Application of an Innovative Precise Integration Method in Solving Equilibrium Equations of Motion for Structural Dynamic Problems

Transcription

1 Applicatio of a Iovative Precise Itegratio Method i Solvig Equilibrium Equatios of Motio for Structural Dyamic Problems Chiu-li Wu Natioal Ceter for Research o Earthquake Egieerig,Taipei City, Taiwa Chig-Chiag Chuag Departmet of Civil Egieerig, Chug Yua Christia Uiversity, Chugli City, Taiwa SUMMARY: A very popular approach to coduct structural dyamic respose aalysis is to first formulate its dyamic equilibrium equatios of motio, ad the employ a step-by-step time itegratio scheme to solve the equatios such that dyamic equilibrium is satisfied at discretized time istats. The selectio of time step size depeds o the features of the time itegratio approach, ad should cosider its umerical stability, desired accuracy, predomiat frequecies of the aalyzed structure as well as the major characteristics of the eteral loadigs. I case of high domiat structural frequecies or dramatic loadig variatios, a sufficietly small time step is usually required to achieve satisfactory umerical accuracy. The authors studied two major cases of employig both force ad mometum equatios of motio together with a so-called Precise Itegratio Method to solve for dyamic structural respose. Compared with most of the covetioal umerical itegratio methods i the literature, the proposed method is foud less isesitive to the selectio of time step size i case of impulsive loadig ad is able to provide superior umerical stability ad accuracy. This study also ivestigates the advatage i the use of a oliear polyomial fuctio i describig the variatio of loadig mometum withi a itegratio time step, which further helps loose the costrait of time step size required for better accuracy. Keywords: precise itegratio, umerical stability, accuracy, high-frequecy dissipatio, spurious resoace 1. INTRODUCTION It is well kow that mechaical goverig equatios describig dyamic egieerig problems basically take the form of partial differetial equatios. To solve the equatios, most umerical solutio methods (e.g., fiite elemet method, fiite differece method, etc.) start with a set of simultaeous ordiary differetial equatios (ODE) through discretized spatial coordiate, ad employ a direct itegratio method to obtai the fial solutio. I the field of structural dyamics, for istace, the origial Newmark s, Wilso-θ, ad Houbolt methods are very popular methods that ca be categorized as implicit methods; o the other had, the modified Newmark s ad cetral differece methods belogs to the group of eplicit methods. I either way, the time step size should be carefully selected i itegratio to esure umerical stability as well as accuracy. Takig eplicit methods for eample, the selectio of the time step size is more rigorous ad usually depeds o the highest domiat frequecy of the structural system beig aalyzed. Such a small time step is usually set to satisfy the requiremet of umerical stability with a tradeoff of icreasig computatioal efforts. I additio, the calculated fudametal periods ad vibratio modes highly deped o the selected structural model that is used to describe a real structural system. I some cases, the higher vibratio modes may ot represet the real respose of the origial structural system, ad will reduce the computatioal accuracy, ad ievitably affect the cofidece i iterpretig umerical results obtaied. As such, it is a commo cocer of may researchers how to develop a reliable itegratio algorithm for solvig simultaeous ODEs ad havig the capability of dissipatig fictitious high frequecy respose at the same time. Subbaraj ad Dokaiish (1989) preseted a thorough review o direct time itegratio methods developed before Oe ca also refer to Hughes (1987) for a umber of popular itegratio

