Shock and Vibration 9 (202) 445 46 445 DOI 0.3233/SAV-202-0685 IOS Prss Optimization dsign of structurs subjctd to transint loads using first and scond drivativs of dynamic displacmnt and strss Qimao Liu a,, Jing Zhang c andliubinyan b a Dpartmnt of Civil and Structural Enginring, Aalto Univrsity, Espoo, Finland b Collg of Civil and Architctur Enginring, Guangxi Univrsity, Nanning, China c Dpartmnt of Mchanical Enginring, Indiana Univrsity-Purdu Univrsity Indianapolis, Indianapolis, IN, USA Rcivd 6 Sptmbr 200 Rvisd 9 July 20 Abstract. This papr dvlopd an ffctiv optimization mthod, i.., gradint-hssian matrix-basd mthod or scond ordr mthod, of fram structurs subjctd to th transint loads. An algorithm of first and scond drivativs of dynamic displacmnt and strss with rspct to dsign variabls is formulatd basd on th Nwmark mthod. Th inquality tim-dpndnt constraint problm is convrtd into a squnc of appropriatly formd tim-indpndnt unconstraind problms using th intgral intrior point pnalty function mthod. Th gradint and Hssian matrixs of th intgral intrior point pnalty functions ar also computd. Thn th Marquardt s mthod is mployd to solv unconstraind problms. Th numrical rsults show that th optimal dsign mthod proposd in this papr can obtain th local optimum dsign of fram structurs and somtims is mor fficint than th augmntd Lagrang multiplir mthod. Kywords: Dynamic rspons optimization, gradint, Hssian matrix, tim-dpndnt constraint. Introduction Dynamic rspons analysis of structurs subjctd to transint loads oftn rquirs much mor dmanding computational cost than static analysis. Thrfor th fficincy of th optimization mthod bcoms critical to dynamic rspons optimization problms. Mathmatically, thr ar thr typs of dynamic rspons optimization mthods: zro ordr mthods (nongradint-basd algorithms), first ordr mthods (gradint-basd algorithms) and scond ordr mthods (gradint-hssian matrix-basd algorithms). Gnrally, th first ordr mthods ar mor fficint than th zro ordr mthods, and th scond ordr mthods ar mor fficint than th first ordr mthods. Zro ordr mthods rquir only th information of dynamic rsponss to construct optimal algorithms. A lot of nongradint-basd algorithms 4] hav bn dvlopd to solv th optimization problm of structurs subjctd to transint loads. First ordr mthods rquir th information of dynamic rsponss and thir first drivativs with rspct to dsign variabls to construct optimal algorithms. Thrfor, it is vry crucial to calculat fficintly th Corrsponding author: Qimao Liu, Dpartmnt of Civil and Structural Enginring, Aalto Univrsity, Espoo, FI-00076 Aalto, Finland. Tl.: +358 9 470 2370; Fax: +358 9 470 23758; E-mail: liuqimao2005@63.com. ISSN 070-9622/2/$27.50 202 IOS Prss and th authors. All rights rsrvd
446 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs first drivativs of structural dynamic rsponss for gradint-basd algorithms. Howvr, many algorithms hav bn dvlopd to calculat dynamic rspons first drivativs with rspct to dsign variabls (also calld snsitivity analysis 5 2]). Today, th gradint-basd algorithms ar th mainstram pattrn 3] to solv th optimization problm of structurs subjctd to transint loads. Much rsarch 4 7] has bn don to solv th structural dynamic rspons optimization problm with th gradint-basd algorithms. Scond ordr mthods rquir th information of dynamic rsponss, thir first and scond drivativs with rspct to dsign variabls to construct optimal algorithms. Th Nwton s mthod and Quasi Nwton s mthod ar th commonly usd scond ordr mthod. Although many rsarchrs hav calculatd dynamic rspons first drivativs with rspct to dsign variabls, thr is littl litratur publishd on dynamic rspons scond drivativ analysis (also calld Hssian matrix analysis). Scond drivativ analysis is mor complicatd than first drivativanalysis. Howvr, th fficincy of th optimization using scond drivativ can b gratly improvd whn th gradint and Hssian matrix can b calculatd fficintly. In this papr, a gradint-hssian matrix-basd algorithm for optimization problm of structurs subjctd to transint loads is dvlopd basd on gradint and Hssian matrix calculations. Th purpos of this papr is to dvlop a gradint-hssian matrix-basd optimization mthod for structurs subjctd to dynamic loads. Th main work of this papr is as follows: First, w formulat an algorithm to calculat dynamic rsponssand thir first and scond drivativs with rspct to dsign variabls. Th algorithm is achivd by dirct diffrntiation and only a singl dynamics analysis, basd on Nwmark-β mthod 8], is rquird. Scond, w formulat th tim-dpndnt structural optimization modl. In this modl, total mass of th structur is th objctiv function. Th dynamic rsponss including strsss of th bam lmnt and nodal displacmnts ar constraints. Third, th tim-dpndnt optimization modl is convrtd into a squnc of appropriatly formd unconstraind intgral mathmatic modls using th intrior pnalty function mthod. Th gradint and Hssian matrixs of th intrior pnalty functions ar also calculatd using th first and scond drivativs of dynamic rsponss. Fourth, Marquardt s mthod 9], a gradint-hssian matrix-basd algorithm, is mployd to solv th unconstraind intgral mathmatic modl. Finally, as th illustration of th dvlopd approach, optimization dsigns of a plan fram subjctd to th horizontal dynamic loads ar dmonstratd. 