Time Integration Schemes in Dynamic Problems

Transcription

1 Internatonal Workshop of Advanced Manufacturng and Automaton (IWAMA 016) Tme Integraton Schemes n Dynamc Problems Effect of Dampng on Numercal Stablty and Accuracy Ashsh Aeran and Hrpa G. Lemu Dept. Mechancal and Structural Engneerng and Materals Scence Unversty of Stavanger, N-4036 Stavanger, Norway ashsh.aeran@us.no, hrpa.g.lemu@us.no nclude shock response from nuclear weapons such as explosves or mpact loadng. Also, problems n whch wave effects such as reflectons and dffractons are mportant falls under the frst category. The nerta problems nclude all other dynamc problems except wave propagaton. The structural response s governed by relatvely small number of low frequency modes. Problem of ths type are often called structural dynamcs problems. The governng equaton n these problems s a second order dfferental equaton [8-9]. Only the structural dynamcs problems wll be dscussed n ths paper. Abstract A great deal of progress has been made n the past several decades towards the understandng and development of tme ntegraton methods n structural dynamcs. These methods nvolves a step by step algorthm for transent analyss of lnear and non-lnear dynamc problems. It s essental to provde a comprehensve survey of varous methodologes used to solve second order dfferental equatons n a sngle artcle. Broadly, the methods nclude drect ntegraton, mode superposton and response spectrum methods among others. The frst two methods uses an ntegraton scheme whle response spectrum method s based on extreme response analyss. Both drect ntegraton and mode superposton method use an ntegraton scheme but the selecton of a partcular method depends on the problem and frequency content of the loadng. A detaled survey of varous ntegraton schemes s presented n ths paper. The stablty and accuracy of these ntegraton schemes has been studed by researches n the past. However, the effect of dampng on the stablty and accuracy of these schemes need to be nvestgated. A sngle degree of freedom system s used to check the stablty and accuracy crtera of varous ntegraton schemes. Also, the effect of dampng on these parameters s studed and results are presented. As mentoned above, structural dynamcs problems nclude solvng a second order dfferental equaton whch s also the equaton of moton. For a lnear elastc system, equaton of moton can be expressed as: Mu (t ) Cu (t ) Ku(t ) R(t ) where M s the dscrete mass matrx, C s the vscous dampng matrx, K s the lnear stffness matrx, R s the external load vector. In general, M, C and K are constant and symmetrc. R= R(t) s a gven contnuous functon of tme t. Mathematcally, (1) represents a coupled system of lnear ordnary dfferental equaton of second order. Keywords: structural dynamcs; ntegraton schemes; dampng; stablty; accuracy In the last several decades, sgnfcant advances have been made n the development and applcaton of tme ntegraton methods for solvng above set of coupled dfferental equatons [10]. Ths was also prmarly made possble due to the parallel development of hgh speed dgtal computers reducng the computatonal tmes and provdng accurate results. Ths has resulted n the development of many commercally avalable software, vz. ANSYS, MARC, NONSAP and etc. [11]. All these software codes are based on varous computatonal methods developed to solve any dynamc problem. I. INTRODUCTION In many engneerng applcatons, t s essental to perform a dynamc analyss n addton to the statc check [1]. Ths means ncluson of the nertal term n the equaton of moton. The nerta term can be neglected only f the loads or dsplacements are appled very slowly []. Some examples where nerta effects are mportant nclude the mpact loadng of the structures where a hgh ntensty load s appled for a short tme or sesmc acton where structure s analyzed for prescrbed ground acceleraton. Offshore structures lke derrcks and flare booms are very crtcal for dynamc loadng due to the dynamc effects of wnd and drllng operatons [34]. Also, tme hstory analyss s requred for predctng the fatgue lves of structures precsely [5]. The objectve of ths paper s to present a revew of varous soluton methods for structural dynamc problems. Soluton methods lke drect ntegraton and modal superposton are based on usng an ntegraton scheme. Varous conventonal as well as recently developed ntegraton schemes are dscussed n ths paper. The use of an ntegraton scheme for a gven problem depends on both the stablty as well as the accuracy of the scheme. These crtera are dscussed n detal. The presence of dampng n the system can have an effect on the stablty of ntegraton scheme beng used and ths has also been nvestgated. Stablty and accuracy results of varous ntegraton schemes are presented for both damped and Dynamc problems can be broadly classfed n two categores based on the effect of the exctaton on the overall structural response: wave propagaton problems and nerta problems [6]. In wave propagaton problems, the behavor at the wave front s of engneerng mportance and ntermedate to hgh frequency structural modes domnate the structural response n these problems [7]. Problems under ths category 016. The authors - Publshed by Atlants Press (1) 13