2 methods i the literature. I additio to fiite differece methods, there are other schemes usig fiite elemet cocept to solve time itegratio problems, e.g., time-discotiuous Galerki method, which is capable of doig itegratio i oe sigle time step with self-startig iitial guess, ad provides higher accuracy ad better umerical stability through dissipative itegratio scheme elimiatig high frequecy oise (Chie et al., 003). I order to solve simultaeous ODEs i a more efficiet maer, Zhog ad Williams (1994) proposed the High Precisio Direct-L (HPD-L) method, which epressed the aalytical homogeeous solutio of the simultaeous ODEs i a fourth order Taylor series epasio to be itegrated usig the famous power-of-two algorithm, ad i the meatime the eteral loadig force was represeted by a segmeted piecewise-liear force history, i which the time step size was maily determied by the degree of oliearity of the eteral force history, while the ifluece of the atural periods of the aalyzed structure o solutio accuracy was foud isigificat. Li et al. (1995) ad She et al. (1995) adopted the same cocept with the loadig force history represeted i the form of the famous Fourier series epasio ad this approach was usually referred to as High Precisio Direct-F (HPD-F) method i the literature, which oftetimes was implemeted through parallel computig techiques to solve dyamic structural problems. Zhog et al. (1996) proposed a so-called Subdomai Precise Time Itegratio Method, which icorporated computatioal efficiecy of the fiite differece method to solve both liear ad oliear dyamic problems. Etedig from Tsai ad Chuag (00) that employed the Precise Direct Itegratio Method (Zhog ad Williams, 1994) i combiatio with umerical kietic dampig to solve static structural problems, this paper adopts Rayleigh dampig as a favorable modificatio for dissipatig spurious high frequecy resposes that oftetimes results from umerical modelig error rather tha real structural behavior. The theoretical backgroud of the Precise Itegratio Method as well as its implemetatio will be itroduced i the followig, ad its umerical stability ad accuracy are thoroughly discussed.. PRECISE INTEGRATION METHOD FOR SOLVING LINEAR DYNAMICS.1. Force equilibrium equatio of motio If we assume that &&, & ad are acceleratio, velocity ad displacemet vectors, respectively, the most structural dyamic problems of egieerig iterest, after spatial discretizatio, ca be epressed i the followig secod order differetial equatio of motio: M && + C& + K = F (.1) i which, M is the system mass matri; C is the system dampig matri; K is the system stiffess matri; F is the eteral force or iput vector applied to the system. Provided that iitial coditios 0 ad & 0 are kow vectors, the goverig equatio ca be rearraged i the followig mathematical form: v=av+ef & c c (.) i which, v = & 0 Ι Ac = Μ Κ -Μ C 0 Ec = -1 M (.3) (.4) (.5)

3 v is the state vector; A c is the time-idepedet system matri; E c is the iput distributio matri which is time-idepedet i this study; F is the eteral force (or iput) vector; I is the idetity matri or uit matri. The solutio ca be obtaied by solvig Eq. (.) with iitial coditios ad epressed as: () A ct A ct t A cs ( ) v t = e v0 + e e E 0 cf s ds (.6) The discrete form of Eq. (.6) ca be epressed as the followig form if the eteral force F is liearly iterpolated betwee two cosecutive time istats: v+1 = Tv + E0F + E1F +1 (.7) 1 1 E0 = Ac T + Ac ( I T ) Ec (.8) E1 = Ac + Ac ( T I) E c (.9) O the other had, a arbitrarily degree of oliear polyomial fuctio ca be used as well; thus, the above epressios for E 0 ad E 1 will take a differet form should the forcig fuctio be oliearly iterpolated betwee two cosecutive time istats. c/ m m c ( ) ( ) A Δ m c t A A T = e = e = e τ (.10) N i which τ = / m. If m is selected as a iteger power of (i.e., m = ) ad a fairly large N value is used (e.g., N = 0 ), the τ = / m will be etremely small such that trucatio error from higher order terms of Taylor series approimatio becomes egligible. With a very small τ, the A epoetial term e τ c ca be approimated by the higher order