Robust Design Optimization of Building Frames using Order Statistics and Local Search

Transcription

1 The Eghth Chna-Japan-Korea Jont Symposum on Optmzaton of Structural and Mechancal Systems May 5-9, 04, Gyeongju, Korea aper No 0059 Robust Desgn Optmzaton of Buldng Frames usng Order Statstcs and Local Search Yuse Tanguch, Maoto Ohsa, Maoto Yamaaa Department of Archtecture, Hroshma Unversty, 79-57, Hgash-Hroshma, Japan Department of Archtecture, Hroshma Unversty, 79-57, Hgash-Hroshma, Japan Correspondng author: Department of Archtecture, Toyo Den Unversty, 0-55, Toyo, Japan Abstract A method combnng pure random search (RS) and local search (LS) s presented for robust desgn optmzaton of buldng frames The robust desgn problem s formulated as a to-stage problem consstng of upper-level optmzaton problem and loer-level antoptmzaton problem The loer-level problem s solved usng RS, and ts stoppng rule s defned based on the order statcs The upper-level problem s solved usng multstart LS, for hch probablstc stoppng rules are nvgated The proposed approach s appled to a buldng frame subjected to a sesmc moton The objectve functon s the total structural volume, and a constrant s gven for the orst value of mamum nterstory drft angle beteen the roof and base The results demonstrate the effectveness of the proposed method Keyords: Uncertanty; To-stage robust desgn problem; Order statstcs; ure random search; Local search Introducton In the desgn process of structures n varous felds of engneerng, uncertanty n the parameters such as materal propertes and geometry should be approprately ncorporated Relablty-based desgn s the most popular approach, f the probablty models of parameters can be approprately assgned Hoever, the varatons of parameters are usually unnon, and only ther bounds may be mated Therefore, the desgn problem turns out to be a to-stage robust desgn problem, here the orst response s found n the loer problem, and the optmal desgn varables are found n the upper problem [,] For a practcal desgn problem of comple responses and constrants, t s dffcult to fnd the globally orst response n the loer problem Furthermore, the probablty for tang the etreme value s very small Therefore, t s desred to develop an approach that does not depend on the model of probablty dstrbuton, and appromates the etreme value th specfed accuracy The authors developed a method for fndng the appromate orst value usng pure random search (RS) [], here the stoppng rule s defned based on order statcs [4] In ths study, e use our approach for the desgn of a buldng frame The upper problem s a combnatoral problem, hch s solved usng a multstart local search (LS) The accuracy of the stoppng rule for RS for the loer problem s frst verfed The number of local optmal soluton n the upper problem s mated based on the sze of attractor that leads to each local optmal soluton usng a seres of determnstc LSs Correspondng author Tel and Fa: , E-mal address: ohsa@hroshma-uacjp To-stage robust desgn problem We frst formulate a to-level optmzaton problem The desgn varable vector s denoted by th ts feasble regon The vector consstng of uncertan parameters of buldng structures s denoted by The objectve functon of the upper problem s denoted by f For smple presentaton of the method, the uncertantes are ncorporated nto only one constrant functon g,, for hch the upper bound g s gven Hence, the robust desgn optmzaton problem s formulated as mnmze f subject to g, g () for all ma The orst value g of the constrant functon g, s obtaned by solvng the follong loer-level problem: g fnd gma ma, subject to () By usng (), e can formulate the upper problem as a robust desgn problem or a orst-case desgn problem: mnmze subject to f g ma g ()

