t d l Stabi l i Iy Pr bk m f r L i nei r M ult istep M a hods, BI T - V . 3, pp . Z74 3 .

H i l ber , H . M . , H ughes, T . 1 . R . , i nd T l y lo r , R . L ., l 977 , Im pr oved N u-meTical Dissi Pal in fm T lm e l nl eg al i OI1 A l go ri l hms in Si m a uri l Dyn am ics,Ear thquake Ensma r tng and St rMetJral Dy lan,i@, v l . S, pp . 283- 292.

H il ber , H . M . . l nd H ughes. T . J . R . . I 97g, c o llocat ion - Dissi p- l i n n d ve hoor r 11rne l nIegr I I i!I Schem es i I1 Sl m a w - l Dyrl I nk s--- Ear thquakeEngiM er ing and Str MCturd Dy nam ig , V . 6, pp . 99- l I g -

H o f f , C . , l nd Pahl , P . 1 ., l g ga , DeveM pmem of l n l m pUd I M a hod wi I l1

Transac l ions oI i he ASME

N umer ic- l Dissi pal i on fr m a Gen er al i zd Singles tep A lg r i l hm f o r S m a ll I alDy nl n1iCi , c omp ttef Af etho4 I m A pp li ed Af er haRi cs and E ng i neer in& Vo l -67, pp . 367- 385.

H o f f , C ., I nd P l hl , P . 1., I% 8 b, --p r an k i l Per f rm M g o f l he e, M el hdl nd Com pri son wi I h Ot ha Di ssip- ti ve A lgo r i thm s in Stm a ur al Dynam ics,Comp uer M ethods m A pp li ed M eekania and E ngineer i ,,g , Vo l . 67 , pp . 87-l l o.

H o f f - C ., H ugh- s. T . 1 . R ., H ul k f 1, G . , M d Pi hl , P . 1.. I 989 . Ex end- dCo mpg i i n f l he H i l ben -H ugh & T l ylm aM et hod I nd the el-M a hod , Com Pli Ief M ethods m A pp l ied M K ha l it s aIId E ,,g ink r ing , V o l - 76 , pp . g7- 93.

H u gh g , T . J . R . , I 9g 7 , T I l e F M i t e E l em e n t M e t h o d -- L i Rea r S t a t ic a, ,d D y

,,a m ie F i ,, i te E I en f e ,, t A n a Iy s i s - P r em i ce-H l l , E n gl ew o o d C U f f s, N t - l er g y .

Jurn al of App|ied Mechanics

H ulbg 1, G . M .. n d C hung , 1 ., I 99 l . --o n the (N n)Im p n ance o f l heSpurious R f T im e I al eg al i I1 A lgori t hms fOf Str uctu ral Dyni m it - sumi n ed tr publi ci t im 1 in C om m un im t ions iR A pp l ied N umer ics l M ethods .

N ew n1i I k . N . M .. I 959 , .A M et hod o f Com pu ll i i n f of Sim a uri l Dy -namics, JOMrnal of the E RgiRet ina M t hanit s D i visi on A SCE , Vo l - 8S, N o .EM 3, pp . 67- 94 .

W ilsm , E . L ., 68, A Comp g ter P rogr am f o r the Dynamic St rt S A naly* isof Undergrod St ri ctu res , SESM RePor i N o . 6g. l , Di vi sion o f SIruct ur i lEn M e i nl nd Sl m a w - l M echi n ics, U ni versi ty o f Ci li for ni- - Berkel ey . C A .

w d , W . L . . Bossak . M . . 1d Zien Kievticz, 0 . C .. I 98 l . A n A lphl M d -i6 Cal ion o f N ewm i r is M a hod -- I Rter ,,at iOIMI l oMn !al f o r N umer ical M et hodsin Eng ineer ing , Vo l - l 5, pp . I 5624 5% .

J UN E 19g3, v |. 6O l 375