Taylor series epasio with satisfactory precisio, takig the fourth order epasio for a eample: ( ) ( ) ( ) 3 4 A e τ c I + Aτ + Aτ /! + Aτ /3! + Aτ /4! = I + T a c c c c (.11) It is oted that the elemets i the additioal matri T a are very small as compared to the idetity matri I. Substitutio of Eq. (.11) back ito Eq. (.10) gives: N ( ) ( ) ( ) ( N 1) ( ) ( N m I + T = I + T = I + T I + T 1) T = (.1) a a a I Eq. (.1), there are N matri multiplicatios of the (I+T a ) term, which ca be epaded ito: ( + ) ( + ) = + + I T I T I T T (.13) a a a a Whe the epoetial matri T is umerically obtaied through computer computatios, Eq. (.13) shall repeatedly multiply itself N times ad this ca be readily accomplished by a do loop computer routie. I such a routie, the umerical result of T a + T a is stored back ito the additioal matri T a each time, ad T a must be stored separately from the idetity matri I durig the computatios for better umerical accuracy istead of beig directly added to the idetity matri I; otherwise, the loss of umerical accuracy will be sigificat due to roud-off errors. This is because the elemets i additioal matri T a are very small as compared to the idetity matri I. After N matri multiplicatios, a

4 the elemets i T a are o loger small umbers such that the additio of T a ad the idetity matri I will have o serious umerical roud-off error. The algorithm give above is called the precise computatio of the epoetial matri as the separate storage of I ad T a durig the computatio process ca ehace the resultig umerical accuracy... Mometum equilibrium equatio of motio If the system M, C ad K remai costat over the time iterval of itegratio, itegratig the force equatio of motio Eq. (.1) with respect to time leads to: M w& + Cw& + Kw = J (.14) i which, w & = & dt ; w & = & dt ; w = dt ; J represets the time itegral of eteral force or the eteral mometum vector durig the time iterval 0 τ t. Eq. (.1) ca be epressed i state space form as follows: w& 0 = w&& M 1 K I M 1 w C w& M J (.15) Eq (.13) ca be solved usig the Precise Itegratio Method with eactly the same procedure previously metioed. 3. NUMERICAL STABILITY AND ACCURACY The Precise Itegratio Method eeds to first assume a N value i order to calculate the amplificatio matri; thus, its umerical stability depeds o the N value used for aalysis. Fig. 3.1(a) shows the relatios betwee spectral radius ρ ad /T usig the fourth-order Taylor series approimatio i case of a udamped SDoF oscillator uder free vibratio. A algorithm is cosidered as stable if the umerical solutio for free vibratio will ot grow without boud for ay arbitrary iitial coditios (i.e., spectral radius ρ 1.) It ca be see that whe the N value icreases, the rage of /T for maitaiig umerical stability becomes wider. Wheever umerical stability is satisfied, the values of spectral radius are mostly close to 1. O the other had, the umerical accuracy of a itegratio method ca be evaluated by its relative period error ( T T)/ T ad fictitious decay of vibratio amplitude. The latter ca be represeted by the so-called algorithmic dampig ratio ξ. The ifluece of both factors depeds o the umerical accuracy of eigevalues calculated from the amplificatio matri of the structural model. Assume the calculated eigevalues of the amplificatio matri for a udamped SDoF oscillator are: ( cosω ± i siω ) ξ ω λ 1,= a ± i b = e (3.1) i which, i = 1, ad b 0 ; Eq. (3.1) also implies: 1 ta ( b a) ( a > 0) 1 ta ( b a) + π ( a < 0) ( a + b ) Δ ω = (3.) l ξ = ω The relative period error is defied as t (3.3)

