HYBRID ANALYSIS METHOD FOR RELIABILITY-BASED DESIGN OPTIMIZATION

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1 Proceedngs of DETC 01 ASME 2001 Desgn Engneerng Techncal Conferences and Computers and Informaton n Engneerng Conference Pttsburgh, Pennsylvana, September 9-12, 2001 DETC2001/DAC HYBRID ANALYSIS METHOD FOR RELIABILITY-BASED DESIGN OPTIMIZATION Kyung K. Cho Center for Computer-Aded Desgn and Department of Mechancal Engneerng College of Engneerng The Unversty of Iowa Iowa Cty, IA 52242, USA kkcho@ccad.uowa.edu ABSTRACT Relablty-Based Desgn Optmzaton (RBDO) nvolves evaluaton of probablstc constrants, whch can be done n two dfferent ways, the Relablty Index Approach (RIA) and the Performance Measure Approach (PMA). It has been reported n the lterature that RIA yelds nstablty for some problems but PMA s robust and effcent n dentfyng a probablstc falure mode n the RBDO process. However, several examples of numercal tests of PMA have also shown nstablty and neffcency n the RBDO process f the Advanced Mean Value (AMV) method, whch s a numercal tool for probablstc constrant evaluaton n PMA, s used, snce t behaves poorly for a concave performance functon, even though t s effectve for a convex performance functon. To overcome dffcultes of the AMV method, the Conjugate Mean Value (CMV) method s proposed n ths paper for the concave performance functon n PMA. However, snce the CMV method exhbts the slow rate of convergence for the convex functon, t s selectvely used for concave-type constrants. That s, once the type of the performance functon s dentfed, ether the AMV method or the CMV method can be adaptvely used for PMA durng the RBDO teraton to evaluate probablstc constrants effectvely. Ths s referred to as the Hybrd Mean Value (HMV) method. The enhanced PMA wth the HMV method s compared to RIA for effectve evaluaton of probablstc constrants n the RBDO process. It s shown that PMA wth a sphercal equalty constrant s easer to solve than RIA wth a complcated equalty constrant n estmatng the probablstc constrant n the RBDO process. NOMENCLATURE X Random parameter; X = [X 1, X 2,, X n ] T x Realzaton of X; x = [x 1, x 2,, x n ] T U Independent standard normal random parameter u Realzaton of U; u = [u 1, u 2,, u n ] T µ Mean of random parameter X d Desgn parameter; d = [d 1, d 2,, d n ] T L U d, d Lower and upper bounds of desgn parameter d P () Probablty functon f X ( x ) Byeng D. Youn Center for Computer-Aded Desgn and Department of Mechancal Engneerng College of Engneerng The Unversty of Iowa Iowa Cty, IA 52242, USA ybd@ccad.uowa.edu Jont Probablty Densty Functon (JPDF) of the random parameter Φ () Standard normal Cumulatve Dstrbuton Functon (CDF) Φ ( ) F G () CDF of the performance functon G(X) β s Safety relablty ndex β s,form Frst order approxmaton of safety relablty ndex β s β t Target relablty ndex G( X ) Performance functon; the desgn s consdered fal f G(X) < 0 G p Probablstc performance measure u G ( U) = 0 Most Probable Falure Pont (MPFP) n frst-order relablty analyss u β = β t Most Probable Pont (MPP) n frst-order nverse relablty analyss u (A)MV MPP usng (advanced) mean value method n PMA u CMV MPP usng conjugate mean value method n PMA u HMV MPP usng hybrd mean value method n PMA n Normalzed steepest descent drecton of performance functon Gabs, Grel Absolute and relatve changes n performance measure ς Crtera for the type of performance functon L( X ) Crack ntaton fatgue lfe L t Target crack ntaton fatgue lfe INTRODUCTION A commonly used desgn optmzaton methodology for engneerng systems comprses determnstc modelng and smulaton-based desgn optmzaton. However, the exstence of uncertantes n physcal quanttes such as manufacturng tolerances, materal propertes, and loads requres a relablty-based approach to 1 Copyrght 2001 by ASME