2 undamped systems. The paper concludes wth dscusson of the results and hghlghtng the mportance of selectng a sutable method for a gven dynamcs problem. II. SOLUTION METHODS FOR DYNAMIC PROBLEMS Mathematcally, Eq. (1) represents a system of lnear dfferental equatons of second order and can be solved by standard procedures for the soluton of dfferental equatons. However, exstng standard procedures for the soluton of general systems of dfferental equatons can be very expensve especally when the order of matrces nvolved s large. In practcal fnte element analyss, only the effectve methods are ncorporated. These effectve methods can be broadly dvded n three categores: (1) Drect ntegraton methods, () Mode superposton method and (3) Response spectrum method. In the followng sectons, the frst two methods are dscussed n some detal. III. DIRECT INTEGRATION METHODS The drect ntegraton methods are tme marchng schemes where the dynamc equlbrum equaton s satsfed at dscrete tme ntervals t apart. Due to ease of applcaton and ther ready usablty n nonlnear problems, these methods have become very popular [1]. The term drect means pror to the numercal ntegraton, no transformaton of the equatons nto a dfferent form s carred out. Any drect ntegraton method s bult on two basc deas. 1. Computng the soluton of equatons of moton at dscrete tme steps. To compute the numercal soluton at specfc tme t, most methods requre the soluton to be specfed at prevous tme step, t -1 [1].. Assumng a varaton of dsplacements, veloctes and acceleratons wthn each tme step, where dfferent forms of these assumed varatons gve rse to dfferent ntegraton methods. The avalable drect ntegraton methods can be broadly subdvded nto two categores: explct methods and mplct methods [13]. Explct methods use the equaton of moton at the tme(s) for whch dsplacements are known, to obtan the soluton at tme t+ t. On the other hand, mplct methods use the equatons at a tme for whch the soluton s unknown, to obtan the response at tme t+ t [14]. The explct methods are more approprate to wave propagaton problems, whle the mplct one s used to nerta problems [10]. A. Explct drect ntegraton methods These methods n general employs fnte dfference methods and are partcularly well suted for short duraton dynamcal problems or wave propagaton problems. In these methods, the equlbrum condtons at tme t, are used to solve for the soluton at tme t+ t. Hence, such methods do not nvolve factorzaton of the stffness matrx n the step by step soluton. Ths also means that there s no necessty to store the stffness matrx f a dagonal mass matrx s used. These methods are computatonally cost effectve compared to mplct methods and less storage s requred. Also, for these methods, computer operatons are relatvely few and are ndependent of the fnte element mesh band or front wdth. Explct ntegraton methods are therefore very effcent for short duraton dynamc problems where stablty as well as accuracy condtons are both ensured. In these problems, the contrbuton of ntermedate to hgh frequency structural modes to the response s mportant. The stablty crtera s generally governed by the hghest frequency of the dscrete system. These stablty and accuracy crtera wll be presented and compared for varous methods n detal. Some of the most commonly used explct ntegraton methods are dscussed n the next secton. These are the second order central dfference method and fourth order Runge-Kutte method. Second Order Central Dfference Method: The second order central dfference method s one of the wdely used explct technques. Ths method s sad to have the maxmum stablty and hghest accuracy among the explct methods [15]. However, ths method s only condtonally stable; that s, the chosen tme step must be smaller than a crtcal tme step to attan a stable soluton. Inablty to handle non-dagonal dampng matrx s another shortcomng of ths method. Advances have been made and procedures have been developed to compute the dagonal mass matrx from the standard consstent mass matrx so that central dfference method can be used effectvely [16-18]. Also, numercal ntegraton technques used to