5 T T T ω ω = 1 = 1 ω ω (3.4) i which, T is the atural period of the structural model; T is the umerical period obtaied from the Precise Itegratio Method; ω is the circular atural frequecy; ω is the circular umerical frequecy obtaied from the Precise Itegratio Method. (a) (b) Figure 3.1. Relatios of spectral radius ρ vs. /T (left) ad ξ vs. /T (right) usig the fourth-order Taylor series approimatio i the Precise Itegratio Method. Figure 3.. Relatio betwee ( T T )/ T ad /T usig the fourth-order Taylor series approimatio i the Precise Itegratio Method. Fig. 3.1(b) shows the relatio betwee algorithmic dampig ratio ξ ad /T correspodig to differet N values uder the fourth-order Taylor series approimatio i case of a udamped SDoF oscillator uder free vibratio. It ca be see from Fig. 3.1(b) that whe N icreases, the algorithmic dampig ratio decreases. Whe N 4, ξ is approimately zero for /T=0~0.5. Fig. 3. shows the relatio betwee the relative period error ( T T)/ T ad /T correspodig to differet N values uder the fourth-order Taylor series approimatio. It ca be see from Fig. 3. that whe N icreases, the relative period error decreases substatially; whe N 1, the relative period error is approimately zero for /T=0~0.5. It should be oted that N=0 correspods to the ordiary itegratio algorithm

6 usig Taylor series epasio itegrated i oe sigle time step, which yields miimal algorithmic dampig ratio ad relative period error oly for /T 0.1 whe the fourth-order Taylor series approimatio is used. It ca be cocluded that the Precise Itegratio Method yields respose results with higher accuracy tha the ordiary itegratio methods if the same time step size is take i the aalyses. I the followig discussios, we will fid that the combiatio of N=0 ad the fourth-order Taylor series approimatio as origially suggested by Zhog et al. (1996) provides a ecellet match with its aalytical couterpart. 4. INCORPORATION OF SPURIOUS HIGH-FREQUENCY DISSIPATING CAPABILITY It is well recogized that the accuracy of atural period ad mode shape estimates largely depeds o the aalytical model used for dyamic respose aalyses; however, eve the best estimates of atural periods ad mode shapes obtaied from a sophisticated structural model may be still slightly deviated from its real life couterpart. As such, oftetimes a umerical model ca oly well represet the first few vibratio modes of the real structure but has larger relative period errors i higher modes; the high frequecy resposes from a umerical model are most likely spurious, ad more or less reduce the accuracy of umerical respose aalyses. Besides, the spurious high frequecy resposes may cause umerical istability. As the spurious high frequecy respose is udesirable ad better be removed, a modificatio is proposed here to eable the Precise Itegratio Method to get rid of the ifluece from fictitious high frequecy respose, ad keeps oly the most trustworthy respose from the first few predomiat modes. To reach this goal, a additioal dampig coefficiet matri C a that is liearly proportioal to system stiffess matri K ad time step size is icorporated ito the Precise Itegratio Method as follows: C = C + Ca = C + α K (4.1) i which, C is the modified system dampig coefficiet matri; C is the origial system dampig coefficiet matri; α is the dampig modificatio factor. A appropriate value is specified for dampig modificatio factor α to dissipate udesirable spurious high frequecy respose. The eigevalues of the amplificatio matri for a SDoF system after icorporatig umerical dampig ca be epressed as follows: ep -ξ ω ± iω λ = 1, ep ξ ω ± ω 1 ξ ξ 1 if 0 ξ < 1 if ξ 1 (4.) (4.3) i which ξ 1 idicates that the origial SDoF system has bee chaged to become critically or overly damped by the additioal umerical dampig, ad therefore o loger represets a vibratory system. Please ote that this mistake is ot acceptable ad should be avoided. The modified system dampig ratio after icorporatig Eq. (4.1) thus takes the followig form: C + α K ξ = = ξ + απ Mω T (4.4) Recall that spectral radius ρ = ma( λ, λ 1 ). If we further assume the SDoF system is free of dampig (i.e., ξ = 0) for simplicity, the substitutio of Eq. (4.4) ito Eqs. (4.) ad (4.3) will yield the followig:

