Sampling-based Approach for Design Optimization in the Presence of Interval Variables

Transcription

1 0 th Word Congress on Structura and Mutdscpnary Optmzaton May 9-4, 03, Orando, orda, USA Sampng-based Approach for Desgn Optmzaton n the Presence of nterva Varabes Davd Yoo and kn Lee Unversty of Connectcut, Storrs, Connectcut, USA, davd.yoo@engr.uconn.edu Unversty of Connectcut, Storrs, Connectcut, USA, ee@engr.uconn.edu. Abstract Ths paper proposes a methodoogy for sampng-based desgn optmzaton n the presence of nterva varabes. Assumng that an accurate surrogate mode s avaabe, the proposed method frst searches the combnaton of nterva varabes for constrants when ony nterva varabes are present or for probabstc constrants when both nterva and random varabes are present. Due to the fact that the combnaton of nterva varabes for probabty of faure does not aways concde wth that for a performance functon, the proposed method drecty uses the probabty of faure to obtan the combnaton of nterva varabes when both nterva and random varabes are present. To cacuate senstvtes of the constrants and probabstc constrants wth respect to nterva varabes by the sampng-based method, behavor of nterva varabes at the case s defned by the Drac deta functon. Then, Monte Caro smuaton s apped to cacuate the constrants and probabstc constrants wth the combnaton of nterva varabes, and ther senstvtes. The mportant mert of the proposed method s that t does not requre gradents of performance functons and transformaton from -space to U-space for reabty anayss after the combnaton of nterva varabes s obtaned, thus there s no appromaton or restrcton n cacuatng senstvtes of constrants or probabstc constrants. Numerca resuts ndcate that the proposed method can search the case probabty of faure wth both effcency and accuracy and that t can perform desgn optmzaton wth mture of random and nterva varabes by utzng the case probabty of faure search.. Keywords: nterva Varabes, Sampng-Based Method, Drac Deta functon, Monte Caro smuaton, Surrogate mode 3. ntroducton eabty anayss and reabty-based desgn optmzaton (BDO) have been deveoped to take uncertanty nto consderaton, and have been successfuy adapted to many engneerng appcatons such as crashworthness of vehce and structura-acoustc system desgn [-0]. The uncertanty s generay categorzed nto aeatory and epstemc uncertantes, where aeatory uncertanty s consdered as rreducbe whereas epstemc uncertanty s reducbe by coectng more data. n case when suffcent amount of data for statstca nformaton s unavaabe, possbty-based (or fuzzy set) methods have utzed membershp functon to mode nsuffcenty coected data [], and adusted standard devaton and correaton coeffcent nvovng confdence ntervas have been utzed to offset an naccurate modeng of data [,3]. When degree of nsuffcency of data s even greater as ony ower and upper bounds of data are avaabe, the methods sted above are not appcabe anymore, thus the dfferent approach s requred. To dea wth data of whch ony ower and upper bounds are avaabe, a method of mut-pont appromaton that evauates the weghtng functon and oca appromatons separatey has been frst deveoped for nterva anayss [4]. Then, the most probabe pont (MPP) based frst-order reabty method (OM) has been utzed for desgn optmzaton wth mture of random and nterva varabes [5]. As bounds of probabty of faure or reabty est n the presence of nterva varabes, desgn optmzaton for the and best cases has been aso deveoped [6], and senstvty anayss consderng bounds of nterva varabes and probabty of faure has been deveoped accordngy [7]. By usng the MPP-based OM, a desgn optmum s very effcenty searched; however t s generay ess accurate for hghy nonnear performance functons and hgh-dmensona nput varabes [8-]. To mprove the accuracy on ths occason, the second order reabty method (SOM) can be apped after the MPP search; however, ts effcency s sacrfced due to the fact that computaton of the Hessan matr s requred by the SOM [-5]. The MPP-based dmenson reducton method (DM) can be aso used for appromatey assessng the reabty of a system, whch s used as a probabstc constrant n BDO [6-8]. n absence of accurate senstvtes of performance functons, the MPP-based reabty anayss or BDO, whch utzes senstvtes of performance functons to fnd the MPP, cannot be drecty used, nstead the sampng-based reabty anayss or BDO can be used [9-3]. Assumng an accurate surrogate mode s gven [3-36], Monte Caro smuaton (MCS) [37] can be apped to fnd a desgn optmum wth affordabe computatona burden. Ths study ntroduces nterva anayss and desgn optmzaton utzng the sampng-based method n the