2 desgn optmzaton [1,2]. Gven the ncreased computatonal capabltes developed durng the last few years, fundamental ssues relatng to the ncluson of quanttatve estmaton of uncertanty have been recently addressed. Technques have been explored whch ncorporate uncertanty durng desgn optmzaton at an affordable computatonal cost. There has been a recent development n the Relablty-Based Desgn Optmzaton (RBDO) ncorporatng probablstc constrants that can be evaluated usng two dfferent approaches, the Relablty Index Approach (RIA) and the Performance Measure Approach (PMA) [3,4]. The evaluaton of a probablstc constrant n the RBDO model s an essental step and thus the probablstc constrant n the RBDO model must be computatonally stable and affordable so that the RBDO process can be effectve. It has been shown that PMA s equvalent to RIA n prescrbng the probablstc constrant [3]. However, these approaches are not equvalent n computatonal robustness n evaluatng probablstc constrants n the RBDO process. That s, RIA may demonstrate nstablty whereas PMA s stable n evaluatng a probablstc constrant [3]. However, several examples of numercal tests of the PMA show neffcency and nstablty n the assessment of a probablstc constrant durng the RBDO process as the result of an neffectve numercal method,.e., the Advanced Mean Value (AMV) method [5,6]. In general, the AMV method exhbts dvergence or slow rate of convergence n addressng a concave performance functon, although t s good for a convex performance functon. Wth respect to a concave performance functon, numercal nstablty as well as neffcency n PMA usng the AMV method hghlghts the need for a stable and effcent computatonal algorthm that utlzes a conjugate drecton, namely, the Conjugate Mean Value (CMV) method. However, the CMV method s computatonally more expensve than the AMV method for a convex performance functon. Consequently, the Hybrd Mean Value (HMV) method s proposed n ths paper to adaptvely select ether the AMV method or the CMV method once the performance functon type s dentfed. It has been noted n Refs. 3 and 4 that the effcency of RIA and PMA to assess the probablstc constrant depends on actveness of the probablstc constrant. The prevous research, however, has not been dealt wth the HMV method proposed n ths paper. Hence, a comparatve study between RIA and PMA from an effcency and robustness perspectve, wth respect to probablstc constrant evaluaton n the RBDO process, s presented n ths paper. It s shown that the conventonal relablty analyss model n RIA causes neffectveness n the RBDO process, whle the nverse relablty analyss model n PMA provdes an effcent and robust RBDO process usng the proposed HMV method. Popular numercal methods for RIA are the HL-RF method [7,8], Modfed HL-RF [8], and Two-Pont Approxmaton (TPA) [9,10]. For PMA, the AMV [5,6] s a popular numercal method. In ths paper, the proposed HMV method wll s used to show effcency and robustness n probablstc constrant assessment for PMA. GENERAL DEFINITION OF RBDO MODEL In the system parameter desgn, the RBDO model [11-14] can be generally defned as Mnmze Cost( d) subject to PG ( ( X) 0) Φ( βt) 0, = 1,2,, np (1) L U n d d d, d R where the cost can be any functon of the desgn vector T T d = [ d ] = µ ( X), X = [ X] ( = 1,2,, n) s the random vector, and the probablstc constrants are descrbed by the performance functon G subject to uncertanty X, ther probablstc models, and ther prescrbed confdence level β t. The statstcal descrpton of the falure of the performance functon G (X) s characterzed by the Cumulatve Dstrbuton Functon (CDF) (0) as F G PG ( ( X ) 0) = FG (0) Φ( βt) (2) where the CDF s descrbed as FG (0) ( ) ( ) 0 1, 1, 2,, = f dx dx G n np X X x = (3) In Eq. (3) f X ( x) s the Jont Probablty Densty Functon (JPDF) of all random parameters. The evaluaton of Eq. (3) requres relablty analyss where the multple ntegraton s nvolved as shown n Eq. (3). Some approxmate probablty ntegraton methods have been developed to provde effcent solutons [1], such as the