7 l ρ ( ω ) α = l ρ 1+ ω l ρ if 0 ξ < 1 if ξ 1 (4.5) (4.6) Fig. 4.1(a) takes a udamped SDoF oscillator uder free vibratio as a eample, ad shows the relatio betwee spectral radius ad /T correspodig to differet α values. Sice the combiatio of N=0 ad the fourth-order Taylor series approimatio as origially suggested by Zhog et al. (1996) provides a good match with the aalytical eact solutio from Eqs. (4.5)-(4.6), oly the umerical results are plotted i Fig. 4.1 for better graphical readability. If the cut-off frequecy for suitable umerical dissipatio ad time step size are both decided, the a appropriate α value ca be readily determied usig Eq. (4.5) or Fig. 4.1(a) to help filter out fictitious high frequecy respose. The α value ca be the fed back ito the modified system matri A c to dissipate udesirable high frequecy resposes. Fig. 4.1(a) suggests that a larger time step would pair with a smaller dampig modificatio factor α while a smaller time step would pair with a larger α value i order to have the same spurious high frequecy dissipatio effects. A much smaller α value tha Eq. (4.5) or Fig. 4.1(a) suggests may lead to isufficiet dissipatio capacity, while a much larger α value may overly damp out trustworthy low frequecy respose from the first few predomiat vibratio modes. The dashed portio of the curves i Fig. 4.1(a) represets ξ 1 ad will erroeously trasform a SDoF oscillator (i.e., uder-damped) ito a o-vibratory system (i.e., critically or overly damped) ad therefore should ot be used. (a) (b) Figure 4.1. Relatios of spectral radius ρ vs. /T (left) ad ( T T )/ T vs. /T (right) usig the fourth-order Taylor series approimatio ad N=0 i the Precise Itegratio Method. The relatio betwee the relative period error ad /T uder differet α values ca be show as Fig. 4.1(b). The curves i Fig. 4.1(b) also agree eceptioally well with the aalytical results calculated from Eq. (3.4). As such, oly the umerical results are plotted i Fig. 4.1(b) for better graphical readability. Fig. 4.1(b) suggests that whe α value icreases, relative period error also icreases. It is therefore suggested that be as large as possible (a larger will ot sigificatly affect the period error too much due to the superior performace from the power-of-two algorithm ad the fourth-order Taylor approimatio ad ca help alleviate computatioal effort) ad the select a suitable α value to dissipate fictitious respose as log as umerical stability ad accuracy requiremet (usually, /T 0.1) are both satisfied. Ultimately, if the values of α ad Δ t are carefully selected, the spurious high frequecy respose from modelig error ca be successfully filtered out, ad accurate respose predictio from the first few predomiat modes ca be assured. The ecellet match betwee umerical results ad eact solutios i Fig. 4.1 facilitates a equatio-based predictio of

8 resultig umerical characteristics whe a combiatio of N=0 ad the fourth-order Taylor series approimatio is employed i the Precise Itegratio Method. The required computatioal effort of a umerical approach is always a major cocer of its users. I case of DOFs = 1000, a 17.4% icrease i computatioal time is observed if N value icreases from 0 to 0, while the computatioal time grows epoetially with the umber of degrees of freedom due to the state space approach employed. I passig, it is metioed that the additioal computatioal effort iduced by the power-of-two algorithm is margial cosiderig the sigificat improvemet i umerical accuracy. 5. ISSUE OF SPURIOUS RESONANCE UNDER FORCED VIBRATIONS The computatioal performace of a time itegratio method for solvig liear dyamics problems is usually evaluated with referece to its capability i accurately obtaiig the homogeeous part of the solutio ad umerical stability; however, the ifluece of eteral loadig has ot bee studied as widely, ad usually limited to the ivestigatio of local trucatio error. Caillo ad Macuso (000, 00) performed accuracy aalysis o time itegratio methods for classically damped MDoF liear dyamic oscillatig systems usig the trasfer fuctio that is epressed i terms of a amplificatio matri ad a loadig vector uder the