Structural Optimization for Transient Response Constraints with Software JIFEX*

Transcription

1 amang Journal o Scinc and Enginring, Vol. 3, No. 3, pp (2) 73 Structural Optimization or ransint Rspons Constraints with Sotwar JIFEX* Yuanxian Gu, Biaosong Chn, Hongwu Zhang and Shutian Liu Stat Ky Laboratory o Structural Analysis or Industrial Equipmnt, Dpt. o Enginring Mchanics, Dalian Univrsity o chnology, Dalian 624, China Abstract h numrical mthods o th structural dsign optimization with transint rspons constraints hav bn studid in th papr. h nw mthods o th rspons analysis and snsitivity analysis or th transint dynamics and th hat conduction constraints with th prcis tim intgration hav bn proposd. Particularly, an icint mthod o snsitivity analysis or nonlinar transint hat conduction is givn. h dsign optimization and init lmnt analysis or gnral structurs with siz and shap variabls and multi-typ constraints ar implmntd in th application sotwar JIFEX. Numrical xampls hav illustratd th ctivnss o th mthods prsntd in th papr and th acility o JIFEX sotwar. Ky Words:structural optimization, snsitivity analysis, transint dynamics, hat transr, tim intgration. Introduction h transint rsponss such as structural dynamics and unstady hat transr ar important problms in structural dsigns and not wll studid in th dsign optimization. hs problms ar not only diicult in tim consuming numrical analysis, but also in th snsitivity analysis particularly. In th solution o linar and nonlinar tim-dpndnt transint rspons problms, thy lad to linar and nonlinar Ordinary Dirntial Equations (ODEs) in tim domain. Among th xisting mthods o solving ODEs, dirct tim-stpping mthods hav bn th most popular and widly advocatd in most sotwar. In particular, th tim dirnc/ θ * -dirnc mthod[2,7,9] plays a dominant rol. As or transint hat conduction analysis, whnθ, th algorithm paramtr, is slctd as,.5, 2/3, and., th orward, cntral (Cran-Nicolson), Galrin, bacward dirnc mthods can b * h projct supportd by th National Natural Scinc Foundation o China (952526, ) and th NKBRSF o China (No. G ). constructd rspctivly. With th dirct itration [2] or Nwton-Raphson [7] itration mthod, th abov mthods can b xtndd to nonlinar analysis. In addition to θ -dirnc mthod, many dirnt approachs hav bn proposd such as hybrid transinit lmnt ormulation[6], rduction mthods[], virtual-puls tim intgral mthodology[5], prcis algorithm[2]. Rcntly, A prcis tim intgration mthod was proposd to structural dynamics [2,22], and applid to othr transint rspons problms [3,6,,3]. h snsitivity analysis giving drivativs inormation o rsponss with rspct to dsign variabls is th basis o dsign optimization, and also valuabl to stochastic analysis and invrs problm. h convntional mthods o snsitivity analysis or transint rsponss hav diicultis in computational icincy and accuracy. Numrous publications [2,5,8,9,,8] hav prsntd th dvlopmnt history o rsarch on hat transr. O ths litraturs, th dirct mthod and th adjoint mthod o snsitivity analysis hav bn proposd basd on continuum or discrt modl. hs authors concntratd on th snsitivity quations drivativ procdur. Bri discussions on solving th quations wr givn in som o th