Frst-Order Relablty Method (FORM) or the asymptotc Second-Order Relablty Method (SORM) wth a rotatonally nvarant measure as the relablty [1,2]. The FORM often provdes adequate accuracy [1,2] and s wdely used for RBDO applcatons. In FORM, relablty analyss requres a transformaton T [15,16] from the orgnal random parameter X to the ndependent and standard normal random parameter U. The performance functon G( X) n X-space can then be mapped onto G(T(X)) G(U) n U-space. As descrbed n Secton 1, the probablstc constrant n Eq. (2) can be further expressed n two dfferent ways through nverse transformatons [3] as: 1 βs = ( Φ ( F G (0))) βt (4) where Gp = FG 1 ( Φ ( βt)) 0 (5) s β and are respectvely called the safety relablty G p ndex and the probablstc performance measure for the probablstc constrant. Equaton (4) s employed to descrbe the probablstc constrant n Eq. (1) usng the relablty ndex,.e., the so-called Relablty Index Approach (RIA). Smlarly, Eq. (5) can replace the probablstc constrant n Eq. (1) wth the performance measure, whch s referred to as the Performance Measure Approach (PMA). Frst-Order Relablty Analyss n RIA In RIA, the frst-order safety relablty ndex β s,form s obtaned usng the FORM by formulatng as an optmzaton problem wth one equalty constrant n U-space, whch s defned as a lmt state functon: mnmze U (6) subject to G ( U ) = 0 where the optmum pont on the falure surface s called the Most Probable Falure Pont (MPFP) u G ( U ) = 0 and thus β s,form = ug( U ) = 0. Ether MPFP search algorthms specfcally developed for the frst-order relablty analyss or general optmzaton algorthms [17] can be used to solve Eq. (6). In ths paper, the HL-RF method s employed to perform relablty analyses n RIA due to ts smplcty and effcency. th 2 Copyrght 2001 by ASME

3 Frst-Order Relablty Analyss n PMA Relablty analyss n PMA can be formulated as the nverse of relablty analyss n RIA. The frst-order probablstc performance measure G p,form s obtaned from a nonlnear optmzaton problem [3] n U-space defned as mnmze G( U) (7) subject to U = βt where the optmum pont on a target relablty surface s dentfed as the Most Probable Pont (MPP) u β = β t wth a prescrbed relablty β t = u β = β t, whch wll be called MPP n the paper. Unlke RIA, only the drecton vector uβ = β u β= β needs to be determned by t explorng the sphercal equalty constrant U = βt. General optmzaton algorthms can be employed to solve the optmzaton problem n Eq. (7). However, the AMV method s well suted for PMA [5,6] due to ts smplcty and effcency. HYBRID RELIABILITY ANALYSIS METHOD FOR PMA It was found that, although the Advanced Mean Value (AMV) method behaves well for a convex performance functon, t exhbts numercal shortcomngs, such as slow convergence or even dvergence, when appled to a concave performance functon. To overcome these dffcultes, the Conjugate Mean Value (CMV) method s proposed n ths paper. However, even though the CMV method always converges, t s neffcent for the convex functon. Consequently, the Hybrd Mean Value (HMV) method s proposed n ths paper to attan both stablty and effcency n the MPP search algorthm n PMA. Advanced Mean Value (AMV) Method Formulaton of the frst-order AMV method begns wth the Mean Value (MV) method, defned as ( ) ( ) MV ( ) where ( ) XG µ UG 0 u = βtn 0 n 0 = = (8) XG( µ ) UG( 0) That s, to mnmze the performance functon G( U) (.e., the cost functon n Eq. (7)), the normalzed steepest descent drecton n0 ( ) s defned at the mean value. The AMV method teratvely updates the drecton vector of the steepest descent method at the probable pont uamv ntally obtaned usng the MV method. Thus, the AMV method can be formulated as (1) 1) u AMV = u MV, u AMV = β t n( u AMV ) (9) where U G( u ( AMV ) nuamv ) = (10) U G( uamv ) As wll be shown, ths method exhbts nstablty and neffcency n solvng a concave functon snce ths method updates the drecton usng only the current MPP. Conjugate