2 ure random search for upper-level problem In ths secton, e summarze the stoppng rule for RS proposed n Ref [] based on order statstcs [4] For smplcty, e omt the desgn varable as gma ma g because e focus on the loer problem A RS s used for solvng problem () We generate an ndependent sample,, n from a unform probablty dstrbuton on, and the correspondng ndependent sample Y g,, Yn gn of the objectve functon values at these ponts Suppose Y j has a probablty dstrbuton F Y The sequence Y,, Yn s arranged n ncreasng order, and the th value s denoted by Yn, such that Y, n Y, n Yn, n The propertes of Y, n,, Yn, n are obtaned by the order statstcs Snce the globally orst value s dffcult to obtan, e use the follong constrant for the loer problem: Yn, g (4) hch ndcates that the th value of the total n samples from RS satsfes the constrant of the upper problem Let and 0, denote preassgned constants, and e choose n and satsfyng I, n (5) here I s the ncomplete beta functon If Eqs (4) and (5) are satsfed, then the follong equaton holds: r Y F g (6) e, e have at least 0 % confdence that at least a proporton of the total n samples s less than g These results ndcate that the number of samples n and the th value n the samples are closely related to the parameters and, and the accuracy of the soluton of RS s ensured by approprately assgnng n and If e assgn the loer bund for n, hch s gven for avodng obtanng the etreme value, the mnmum value for the prescrbed confdence s found by solvng the follong optmzaton problem: mnmze n subject to I, n n 4 Local search for loer-level problem In ths secton, e summarze the stoppng rules for LSs for solvng the upper robust desgn optmzaton problem, hch s a mnmzaton problem The follong determnstc algorthm of LS s used: (7) Algorthm (Local Search) Sample an ntal random pont 0 from a unform probablty dstrbuton Set 0 Enumerate all N neghborhood solutons of, denoted by (,, N), and compute f ( ) mn Select the b soluton, hch has the small value of f ( ) mn mn 4 If f( ) f( ), let,, and go to ; otherse, output as a local optmal soluton and termnate the Algorthm Suppose e obtan local optmal solutons,, by carryng out LS t tmes from randomly generated ntal solutons The number of LSs that fnd s denoted by n, e, n n t Defne X as attractor or regon of attracton [5] of, hch s the set of solutons that leads to by carryng out LS The rato of the sze s of attractor X to number of all the feasble solutons s denoted by c If s s defned as the number of ntal solutons leadng to, then s n and s s t hold Suppose there h local optmal solutons, hch are not non a pror Then, c c h s satsfed Boender and Kan [6] derved the follong mate of the number of local optmal solutons based on Baysan approach, here c s supposed to be unformly dstrbuted beteen 0 and satsfyng c c h, and s obtaned as the mean value of posteror mate of h t ( ) () t Based on Eq (), e can use the follong stoppng rule of multstart LS: Rule : Termnate multstart LS f 05 s satsfed Let a denote the number of local optmal solutons that have not been found after carryng out LSs t tmes; e, h = + a Snce e use a determnstc algorthm for LS, the follong to cases are consdered for matng s,, s and c,, c C-: The attractor X conssts of the ntal solutons that reaches ; e, s n C-: The attractor X conssts of all feasble solutons along the path beteen the ntal soluton and Furthermore, the szes s,, sh of attractors X,, Xh of the solutons that have not been found are mated by one of the follong methods:

3 C-mean: s,, sh are equal to the mean value of s,, s C-mn: s,, sh are equal to the mnmum value of s,, s For eample, C--mean denotes that the methods C- and C- mean are used ( ) Let c h denote the rato of s of X hen there ets h local optmal solutons, hch are mated by dvdng s by the sum of s,, sh Then the lelhood ( h) ( n,, n) for n tmes fndng (,, ) n t trals s computed as h C h n ( c, ( n n t) (9) t, ) here C t, s a coeffcent that vanshes n the follong equatons Based on Eq (9), e can use the follong stoppng rule of multstart LS: Rule : Termnate multstart LS f ( j) / ( ) s smaller than a specfed small value, here j s a specfed value Fnally, suppose e fnd c ( h) usng the combnaton of (C- or C-) and (C-mean or Cmn) Then, the probablty of mssng the (+)th soluton n t trals s computed as ( h) ( h) t ( c ) () Based on Eq (), e can use the follong stoppng rule of multstart LS: Rule : Termnate multstart LS f value (h) s less than a prescrbed small For the case h, the follong relaton holds: c ( ) c () ( ) ( ) c Therefore, t s easly seen that ( ) ( ) c ( ) ( ) ( ) c c ( ) c t ( ) ( ) c t () s satsfed as confrmed n the follong eamples Therefore, e nvgate only Rules and n the eamples 4 Eample of mathematcal problem for LS The number of local optmal solutons s mated for a t functon called Sheel- [5], hch s defned as f T b A A here b s the th component of b, A, and , () A s the th ro of 0, () A c, 4 Ths functon has optmal solutons To solve ths problem as a combnatoral problem, each component of s supposed to have dscrete values th unform nterval of 0 beteen the loer bound 0 and the upper bound The number of neghborhood solutons s 4 ncludng the current soluton, because there are four varables and three patterns, 0, + for the ncrement of each varable Table Ten solutons of problem Sheel- for LS- Soluton No f ( ) s (C-) Soluton No f ( ) s (C-)