2 74 Yuanxian Gu t al. abov paprs by using th tim-dirnc intgration mthod ( θ -dirnc intgration mthod). In act, how to solv th snsitivity quations is also important. It will ma a grat ct on th snsitivity prcision and algorithm icincy. In this papr, th Prcis im Intgration (PI) mthod [2,22] is xtndd to th solution o linar and nonlinar transint hat conduction problms. Particularly, nw algorithms ar proposd to solv th snsitivity quations o linar and nonlinar transint hat conduction, and structural transint dynamics problms with PI mthod. h numrical advantags o PI mthod such as unconditionally stabl, high rsolution, and adaptiv to sti problm hav improvd th snsitivity accuracy noticably and avoidd th numrical oscillations. h nw mthods o rsponss analysis and snsitivity analysis or transint constraints ar implmntd in th gnral purposd sotwar JIFEX [4,7] or structural init lmnt analysis and dsign optimization. Numrical xampls ar givn in th papr to illustrat th accuracy o nw mthods and application acilitis o sotwar JIFEX. 2. Prcis im Intgration or ransint Rsponss 2. Prcis im Intgration Mthod and Structural ransint Dynamics h PI mthod was proposd by Zhong [2,22] or th solution o structural dynamic quation with high prcision and numrical stability. Firstly, th PI mthod is prsntd or th problm o structural transint dynamics. h motion quation o structural transint dynamics can b xprssd in th init lmnt discrt orm as blow () t M & x Gx& Kx () Initial condition x ( ) x and x &( ) x& ar givn. h M, G, K ar tim-invariant mass, damping and stinss matrics with n n dimnsion rspctivly, and () t is xtrnal load vctor. In gnral, (t) {} and Eq.(l) is non-homognous. Lt us din p Mx& Gx / 2, q x (2) thn, Eq.(l) bcoms th ollowing orms v & Hv r (3a) q v, p M G / 2, H GM / 2, r GM G/4 K, M (3b) whr H is a matrix with 2 n 2n dimnsion. h gnral solution {v} o Eq.(3) is v ( t) xp( Ht) v t xp( H ( t s) ) r( s) ds, (4) whr v ar th initial conditions. hn th solution o th systm at tim t is v xp t ( Ht ) v [ H ( t s) ] r()s s d xp Dnoting tim stp lngth solution at tim t is v xp τ t t ( Ht ) v [ H ( t s) ] r()s s d t xp (5), th (6) By dducing th rlationship o v and v, th intgration schm o th PI can b obtaind v xp ( Hτ ) v [ H ( τ s) ] r( t s) ds τ xp h matrix xponntial is dind as (7) A xp ( H τ ) (8) I th non-homognous vctor is linar unction in th tim intrval (t,t ), such that

3 Structural Optimization or ransint Rspons Constrains with Sotwar JIFEX 75 () t r r ( t ) r (9) thn, th solutionν can b xprssd as blow v A v H t [ H ( r H r )] [ r H r r ] τ () Mor approximation mthods to dal with othr inds o non-homognous vctors can b ound in litraturs [6,3]. h calculation o xponntial matrix A is a y point in PI mthod. hr ar som algorithms o xponntial matrix calculation [4]. h 2 N algorithm is usd in PI mthod with suprposition o xponntial unction and scond ordr aylor xpansion, A xp I m ( H τ ) [ xp( H τ / m) ] 2 H t ( H t) / 2 I Aa () It is notd that t is a vry small, or xampl, i ta N 2, m 2 N 48576, thn t τ m is an xtrmly small tim intrval. Howvr, it is sam or A a compard to th idntity matrix I, and thus algorithm is particularly dsignd to xcut th addition o matrix A a irstly, and add thm to I inally. or ( itr ; itr < N; itr ) (2) A 2A A A ; Whn th loop is ovr, A a a A I (3) 2.2 Linar ransint Hat Conduction Problm Atr spac discrtization by numrical tchniqus such as init lmnt, init dirnc, boundary lmnts tc, th linar transint hat conduction quations ar as ollow. M & K R (4) whr M, K, and R ar th capacity matrix, th conduction matrix, th tmpratur vctor and hat load vctor. h PI mthod can b xtndd to th transint hat conduction problm by adopting th ollowing transormation, & - - H