2. Calculation of first and scond drivativs This sction prsnts two subjcts. Sction 2. introducs th calculations of first and scond drivativs of dynamic rsponss (i.., nodal displacmnt and normal strss) with rspct to dsign variabls of structurs. Sction 2.2 discusss how to calculat first and scond drivativs of structural mass with rspct to structural dsign variabls. 2.. Calculation of first and scond drivativs of dynamic displacmnt and strss Th first and scond drivativs of dynamic rspons of structurs ar th prrquisits of an fficint gradint- Hssian matrix-basd algorithm. W will show hr that, in a singl structural dynamic analysis, th first and scond drivativs of a dynamic rspons can b drivd simultanously. Plan bam lmnt is usd xtnsivly in nginring. Thrfor, th dynamic optimal dsign mthod is dmonstratd with th plan bam lmnt in this work. Th lmnt cross-sction shown in Fig. is H structural-stl shap. Th dsign variabls of lmnt ar d, d 2, d 3 and d 4. Point c is th cntroid of th lmnt cross-sction. Axs y and z ar th principal cntroidal axs. Th cross-sctional ara is A =2d d 3 + d 2 d 4 () Th cross-sctional momnt of inrtia is Iz = d 4 d3 2 2 +2 d d 3 3 ( ) ] d 2 + 3 + d 2 d 2 2 d 3 (2) Th farthst distanc y from th nutral axis is
Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs 447 Fig.. Elmnt cross-sction and lmnt dsign variabls. ( ) = ± d 3 + d 2 2 Th first momnt of th shadd ara with rspct to th nutral axis is Sz = d d d 3 + d 2 3 + d 2 d 2 2 d 2 4 4 Th structur is dividd with N lmnts. Th dsign variabl vctor is dfind as. (3) (4) d = d,d 2,d 3,d 4,,d,d 2,d 3,d 4,,d N,d N 2,d N 3,d N ] T 4 (5) Considr th quations of motion for a linar systm subjctd to dynamic forcs Mẍ + Cẋ + Kx = f (t) (6) with th following initial conditions: { x (0) = x0 (7) ẋ (0) = ẋ 0 whr K, M, andc ar stiffnss matrix, mass matrix, and damping matrix, rspctivly. x (t), ẋ (t) and ẍ (t) ar unknown nodal displacmnt, vlocity and acclration vctors, and f(t) is th load vctor. Suppos that th dynamic loads act on th nods. Rayligh damping is usd in this work, th structural damping matrix is C = α M + α 2 K (8) whr ( ) 2 ς ω ς2 ω 2 α = ω 2 ω2 2 (9) α 2 = 2(ς 2ω 2 ς ω ) ω 2 2 ω2 (0) whr ω and ω 2 ar th first and scond natural frquncy of th structur, rspctivly. ς and ς 2 ar th damping ratio. In this work, ς = ς 2 =0.02. α and α 2 ar constants. Equations (6) and (7) must b satisfid for all tim priod t 0, Γ]. Γ is th duration of th dynamic loads. In practic, th solution of this initial-boundary-valu problm (IBVP) rquirs intgration through tim. This is achivd numrically by discrtising in tim th continuous tmporal drivativs that appar in th quation. Any on of th tim intgration procdurs can b usd for this purpos. Th most widly usd family of dirct tim intgration mthods for solving Eq. (6) is th Nwmark family of mthods. Th Nwmark mthod can b formulatd by considring quilibrium at any discrt tim t +Δt, and is givn by th following quation: Mẍ +Cẋ +Kx =f ()
448 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Th nodal displacmnts, nodal acclrations and nodal vlocitis can b achivd at any tim t +Δt by solving Eq. (). Prhaps th most widly usd dirct mthod for th quation of motion (6) is th Nwmark mthod, an implicit tchniqu, which consists of th following finit diffrnc assumptions with rgard to th volution of th approximat solution: ( ) ] x =x (t)+δtẋ (t)+δt 2 2 β ẍ (t)+βẍ (2) ẋ =ẋ (t)+δt ( δ) ẍ (t)+δẍ ] (3) whr any particular choic of th paramtrs β and δ dtrmins th stability and accuracy charactristics of th solution. In this work, paramtrs δ 0.5 and β =0.25 (0.5+δ) 2.Wdfin th intgral constants: a 0 = βδt, ( 2 a = δ βδt, a 2 = βδt, a 3 = 2β, a 4 = δ β,a 5 = Δt δ 2 β ), 2 a 6 =Δt ( δ), a 7 = δδt. Th paramtrs β and δ will b rplacd by thos constants in th following formulas. In addition to Eqs (2) and (3) th quilibrium Eq. () at tim station t +Δt is considrd. This way a systm of quations is formd for th dtrmination of th thr unknowns x, ẋ and ẍ, assuming that th displacmnt, vlocity, and