assumptio of statioary disturbace. The MDoF oscillatig system usually ca be decomposed ito a group of SDoF oscillators usig modal decompositio if a classical dampig is assumed. Whe compared with the eact trasfer fuctio, umerical accuracy ad spurious resoace resulted from a time itegratio method due to the ifluece of the eteral loadig ca be the evaluated. As a statioary eteral ecitatio ca always be represeted by a superpositio of siusoidal waveforms usig epoetial Fourier series, the trasfer fuctio of a viscously damped liear SDoF oscillatig system betwee a statioary acceleratio ecitatio f ( t) = ep( i Ωt) ad the resulted system displacemet respose ca be the aalytically epressed i the followig form accordig to radom vibratio theory: H 1 = (5.1) ω Ω + iξω Ω O the other had, all umerical time itegratio methods have their ow umerically resulted trasfer fuctios that may deviate from the eact form of Eq. (5.1). If we assume the relatio betwee the eteral loadig ad the respose vector cotaiig displacemet ad velocity at a particular time istat t is of the followig vector form: H = e & H ( iω) & (5.) H ad H & correspod to the trasfer fuctios for displacemet ad velocity respose of the SDoF structural system, respectively. If we substitute Eq. (5.) ito Eq. (.7), the followig epressio for trasfer fuctio is the obtaied from the eplicit recursive equatio of the Precise Itegratio Method: i Ω 1 [ e I T ] [ ] H 1 = E 0 E 1 i H & e Ω (5.3) Eq. (5.3) ca be employed to calculate the trasfer fuctios for the Precise Itegratio Method. Similarly, both the trasfer fuctios resulted from the origial Newmark s ad the cetral differece methods ca be obtaied followig the same procedure. If the forcig-to-system frequecy ratio ν is defied as the eteral harmoic ecitatio frequecy Ω divided by the system frequecy ω, the the orm of the umerical trasfer fuctio ormalized with respect to that of the eact trasfer fuctio

9 deoted by H / H ca be employed to evaluate umerical accuracy ad spurious resoace for umerical itegratio methods. It should be oted that i case of H / H = 1, the umerical time itegratio method yields perfectly accurate result as the aalytical eact solutio. Fig. 5.1 compares umerical accuracy ad spurious resoace of several itegratio methods, icludig the Precise Itegratio Method, Newmark s method ad the cetral differece method, withi 0 ν. 5 by assumig a udamped SDoF system of a atural circular frequecy ω = rad/sec (i.e., atural period T = 3.14sec). All three methods use a time step Δ t =0.314sec =0.1T. The parameters γ = 0.5 ad β = 0.5 are assumed i the origial Newmarks method, correspodig to the average acceleratio method. Whe the forcig-to-system frequecy ratio ν is much smaller tha 1 (i particular, smaller tha 0.7), all three itegratio methods yield respose fairly close to the eact solutio; however, both the cetral differece ad the origial Newmark s (average acceleratio) methods show ufavorable spurious resoace with abrupt vertical spikes whe the forcig-to-system frequecy ratio ν is i the viciity of 1. Whe the forcig-to-system frequecy ratio ν is larger tha the value causig spurious resoace i the cetral differece ad the origial Newmark s methods, the umerical-to-eact trasfer fuctio ratio starts to deviate away from 1. For larger time step sizes tested i this study, the spurious resoace problem becomes catastrophically worse, ad ca result i a large half-power badwidth at spurious resoace such that the resultig umerical accuracy substatially degrades ad becomes uacceptable. The accuracy of the Precise Itegratio Method also degrades if a ecessively large time step is take (e.g., Δ t >0.3T). Fig. 5.(a) plots the umerical accuracy of the Precise Itegratio Method with the forcig fuctio liearly iterpolated withi a itegratio time step. The curves show cover a good spectrum of Δ t /T values (1/100, 1/40, 1/0, ad 1/10), correspodig to ω = 0., 0.5, 1, ad rad/sec. The parametric study idicates that although spurious resoace does ot eist i the Precise Itegratio Method, its overall umerical accuracy degrades whe the structural ad/or forcig frequecy grows higher. Fig. 5.