2 presence of ony nterva varabes and n the presence of both random and nterva varabes. Due to the presence of nterva varabes, obtanng the combnaton of nterva varabes for both constrants and probabstc constrants s nvoved [5]. When both random and nterva varabes are present, the combnaton of nterva varabes for probabty of faure s drecty searched usng the probabty of faure and ts senstvty snce the desgn pont where the case probabty of faure occurs does not aways concde wth that for the case performance functon; t s hghy key as many studes have assumed, however not aways. To evauate senstvtes of probabty of faure wth respect to nterva varabes, the Drac deta functon s utzed to defne behavor of the nterva varabes at the case [38-4]. Assumng an accurate surrogate mode s gven, one mert of the proposed method ests not ony durng the case probabty of faure search but aso durng reabty anayss after the case probabty of faure search. The case probabty of faure search, whch w be epaned n Secton 4., utzes a vector of nterva varabes nstead of ndvdua components of the vector, and t thus promses effcency. Aso, t resoves the probem that the case probabty of faure does not aways occur where the case performance occurs. Durng the reabty anayss after the case probabty of faure search, another mert of the proposed method s that t does not make further appromatons snce t does not requre gradents of the performance functon and transformaton of desgn varabes from -space to U-space, thus there s no appromaton or restrcton n cacuatng the senstvtes of constrants or probabstc constrants [30]. The paper s organzed as foow. Secton 4 brefy revews the sampng-based BDO. Secton 5 epans desgn optmzaton wth nterva varabes ony, ncudng the agorthm to obtan the combnaton of nterva varabes for a performance functon, mathematca dervaton for senstvtes of each constrant wth respect to nterva varabes by defnng behavor of the nterva varabes at the case usng the Drac deta functon, and ther computaton by the MCS. Secton 6 epans the sampng-based desgn optmzaton wth both random and nterva varabes ncudng detas to obtan the combnaton of nterva varabes for probabty of faure and computaton of probabstc constrants and ther senstvtes by the MCS. Secton 7 ustrates search for the combnaton of nterva varabes for probabty of faure and desgn optmzaton wth random and nterva varabes usng numerca eampes. Secton 8 summarzes and concudes the paper wth dscusson of future research. 4. evew of Sampng-based BDO 4. ormuaton of BDO The mathematca formuaton of BDO s epressed as mnmze cost( d) tar subect to P G 0 P,,..., NC L U ndv d d d d,, and N () where d ( ) s the desgn vector, whch s the mean vaue of the N-dmensona random vector T,,... N ; P s the target probabty of faure for the th constrant; and NC, ndv, and N are the tar number of probabstc constrants, desgn varabes, and random varabes, respectvey [30]. To carry out BDO usng Eq. (), the probabstc constrants and ther senstvtes must be evauated. evews on the reabty and ts senstvty anayses are epaned n Sectons. and.3, respectvey. 4. Probabty of aure The probabty of faure wth random varabes, denoted by P, s defned usng a mut-dmensona ntegra ; P P f d E () N where s a matr of dstrbuton parameters, whch ncudes mean ( ) and/or standard devaton ( ) of P represents a probabty measure; Ω s defned as a faure set; f ; ; and functon (PD) of ndcator functon and defned as E represents the epectaton operator [30,4]. ; s a ont probabty densty n Eq. () s caed an

3 ,. (3) 0, otherwse 4.3 Senstvty of Probabty of aure Wth the four reguarty condtons satsfed, whch are aso epaned n deta n ef. [30], takng the parta dervatve of Eq. () wth respect to yeds P f ; d (4) N and the dfferenta and ntegra operators can be nterchanged due to the 4 th reguarty condton n ef. [30] and the Lebesgue domnated convergence theorem [43,44] gvng P E n f ;. (5) The parta dervatve of the og functon of the ont PD n Eq. (5) wth respect to s known as the frst-order score functon [30] for and s denoted as () s n f ; ;. (6) To derve the senstvty of the probabty of faure n Eq. (), t s requred to know the frst-order score functon n Eq. (6), whch s obtaned usng the foowng equaton for ndependent random varabes where ; () s n f ; n f ; ; (7) f s the margna PD correspondng to the th random varabe foowng equaton for correated random varabes () s cu v n f ;, and obtaned usng the n f ; n, ; ; (8) where c s a copua densty functon, u ; and v ; are margna CDs for and respectvey, and s the correaton coeffcent between and [30]. The nformaton of margna PDs, CDs, and commony used copua densty functons s sted n deta n ef. [30]. 4.4 Smpfcaton of the nonnear characterstcs of vehce behavor The MCS can be apped to cacuate the probabstc constrants n Eq. () and ther senstvtes. Denotng a surrogate mode for the th constrant functon wth random varabes as ˆ G Eq. () can be cacuated as K k,, the probabstc constrants n K ( k ) tar P P G 0 ˆ P (9) where K s the MCS sampe sze, ( k ) s the k th reazaton of, and the faure set ˆ for the surrogate mode 3