compute the mass matrx are modfed further to generate better and accurate dagonal mass matrces [19]. The convergence of a dagonal mass approxmaton has also been proved [0]. However, some errors are ntroduced whle computng the dagonal mass matrx from standard mass matrx and have been examned [1]. Accordng to [], errors ntroduced by the lumped masses and the central dfference operator tend to be compensator. So the use of dagonal mass matrces n explct ntegraton methods s desrable both for accuracy and computatonal effcency. Fourth Order Runge-Kutte Method: The fourth order Runge-Kutte method was proposed n the begnnng of 19th century. The method has been extensvely used n the past for solvng ordnary dfferental equatons. A detaled survey of ths method along wth others s gven by [3]. Ths one step algorthm has several desrable features lke () the method s self-startng () the tme step can be easly changed () explct n nature and hence negates the need of teraton n nonlnear problems (v) the model s a 4th order method and possesses a weak nstablty only. However, n use of ths method, acceleraton vector must be computed four tmes per tme step. Due to ths, the computatonal tme requred for the soluton of a problem can be large compared to other numercal ntegraton methods. Also, the method does not provde any estmate of the resdual errors. Ths method was modfed further and several modfed methods were developed. Some of these methods nclude Runge-kutta-Fehlberg methods of order 1 to 3 [4], adaptve Runge-Kuta method [5], among others. These mproved methods have better stablty propertes are equpped wth automatc step control based on the local error estmates [6]. Recently Proposed Explct Methods: Chang [7] recently proposed a new famly of explct methods whose numercal 14

3 propertes for lnear elastc systems are exactly the same as those of the Newmark famly method. For ths subfamly, the possblty of uncondtonal stablty and second-order accuracy enables usng a large tme step and nvolves no teratve procedure. The method has proven very effcent for solvng general structural dynamc problems where the responses are domnated by low frequency modes. The method s computatonally more effectve compared to other condtonally stable explct methods where the step sze s lmted. Ths method was extended to nonlnear systems and a famly of non-teratve schemes for nonlnear dynamc problems was proposed [8]. Reference [9] also proposed a famly of uncondtonally stable explct drect ntegraton algorthms wth controlled numercal energy dsspaton. These algorthms are uncondtonally stable for lnear elastc and stffness softenng type nonlnear systems. B. Implct drect ntegraton methods The mplct methods fnd ther strength n the areas where explct methods are not so effectve. These methods are most effectve for structural dynamc problems n whch structural response s controlled by a relatvely small number of low frequency modes [30]. Also, problems wth complex structural geometres can be solved usng these algorthms. In these methods, the soluton for the dsplacements at tme t+ t nvolves solvng the stffness matrx at each tme step. Ths may lead to hgh computatonal effort and larger storage requrement compared to explct methods. However, unlke explct methods, these methods are uncondtonally stable and permts large tme steps. Wth the advancement of hgh speed computers, the uncondtonally stablty crtera provde a bg advantage. Some of the commonly used effectve mplct drect ntegraton methods are presented n bref. The Newmark Famly of Methods: These represent the most commonly used mplct methods for solvng the equaton of moton n a dynamc problem. The methods are based on the followng equatons. u The parameters β and γ defne the varaton of acceleraton over a tme step and determne the stablty and accuracy characterstcs of the method. Typcal selecton of γ s 1/, and 1/ 6 1/ 4 s satsfactory from all ponts of vew, ncludng the accuracy as shown n Table I. TABLE I. u 1 u (( 1 ) t) u u 1 1 u u ((0.5 )( t) ) u ( ( t) ) u 1 PARAMETERS FOR NERMARK METHODS Method Type β γ Stablty Average acceleraton Implct 1/4 1/ uncondtonal Lnear acceleraton Implct 1/6 1/ condtonal Central dfference Explct 0 1/ condtonal Wlson-θ method: Ths method s based on the assumpton that the acceleraton vares lnearly durng the tme nterval t to t+θ t. For θ = 1, ths method reduces to the lnear acceleraton method of the Newmark famly of methods. However, unlke the former, ths method s only condtonally stable. It s noted () (3) that n lnear problems, the method s uncondtonally stable for Hence, θ = 1.40 s usually used. IV. MODE SUPERPOSITION METHODS It s seen that the number of matrx operatons