r, H -M K; r M R (5) a a a I matrix R varis linarly within th tim intrval as, R R R r R ( t t ), R R( t ) ( R( t ) R( t ))/ τ M R, r M R, (6) (7) h intgration ormulations o th PI mthod ar A H A xp [ ( )] H r H r [ r H r rτ ], ( Hτ ) (8) whrτ is th tim stp lngth. hr ar som simpliications or hat conduction, bcaus o H ( M K ) K M (9) thn, th PI or linar transint hat conduction is A K [ K ( R MK R )] [ R MK R R ] τ (2) h product o K - and a vctor can b carrid out by LDL dcomposition and bac substitution o th vctor. h abov cods ar usually ord in th convntional init lmnt program. It is no nd to program th additional cod or invrs computation o matrix H. h constant, linar, and sinusoidal approximations all availabl to dal with th non-homognous vctor R. 2.. Nonlinar ransint Hat Conduction Problm In gnral, th hat transr is nonlinar. h nonlinaritis usually includ th tmpratur-dpndnt matrial paramtrs and th radiation boundary condition. h transint nonlinar quations ar ( ) K ( ) R M & (2) whr th symbolic manings ar th sam as bor, but thy ar not constant and dpndd on tmpratur. As or init lmnt mthods, th rlativ matrixs ar

4 76 Yuanxian Gu t al. M ( ) Ω ( ) c( ) ρ N NdΩ (22) ( ) ( N ) ( )( N ) K d Ω (23) Ω whr N dnots th shap unction o lmnts, and Σ dnots th assmbling o lmnts. It can b sn that th dnsity ρ, spciic hat c, hat conductivity ar all th unctions o tmpratur. h right hand trm R can b dcomposd as two parts R R (24) R r h linar part R is indpndnt o tmpratur. h radiation part R r, is writtn as R r s s σεn σεn 4 4 ( ) m i a ds N ii 4 4 a ds (25) whr σε is th product o Stan-Boltzman constant and th missivity o structural surac, a is th ambint tmpratur. h abov ormula can b valuatd by Guass intgration so as to rduc th complxity. Som transormations ar ndd to solv th nonlinar quations by th PI mthod. Eq.(2) is rwrittn as hn, ( M ( q) M M ) & ( K ( ) K K ) R ( M ( ) M ) ( ) K M & K R & ( K ) (26) (27) Employing th standard ormula o PI or th abov quations whr & H M R (28) H M K, (29) R R - ( M( ) M ) & ( K( ) K ) h introducd matrix in th abov transormations can ta th orm as ( ), K K ( ) M (3) M It mans that M, K ar th initial hat capacity matrix, hat conduction matrix rspctivly. h objctiv o adopting th transormations is to ma th matrix H constant, thn th computational xpns can b rducd whn carrying out th prcis computation o matrix xponntial. hs transormations don't introduc any artiicial approximations and th transormd quations ar idntical to th original ons. In practic, bcaus dnsity and spciic hat vary with tim smoothly, thy can b viwd as constant. Hnc th icincy can b improvd. h analytical solution o Eq.(28) is obtaind according to Eq.(7) A τ xp ( H ( τ s) ) M R( t s, ) ds (3) h approximations may b applid. In ordr to simpliy th wor, th ollowing prdictor-corrctor algorithm is rcommndd. Firstly th trm R is dividd as two parts, R R E R I (32) whr th R E is th xplicit part and is only dpndd on tim and causd by hat sourc, convction, hat lux, prscribd boundary tmpratur, and tc. R I is th implicit part rlatd to both tim and stat variabl causd by radiation, tmpratur-dpndnt matrial paramtr. As or xplicit part, Eq.(8) is mployd, as or implicit part, th ollowing algorithm calld Prdictor-Corrctor is prormd, () Prdictor. Ltting n th countr, whn n, supposing that R is constant within th tim intrval R I R I t, (33) ( ) Atr substituting it into Eq.(3), th prdiction tmpratur q, at t is obtaind.