4 s defned as ˆ G ˆ : 0 [30]. Senstvtes of the probabstc constrants n Eq. () are cacuated usng the score functon as P ; (0) K ( k) () ( k) ˆ s K k () ( k where s ) ; s obtaned usng Eqs. (7) and (8) for ndependent and correated random varabes, respectvey. 5. Desgn Optmzaton wth nterva Varabes 5. ormuaton of Desgn Optmzaton wth nterva Varabes The mathematca formuaton of desgn optmzaton wth nterva varabes ony s epressed as mnmze cost( d), subect to G 0,,..., NC () L U ndv d d d d,, and N where d s the desgn vector, whch s the md-pont of the N-dmensona nterva vector T,,,..., N where N s the number of nterva varabes. n Eq. () s the case nterva varabes for the th constrant, whch s obtaned by sovng the optmzaton probem to where s the nterva ength of s not avaabe, mamze G subect to,..., for N (). t shoud be noted that as any statstca nformaton of an nterva varabe, must be consdered for the desgn optmzaton. To carry out the desgn optmzaton wth nterva varabes usng Eq. (), constrants wth the case nterva varabes, namey the case constrants or the case performance, and ther senstvtes must be evauated. Each of the case constrant s obtaned by the case performance search that soves Eq. () and w be epaned n Secton 3., and senstvty anayss of each of the case constrant and ts cacuaton are epaned n Secton 3.3. t s assumed n ths study that gradents of performance functons are not avaabe; however t can be drecty used f avaabe. 5. Agorthm Searchng for Worst Case Performance The agorthm epaned n ths secton searches the case performance, and the agorthm was orgnay deveoped by Lu et a. n ef. for the mama possbty search (MPS) for possbty-based desgn optmzaton. An mportant mert of the proposed agorthm s that t utzes a vector of nterva varabes and a vector of senstvtes of a performance functon wth respect to a nterva varabes, thus ts effcency s not affected by the dmenson of the nterva varabes. The agorthm for the case performance search s summarzed as foowng, whch s aso shown n the fowchart n g.. Step. Normaze nterva varabes usng Z Z (3) or such that Z

5 Step. Set the teraton counter k = 0 wth the convergence parameter = 0-3 (0). Set =. Let Z 0. Cacuate the performance G (0) (0) Z and the senstvty G Z (0) (0) (0) to obtan G Z. Let the drecton vector be G ( k) ( k) Step 3. Search the net pont as 0.5sgn ( ) Step 4. Cacuate the performance G k ( k ) Z and ts senstvty G d Z.. t s epaned n Secton 3.3 how Z d where 0.5 s obtaned from Step. Let k = k +. Z. Let a conugate drecton vector ( k ) k ) k d G Z d ) where G ( k) / G ( k). k ) sgn sgn k G Z Z ), t s the case and go to Step. Step 5. ( f ) ( ) ( ) G Z GZ, et = k and go to Step 3. Otherwse, go to Step 6 wth Z, G ( ) ( ) G Z. Z Z f Z and f behavor of the performance functon s not monotonc wthn an nterva doman, n other words, f any component of the case nterva vector does not occur at the verte of ts nterva doman, nterpoaton agorthm must be addtonay apped to obtan more accurate case performance []. ( Step 6. Let = 0 and a drecton vector be ) ( ) G Step 7. d Z. ( ) Cacuate the new pont Z k on the boundary of the doman from the start pont search drecton () d. Let k = k +. ( ) Step 8. Cacuate the performance G k ( ) Z and ts senstvty G Z k ( k ) sgn G Z / Z sgn Z, for Z 0.5 ( k ) GZ / Z, for Z 0.5 then t s the case and go to Step. Otherwse, go to Step 9. ( ) Step 9. Use G ( ) Z, G k ( ) Z, G ( ) Z, and G k. f ( ) Z aong the Z to construct the thrd order poynoma f() t ( ) on the straght ne between Z ( ) and Z k where t s the parameter for the ne. Cacuate the mamum pont t * ( ) for ths poynoma. Let Z k be the pont on the ne correspondng to t *. Let k = k +. ( k ) G ( k ) G Z. Check the convergence crtera Step 0. Cacuate the performance Z and ts senstvty Step. usng the equaton n Step 8. f converged, t s the case and go to Step. Otherwse, et the new ( ) ( k ) ( d G Z d ) where β s gven conugate drecton vector be ( k) ( k) by G G De-normaze Z / Z. Let = k, = +, and go to Step 7., Z by Eq. (3) n Step to obtan,. The proposed agorthm requres evauaton of senstvtes of a performance functon wth respect to nterva varabes. When gradents of the performance functon are not avaabe, senstvtes of each performance functon wth respect to nterva varabes can be cacuated by the sampng-based method, and dervaton of the senstvtes of the performance functon wth respect to nterva varabes and ts cacuaton are epaned n Secton