(b) reports the advatage of employig a secod-degree polyomial i describig the variatio of loadig mometum withi itegratio time steps, which shows substatially improved accuracy. Numerical eample study shows that the use of oliear polyomials ca further loose the costrait of time step size selectio but still keeps the desired level of umerical accuracy at the same time. Figure 5.1. H / H vs. ν relatios obtaied from the cetral differece, the origial Newmark s ad the Precise Itegratio Methods (ξ = 0, =0.314sec= 0.1T). 7. CONCLUSIONS A eplicit time itegratio method capable of dissipatig spurious high frequecy resposes is proposed i the study, which is a modified versio of the origial Precise Itegratio Method. From umerical stability ad accuracy aalyses, it is foud that whe the N value reaches a certai threshold (e.g., N 0) ad the fourth-order Taylor series approimatio is employed, the Precise Itegratio Method becomes almost ucoditioally stable over a wide rage of frequecies ad its accuracy is

10 better tha most commoly used itegratio methods i the egieerig commuity. The power-of-two algorithm ad Taylor series epasio are both well kow mathematical cocept i tetbooks, so the authors trust that its ed users will be able to get familiar with the approach more easily compared with other state-of-the-art methods havig bee proposed i the literature, while the Precise Itegratio Method still provides competitively ecellet umerical stability ad accuracy. The et missio of the study i the ear future is to take a step further to ivestigate the possibility of etedig to materially oliear systems ad carefully evaluate its umerical performace. (a) (b) Figure 5.. H / H vs. ν relatios obtaied from force (left) ad mometum (right) equilibrium of the Precise Itegratio Method uder various structural atural frequecies (=0.314sec, ω =1~5 rad/sec). AKCNOWLEDGEMENT The study reported here was fuded i part by the Natioal Sciece Coucil of Taiwa uder grat umber NSC95-1-E This fiacial support is gratefully ackowledged. All opiios epressed i this paper are solely those of the authors ad do ot ecessarily represet the views of the sposor. REFERENCES Caillo, V. ad Macuso, M. (00). Spurious resoaces i umerical time itegratio methods for liear dyamics. J. of Soud ad Vibratio 38: 3, Caillo, V. ad Macuso, M. (00). O the behavior of dissipative time itegratio methods ear the resoace coditio. J. of Soud ad Vibratio 49: 3, Chie, C.C., Che, Y.H., Chuag C.C. (003). Dual reciprocity BEM aalysis of -D trasiet elastodyamic problems by time-discotiuous Galerki FEM, Egieerig Aalysis with Boudary Elemets, 7, Hughes, T.J.R. (1987). The fiite elemet method: liear static ad dyamic fiite elemet aalysis, Pretice-Hall, Eglewood Cliffs, NJ. Li, J., She, W., Williams, F.W. (1995). A high precisio direct itegratio scheme for structures subjected to trasiet dyamic loadig, Comput Struct 56, She, W., Li, J., Williams, F.W. (1995). Parallel computig for high precisio direct itegratio method. Comput Meth Appl Mech Egrg 16, Subbaraj, K. ad Dokaiish, M.A. (1989). A survey of direct time-itegratio method i computatioal structural dyamic - Ι. Eplicit methods, Comput Struct 3, Subbaraj, K. ad Dokaiish, M.A. (1989). A survey of direct time-itegratio method i computatioal structural dyamic - II. Implicit methods, Comput Struct 3, Tsai, M.C. ad Chuag, C.C. (00). A applicatio of precisio itegratio method i structure static ad dyamic problems, i: Proceedigs of 6th Natioal Coferece o Structural Egieerig, Taiwa, Paper No. F19 (i Chiese.) Zhog, W.X. ad Williams, F.W. (1994). A precise time step itegratio method. Proceedigs of the Istitutio of Mechaical Egieers. 08: Zhog, W.X., Zhu, J., Zhog, X.X. (1996). O a ew time itegratio method for solvig time depedet partial differetial equatios. Comput Meth Appl Mech Egrg 130,