5 Structural Optimization or ransint Rspons Constrains with Sotwar JIFEX 77 (2) Corrctor. Using th prdiction tmpratur, ltting nn, linar approximation is adoptd. R I R s I ( t, ) ( RI ( t,, n ) RI ( t, )) [, τ ] s / τ, (34) Atr substituting it into Eq.(3) again, th nw prdiction tmpratur,n is obtaind. h abov ormula can b xcutd conscutivly until th two prdictions satisy th norm condition < δ (35), n, n whr δ is th givn rror tolranc. Bcaus th linar approximation is mployd in th abov ormula, this prdictor-corrctor can achiv scond ordr prcision. h algorithm is asy to implmntd in program. O cours, highr prcision algorithm can b construct by using highr prcision approximations and th algorithm is still stabl bcaus o high prcis computation o matrix xponntial which guarants th stability. 3. Snsitivity Analysis or ransint Rsponss with PI Mthod h snsitivity analysis mthods or structural transint rsponss, i.. th linar dynamics, th linar and th nonlinar hat conduction, hav bn proposd on th basis o prcis tim intgration. h nw mthods ar mor accurat compard to convntional tim dirnc mthods such as Nwmar and 6-dirnc mthods. 3. Snsitivity Analysis o Structural ransint Dynamics Dirntiating Eq.(3a) with rspct to dsign variabl α, w hav th snsitivity quation o gnral vctor ν as ollowing dv& dv H R, dh dr R v (36) It is notd that Eq.(36) is sam as Eq.(3a).Maing th linar approximation to th right-hand vctor in Eq.(36) sam as that in Eq.(3a),thn, th Eq.(36) is transormd as blow dv& dv H R R dr( t ) dh( t ) R v dh ( t ) v v t R ( t ) t (37a) ; whnt t (37b) h gnral solution o Eq.(37) similar to that o Eq.() can b obtaind as blow dv dv A ( I ) H ( R H R ) H τ R (38) With th initial condition o original dynamics problm dx dx&, (39) th initial condition and irst stp o Eq.(38) is obtaind as dv, dv H ( ) ( A I ) H R ( t ) H R ( t ) R ( t )( t t ) (4) Now, w hav th prcis tim intgration ormulations Eq.s(38) and Eq.(4) to calculat th drivativs o vctor v (,,...,n). It is clar rom th dinitions o Eq.(2) and (3) that th drivativs o dynamic displacmnts x just is th irst hal part o drivativs o gnral vctor v.sam as th gnral procdurs o snsitivity analysis or transint rsponss, th Eq.(38) is computd simultanously with Eq.(). With computing th matrix xponntial irstly, Eq.()and (38) ar icint xplicit multiply calculation. his is a atur o th nw snsitivity analysis algorithm with th PI mthod. Sinc th matrix H is composd o M, K and G and ths matrics ar tim indpndnt, h drivativs o H can b computd onc bor th calculation o Eq.s() and (38). In th abov prsntations, th xtrnal load vctor r is assumd to b th linar

6 78 Yuanxian Gu t al. non-homognous. For othr inds o non-homognous xtrnal load vctor o Eq.(l), som othr approximation tratmnt mthods hav bn proposd in litraturs[6,3]. Similarly, th right-hand vctor R in th snsitivity quation Eq.(36) is approximatd linarly as Eq.(37). h othr approximation procdurs availabl or th nw snsitivity analysis algorithm with th PI mthod can b drivd asily. According to th dinitions in Eq.(3), th scond hal part o d v / is th drivativ o gnralizd vctor p, d p /. hror, th drivativs o vlocity vctor x& can b computd with Eq.(2) as blow dx& M dp dm dg dx x. 5 x G (4) & h ormulation calculating th drivativs o acclration vctor & x& can b obtaion orm th motion qustion o structural dynamics Eq.(l) as blow d&& x M G M d d d x&& x& dx dk dx G & x K (42) In summary, th nw snsitivity analysis algorithm with th PI mthod is irst to calculat th matrics A and H - and th drivativs o M, K, G, and H, r, thn calculat th drivativs o gnralizd stat vctor v and displacmnts by Eq.(38), drivativs o vlocity x& & by Eq.(4), and drivativs o acclration x by Eq.(42). o gnral cas, th drivativs o th load vctor (t) has bn considrd hr although loads ar normally variabl indpndnt. 3.2 Snsitivity Analysis o Linar ransint Hat Conduction h snsitivity quation o linar hat conduction can b obtaind dirntiating Eq.(4) as M & ' K' R' M' & K ' (43) whr d dm dk dr ', M ', K ', R' Eq.(43) is vry similar to th original on Eq.(4) xcpt or th right hand trms, so th sam solution algorithm can b applid. h M ', K ', R ' ar drivativs o capacity matrix, conduction matrix, hat load vctor with rspct to dsign variabl which ar tim- and tmpraturindpndnt. hus, thy ar computd only onc with analytical or init dirnc mthod. Bor solving th abov quations, th tmpratur and th tmpratur rat in tim hav bn obtaind. hn th snsitivity quations is transormd as ' & H' M R' M' &, H K' M K, (44) It shows that i PI is applid to solv th snsitivity quations, th xponntial matrix ndn't to comput again. Furthrmor, it is assumd that th tmpratur and its rat in tim vary linarly within th tim intrval. ( t t h ) ( ) ( ( h, h [, τ ] ( t ) R' ( t ) M' & K' t ) ( t )) h / τ (45) h intgration ormulation or snsitivity analysis is ' A K ' [ K ( MK ) ] [ MK ] 3.3 Snsitivity Analysis o Nonlinar ransint Hat Conduction τ (46) By dirntiating nonlinar hat conduction, th snsitivity quations ar M ' & K ' R ', ' ( M' ( ) M ) & K' ( ) (47) ' ( K ) Sinc th right-hand trm dpnds on tmpratur, th snsitivity quation is nonlinar too. h problm is how to valuat. As th sam bor, it can b dcomposd into two parts: th xplicit part and th implicit part.