6 nterpoaton nta nterva Varabes Step 5. Store nterva Varabes, Performance, and Senstvty Step. Normaze nterva Varabes Yes Step 6. Cacuate Drecton Vector Step. Cacuate Performance and Senstvty Step 3. Search the Net Pont by Drecton Vector No Check Convergence n Step 5. Step 7. Search the Net Pont by Drecton Vector on the Boundary on the Doman Check Convergence n Step 8. Step 4. Cacuate Performance and Senstvty Check Convergence n Step 4. Yes No Yes Step. De-Normaze nterva Varabes Step 9. Search the Net Pont on the Lne between the Pont n Step 5 and the Pont n Step 7. Check Convergence n Step 0. Yes No No Worst Case nterva Varabes gure. owchart for Worst Case Performance Search 5.3 Senstvty Anayss of Worst Case Performance uncton and ts Cacuaton The behavor of any pont wthn the nterva of an nterva varabe can be epressed usng the Drac deta functon () [45] as 0, 0, 0, (4) and shftng Eq. (4) by the case of denoted as, yeds whch s constraned to satsfy the dentty,, 0,,, (5) Aso, the property of the Drac deta functon [45] yeds Usng Eqs. (4)~(7) and assumng, d. (6) G, d G,. (7) G s a contnuousy dfferentabe functon of any rea number, senstvty 6

7 of the case performance functon wth respect to the th case nterva varabe n genera dmenson becomes G,,, G,, d G d, N N (8) N,, where. Based on the defnton of the Drac deta functon, behavor of a snge nterva varabe at ts case,, can be treated as a Gaussan norma dstrbuton wth of and approachng to 0, whch mpes,, 0.5 / = m e m f. (9) 0 0 Equaton (9) s verfed n ths secton frst. Consder senstvty of an one-dmensona performance functon, G wth respect to the case of nterva varabe, whch by usng Eq. (8) becomes G d. (0), G, G n Eq. (0) usng the Tayor seres epanson at ( m), where m,,, can be epressed as ( m), G m m m () G a m!,, m0 m0 a G / m!. Usng Eqs. (9) and () and the score functon epaned n Secton.3, the rght hand sde of Eq. (0) s evauated as,,, m G d a, m m f d () m0 Usng the epectaton operator, the Eq. () s further smpfed as 0 m m m,,, G d E m a E a, a 0 0 m0 (3) where, E p, p p 0 f p s odd and E p!! f p s even accordng to the property of centra moments of a norma dstrbuton. Usng Eq. (), the eft hand sde of Eq. (0) s evauated as,, G am G m0,, m,, m amm a. (4) m, The dentca resuts n Eqs. (3) and (4) demonstrate the vadty of treatng behavor of, Gaussan norma dstrbuton wth of and approachng to 0. at, as a 7

8 nay, usng Eq. (9), Eq. (8) s further deveoped as,,, G d E G,, m 0 N G (5) and senstvty of the case constrant wth respect to desgn pont where,, N n Eq. () becomes G G (6), N,,, EG m 0,, /, f /,, / 0, f /. (7) Addtonay, t s noted that the Drac deta functon can be aso appcabe to defne behavor of determnstc varabes when senstvtes of performance functons wth respect to determnstc varabes are not avaabe. Thus, n the presence of determnstc varabes, the proposed sampng-based method can be apped to evauate senstvtes of performance functons even when gradents of the performance functons are not obtanabe. Gˆ. The MCS can be apped to Denote a surrogate mode for the constrant functon wth nterva varabes as cacuate senstvty of a performance functon wth respect to the th case nterva varabe durng the case performance search n Secton 3. usng Eq. (5) as where s for,, ˆ, G G ˆ (8), () can be cacuated usng Eqs. (6) and (7) as The desred vaue of K ( k ),, ( k ) G,, K k, comng from n Eq. (6), and senstvtes of the case constrants n Eq., ˆ, G G ˆ. (9) N K ( k ),,, ( k ),,, G K k,,, used n Eqs. (8) and (9) for the sampng-based method s determned through the foowng smuaton anayss. Durng the smuaton anayss, rato of to consdered snce depends on or nstead of ust s, and senstvty of a performance functon G wth respect to s cacuated by the MCS whe changes from 0. to 0.00 n descendng order. The resut s then compared to the true senstvty, whch s anaytcay obtaned as. rom the resut shown n gure, t s demonstrated that for the desred vaue of. rom the neggbe amount of error ess than 0.3% n gure, vadty of cacuatng senstvtes of a performance functon wth respect to nterva varabes usng the sampng-based method wth a very sma standard devaton can be aso shown. 8