requred n a drect ntegraton soluton s drectly proportonal to the number of tme steps used n the soluton procedure. Use of a drect ntegraton method can be effectve for a relatvely short duraton nvolvng fewer tme steps. For large duraton problems, t s desrable to carry out the ntegraton process by frst transformng the equlbrum equatons n Eq. (1) nto a form n whch the step-by-step soluton s more effectve and less costly. Mode superposton methods are such methods of transformng equlbrum equatons from global coordnates to modal coordnates and then use drect ntegraton methods to solve these smplfed equatons. Mode-superposton analyss s an effcent tool for the evaluaton of structural response wth many degrees of freedom. The generalzed equaton of moton n modal coordnates for a damped system can be wrtten as (4). X T CX where the columns n are the mass normalzed egenvectors (free vbraton modes) ϕ 1, ϕ, ϕ 3 ϕ n and Ω s a dagonal matrx lstng the egenvalues ( frequences squared). Equaton (4) conssts of n uncoupled equatons whch can be solved exactly usng the Duhamel ntegral. Alternatvely, any drect ntegraton numercal method can be used. Snce the perods of vbraton are known, a tme step t can be chosen n the step-by-step ntegraton n order to obtan a requred level of accuracy. The equlbrum equaton reduces to n equatons of the form: The above equaton reduces to Eq. (1) and represents governng moton of the sngle degree of freedom system wth no dampng. For the soluton, as explaned earler, the response can be obtaned by summaton of the response n each mode. Effectveness of mode-superposton methods: The dea behnd mode-superposton soluton of a dynamc problem s that frequently only a small fracton of the total number of decoupled equatons needs to be consdered n order to obtan a good approxmate soluton of the equlbrum equaton. Most frequently, only the frst p equlbrum equatons need to be solved.e. only the frst p modes out of n are governng where p << n. Ths means that only p equatons out of n need to be solved and total response n the p modes can then be wrtten as gven by Eq. (6). U p p (4) (5) x (6) 1 X It s also seen that a typcal fnte element procedures approxmates the lowest frequences more precsely and lttle or no accuracy can be expected n approxmatng the hgher T R( t) x x x r 15

4 frequences and mode shapes. Therefore, the use of lower mportant modes for a system s justfable. The fact that only few modes may need to be consdered to arrve at a good approxmaton soluton makes mode superposton method superor to drect ntegraton methods. By solvng only p equatons out of n, both computatonal effort as well as cost can be saved. In summary, assumng that the decoupled equatons have been solved accurately, the errors n a mode superposton analyss usng p < n are due to the fact that not enough modes have been used, whereas the errors n a drect ntegraton analyss arse because of the use of a too large tme step. V. ANALYSIS OF INTEGRATION SCHEMES The soluton of dynamc equlbrum equatons can be solved ether by drect ntegraton or mode superposton method each of whch uses an ntegraton scheme. The cost of usng any ntegraton method depends on the tme step and number of steps requred for soluton. The chosen tme step should be small enough to guarantee desrable accuracy but at the same tme t should not be too small for cost reasons. Therefore, selecton of tme step s very mportant n any ntegraton scheme. Ths selecton s generally governed by two crtera whch are namely the stablty and accuracy crtera. In case of drect ntegraton or mode superposton, the basc equaton can be denoted by (7). x x x r (7) It s therefore satsfactory to study the stablty and accuracy crtera for above typcal equaton. A. Stablty Analyss The am n numercal ntegraton of any dynamc problem s to fnd an approxmate soluton to the actual dynamc response of the structure. In order to predct the response accurately, the equlbrum equatons (1) must be ntegrated to hgh precson. Ths means that all uncoupled n equatons of the form of Eq. (7) need to be ntegrated accurately. For the stablty analyss, matrx A and vector L are defned as the ntegraton approxmaton and load operators. These quanttes can be determned for any ntegraton method and are documented well for each ntegraton scheme. The spectral n 1 decomposton of matrx A s gven by A PJ P where P s the matrx of egenvectors of A, and J s the Jordan canoncal form of A wth egenvalues λ of A on ts dagonal. The spectral radus of matrx A s defned as ρ (A) and s gven as (8). 