7 Structural Optimization or ransint Rspons Constrains with Sotwar JIFEX 79 (48) E I whr E is th xplicit part which is indpndnt o tmpratur causd by th snsitivity o hat sourc, convction, hat lux, prscribd boundary tmpratur with rspct to dsign paramtr. h xplicit part can b calculatd by Eq.(8) irstly. I is th implicit part which is dpnd on tmpratur, and dduc with init dirnc mthod at tim t, R ' ( t, α α, α ) R ( t, ) I K I I α (49) whr α is th prturbation o th dsign variabl. h prdictor-corrctor algorithm is rcommndd.. Prdictor. Lt n is th countr o itration, whn n,, ( t ) I (5) Atr substituting irst prdiction o 2. Corrctor. Lt n n,, into Eq.(47), w gt th I ', by th PI mthod. ( ) ( t, ) ( t,, n ) ( t, ) h, h [, τ ] h τ (5) Also substituting, into Eq.(47), w gt th ' corrctiv valu o, n. h abov corrctor ormulas can b also xcutd by itration. h convrgnc condition can b < δ (52), n, n It can b ralizd that th procss o snsitivity analysis is vry similar to th on o solving hat transr quations. his is vry bnit or implmntation. 4. Optimization Modling and Solution Algorithm Implmntd in JIFEX JIFEX is a nw init lmnt sotwar or structural analysis and dsign optimization on th MS Windows95/N platorms. JIFEX posssss mor powrul capabilitis o init lmnt analysis dsign optimization, structural init lmnt modling basd on th CAD pacag AutoCAD, and graphics visualization. h dsign optimization o JIFEX ors th usrs many acilitis to din thir optimal modl. In th optimal modl, th objctiv unction and th constraint unctions can b th structural wight, th strngth, th stinss, th vibration bhavior, th bucling stability. h dsign variabls considrd in th dsign optimization ar siz variabls and shap variabls. h siz variabls includ th thicnss o shll, plat and mmbran lmnts, cross sctional ara o th bar lmnt, and cross sctional gomtric paramtrs o th bam lmnt. h variabls o composit lmnts, i.. layr thicnss o laminat plat, ply orintation, cor hight o honycomb sandwich plat, ar also classiid as siz variabls. Shap variabls ar th natural variabls, i.. gomtric paramtrs o th splin and quadratic curvs and th surac o th structural boundary, as wll as nod coordinats. Furthrmor, JIFEX can conduct icint structural snsitivity analysis which provid th usrs with important inormation or optimal dsign, rliability analysis and invrs problm rsarch. With th hlp o JIFEX, th usrs can complt th optimal dsign o complx built-up structur. JIFEX has bn nhancd rcntly. h nw snsitivity analysis tchniqus introducd abov or transint structur, linar and nonlinar hat conduction and thrmal structural rsponss ar implmntd. h optimal modl can includ th transint rsponss (such as th transint displacmnt, th vlocity, th acclration), th tmpratur and thrmal rsponss (such as th thrmal strss and thrmal displacmnt). So th sotwar can ulil mor complicatd problm o structural optimization. h optimization algorithms o th program JIFEX ar improvd squntial linar programming (SLP) and squntial quadratic programming (SQP). h gnral mathmatical modling o structural dsign optimization problms is as ollow