9 3.5 (% error) gure. Error of Senstvty as changes from 0.00 to Desgn Optmzaton wth andom and nterva Varabes 6. ormuaton of Desgn Optmzaton wth Mture of andom and nterva Varabes The mathematca formuaton of desgn optmzaton wth mture of random and nterva varabes s epressed as mnmze cost( d) where,..., N, N,..., NN, tar subect to P P G, 0,,..., NC P L U ndv N d d d d d s the desgn vector;,,, and (30) N, n Eq. (30) s the case nterva varabes for the th probabstc constrant, whch s obtaned by sovng the optmzaton probem to mamze PG 0 subect to for,..., N. (3) To carry out the desgn optmzaton wth nterva varabes usng Eq. (30), probabstc constrants wth the case nterva varabes, namey the case probabstc constrants or the case probabty of faure, and ther senstvtes must be evauated. Each of the case probabstc constrant s obtaned by the case probabty of faure search that soves Eq. (3) and w be epaned n Secton 4., and senstvty anayss of each of the case probabstc constrant and ts cacuaton are epaned n Secton 4.3. t shoud be noted that the case probabty of faure does not aways occur at the pont where the case performance occurs, whch s demonstrated wth an eampe n Secton 4.. Thus, by appyng an agorthm for the case performance search n Secton 3. by drecty utzng probabty of faure and ts senstvty n repacement of performance vaue and ts senstvty, the probem ponted out n the prevous sentence can be resoved. 6. Worst Case Probabty of aure The case probabty of faure wth random and nterva varabes, denoted by P, s defned usng Eq. (30) and a mut-dmensona ntegra as NN,, P, f d d E., (3) The case probabty of faure n Eq. (3) s obtaned usng the agorthm for the case performance search epaned n Secton 3. by utzng probabty of faure and ts senstvty n repacement of the 9

10 - -5 performance functon and ts senstvty. Dervaton of the senstvty of the probabty of faure wth respect to the case nterva varabes and ts cacuaton are epaned n Secton 4.3. Usuay, the case probabty of faure occurs at the case performance, so conventonay the case probabty of faure s cacuated by evauatng the probabty of faure at the case performance [5-7]. However, ths s not aways the case and the foowng eampe demonstrates t. Consder a D hghy nonnear poynoma functon, G (33) 3 4 ( ) (W 6) (W 6) 0.6 (W 6) Z Tabe. Property of nput Varabes Varabes Types Dstrbuton Parameters nterva N/A andom Norma.5 W where Z As shown n Tabe, respectvey. The md-pont and nterva ength of and are nterva and random varabes, are 6.5 and 3, respectvey. The mean and standard devaton of are.5 and, respectvey. Then, s dvded nto 00 sub-ntervas, for each of whch, the performance functons and probabty of faures are evauated. or the evauaton of the probabty of faure, MCS sampe are used for each sub-nterva The case performance The case probabty of faure 5-4 G = = gure 3. Worst Case Performance and Worst Case Probabty of aure As shown n g. 3, the case probabty of faure does not occur where the case performance occurs. The case probabty of faure occurs at / 8 where performance and probabty of faure are and 0.48, respectvey, whe the case performance occurs at 5, where the performance and probabty of faure are.547 and 0., respectvey. Thus, ths study suggests usng the agorthm for the case probabty faure search drecty nstead of obtanng the case probabty of faure by cacuatng the probabty of faure where the case performance occurs. The MCS can be apped to cacuate probabty of faure durng the case probabty of faure search. Denotng the surrogate mode for constrant functons wth random and nterva varabes as Ĝ,, the probabty of faure durng the case probabty of faure search can be cacuated usng Eq. (3) as P P G P (34) K ( k 0 ) tar ˆ,,, K, k 0

11 where the faure set ˆ for the surrogate mode s defned as ˆ, G ˆ : 0,. 6.3 Senstvty Anayss of Worst Case Probabty of aure and ts Cacuaton Takng parta dervatve of Eq. (3) wth respect to the th case nterva varabe yeds N P,, E,. (35), Then, takng parta dervatve of Eq. (3) wth respect to the md-pont of the th nterva varabe usng Eq. (35) yeds N, N, P P,, E,, (36),, where s obtaned from Eq. (7). Takng parta dervatve of Eq. (3) wth respect to the mean of the th random varabe yeds P NN n f ;,, f ; d d. (37) Usng Eq. (7), Eq. (37) s further smpfed as n f ; n f ; ;. (38) P,, f d E,, NN The MCS can be apped to cacuate senstvty of the probabty of faure wth respect to the th case nterva varabe durng the case probabty of faure search n Secton 4. based on Eq. (35) as P,. (39) K ( k ), ( k ),, ˆ K, Senstvtes of the case probabstc constrants n Eq. (30) wth respect to the th nterva varabe at the md-pont as P (40) N K ( k ),, ( k ),,,, ˆ, K k,,, based on Eq. (36). Senstvtes of the case probabstc constrants n Eq. (30) wth respect to the th random varabe at the mean pont are cacuated as P K ( k), () ( k) ˆ s K k, ; (4) based on Eq. (38). 7. Numerca Eampes