4 Seres of LSs denoted by LS-, LS-7 s carred out from randomly generated 7 ntal solutons Ten optmal solutons are found before satsfyng Rule n 6 cases All solutons and correspondng szes of attractors for LS- are lsted n Table The number of trals mstop for satsfyng Rule, and the number of trals m for obtanng all local optmal solutons are lsted n Table Table Number of steps m stop for satsfyng Rule and m hen all solutons are found for LS-,, LS-7 LS No m stop 9 m () () () () Table Ratos / and / for problem Sheel- at m stop trals of LS- () () () () / / () () C--mean C--mn C--mean C--mn () () () () () Ratos / and / evaluated at mstop are lsted n Table We can see from the () () () table that values of / ( ) are very small f C-mean s used; hoever, they are not suffcently small f C- () () () mn s used By contrast, ratos / ( ) evaluated at the step m usng C-mean seems to be approprate as shon n Table 4 Therefore, a less strct stoppng rule s desred 5 Eample of robust desgn of a buldng frame 5 Descrpton of model Effectveness of the proposed robust desgn method s nvgated for a sesmc desgn problem of a 4-story plane shear frame model as shon n Fg All columns have the same secton C Beams are classfed nto groups, hch consst of G of nd and nd floors and G of 4th floor and roof We choose each soluton from predefned secton n Table 5 We use frame analyss softare OpenSees Ver 4 [7] for tme hstory response analyss The materal s steel th Young s modulus E = 05 Ga lastc hnges of length 0 m can at both ends of members, hch s modeled as a fber secton th nematc hardenng rato 00E We use an artfcal ground motons as shon n Fg, hch s compatble to the acceleraton response spectrum n Table 5 The duraton s 0 sec, and the tme step 00 sec Table 5 Target acceleraton response spectrum (dampng factor = 005) erod (s) T T T Acceleraton (m/s ) T 0 7/T () () () () Table 4 Ratos / and / for problem Sheel- at m trals of LS- () () () () / / () () C--mean C--mn C--mean C--mn Fg A 4-story shear frame model Fg Hstores of and for LS- of Sheel- Hstores of and are plotted for LS- n Fg, hch shos that s a good upper bound for ; hoever, the convergence of to s very slo also for LS-, 4, 6, and 7 Fg Sesmc moton 4

5 Table 6 Lst of avalable sectons lst Column B B -4 0 H H H H H H H H H H H H H H H H H H H H than 900 n 5 trals Table 7 Numbers and ratos of parameters sets satsfyng g( ) Y6,65 among randomly generated 00 sets, and the etreme values for four RS-,, RS-4 RS No 4 Y 6, Y 65, Verfcaton by 00 samples Number of parameter sets satsfyng g( ) Y 6,65 Y 00, The desgn varable vector represents the cross-sectons of beams and columns For eample, the jth secton n Table 6 s chosen for the th varable f j, here,, and correspond to Column, B, and B, respectvely The objectve functon s the total structural volume V ( ) that s to be mnmzed Constrants are gven so that the orst value of the mamum nterstory drft angle ma (, Θ ) beteen the roof and base does not eceed the upper bound 00 Uncertanty s gven for the yeld stresses C and G, respectvely, of columns and beams as 5 5 Θ (, ) (4) 0 0 (,) C, G The problem of robust desgn optmzaton s defned as follos: mnmze f( ) V( ) subject to g(, Θ) ma (, Θ ) 00 (5) {,,, }, (,, ) 5 Verfcaton of etreme value th specfed accuracy The uncertan parameters representng the yeld stresses are assumed to dstrbute unformly beteen the upper and loer bounds, because the formulatons n Sec do not depend on the types of dstrbuton By solvng problem (7) for and 09, e obtan n 65 and 6, hch means that the 6nd small value among objectve values of 65 parameter sets generated by RS s regarded as the appromate orst value of nterstory drft angle Table 7 shos the results of four sets RS-,, RS-4 of RSs of 65 trals for the desgn (,, ) (5,,5) The number of parameter sets satsfyng g( Θ ) Y6,65 among randomly generated 00 sets s also lsted It s confrmed that the numbers are not less than The results of 50 sets of 00 trals are lsted n Table Note that the number of parameter sets satsfyng g( ) Y6,65 s less Table Number of parameter sets satsfyng g( ) Y6,65 among 50 sets of 00 trals Optmzaton results We carry out to sets of mutstart LSs th t = 0 and 50, respectvely Eght solutons found by 0 trals are lsted n (9) () () () () Table 9 The values of /, /, and are shon n Table 9The objectve values and the szes of attractors are also lsted The results of t = 50 trals from dfferent ntal random seed from t = 0 are also lsted n Tables and, here soluton are found for ths case Note that the LS used here s not completely determnstc, because the loer problem nvolves uncertanly It s seen from Tables 9 and that the solutons 6 and 7 n Table 9 do not n Table ; therefore, there at least local optmal solutons It can be confrmed from Tables 9 ( ) ( ) and that / and ( ) have the same value f C-mean s used Table 9 Eght solutons found by 0 trals C G G V() s Table Ratos (9) () () () / and / for t 0 (9) () () () () / / () 5