Mean Value (CMV) Method When appled for a concave functon, the AMV method tends to be slow n the rate of convergence and/or dvergent due to a lack of updated nformaton durng the teratve relablty analyss. These t knds of dffcultes can be overcome by usng both the current and prevous MPP nformaton as appled n the proposed Conjugate Mean Value (CMV) method. The new search drecton s obtaned by ( k 2) ( k 1) combnng nu ( CMV ), nu ( CMV), and nu ( CMV) wth an equal weght, such that t s drected towards the dagonal of the three consecutve steepest descent drectons. That s, (0) (1) (1) (2) (2) ucmv = 0, ucmv = uamv, ucmv = uamv, ( k 1) ( k 2) ( k + 1) nu ( CMV) + nu ( CMV) + nu ( CMV ) (11) ucmv = βt for k 2 ( k 1) ( k 2) nu ( CMV) + nu ( CMV) + nu ( CMV ) where U G( ucmv) nu ( CMV ) = U G( ucmv). (12) Consequently, the conjugate steepest descent drecton sgnfcantly mproves the rate of convergence, as well as the stablty, compared to the AMV method for the concave performance functon. However, as wll be seen n the next secton, the proposed CMV method s neffcent for the convex functon. Example 1: Convex Performance Functon A convex functon s gven as G( X ) = exp( X1 7) X (13) where X represents the ndependent random varables wth X ~ N(6.0,0.8), = 1, 2 and the target relablty ndex s set to β t = 3.0. As shown n Fg. 1, the constrant n Eq. (7) s always satsfed and the performance functon around the MPP s convex wth respect to the orgn of U-space. The AMV method demonstrates good convergence behavor for the convex functon snce the steepest descent drecton nu ( AMV ) of the response gradually approaches to the MPP, as shown n Fg. 1(a). In Table 1, the convergence rate of the AMV method s faster than that of the CMV method for the convex functon because the conjugate steepest descent drecton tends to reduce the rate of convergence for the convex functon. Thus, for the convex performance functon, the AMV method performs better than the CMV method. Table 1. MPP Hstory for Convex Performance Functon AMV CMV X 1 X 2 G X 1 X 2 G Copyrght 2001 by ASME

4 ( a ) AMV Method ( a ) AMV Method ( b ) CMV Method Fgure 1. MPP Search for Convex Performance Functon Example 2: Concave Performance Functon 1 Consder the concave performance functon G( X ) = [exp(0.8x1 1.2) + exp(0.7x2 0.6) 5]/10 (14) where X represents an ndependent random vector wth X1 ~ N(4.0,0.8) and X2 ~ N(5.0,0.8) and the target relablty ndex s set to β t = 3.0. As shown n Fg. 2, the performance functon around the MPP s concave wth respect to the orgn of U-space. The AMV method appled to the concave response dverges as a result of the oscllaton observed n Fg. 2(a). As shown n Table 2, after 34 th teraton, oscllaton occurs n frst-order relablty analyss due to the cyclc behavor of the steepest descent drectons,.e., ( k 2) 1) ( k 1) nu ( AMV ) = nu ( AMV ) and nu ( AMV ) = nu ( AMV). Ths example shows that, unlke the convex functon, the AMV method does not converge for the concave functon. As presented n Table 2, the CMV method appled to the PMA s stable when handlng the concave functon by usng the conjugate steepest drecton. Example 3: Concave Performance Functon 2 A dfferent stuaton usng another concave functon s presented 2 G( X ) = 0.3X1 X2 X X1+ 1 (15) where X represents the ndependent random varables wth X1 ~ N(1.3,0.55) and X2 ~ N(1.0,0.55) and the target relablty of β t = 3.0 s used. ( b ) CMV Method Fgure 2. MPP Search for Concave Performance Functon 1 Table 2. MPP Hstory for Concave Performance Functon 1 AMV CMV X 1 X 2 G X 1 X 2 G Dverged Although the AMV method has converged n ths case, t requres substantally more teratons than the CMV method. Smlar to Example 2, the slow rate of convergence s the result of oscllatng behavor of relablty teratons when usng the AMV method. Based on the prevous examples, t can be concluded that the AMV method ether dverges or performs poorly compared to the 4 Copyrght 2001 by ASME