8 8 Yuanxian Gu t al. s.t. g x il j min ( X ) ( X ) j (,2...m) x x i (,2...n) i iu (53) whr (X) is th objctiv unction, n is th numbr o dsign variabls x i, m is th numbr o constrains unctions g j (X), x iu, and x il ar uppr bounds and lowr bounds o th variabls rspctivly. h objctiv unction and constraint unctions may b wight, displacmnts, strsss, vibration rquncis, dynamic rsponss, bucling loads, and othr bhaviors o structurs. In th SLP algorithm, th objctiv and constraints unctions ar approximatd with linar xtnsions at th currnt dsign point o th optimization itration. hn th original problm (53) is transormd into th ollowing linar programming problm. s.t. g min j ( X ) ( X ) ( X ) g ( X ) x il x i j x iu X X whr th ( ) and g ( ) X j X (54) ar drivativ gradints o th objctiv unction and constraint unctions rspctivly. h linar programming (54) is solvd with th Lam pivot algorithm to ind a nw dsign. hs approximation and solution procdur ar rpatd until th convrgnc is rachd. In th SQP algorithm, th constraints ar th sam as in Eq.(54), and th objctiv unction is approximatd with scond ordr xtnsion as ollows 2 ( X ) ( X ) X X ( X ) X 2 hn th problm is transormd into a quadratic programming problm, and is also solvd by th Lam pivot algorithm. h SQP algorithm is usd whn th objctiv unction is structural wight sinc its scond ordr drivativs ar asy to comput. Whil th SLP algorithm is usd or othr cass with objctivs o bucling load, strss, vibration rquncy and so on. o ovrcom th rror problm causd in th linar approximation o constraint unctions and to nsur itration convrgnc, th approximat lin sarch and adaptiv mov limit approachs hav bn mployd. h basic SLP and SQP algorithms mntiond abov ar improvd to b a two-stag solution procdur or ach itration: (a) solving linar or quadratic programming problm to obtain a nw dsign X, and (b) inding th accptd nw dsign X nw X βd by th approximat lin sarch along th dirction o d X X h approximat lin sarch is basd on th ollowing Goldstin critrion. β ( η) F( X < η <. 5 F( X ) d βd) F( X ) βη F( X ) d (55) whr β is th stp lngth o th lin sarch with initial valu., and η is a prscribd constant actor. F(X) is th original objctiv unction or a modiid multi-critrion objctiv unction. In th latr cas, th algorithm is xtndd to dal with th multipl critrion dsign optimizations and solv th unasibl dsign problms. h Goldstin critrion mans that th dsign satisying condition (55) is bttr in som dgr than th old dsign X, though it may not b th optimum. his way, it mas th objctiv unction F(X) dcras stp by stp and th itration convrgnc is stabl. Starting rom X and with th initial stp lngth β l., th approximat lin sarch is carrid out with th ollowing thr conditions and inishd within a w stps. (a) I th Goldstin critrion (55) is satisid, thn th X nw X βd is accptd as a nw dsign and th lin sarch stops. (b) I F( X o β d)-f( X ) > βη F( X )d, thn rduc th stp lngth β to continu th lin sarch, and rduc th mov limit or nxt itration; (c) I F( X o β d)-f( X ) < ( X ), β ( η) F d thn ta th X nw X βd as a nw dsign and stop th lin sarch. In this cas, th mov limit o th nxt itration can b nlargd. 5. Numrical Exampls Exampl. Snsitivity analysis and dsign optimization on transint dynamic rspons or 72-bar truss structur. h structur is shown in

9 Structural Optimization or ransint Rspons Constrains with Sotwar JIFEX 8 Figur (a). h structural lngths in x and y dirction rspctivly ar 3m, and th hight o ach lvl is 5m. h matrial Young's modulus E2 l g/m 2 mass dnsity ρ 78g/m 3. h cross sction ara o all bars is.5m 2. h load P x acts at th top-lvl, nod 2, 3, 4, and 5, in th x-dirction and its variation with tim is shown in Figur l (b). h calculation is carrid out during tim intrval t sc. h maximum dynamic displacmnt rspons computd by th PI mthod with tim stp Δt. sc is u m at tim t 2.66sc, and that by Nwmar tim dirnc mthod with tim stp Δt.sc is u m at tim t 2.64sc. All o th 72 bars ar dividd into 6 dsign variabls with variabl-lin tchniqu. hs 6 dsign variabls and thir lind bars ar dind as ollowing 6 groups. 