6 C--mean C--mn C--mean C--mn Table Eleven solutons found by 50 trals C G 6 G V() s C G G V() s Table Ratos of (4) () / and / for t 50 () () () () () (4) () () / / C--mean C--mn C--mean C--mn Table Estmated and obtaned numbers of local optmal solutons for t = 0 and 50 t Fg4 Hstores of and for t 50 We can see from these results that () for t = 50 s () smaller than for t = 0, f C-mean s used The mated and obtaned numbers of local optmal solutons for t = 0 and 50 are lsted n Table, hch shos that s larger than by about 6 for both t = 0 and 50 Therefore, ehbts slo convergence also n ths eample For Rule, (4) () the value of / for C--mean has suffcently small value Therefore, through further nvgaton for other problems, e may use Rule for the stoppng rule of LSs 6 Conclusons A robust desgn optmzaton problem has been formulated as a to-stage optmzaton problem The orst response s found n the loer problem, and the locally optmal desgn varables s found n the upper problem In the loer problem, the appromate orst value s found usng the pure random search and order statstcs The accuracy of the results have been confrmed n the eample of a four-story shear frame subjected to sesmc motons, here the mamum average drft angle s consdered as the representatve response The upper problem s solved usng a multstart local search, here the varables are supposed to tae dscrete values Three stoppng rules and formula for matng number of local solutons are compared n a mathematcal problem It has been confrmed that the frst rule proposed n Ref [6] s a lttle conservatve The second rule proposed n ths study may also be used after further nvgaton for other problems It has been shon n the eample of robust desgn of shear frame that there are multple local optmal solutons; hoever, applcaton of stoppng rules to robust desgn s dffcult, because computatonal cost for each local search s very large, and e cannot fnd many local optmal solutons Acnoledgment Ths or s partly supported by Grant-n-Ad for Scentfc Research of JSS (No ) References [] I Elshaoff and M Ohsa, Optmzaton and Ant- Optmzaton of Structures under Uncertanty, Imperal College ress, 0 [] M Ohsa and M Katsura, A random samplng approach to orst-case desgn of structures, Struct Multdsc Optm, Vol 46, pp 7-9, 0 [] M Yamaaa and M Ohsa, Worst-case desgn of structures usng stoppng rules n -adaptve random samplng approach, roc th World Congress of Structural and Multdscplnary Optmzaton (WCSMO), Orlando, aper No 54, 0 [4] H A Davd and N N Haady, Order Statstcs, rd Ed, John Wley & Sons, Inc, 00 [5] I E Lagars and I G Tsoulos, Stoppng rules for boconstraned stochastc global optmzaton, Appl Math Comp, Vol 97, pp 6-6, 00 [6] C G E Boender and A H G Rnnooy Kan, Baysan stoppng rules for multstart global optmzaton methods, Mathematcal rogrammng, Vol 7, pp 59-0, 97 [7] Open System for Earthquae Engneerng Smulaton (OpenSees), EERC, USB, 6