12 Numerca studes are carred out n ths secton to verfy the agorthm that searches the case probabty of faure n Secton 4 for both ow-dmensona and hgh-dmensona cases. Aso, desgn optmzaton wth mture of nterva and random varabes that utzes the case probabty of faure search s carred out. 7. Worst Case Probabty of aure Search for Two-Dmensona nputs n ths numerca eampe, the agorthm that searches the case probabty of faure s apped to a two-dmensona case, and one of nput varabes s an nterva and the other s a random. Consder a nonnear performance functon gven as G (4) Tabe. Property of nput Varabes Varabes Types Dstrbuton Parameters nterva NA 0.5 andom Norma. As shown n Tabe, s an nterva varabe wth ts md-pont at 0.5 and ts nterva ength of, and s a normay dstrbuted random varabe wth ts mean at. and ts standard devaton of G = 0 G = G = P =.00 The case probabty of faure gure 4. Search Hstory of Worst Case Probabty of aure Tabe 3. Search Hstory of Worst Case Probabty of aure teraton P P / Wth the gven property of these nput varabes and the performance functon n Eq. (4), the case probabty of faure s obtaned usng the case probabty of faure search epaned n Secton 4.. The resuts are shown both n Tabe 3 and g. 4. The case nterva varabes at the 4 th teraton n Tabe 3 s obtaned by an nterpoaton of two case nterva varabes canddates at the nd and the 3 rd teraton durng Step 9 of the case probabty of faure search. The obtaned resut s compared wth the resut obtaned by 7 dvdng the nterva doman nto 00 sub-ntervas and performng the MCS wth 50 sampes for a sub-ntervas, whch s shown n Tabe 4.

13 Methods Tabe 4. Comparson of esuts Obtaned by Dfferent Methods P Number of MCS Proposed Agorthm Performng MCS for a 00 sub-ntervas n terms of effcency, the proposed agorthm requres (4teratons) (MCS/teraton) = 4MCSs, and performng the MCS for a 00 sub-dvded ntervas requres (00sub-ntervas) (MCS/sub-nterva) = 00MCSs. Thus, the proposed agorthm s 5 tmes more effcent than the crude MCS whe mantanng accuracy n ths eampe. 7. Worst Case Probabty of aure Search for Hgh-Dmensona nputs gure 5. Schematc Dagram of Cantever Tube n ths numerca eampe, the agorthm that searches the case probabty of faure s apped to a hgh-dmensona case where of nput varabes are nterva and 9 of them are random varabes. Consder the cantever tube shown n g. 5 subected to eterna forces,, and P, and torson T [6]. The performance functon s defned as the dfference between the yed strength S y and the mamum stress, namey, where G g S (43) ma y ma s the mamum von Mses stress on the top surface of the tube at the orgn, whch s gven by y (44) ma 3 z where the norma stress s obtaned P sn sn L cos L cos d 4 4 d d t d d t 4 64 (45) and the shear stress z s obtaned as z Td d d t , (46) respectvey. The property of random and nterva varabes are gven n Tabes 5 and 6, respectvey. As shown n Tabes 5 and 6, nne random varabes ~ havng varous dstrbutons and two nterva varabes 9 0 and havng the dentca nterva ength at dfferent md-ponts are used as nput varabes. Wth the property of nput varabes and the performance functon n Eq. (43), the case probabty of faure 7 s obtaned usng the case probabty of faure search. The MCS wth 5 0 sampes s tred for every 4 teraton, and the toerance of 0 3 nstead of 0 s set for ths eampe snce the senstvty of probabty of faure wth respect to both nterva varabes s ess than 0 throughout the nterva doman. By usng the 3

14 Tabe 5. Property of andom Varabes Varabes Parameter Parameter Dstrbuton 5 mm (mean) 0. mm (std* ) Norma ( ) d 4 mm (mean) 4 mm (mean) Norma ( L ) 9.75 mm (b ** ) 0.5 mm (ub***) Unform 3 ( L ) mm (b) 60.5 mm (ub) Unform 4 ( ) 3.0 kn (mean) 0.3 kn (std) Norma 5 ( ) 3.0 kn (mean) 0.3 kn (std) Norma 6 P.0 kn (mean). kn (std) Gumbe ( ) 7 ( ) 8 T 90.0 Nm (mean) 9.0 Nm (std) Norma ( ) 9 S y 33.7 MPa (mean).0 MPa (std) Norma *: std-standard devaton **: b ower bound of a unform dstrbuton ***: ub upper bound of a unform dstrbuton Tabe 6. Property of nterva Varabes Varabes ( ) 0 ( ) Parameters 0 5, 0 0 0, 0 proposed agorthm, the case probabty of faure s obtaned n 8 teratons ncudng the one wth the nterpoaton and the dscard one. n Tabe 7, snce the probabty of faure at the 4 th teraton s smaer than that at the 3 rd teraton, t s dscarded durng the Step 5 of the case probabty of faure search n Secton 4.. Search hstory s shown n both Tabe 7 and g. 6. The case probabty of faure s obtaned as and the case nterva varabes are obtaned as [3.993, 7.887]. The obtaned resut s then compared wth the 7 resut obtaned by dvdng both nterva domans nto 00 sub-ntervas and performng MCS wth 50 sampes for a combnatons of sub-ntervas. teraton ( ) 0 Tabe 7. Search Hstory of Worst Case Probabty of aure ( ) P P / 0 P / E E E E E E-04 4 * E E-03 3' E E E E-04 4' E-04.36E E-05.6E-07 *: Dscarded durng Step 5 of Worst Case Probabty of aure Search ' : Utzed for nterpoaton The resut of the comparson s shown n Tabe 8. n terms of effcency, the proposed agorthm requres (8teratons) (MCS/teraton) = 8MCSs, and performng MCS for a combnatons of 00 sub-ntervas requres (00 00combnatons) (MCS/combnaton) = 0000MCSs. Thus, the proposed agorthm s 50 tmes more effcent whe mantanng accuracy n ths eampe. As suggested by the current and the prevous eampes, the more nterva varabes there are, the ess effcent performng the crude MCS eponentay becomes. n genera dmenson, performng the MCS for a combnatons of 00 sub-ntervas of every nterva varabe requres (00 N 4