4 group vrtical bars o lvl,2,3,4 (th top lvl is lvl ); 4 group x-y plain horizontal dg-bars o lvl, 2, 3, 4; 4 group vrtical-sw bars o lvl, 2, 3, 4; 4 group x-y plain horizontal cross-bars o lvl,2,3,4. h drivativs o th maximum dynamic displacmnt u 2 (t 2.66sc) ar computd with th PI mthod and th Nwmar mthod rspctivly. h computational rsults ar listd in abl. o chc th snsitivity accuracy, th rsults computd by th init dirnc mthod (dnotd by FDM) and rlativ rrors o th PI mthod compard to th init dirnc ar also givn in abl. It is shown that th PI mthod givs quit accurat rsults o snsitivity drivativs or structural transint dynamic rsponss. Dsign optimization o 72-bar truss structur with transint dynamic dormation constraint. h structural data and dsign variabls ar sam as thos o th xampl 2. h objctiv is th structural wight, and constrains ar dind that th maximum dynamic displacmnt rspons is limitd blow.5m. h low-bound and up-bound o dsign variabls ar givn as.lm 2 and.2m 2 rspctivly. h tim intrval t sc is considrd, and th damping with coicints αβ. is tan into account. h prcis tim intgration mthod is mployd in th dynamic analysis and th snsitivity analysis or th transint dynamics rsponss. h optimum dsign obtaind atr itrations is that th structural wight is rducd rom th initial dsign o g to 932.g with 22.6% dcrasing, and th constraint is satisid. h variabl valus o optimum dsign and thir cass to constraint bounds ar givn in abl 2. h matrial distribution among bars is rasonabl. Figur. 72-bar truss structur abl. Drivativs o dynamic displacmnt u 2 (t2.66 sc) D.V. PI Nwmar FDM Error (%) Exampl 2. Snsitivity analysis o linar transint hat conduction. A slab o thicnss L is initially at a uniorm tmpratur i 2.. Suddnly at tim t, th tmpratur o both surac ar lowrd to and maintaind at. or all t >. h analytical solution o tmpratur through th thicnss and th snsitivity o tmpratur with rspct to thrmal capacity c ar ( x, t) 4 ( ) 2 α ( nπ / L ) i π nπ L t sin x n,3,5... n

10 82 Yuanxian Gu t al. ( x, t) 4( ) d dc i π 2 2 α ( nπ / L ) t π nπ n t sin x n, 3, 5... n ρ cl L Whr α/cρ,, p, c2.. wnty linar two-nod lmnts ar mployd. h PI mthod and thθ-dirnc mthod ar usd rspctivly to solv this tim-dpndnt problm. In tim dirnc, th algorithm paramtr θ is slctd as.5 or.667. h Figurs 2(a) and 2(b) giv th comparativ rsults or th tmpratur distribution. h PI mthod givs a good agrmnt with th analytical solution, whil th tim dirnc mthod shows numrical oscillations at th both nds spcially whn θ.5. Figur2 2(c) and 2(d) ar th comparativ snsitivity calculation rsults. h PI mthod is th bst on agring with th analytical solution. h tim dirnc producs strong oscillation and its rsult at som nods loss applicability bcaus o grat rrors. Although whn θ.667 in Figur 2(d), th tim dirnc givs a good rsults, it ails du to strong numrical oscillation shown in Figur 2() whn th tim stp lngth incrass. h PI mthod still shows a xcllnt agrmnt whn tim stp lngth incras, shown in Figur 2(). h rrors o th PI mthod ar causd by () th init lmnt modl rror, and (2) th algorithm rror. In snsitivity analysis with th PI mthod, th variation o nod tmpratur and tmpratur rat in tim intrval ar approximatd to b linar. Exampl 3. Snsitivity analysis o 2D linar transint hat conduction. h sam rctangular plat shown in Figur 3 is usd. h paramtr ar as ollows: thicnss h., spciic hat c2., and tmpratur-dpndnt hat conductivity