1,... The stablty crtera s gven as ( A) max (8) (a) If all egenvalues are dfferent, then ( A) 1. (b) If A contans multple egenvalues, then all such egenvalues should be smaller than 1. It s noted that the spectral rad and therefore the stablty of the ntegraton methods depend on the tme rato t/t, the dampng rato ξ and the ntegraton parameters used. Therefore, for a gven t/t and ξ, t s possble n the Wlson θ method and n the Newmark method to vary the parameters θ and α, δ respectvely to obtan optmum stablty and accuracy characterstcs. Fg. 1 shows the stablty characterstcs for varous ntegraton schemes. It can be seen that the central dfference method s only condtonally stable and the Newmark, Wlson θ and Houbolt methods are uncondtonally stable. The stablty characterstcs of Wlson ntegraton scheme wth varyng values of ts parameter θ s shown n Fg.. It s noted that the scheme s only condtonally stable for θ values of 0.5 and 1.0 whle t s uncondtonally stable for values 1.40 and more. It s therefore desrable to study the stablty of the Wlson operator, as a functon of θ for dfferent values of t/t for both damped and undamped cases. The results are shown n Fg. 3. It can be seen that for an undamped system, the method s uncondtonally stable for any t/t rato provded However, presence of dampng n the system allows choosng a lower θ value and method s uncondtonally stable for The Wlson scheme works well wth θ=1.40 as shown above. The method s found to retan ts uncondtonal stablty when dampng s present as shown n Fg. 4. The central dfference method s a condtonally stable method; that s, the chosen tme step must be smaller than a crtcal tme step to attan a stable soluton. The stablty condton and crtcal tme step s gven by (9). T t t cr (9) where T s the smallest natural perod of the structure correspondng to the hghest frequency mode. Fg. 1. Spectral rad of ntegraton schemes, ξ = 0 16

5 Fg.. Spectral rad of Wlson-θ for varyng θ, ξ = 0 Fg. 5. Spectral rad of central dfference method for damped cases Fg. 3. Spectral rad of Wlson-θ scheme as a functon of θ Fg. 6. Spectral rad of Newmark method for varous sets of parameters Fg. 4. Spectral rad of Wlson-θ for damped cases, θ = 1.40 For undamped system, the method s stable for t/t = 1/ = as shown n Fg. 5. It s also shown that the method remans stable when dampng s present n the system. For the Newmark method, the two parameters γ and β can be vared to obtan optmum stablty and accuracy. The ntegraton scheme s uncondtonally stable provded that 0. 5 and 0.5( 0.5). The method correspondng to γ = 0.5 and β = 0.5 has the most desrable accuracy characterstcs. The stablty characterstcs of ths methods wth dfferent sets of parameters s shown n Fg. 6. It s also desrable to observe the stablty characterstc of ths ntegraton scheme when dampng s present n the system. The effect of dampng on numercal stablty for some sets of parameters s shown n Fg. 7 to Fg. 9. The results are shown for three sets of the parameters. It s concluded that the Newmark ntegraton scheme retan ts uncondtonal stablty under dampng effects. It s therefore sutable to use ths ntegraton scheme for damped systems as well. The ntegraton schemes dscussed so far showed that the stablty characterstcs for an ntegraton scheme are retaned under the dampng effects. The stablty characterstcs of Houbolt method s shown n Fg. 10. It can be observed that lke other mplct ntegraton schemes, ths method s also uncondtonally stable for undamped system. However, t s observed that the presence of dampng ntroduces nstablty n the method for a smaller tme step t. The method remans stable for hgher tme step values. It s therefore recommended not to use ths ntegraton scheme n damped systems for solvng short duraton dynamc problems as t mght gve naccurate results. 17

6 Fg. 10. Spectral rad of Houbolt method for damped cases Fg. 7. Spectral rad of Newmark method: damped γ = 1/ and β = 1/4 Fg. 8. Spectral rad of Newmark method: damped γ = 1/ and β = 1/ Fg. 9. Spectral rad of Newmark method: damped γ = 11/0, β = 3/10 B. Accuracy Analyss Along wth the stablty, the accuracy of the soluton s very mportant. The choce of an ntegraton scheme s governed by the cost of the soluton whch n turn depends on the number of steps requred n the ntegraton. The drect ntegraton of the equlbrum equatons n Eq. (1) s equvalent to ntegratng smultaneously all n decoupled equatons of the form of Eq. (7). Therefore, accuracy of soluton of (1) can be studed by assessng the accuracy obtaned n the soluton of (7) as a functon of t/t. The accuracy analyss s explaned on a smple ntal value problem defned by 0 x x 0 0 x 1.0 ; x 0. 