15 8 G = ' G = 0 4* 3 G = ' P =0.0 0 G 3 = 0.03 =0.0 The case probabty of faure gure 6. Search Hstory of Worst Case Probabty of aure combnatons) (MCS/combnaton) = (0) N MCSs. On the other hand, the proposed agorthm requres smar number of MCSs regardess of dmenson of nterva varabes snce t utzes a vector of nterva varabes and ts senstvty vector nstead of ther ndvdua components. Tabe 8. Comparson of esuts Obtaned by Dfferent Methods Methods Worst Case nterva Varabe Probabty of aure Number of MCSs Proposed Agorthm [ ] Performng MCS for a combnatons of 00 sub-ntervas [ ] Desgn Optmzaton wth Mture of andom and nterva nputs Ths numerca eampe shows the desgn optmzaton wth mture of random and nterva varabes, utzng the case probabty of faure search. Consder a D mathematca desgn optmzaton probem, whch s formuated to mnmze where three constrants are gven by d C d d, tar subect to P P G 0.75%, ~ 3 d, P L U d d d d,,, and (47) G G G (48) The propertes of two nput varabes, one nterva and one random varabe, are shown n Tabe 9. As shown n Eq. tar (47), the target probabty of faure P s set to.75% for a constrants. g. 7 shows the optmum desgn of the sampng-based desgn optmzaton wth nterva and random varabes. As can be seen n g. 7, the determnstc desgn optmum (d dopt ) was frst searched to enhance effcency of the 5

16 Tabe 9. Property of nput Varabes L nput Varabes Varabe Types d O d U d Parameters nterva andom desgn optmzaton procedure. n g. 7, the dotted bo ustrated around the desgn optmum (d opt ) shows the tar ont range of and. Wth P of.75%, aowed tota range of dstrbuton of becomes 4.6, and wth. of, sze of the dotted bo becomes..6. Wth the dotted bo around d opt t s easy dentfed that d opt s the desred optmum as vertces of the bo are rght on two actve constrants, G G. P and P occur on the eft and the rght bounds of 0 are 0.03 and 0.08, respectvey, whch are very cose to P tar, respectvey where 0 P and and P. Desgn search hstory and number of teratons taken to obtan the case probabty of faure at each desgn are shown n Tabe 0. One MCS for each teraton s used to obtan P and whe appyng the case probabty of faure search epaned P n Secton 4.. At the nd teraton durng the desgn search, P does not behave monotonc wthn the doman of the nterva varabe, thus the nterpoaton agorthm s apped to fnd more accurate case probabty of faure, whch s why 5 teratons are taken to obtan P. Overa teratons taken to obtan P s around for each desgn search snce concuded P behaves monotonc most tmes wthn the doman of the nterva varabe. Thus t s G 3 () = 0 4 G () = 0 d opt d dopt G () = gure 7. Optmum Desgn of Sampng-Based Desgn Optmzaton wth andom and nterva Varabes that the computatona burden to obtan P n ths eampe s affordabe. Tabe 0. Desgn Search Hstory and Number of teratons for Worst Case Probabty of aure Desgn Pont # teratons # teratons # teratons teraton (d, d ) for P for P for P (3.39,.0639) (4.384, ) 5 3 (3.84,.9765) 4 (.8478, 3.096) 5 (3.377, 3.037) 6 (3.3407, 3.005) 7 (3.374, 3.) 8 (3.3838, 3.74) 3 6