acclration vctors at th prvious tim station t hav alrady bn computd. Thus, th solution for th displacmnt vctor is K x =F (4) whr K = K + a 0 M + a C (5) and F =f +M a 0 x (t)+a 2 ẋ (t)+a 3 ẍ (t)] + C a x (t)+a 4 ẋ (t)+a 5 ẍ (t)] (6) Th acclrations, ẍ, which ar rquird for th computations at th nxt tim station, can b computd as ẍ =a 0 x x (t)] a 2 ẋ (t) a 3 ẍ (t) (7) whil th vlocitis, ẋ, can b obtaind dirctly from quation (3), as follows. ẋ =ẋ (t)+a 6 ẍ (t)+a 7 ẍ (8) Whn th dynamic loads act on th nods, w only nd to dtrmin th intrnal forcs and intrnal coupls at th cntr and two nds of th lmnt. Th lmnt nodal forc vctor, F,is F = K T δ (9) whr K is th lmnt stiffnss matrix in a local coordinat systm, T is th lmnt coordinat transformation matrix, δ is th lmnt nodal displacmnt vctor in a local coordinat systm. Th intrnal forc vctor at th cntr of lmnt, F M,is F M = F F 2 l 2 F 2 F 3 ] T (20) whr l is th lngth of lmnt. Th thr trms, i.., F, F 2 and l F 2 2 F 3, ar axial forc, shar forc and bnding momnt, rspctivly. F, F 2 and F 3 ar th first, scond and third trm of th lmnt nodal forc vctor, F, rspctivly. Th imum strss at th cntr and two nds, i.. th i nd and th j nd, of th lmnt ar calculatd as follows. Th imum normal strss at th i nd of th lmnt is σix = F A + F 3 y m (2)
Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs 449 Th imum normal strss at th j nd of th lmnt is σjx = F 4 A + F 6 y m (22) whr F 4 and F 6 ar th fourth and sixth trm of th lmnt nodal forc vctor, F, rspctivly. Th imum normal strss at th cntr of th lmnt is σmx = F M A + F M3 y m (23) whr F M and F M3 ar th first and third trm of th intrnal forc vctor at th cntr of lmnt, F M, rspctivly. Th imum shar strss at th i nd of th lmnt is τixy = F 2 Sz d (24) 4 I z Th imum shar strss at th j nd of th lmnt is τjxy = F 5 S d 4 I z whr F 5 is th fifth trm of th lmnt nodal forc vctor, F. Th imum shar strss at th cntr of th lmnt is τmxy = F M2 S d 4 I z whr F M2 is th scond trm of th intrnal forc vctor at th cntr of lmnt, F M. (25) (26) 2... Formulas for first drivativs of dynamic displacmnt and strss Now w will driv th formulas for th first drivativs of dynamic rspons, i.., dynamic displacmnt and strss. Diffrntiating Eq. (4) with rspct to th dsign variabl d i,whav x K = F K x (27) whr K and = K + a 0 M + a C (28) F a 0 x (t)+a 2 ẋ (t)+a 3 ẍ (t)] + M = M + C a x (t)+a 4 ẋ (t)+a 5 ẍ (t)] + C a 0 x (t) ẋ (t) + a 2 ] x (t) ẋ (t) ẍ (t) a + a 4 + a 5 ] ẍ (t) + a 3 Aftr th first drivativs of displacmnt vctor at tim t +Δt is obtaind from Eq. (27), through diffrntiating Eq. (7) with rspct to dsign variabls d i,whav ] ẍ x x (t) ẋ (t) ẍ (t) = a 0 a 2 a 3 (30) Thn, th first drivativs of acclration vctor at tim t +Δt is obtaind from Eq. (30). Diffrntiating Eq. (8) with rspct to dsign variabl d i,whav ẋ = ẋ (t) + a 6 ẍ (t) (29) + a 7 ẍ (3)
450 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Diffrntiating Eq. (9) with rspct to th dsign variabl d i,whav F whr δ (t+δt) = K T δ + K T δ (32) can b calculatd from x(t+δt). Diffrntiating Eq. (20) with rspct to th dsign variabl d i,whav F M = d F (t+δt) i F 2 (t+δt) l 2 F 2 (t+δt) F ] T 3 (t+δt) (33) Diffrntiating Eq. (2) with rspct to th dsign variabl d i,whav σix = F + F 3 A F + F 3 A A 2 F 3 y m EI 2 z Diffrntiating Eq. (22) with rspct to th dsign variabl d i,whav σjx = F 4 + F 6 A F 4 + F 6 A A 2 F 6 y m EI 2 z Diffrntiating Eq. (23) with rspct to th dsign variabl d i,whav σmx = F M A F M A 2 A I z (34) I z (35) + F M3 + F M3 F M3 y m Iz 2 (36) Diffrntiating Eq. (24) with rspct to th dsign variabl d i,whav τixy = F 2 S d 4 I z + S F 2 d 4 I z d 4 F 2 S d 2 I z F 2 4 I z S d 4 I2 z Diffrntiating Eq. (25) with rspct to th dsign variabl d i,whav τjxy = F 5 d 4 F 5 S d 4 I z S d 2 I z F 5 4 I z + S F 5 d 4 I z S d 4 I2 z Diffrntiating Eq. (26) with rspct to th dsign variabl d i,whav τmxy d 4 F M2 = F M2 S d 4 I z S d 2 I z F M2 4 I z + S F M2 d 4 I z S d 4 I2 z (37) (38) (39)
Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs 45 2..2. Formulas for scond drivativs of dynamic displacmnt and strss Now w will driv th formulas for th scond drivativs of dynamic rspons, i.., dynamic displacmnt and strss. Furthr diffrntiating Eq. (27) with rspct to th dsign variabl d j,whav K 2 x whr 2 K = and 2 F = 2 F 2 K x K x K x (40) 2 K 2 M 2 C + a 0 + a (4) x (t) ẋ (t) ẍ (t) a 0 + a 2 + a 3 ] = 2 M a 0 x (t)+a 2 ẋ (t)+a 3 ẍ (t)] + M + M ] x (t) ẋ (t) ẍ (t) 2 x (t) 2 ẋ (t) 2 ẍ (t) a 0 + a 2 + a 3 + M a 0 + a 2 + a 3 + 2 C a x (t)+a 4 ẋ (t)+a 5 ẍ (t)] + C ] x (t) ẋ (t) ẍ (t) a + a 4 + a 5 + C ] x (t) ẋ (t) ẍ (t) 2 x (t) 2 ẋ (t) 2 ] ẍ (t) a + a 4 + a 5 + C a + a 4 + a 5 Aftr 2 x(t+δt) is computd from Eq. (40), with furthr diffrntiating Eq. (30) with rspct to th dsign variabls d j,whav 2 ẍ 2 ] x = a 0 2 x (t) 2 ẋ (t) 2 ẍ (t) a 2 a 3 (43) Thn 2 ẍ(t+δt) is obtaind from Eq. (43). Furthr diffrntiating Eq. (3) with rspct to th dsign variabls d j, w obtain 2 ẋ = 2 ẋ (t) + a 6 2 ẍ (t) + a 7 2 ẍ (44) Furthr diffrntiating Eq. (32) with rspct to th dsign variabl d j,whav 2 F (t+δt) 2 = K T δ + K (45) + K T δ (t+δt) + K T 2 δ (t+δt) T δ (t+δt) Furthr diffrntiating Eq. (33) with rspct to th dsign variabl d j, w obtain 2 F M = F 2 (t+δt) 2 F 2 (t+δt) l 2 2 F 2 (t+δt) F ] T 2 3 (t+δt) (46) Furthr diffrntiating Eq. (34) with rspct to th dsign variabl d j,whav 2 σix + 2 F A A A 3 F 3 F 3 = 2 F A F A 2 2 2 F A 2 Iz + F 3 Iz y m F 3 A F A 2 2 A + 2 F 3 EI z y m Iz 2 + I z I z F 3 EI 2 z A + F 3 y m + F 3 EI z ] 2 (42) (47) 2 F 3 y m Iz EI 3 F 3 y m 2 Iz EI 2 z z
452 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Furthr diffrntiating Eq. (35) with rspct to th dsign variabl d j, w obtain 2 σjx + 2 F 4 F 6 = 2 F 4 A A A A 3 F 4 F 6 2 2 I z + F 6 I z y m A 2 F 4 A 2 A 2 A + 2 F 6 I z F 6 F 6 EI 2 z F 4 A 2 + F 6 2 + F 6 Iz + I z 2 F 6 EI 3 Furthr diffrntiating Eq. (36) with rspct to th dsign variabl d j,whav 2 σmx = 2 F M A F M A 2 z A + 2 F M A A A 3 F M 2 A A 2 + 2 F M3 F M3 (t+δt) F M3 2 2 Iz + F M3 (t+δt) Iz y m F M3 EI z EI 2 z I z F M3 (t+δt) EI 2 I z z A EI z 2 (48) Iz F 6 2 Iz EI 2 F M A 2 y m z A + F M3 + F M3 (t+δt) 2 + I z 2 F M3 y m Iz 3 F M3 y m 2 Iz 2 (49) Furthr diffrntiating Eq. (37) with rspct to th dsign variabl d j, w obtain 2 τixy F 2 Iz S = 2 F 2 d 4 I2 z S d 4 F 2 d 2 4 I z d 4 F 2 Sz d 2 d 4 4 I z + d 4 Iz F 2 Sz d 2 4 I2 z Sz d + F 2 4 I z + 2 Sz F 2 d 4 I z S Iz S S d 4 I z + S F 2 F 2 d 4 I z F 2 d 2 d 4 F 2 S 4 I2 d 2 z F 2 d 2 + d 4 d 4 2 F 2 4 I z 2 Iz F 2 S d 4 I2 z 4 I z EI z d 4 Sz d 2 S d 3 4 I z I z F 2 S d 4 I2 z I z Sz F 2 d + I z d 4 F 2 S 4 I2 z d 2 + I z Iz 2 F 2 S 4 I2 z d 4 I3 z Furthr diffrntiating Eq. (38) with rspct to th dsign variabl d j,whav 2 τjxy F 5 Iz S = 2 F 5 d 4 I2 z S d 4 I z + F 5 + 2 Sz F 5 d 4 I z S d 4 I z + S F 5 4 I z F 5 d 4 Sz d 2 d 4 I z 4 I z (50)
Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs 453 S d 4 d 4 F 5 + d 4 I z F 5 F 5 d 2 4 I z S Iz F 5 Sz d 2 d 4 4 I z S d 2 4 I2 z S 2 I z F 5 d 4 I2 z 2 d 4 F 5 F 5 d 2 + d 4 d 4 2 F 5 4 I z S d I z F 5 4 I2 z S d 2 4 I z S d 3 4 I z S d 4 I2 z I z Sz F 5 d + I z d 4 F 5 Sz 4 I2 z d 2 + I z Iz 2 F 5 Sz 4 I2 z d 4 I3 z Furthr diffrntiating Eq. (39) with rspct to th dsign variabl d j, w obtain 2 τ Mxy (t+δt) = 2 F M2 (t+δt) F M2 Iz S S d 4 F M2 d 4 I2 z d 2 4 I z d 4 F M2 + d 4 I z F M2 I z S Sz d 4 I z + F M2 (t+δt) + 2 Sz F M2 d 4 I z S Iz F M2 Sz d 2 d 4 4 I z S d 2 4 I2 z S 2 I z F M2 F M2 d + I z d 4 F M2 4 I2 z d 4 I2 z S d 4 I z + S F M2 (t+δt) d 4 F M2 2 d 4 F M2 F M2 d 2 + d 4 d 4 2 F M2 4 I z S d I z 4 I2 z S d 2 4 I2 z d 4 I z S d 2 4 I z S d 3 4 I z F M2 + I z I z 2 F M2 Sz d 2 S d 4 I2 z S d 4 I3 z 2..3. Computation procdur of th first and scond drivativs of dynamic displacmnt and strss In this work, w suppos that th initial conditions ar x (0) = 0, ẋ (0) = 0, ẍ (0) = 0, and th dynamic rsponss (nodal displacmnts and strsss), thir first and scond drivativs with rspct to dsign variabls ar qual to zro. Sction 2..2 givs th formulas for calculating th first and scond drivativs of th dynamic rspons. This sction provids th dtaild computation procdur as follows. Procdur of calculating dynamic rspons first and scond drivativs: Stp Initial calculations: Stp. x (0) = 0,ẋ (0) = 0,ẍ (0) = 0,σix (0) = 0,σ jx (0) = 0,σ Mx (0) = 0,τ ixy (0) = 0,τ jxy (0) = 0, τ Mxy (0) = 0. Stp.2 x(0) =0, ẋ(0) τ jxy (0) =0, τ Mxy (0) =0. Stp.3 2 x(0) =0, 2 ẋ(0) =0, ẍ(0) =0, σ ix (0) =0, σ jx (0) =0, σ Mx (0) =0, τ ixy (0) =0, 2 τ ixy (0) =0, 2 τ jxy (0) =0, 2 τ Mxy (0) =0. =0, 2 ẍ(0) =0, 2 σ ix (0) =0, 2 σ jx (0) =0, 2 σ Mx (0) =0, 4 I z (5) (52)
454 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Stp.4 St Δt. Stp.5 Comput intgral constants: a 0 = βδt 2,a = δ βδt,a 2 = βδt,a 3 = 2β,a 4 = δ β,a 5 = Δt 2 a 7 = δδt. whr intgral paramtrs δ 0.5 and β =0.25 (0.5+δ) 2. Stp.6 Solv Eq. (5) K. ( ) δ β 2,a 6 =Δt ( δ), Stp 2 Calculations for ach tim stp t +Δt: Stp 2. Solv Eq. (4) x. Stp 2.2 Solv Eq. (7) ẍ, solv Eq. (2) σix, solv Eq. (22) σ jx, solv Eq. (23) σmx, solv Eq. (24) τ ixy, solv Eq. (25) τ jxy, and solv Eq. (26) τmxy. Stp 2.3 Solv Eq. (8) ẋ. Stp 2.4. Solv Eq. (27) x(t+δt). Stp 2.5 Solv Eq. (30) ẍ(t+δt) solv Eq. (35) σ jx (t+δt) solv Eq. (37) τ ixy (t+δt) and solv Eq. (39) τ Mxy (t+δt). Stp 2.6 Solv Eq. (3) ẋ(t+δt). Stp 2.7 Solv