l... h surac is subjctd to radiation, th radiation coicint σε 7 and th ambint tmpratur a.. h boundary condition ar (x)(y)5.. h initial tmpratur in th plat is 5.. h dsign paramtr is th thicnss o th plat. h rsults by th init dirnc mthod notd with FDM ar obtaind with τ.. h rsults by th PI mthod ar got with thr dirnt tim stp lngths τ.,.25,.5. h tmpratur drivativs o ight nods at thr tim -points t.5,.3, and.5 ar givn in abl 3. It shows that as or th nonlinar hat conduction problms snsitivity analysis with th PI mthod can giv good rsults with high prcision. abl 2. Dsign variabl valus D.V. Opt. Cas. Low-bound 2.2 Up-bound 3.2 Up-bound 4.2 Up-bound 5. Low-bound 6. Low-bound 7. Low-bound 8. Low-bound 9.3 Intrmdiat.6 Intrmdiat.6 Intrmdiat 2.6 Intrmdiat 3. Low-bound 4. Low-bound 5. Low-bound 6. Low-bound

11 Structural Optimization or ransint Rspons Constrains with Sotwar JIFEX 83 Exampl 4. h xampl 4 is strss optimization problm subjct to both thrmal load and mchanical load. A squar plat has a hol at th cntr, and its init lmnt msh o quartr part is shown in Figur 4. h lngth o squar plat is 5.. h matrial paramtrs ar κ 2.58xl 3,ρc 6.2x 5 α., E 2.x 6 μ.3. hr is no hat sourc within th body o structur. h initial condition is that th tmpratur o all nods is 2. h boundary conditions ar that th tmpratur is givn b 2 at all xtrnal dgs, and thr is a hat low on th intrnal hol dg with th dnsity q -.x 6 h dsign objctiv is to minimiz th maximum von Miss strsss by mans o changing th shap o th hol. h dsign variabls ar th coordinats o control nods, 2, 3 in Figur 4, i.. yl, x2, y2, x3. h initial valus o dsign variabls ar.,.5,.5, and.. h shap o th hol is intrpolatd to b th quadratic curv, and th ara o th hol is constraind not changd. Figur 2. mpratur and Snsitivity analysis or linar transint hat conduction

12 84 Yuanxian Gu t al. is provd to b symmtry by adopting som transormations. h symmtry can improv th computational icincy o th algorithm gratly. In conjunction with smi-analytical mthod, th linar snsitivity quations ar solvd by th PI mthod dirctly. h Prdictor-Corrctor mthod is usd to solv th nonlinar snsitivity quations. Numrical xampls show that th PI mthod can giv high prcis rsults and avoid th numrical oscillation. It is concludd that th PI mthod is accurat or th snsitivity analysis o linar and nonlinar transint hat conduction problms. h ollowing our dirnt load cass hav bn studid: () thrmal load only, (2) thrmal load and mchanical load Q x 4 acting at dg CD, Q x Q y 4 at DE, (3) thrmal load and mchanical load Q x 4 at dg CD, (4) thrmal load and mchanical load acting at Q x 4 dg CD, Q 2 at DE. h shap o hol and strss distribution o th initial dsign and optimum dsign ar shown in Figurs 5, 6, 7 and 8. h maximum strsss hav bn rducd in grat dal by shap optimization. h optimization rsults o thr load cass, maximum von Miss strsss o initial dsign and optimum dsign, th strss rducing ratio, and th optimum valus o dsign variabls, ar givn in abl 4. Compard to th thortical solutions o th cass with only mchanical load, th solutions obtaind hr ar corrct in gnral and mor intrsting or th thrmal dsign o structurs. 6. Conclusions In this papr, th Prcis im Intgration mthod is proposd to solv th snsitivity quations o linar and nonlinar transint hat conduction problms. Dtaild ormulas o th PI mthod ar givn, and numrical approximations and rror analysis or nonlinar problm ar studid.h matrix xponntial o hat conduction y

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