17 8. Concusons Sampng-based desgn optmzatons wth ony nterva varabes and wth both nterva and random varabes are deveoped n ths study. or the desgn optmzaton wth nterva varabes ony, each of the case constrant s evauated by the deveoped case performance search where nterva and senstvty vector are utzed, thus effcency s promsed regardess of the dmenson of the nterva varabes. t s assumed that gradents of performance functons are not avaabe n ths study. Therefore, senstvtes of a performance functon wth respect to nterva varabes are derved by defnng behavor of the nterva varabes at the case by the Drac deta functon to cacuate t by the sampng-based method. Through the smuaton anayss, desred vaue of standard devaton for the nterva varabes at the case s determned, and the error of the resut turns out to be neggbe at the desred vaue. Usng the obtaned vaue, the senstvtes of each of the case constrants both at the case and desgn ponts are cacuated by the MCS. or the desgn optmzaton wth random and nterva varabes, the case probabstc constrants are evauated by the case probabty of faure search. Snce probabty of faure does not aways occur where the case performance occurs as demonstrated n ths study, the case probabty of faure s obtaned by drecty usng the probabty of faure and ts senstvty. Smary to desgn optmzaton wth nterva varabes ony, the senstvtes of the probabty of faure both at the case and desgn ponts are derved, whch are then cacuated by the MCS. Numerca eampes show the case probabty of faure s obtaned effcenty for both ow and hgh-dmensona nputs regardess of the dmenson of the nterva varabes and the desgn optmzaton wth random and nterva varabes s successfuy carred out wth effcency utzng the case probabty of faure search. 9. eferences [] Youn, B.D., Cho, K.K., Yang,.J., and Gu, L., eabty-based Desgn Optmzaton for Crashworthness of Vehce Sde mpact, Structura and Mutdscpnary Optmzaton, Vo. 6, No. 3-4, pp. 7-83, 004. [] Youn, B.D., Cho, k.k., and Y, K., Performance Moment ntegraton (PM) Method for Quaty Assessment n eabty-based obust Optmzaton, Mechancs Based Desgn of Structures and Machnes, Vo. 33, No., pp. 85-3, 005. [3] Youn, B.D., Cho, K.K., and Tang, J., Structura Durabty Desgn Optmzaton and ts eabty Assessment, nternatona Journa of Product Deveopment, Vo., Nos. 3/4, pp , 005. [4] Jung, B.C., Lee, D., Youn, B.D., Lee, S., A Statstca Characterzaton Method for Dampng Matera Propertes and ts Appcaton to Structura-Acoustc System Desgn, Journa of Mechanca Scence and Technoogy, Vo. 5, No. 8, pp , 0. [5] Daskewcz, M.J., German, B.J., Takahash, T.T., Donovan, S., and Shaanan, A., Effects of dscpnary uncertanty on mut-obectve optmzaton n arcraft conceptua desgn, Structura and Mutdscpnary Optmzaton, Vo. 44, No. 6, pp , 0. [6] Acar, E. and Soank, K., System eabty-based Vehce Desgn for Crashworthness and Effects of Varous Uncertanty educton Measures, Structura and Mutdscpnary Optmzaton, Vo. 39, No. 3, pp. 3-35, 009. [7] Snha, K., eabty-based Mutobectve Optmzaton for Automotve Crashworthness and Occupant Safety, Structura and Mutdscpnary Optmzaton, Vo. 33, No. 3, pp , 007. [8] rangopo, D.M., and Maute, K., Lfe-Cyce eabty-based Optmzaton of Cv and Aerospace Structures, Computers & Structures, Vo. 8, No. 7, pp , 003. [9] Lan, Y., Km, N.H., eabty-based Desgn Optmzaton of a Transonc Compressor, AAA ourna, Vo. 44, No., pp , 006. [0] Dong, J., Cho, K.K., Vahopouos, N., Wang, A., and Zhang, W., Desgn Senstvty Anayss and Optmzaton of Hgh requency adaton Probems Usng Energy nte Eement and Energy Boundary Eement Methods, AAA Journa, Vo. 45, No. 6, pp , 007. [] Du, L., Cho, K.K., and Youn, B.D., An nverse Possbty Anayss Method for Possbty-Based Desgn Optmzaton, The Amercan nsttute of Aeronautcs and Astronauts, Vo. 44, No., pp , 006. [] Noh, Y., Cho, K.K., Lee,., Gorsch, D., and Lamb, D., eabty-based Desgn Optmzaton wth Confdence Leve for Non-Gaussan Dstrbutons Usng Bootstrap Method, Journa of Mechanca Desgn, Vo. 33, No. 9, pp ~, 0. [3] Noh, Y., Cho, K.K., and Lee,., eabty-based Desgn Optmzaton wth Confdence Leve Under nput Mode Uncertanty due to Lmted Test Data, Structura and Mutdscpnary Optmzaton, Vo. 43, No. 4, pp , 0. [4] Penmetsa,.C. and Grandh.V., Effcent Estmaton of Structura eabty for Probems wth Uncertan ntervas, Computers & Structures, Vo. 80, No., pp. 03-, 00. 7

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