0 ; 0 (10) x The exact soluton for Eq. (10) s gven as x = cos ωt. The Newmark and Wlson θ methods can be drectly used wth the ntal condtons gven n Eq. (10). The numercal soluton obtaned usng these ntegraton schemes s shown n Fg. 11 and Fg. 1 respectvely. The natural perod of the system, T s fxed as 10 seconds and the tme step used n the ntegraton scheme s vared. The results obtaned usng Newmark scheme shows ncrease n the tme perod wth ncrease n the step sze (or t/t). Also, a very slght decrease n the ampltude s also observed. However both the perod elongaton and ampltude decay are sgnfcant when Wlson ntegraton scheme s used. From above two fgures, t s demonstrated that the errors n any ntegraton scheme can be measured n terms of two parameters namely the perod elongaton and ampltude decay. Fg. 13 and Fg. 14 shows the percentage perod elongatons and ampltude decays n the mplct ntegraton schemes as a functon of t/t. It can be seen that the numercal ntegraton usng any of the methods s accurate when t/t s smaller than about However, when ths rato s hgher, varous ntegraton schemes exhbt dfferent characterstcs. It can be seen that for a gven rato t/t, the Wlson θ method wth θ = 1.4 ntroduces less ampltude decay and perod elongaton than the Houbolt method and the Newmark constant average acceleraton method ntroduces only perod elongaton and no ampltude decay. Whle usng one of the uncondtonally stable schemes, the tme step t can be much larger (compared to central dfference method) and should only be small enough that the response n all modes that sgnfcantly contrbute to the total structural response s calculated accurately. In ths way, stablty as well as accuracy crtera are both met. The other modal responses of hgher frequency are not evaluated accurately, but the errors are unmportant because the response measured n those modes s neglgble and does not grow artfcally. 18

7 Fg. 11. Numercal soluton obtaned for Eq. (10) usng Newmark scheme Fg. 1. Numercal soluton obtaned for (10) usng Wlson scheme, θ=1.40 VI. Fg. 13. Percentage perod elongaton for varous ntegraton schemes DIRECT INTEGRATION VERSUS MODE SUPERPOSITION A dynamc equaton represented n Eq. (1) can be solved ether usng drect ntegraton methods or mode superposton method. Both of these methods use an ntegraton scheme n whch the hgh frequency response s fltered out of the soluton. The drect ntegraton method s equvalent to a mode superposton analyss n whch all egenvalues and vectors have been calculated and uncoupled equatons n Eq. (7) are ntegrated wth a common tme step t. Fg. 14. Percentage ampltude decay for varous ntegraton schemes For ths method, the ntegraton s accurate for those modes for whch t/t s small, but the response n the modes for whch t/t s large s elmnated by the artfcal dampng. Therefore, the drect ntegraton s qute equvalent to a mode superposton analyss n whch only the lowest modes of the system are consdered. The number of modes to be ncluded n the analyss depends on the tme step t and the dstrbuton of the perods. It s noted that the drect ntegraton method s the most effectve when all mportant perods of the system are clustered together, when tme step that s based on the smallest natural perod s chosen. For system wth natural perods far apart, t s recommended to use mode superposton method. In ths case, a separate sutable tme step can be chosen for each of the n uncoupled equatons. The number of modes to be consdered n mode superposton method depends on the load dstrbuton and frequency content of the loadng. VII. DISCUSSION AND CONCLUSIONS A comprehensve survey of varous methods used to solve a dynamc problem s presented n ths artcle. The methods used to solve an nertal dynamc problem, namely, the drect ntegraton methods, mode-superposton method and the response spectrum method are revewed. The frst two methods along wth ther advantages and dsadvantages are dscussed n ths paper. Both methods use an ntegraton scheme for solvng dfferental equatons. The stablty and accuracy of some of these ntegraton schemes s dscussed. The drect ntegraton methods are the most general soluton methods for dynamc analyss and equlbrum equatons are solved usng a step-by-step procedure. Mode superposton method s another powerful tool to solve dynamc problems by reduced computatonal effort compared to drect ntegraton methods. The reducton n the computatonal effort/ cost s due to the transformaton of equlbrum equatons from the global coordnate system to modal coordnate system. Usng ths transformaton, uncoupled equatons of equlbrum are obtaned whch can be easly solved smlar to a sngle DOF system. The stablty and accuracy of these ntegraton schemes are also dscussed. The most mportant step n any scheme s the selecton of the tme step. Too small tme step can lead to huge 19

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