Eq. (40) 2 x(t+δt)., solv Eq. (34) σ ix (t+δt), solv Eq. (36) σ Mx (t+δt),,, solv Eq. (38) τ jxy (t+δt), Stp 2.8 Solv Eq. (43) 2 ẍ(t+δt), solv Eq. (47) 2 σix (t+δt), solv Eq. (48) 2 σ jx (t+δt), solv Eq. (49) 2 σ Mx (t+δt), solv Eq. (50) 2 τixy (t+δt), solv Eq. (5) 2 τjxy (t+δt), and solv Eq. (52) 2 τ Mxy (t+δt). Stp 2.9 Solv Eq. (44) 2 ẋ(t+δt). Stp 3 Rptition for th nxt tim stp. Rplac t by t +Δt and implmnt stps 2. to 2.9 for th nxt tim stp. 2.2. First and scond drivativs of structural mass W us structural mass as th objctiv function in optimization. Th objctiv function or th structural mass can b xprssd as w (d) = N ρa l = Th first drivativs of th structural mass ar obtaind by diffrntiating Eq. (53) with rspct to th dsign variabls, w (d) N A = ρl (54) = Th scond drivativs of th structural mass can b obtaind by furthr diffrntiating Eq. (54) with rspct to th dsign variabls, (53)
Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs 455 2 w (d) = N 2 A ρl = (55) 3. Optimization mathmatical modl In gnral, w aim to minimiz th total mass w (d) of th structur. At th sam tim, th normal strsss, shar strsss, and nodal displacmnts should satisfy th constraints in th duration of transint loads. W formulat th optimization problm of th structurs subjctd to transint loads as follows. Find d min w (d) s.t. σix + (d,t) ( =, 2,,N,t 0, Γ]) σ ix (d,t) ( =, 2,,N,t 0, Γ]) σ + jx (d,t) ( =, 2,,N,t 0, Γ]) σ jx (d,t) ( =, 2,,N,t 0, Γ]) σ + Mx (d,t) ( =, 2,,N,t 0, Γ]) σ Mx (d,t) ( =, 2,,N,t 0, Γ]) τ] τixy (d,t) τ] ( =, 2,,N,t 0, Γ]) τ] τjxy (d,t) τ] ( =, 2,,N,t 0, Γ]) τ] τmxy (d,t) τ] ( =, 2,,N,t 0, Γ]) x k ] x k (d,t) x k ] (k =, 2,,N f ) d J d J d J (J =, 2,, 4N) (56) whr + and ar tnsion and comprssion, rspctivly. is allowabl normal strss. τ] is allowabl shar strss. Γ is th duration of transint loads. N f is th numbr of dgr of frdom. x k ] is allowabl displacmnt on th kth dgr of frdom. d J is th lowr limit of th Jth dsign variabl. dj is th uppr limit of th Jth dsign variabl. Normalizing th constraints of th optimal modl Eq. (56), w obtain th nw quivalnt mathmatic modl as follows. Find d min w (d) s.t. g (d,t)= σ+ ix (d,t) 0( =, 2,,N; t 0, Γ]) g +N (d,t)= σ+ ix (d,t) 0( =, 2,,N; t 0, Γ]) g +2N (d,t)= σ ix (d,t) 0( =, 2,,N; t 0, Γ]) g +3N (d,t)= σ ix (d,t) 0( =, 2,,N; t 0, Γ]) g +4N (d,t)= σ+ jx (d,t) 0( =, 2,,N; t 0, Γ]) g +5N (d,t)= σ+ jx (d,t) 0( =, 2,,N; t 0, Γ])
456 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs g +6N (d,t)= σ jx (d,t) 0( =, 2,,N; t 0, Γ]) g +7N (d,t)= σ jx (d,t) 0( =, 2,,N; t 0, Γ]) g +8N (d,t)= σ+ Mx(d,t) 0( =, 2,,N; t 0, Γ]) g +9N (d,t)= σ+ Mx(d,t) 0( =, 2,,N; t 0, Γ]) g +0N (d,t)= σ Mx(d,t) 0( =, 2,,N; t 0, Γ]) g +N (d,t)= σ Mx(d,t) 0( =, 2,,N; t 0, Γ]) g +2N (d,t)= τ ixy(d,t) τ ] 0( =, 2,,N; t 0, Γ]) g +3N (d,t)= τ ixy(d,t) τ ] 0( =, 2,,N; t 0, Γ]) g +4N (d,t)= τ jxy(d,t) τ ] 0( =, 2,,N; t 0, Γ]) g +5N (d,t)= τ jxy(d,t) τ ] 0( =, 2,,N; t 0, Γ]) g +6N (d,t)= τ Mxy(d,t) τ ] 0( =, 2,,N; t 0, Γ]) g +7N (d,t)= τ Mxy(d,t) τ ] 0( =, 2,,N; t 0, Γ]) g k+8n (d,t)= x k(d,t) x k ] 0(k =, 2,,N f ) g k+nf +8N (d,t)= x k(d,t) x k ] 0(k =, 2,,N f ) g J (d) = dj d J 0(J =, 2,, 4N) g J+4N (d) = dj d J 0(J =, 2,, 4N) (57) 4. Transformation of mathmatical modl Th ky fatur of th transformation mthod is to transform a constraind problm into an unconstraind problm. Thus, w minimiz only on function in th transformation mthod. This is an attractiv aspct in that many tim-dpndnt constraints and an objctiv function can b mrgd into a singl function. Th rprsntativs of th transformation mthod ar th augmntd Lagrang multiplir mthod 6] and th xtrior pnalty function mthod 20] in structur optimal dsign undr dynamic loads. Howvr, th augmntd Lagrang multiplir function and th xtrior pnalty function ar discontinuous functions, thrfor, th gradint and Hssian matrix calculations of ths functions ar difficult whn th dirct diffrntiation mthod is mployd to obtain th first and scond drivativs. Compard to th augmntd Lagrang multiplir function and th xtrior pnalty function, th intrior pnalty function is a continuous function, so th gradint and Hssian matrix calculations of th intrior pnalty function ar rlativly asy whn th dirct diffrntiation mthod is usd to obtain th first and scond drivativs. Th intrior pnalty function mthod rquirs a fasibl initial dsign point. Typically, it may b difficult to obtain a fasibl initial dsign in a complx problm. Howvr, in structural optimization problms, a fasibl dsign point can b found in th structurs with th larg cross-sctional aras. Thrfor, in this papr th intrior pnalty function is mployd to transform th inquality constraint optimization problm.
Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs 457 4.. Intrior pnalty function mthod In this papr th inquality constraint optimal mathmatic modl Eq. (57) is convrtd to a squnc of appropriatly formd unconstraind intgral mathmatic modl using th intrior pnalty function mthod. Th intrior pnalty function mthod is adoptd as follows. 2N f +8N Γ P (d,r k )=w (d) r k 8N g J (d,t) dt + (58) g I (d) J= 0 In Eq. (58), pnalty paramtr r k is a squnc of numbrs which ar dgrssiv. Whn r k 0, th minimum of pnalty function P (d,r k ) approachs th minimum of th constraint problm. So th solution of th inquality constraint optimization modl, Eq. (57), is transformd into a squnc of unconstraint problms: Find d 2N f +8N Γ min P (d,r k )=w (d) r k 8N g J (d,t) dt + (59) g I (d) J= 0 Initial pnalty paramtr r can b calculatd by th following quation, 2N f +8N Γ r 8N g J (d 0,t) dt + = p 0 g I (d 0 ) 00 w (d 0) (60) J= 0 I= whr d 0 is th initial dsign point and p 0 = 50. In this work, w choos p 0 = 50. Th pnalty paramtr r k dcrass according to th following rul: r k+ = r k (6) c whr c =0 50 and c = 0 in this work. k is th numbr of pnalty paramtr which will b usd in th procss of sarch. 4.2. Calculation of gradint and Hssian matrix of intrior pnalty function Now w calculat th first and scond drivativs of th pnalty function with rspct to th structural dsign variabls. Th tim stp and duration of dynamic loads ar Δt and Γ, rspctivly. Lt a = Γ Δt. Th first drivativs of pnalty function can b obtaind by diffrntiating Eq. (58) with rspct to th dsign variabl d i, P (d,r k ) = w (d) + r k 2N f +8N Γ J= 0 I= I= g J (d,t) 8N gj 2 (d,t) dt + I= g I (d) gi 2 (d) (62) Th scond drivativs of pnalty function is calculatd by furthr diffrntiating Eq. (62) with rspct to th dsign variabl d j, 2 P (d,r k ) 0 = 2 w (d) + r k 2N f +8N J= Γ 0 2 g J (d,t) gj 3 (d,t) g J (d,t) + 2 ] g J (d,t) gj 2 (d,t) dt 8N 2 g I (d) g I (d) + r k g 3 I= I (d) (63) Th intgral trms in Eqs (58), (59), (60), (62), (63) ar computd by using th trapzoidal form intgral formula: Γ a g J (d,t) dt = ] Δt 2 g J (d,zδt) + (64) g J (d, (z +)Δt) z=0
458 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Γ 0 g J (d,t) gj 2 (d,t) a a dt= z=0 Δt 2 Γ 2 g J (d,t) g J (d,t) 0 gj 3 (d,t) + { Δt 2 g J (d,zδt) 2 g 3 z=0 J (d,zδt) 2 g J (d, (z +)Δt) gj 3 (d, (z +)Δt) g 2 J (d,zδt) g J (d,zδt) + 2 ] g J (d,t) gj 2 (d,t) dt = g J (d,zδt) + g 2 c (d,zδt) g J (d, (z +)Δt) + ] g J (d, (z+)δt) gj 2 (d, (z+)δt) (65) 2 ] g J (d,zδt) + (66) ]} 2 g J (d, (z +)Δt) gj 2 (d, (z +)Δt) Th first and scond drivativs of th pnalty function with rspct to th structural variabls ar calculatd. Thn th gradint and Hssian matrix can b achivd. 5. Solving optimization problms Th inquality constraint optimization modl, Eq. (57), is convrtd into a squnc of th appropriatly formd unconstraind intgral modl, Eq. (59). Marquardt s mthod, a gradint-hssian matrix-basd algorithm, is adoptd to solv th unconstraind problm, taking advantags that th gradint and Hssian matrixs ar fully usd in this optimal mthod. Marquardt s mthod combins Cauchy s and Nwton s mthods in a convnint mannr that xploits th strngths of both but dos rquir scond-ordr information. Th major mrit of Marquardt s mthod is its simplicity, dscnt proprty, xcllnt convrgnc rat nar th optimum, and absnc of a lin sarch. Basd on Marquardt s mthod, th computation procdur of solving th mathmatic modl Eq. (56) is as follows. Th computr procdur of solving th mathmatic modl Eq. (56): Stp. Chos th initial fasibl dsign point d 0, calculat r by solving Eq. (60), dfin convrgnc critrion ε,ltk =. Stp 2. Start from dsign point d k, solv th mathmatic modl Eq. (59) with Marquardt s mthod to obtain th optimum dsign d k.th stps of solving th mathmatic modl Eq. (59) with Marquardt s mthod is from Stp 2.. to Stp 2..: Stp 2.. Lt d (0) k = d k. Dfin M I = imum numbr of itrations allowd ε 2 =convrgnc critrion I =idntity matrix Stp 2.2. St i =0. λ (0) ( =0 5. ) Stp 2.3. Calculat P d (i) k,r k. ( Stp 2.4. Is P d (i) k k),r ε2? Ys: Go to stp 2.. No: Continu. Stp 2.5. Is i M I? Ys: Go to stp 2.. No: Continu. Stp 2.6. Calculat S Stp 2.7. St d (i+) k ( ) d (i) k = ( k + S = d(i) ( d (i+) k,r k Stp 2.8. Is P Ys: Go to stp 2.9. ) <P 2 P d (i) k ). ( d (i) k,r k ( ) ] ( ) d (i) k,r k + λ (i) I P d (i) k,r k. )?
Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs 459 No:Gotostp2.0. Stp 2.9. St λ (i+) = 2 λ(i) and i = i +.Gotostp2.3. Stp 2.0. St λ (i) =2λ (i).gotostp2.6. Stp 2.. St d k = d (i) k. Go to stp 3. Stp 3. Is P (d k,r k ) P (d k,r k ) ε? Ys: d k is th bst dsign. Printrsultandstop. No: Continu. Stp 4. Calculat r k+ by solving Eq. (6), k = k +.Gotostp2. 6. Numrical xampl In this sction, optimization dsign of th plan fram shown in Fig. 2 is prformd using th mthod proposd in this papr. Th plan fram is dividd with thr plan bam lmnts. Th cross-sction of th lmnt is H structural-stl shap, shown in Fig.. W us typical stl matrials proprtis, i.., lastic modulus E = 20 GPa and matrial dnsity ρ = 7850 kg/m 3, for all lmnts. Th structural damping ratios ar ς = 0.02, ς 2 = 0.02. Th horizontal dynamic load, F h (t) = 500 sin 3π 4 t (kn), acts on th nod 2. A duration of 3 s and an incrmntal tim stp of 0.0 s ar considrd in optimization procdurs. Th convrgnc critrion: ε = 0 3, ε 2 = 0 3 and th imum numbr of itrations allowd: M I = 5. Numrical tsts show that ths critria ar sufficint for achiving convrgnc in a rasonabl tim. Fig. 2. Plan fram. In th optimal mathmatic modl Eq. (57), th allowabl normal strss =200 MPa and allowabl shar strss τ] =00 MPa; th nodal allowabl displacmnts ar: x ]=0.00 m, x 2 ]=0.00 m, x 3 ]=0. rad, x 4 ] = 0.00 m, x 5 ] = 0.00 m, x 6 ] = 0. rad; th dsign variabl vctor is dfind as: d = d,d 2, d 3,d 4,d2,d2 2,d2 3,d2 4,d3,d3 2,d3 3,d3 4 ]T. Th dsign spac is shown in Tabl. Tabl Dsign spac of fram (unit: mm) Dsign variabls d d 2 d 3 d 4 d 2 d 2 2 d 2 3 d 2 4 d 3 d 3 2 d 3 3 d 3 4 Lowr limit 00 00 4 4 00 00 4 4 00 00 4 4 Uppr limit 500 000 70 70 500 000 70 70 500 000 70 70 Th optimum dsigns of th plan fram ar sarchd from two diffrnt initial fasibl dsign points. Squncs of optimum dsigns ar shown in Tabls 2 and 3. From th optimum rsults shown in Tabls 2 and 3, th masss of th optimum dsigns approach to th final targts, 26 kg and 6 kg, from th initial fasibl dsign (i.., 53 kg) and 2 (i.., 45 kg), rspctivly. It indicats that th optimization mthod prsntd in this papr is ffctiv. Howvr, th optimum dsigns ar diffrnt if th initial dsigns ar not sam. Thrfor, th optimum dsigns obtaind with th optimization mthod in this papr ar local solutions, but not global solutions. W should find th diffrnt local solutions from as many initial dsigns as possibl. Thn w can choos th bst dsign from th local dsigns as th ffctiv dsign.
460 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Tabl 2 Initial fasibl dsign and optimum dsign (unit: mm) Dsign variabls d d 2 d 3 d 4 d 2 d 2 2 d 2 3 d 2 4 d 3 d 3 2 d 3 3 d 3 4 Mass/kg Tim/s Initial dsign 450 900 60 60 450 900 60 60 450 900 60 60 53 0 Mthod in this papr r = 25.4 342 80 45 46 263 785 42 39 294 794 45 44 63 2392 r 2 = 25.4 34 800 33 34 26 785 23 292 793 34 29 26 4770 r 3 = 2.54 34 800 33 34 26 785 23 292 793 34 29 26 749 Augmntd Lagrang multiplir 363 820 29 32 276 779 8 0 300 80 38 27 24 8426 Tabl 3 Initial fasibl dsign 2 and optimum dsign (unit: mm) Dsign variabls d d 2 d 3 d 4 d 2 d 2 2 d 2 3 d 2 4 d 3 d 3 2 d 3 3 d 3 4 Mass/kg Tim/s Initial dsign 2 300 600 40 40 300 600 40 40 300 600 40 40 45 0 Mthod in this papr r = 96.33 303 600 30 37 298 599 8 8 298 599 27 27 24 28 r 2 = 9.633 302 600 27 35 298 599 8 8 298 599 27 27 22 4235 r 3 = 0.9633 302 600 25 33 298 599 8 8 298 599 22 22 6 6558 r 4 = 0.09633 302 600 25 33 298 599 8 8 298 599 22 22 6 8896 Augmntd Lagrang multiplir 300 596 26 3 298 598 8 7 294 593 25 27 6 042 All simulations ar prformd on a prsonal computr with Window XP oprating systm. Th computr has a Pntium(R) 4 CPU, and frquncy of th CPU is 2.8 GHz. Th computr also has 52MB RAM. Th itration courss shown in Fig. 3 indicat that th algorithm is convrgnt. From initial fasibl dsign, th computational tim of achiving th local optimum dsign is about 749 s with th mthod in this papr, and about 8426 s with th Augmntd Lagrang multiplir mthod. From initial fasibl dsign 2, th computational tim of achiving th local optimum dsign is about 8896 s with th mthod in this papr, and about 042 s with th Augmntd Lagrang multiplir mthod. Th computational tim shows that th mthod proposd in this papr is somtims mor fficint than th Augmntd Lagrang multiplir mthod. Mass/kg 80 60 40 20 00 80 60 40 20 0 0 000 2000 3000 4000 5000 6000 7000 8000 9000 0000 Tim/s Fig. 3. Itration courss. Initial dsign Initial dsign 2 It should b notd that th gradint and Hssian matrix calculation is vry difficult for th dynamic optimization problm bcaus it is difficult to calculat th dynamic rspons first and scond drivativs. Gnrally, th gradint and Hssian matrix calculation rquirs much computational tim that can not b accptd in th structural optimization. Howvr, in this work w dvlop an algorithm, only a singl dynamics analysis is rquird, to obtain th gradint and Hssian matrix. In addition, w us th intrior pnalty function mthod to transform a constraind problm into an unconstraind problm. So many tim-dpndnt constraints and an objctiv function can b mrgd into a squnc of appropriatly formd unconstraind intgral singl tim-indpndnt functions. Thos unconstraind intgral singl tim-indpndnt functions ar continuous, so th gradint and Hssian matrix calculations ar asir than othr discontinuous transform functions, th augmntd Lagrang multiplir function and xtrior pnalty function.
Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs 46 7. Conclusions In this papr, w dvlopd an optimization mthod of structurs subjctd to transint loads. Th conclusions ar as follows. () An algorithm is formulatd to calculat structural dynamic rsponss, i.., nodal displacmnts and strsss, and thir first and scond drivativs with rspct to dsign variabls. Th algorithm is achivd by dirct diffrntiation and only a singl dynamics analysis basd on Nwmark-β mthod is rquird. (2) Th intrior pnalty function mthod is usd to transform a constraind problm into an unconstraind problm. So many tim-dpndnt constraints and an objctiv function can b mrgd into a squnc of appropriatly formd unconstraind intgral singl tim-indpndnt functions. Thos unconstraind intgral singl tim-indpndnt functions ar continuous, so th gradint and Hssian matrix calculations ar asir than othr discontinuous transform functions, th augmntd Lagrang multiplir function and xtrior pnalty function. (3) Th gradint-hssian matrix-basd optimization mthod prsntd in this papr has th charactristics of its simplicity, dscnt proprty, xcllnt convrgnc rat nar th optimum, and absnc of a lin sarch. (4) Th numrical rsults show that th optimum dsigns obtaind with th optimization mthod prsntd in this papr ar local solutions, but not global solutions and that th optimization mthod is ffctiv. (5) Th numrical rsults also show that th optimal dsign mthod proposd in this papr is somtims mor fficint than th augmntd Lagrang multiplir mthod. Somtims, it dpnds on th chosn initial dsign. Rfrncs ] F.Y. Kocr and J.S. Arora, Optimal dsign of h-fram transmission pols for arthquak loading, Journal of Structur Enginring 25 (999), 299 308. 2] A.E. Baumal, J.J. McPh and P.H. Calamai, Application of gntic algorithms to th dsign optimization of an activ vhicl suspnsion systm, Computr Mthods in Applid Mchanics and Enginring 63 (998), 87 94. 3] C.P. Pantlids and S.R. Tsan, Optimal dsign of dynamically constraints structurs, Computrs and Structurs 62 (997), 4 50. 4] I. Buchr, Paramtric optimization of structurs undr combind bas motion dirct forcs and static loading, Journal of Vibration and Acoustics Transactions of th ASME 24 (2002), 32 40. 5] F. van Kuln, R.T. Haftka and N.H. Kim, Rviw of options for structural dsign snsitivity analysis. Part : Linar systms, Computr Mthods in Applid Mchanics and Enginring 94 (2005), 323 3243. 6] C.C. Hsih and J.S. Arora, Dsign snsitivity analysis and optimization of dynamic rspons, Computr Mthods in Applid Mchanics and Enginring 43 (984), 95 29. 7] J.L. Chn and J.S. Ho, Dirct variational mthod for sizing dsign snsitivity analysis of bam and fram structurs, Computrs and Structurs 42 (992), 503 509. 8] K. Kulkarni and A.K. Noor, Snsitivity analysis for th dynamic rspons of viscoplastic shlls of rvolution, Computrs and Structurs 55 (995), 955 969. 9] M. Bogomolni, U. Kirsch and I. Shinman, Efficint dsign snsitivitis of structurs subjctd to dynamic loading, Intrnational Journal of Solids and Structurs 43 (2006), 5485 5500. 0] U. Kirsch, M. Bogomolni and F. van Kuln, Efficint finit-diffrnc dsign-snsitivitis, AIAA Journal 43 (2005), 399 405. ] U. Kirsch and P.Y. Papalambros, Accurat displacmnt drivativs for structural optimization using approximat ranalysis, Computr Mthods in Applid Mchanics and Enginring 90 (200), 3945 3956. 2] K.W. L and G.J. Park, Accuracy tst of snsitivity analysis in th smi-analytic mthod with rspct to configuration variabls, Computrs and Structurs 63 (997), 39 448. 3] B.S. Kang, G.J. Park and J.S. Arora, A rviw of optimization of structurs subjctd to transint loads, Structural and Multidisciplinary Optimization 3 (2006), 8 95. 4] J.S. Arora and J.E.B. Cardoso, A dsign snsitivity analysis principl and its implmntation into ADINA, Computrs and Structurs 32 (989), 69 705. 5] C.C. Hsih and J.S. Arora, A hybrid formulation for tratmnt of point-wis stat variabl constraints in dynamic rspons optimization, Computr Mthods in Applid Mchanics and Enginring 48 (985), 7 89. 6] A.I. Chahand and J.S. Arora, Dvlopmnt of a multiplir mthod for dynamic rspons optimization problm, Structural Optimization 6 (993), 69 78. 7] C.P. Pantlids and S.R. Tsan, Optimal dsign of dynamically constraints structurs, Computrs and Structurs 62 (997), 4 50. 8] N.M. Nwmark, A Mthod of Computation for structural dynamics, Journal of Enginr Mchanics Division 85 (959), 67 94. 9] G.V. Rklaitis, A. Ravindran and K.M. Ragsdll, Enginring Optimization Mthods and Applications, John Wily and Sons, Nw York, 983. 20] J.H. Cassis and L.A. Schmit, Optimum structural dsign with dynamic constraints, Journal of Structural Enginring Procdings ASCE 02 (976), 2053 207.