5 CMV method, for the concave performance functon. Thus, a desrable approach s to select ether the AMV or CMV methods once the type of performance functon has been determned to acheve the most effcent and robust evaluaton of probablstc constrant, as dscussed n the followng secton. ( a ) AMV Method ( b ) CMV Method Fgure 3. MPP Search for Concave Performance Functon 2 Table 3. MPP Hstory for Concave Performance Functon 2 AMV CMV X 1 X 2 G X 1 X 2 G Hybrd Mean Value (HMV) Method To select an approprate MPP search method, the type of performance functon must be frst dentfed. In ths paper, the functon type crtera s proposed by employng the steepest descent drectons at the three consecutve teratons as follows 1) 1) ( k 1) ς = ( n n ) ( n n ) ( k + 1) sgn( ς ) > 0 : 1) Convex type at uhmv w.r.t. desgn d 0 : ( k + 1) Concave type at uhmv w.r.t. desgn d 1) where ς s the crteron for the performance functon type at the (16) k+1 th step and n s the steepest descent drecton for a performance functon at the MPP uhmv at the k th teraton. Once the performance functon type s defned, one of two numercal algorthms, AMV and CMV, s adaptvely selected for the MPP search. The proposed numercal procedure s therefore denoted as the Hybrd Mean Value (HMV) method, and s summarzed as: Step 1. Set the teraton counter k=0. Select the convergence parameter ε. Compute the steepest descent drecton of the performance functon n U-space where (0) (0) U G( uhmv) nu ( HMV ) = (0) U G( uhmv) (0) where uhmv = 0 (orgn n U -space) Step 2. If the performance functon type s convex or k < 3, calculate the MPP usng the AMV method (note that Step 2 of AMV method s the same as that of HMV method when k < 3 ) as 1) uhmv = β t n( u HMV) If the performance functon s concave and k 3, compute the MPP usng the CMV method as ( k 1) ( k 2) ( k + 1) nu ( HMV) + nu ( HMV) + nu ( HMV ) uhmv = β t ( k 1) ( k 2) nu ( HMV) + nu ( HMV) + nu ( HMV ) U G( u where HMV) nu ( HMV ) = U G( uhmv) Step 3. Calculate the performance ndex ( k + 1) G( u HMV) ( k 1) β + at the new MPP ( k + u HMV 1) ( k + 1) (k+ 1) (k+ 1) ( t Grel Gabs ) and the relablty. Check to see f max β β,, ε where 1) ( k + 1) G( uhmv) G( uhmv) Grel = ( k + 1) G( uhmv) 1) 1) Gabs = G( uhmv) G( u HMV) and If the convergence crtera hold, then stop. Otherwse, go to Step 4. ( k 1) Step 4. Compute the gradent U G + ( u HMV) of the performance 1) functon and check the crtera ς for performance functon type. Set k = k + 1 and return to Step 2. 5 Copyrght 2001 by ASME

6 Example 4: Relablty Analyss of Analytcal Examples The numercal algorthm proposed n Secton 3.3 was appled to the prevous three examples. For the frst example, the proposed numercal algorthm dentfes ς as postve, hence the AMV method was then used to search for the MPP and requred 6 teratons. For the second and thrd examples, the values of ς were dentfed as negatve and the CMV method was utlzed for the MPP search. In conjuncton wth the numercal algorthm presented n Secton 3.3, the HMV method performed qute well for any type of performance functon. Example 5: Relablty Analyss of Durablty Model A roadarm from a mltary tracked vehcle shown n Fg. 4 s employed to demonstrate the effectveness of the HMV method for a large-scale problem. Relablty analyss for ths example nvolves the crack ntaton fatgue lfe performance measure. A 17-body dynamcs model s created to drve the tracked vehcle on the Aberdeen Provng Ground 4 (APG4) at a constant speed of 20 mles per hour forward (postve X 2 ) [13,18]. A 20-second dynamc smulaton s performed wth a maxmum ntegraton tme step of 0.05-second usng the dynamc analyss package DADS [19]. Three hundred and ten 20-node soparametrc fnte elements, STIF95, and four beam elements, STIF4, of ANSYS are used for the roadarm fnte element model shown n Fg. 5. The roadarm s made of S4340 steel wth materal propertes of Young s modulus E= ps and Posson s rato ν=0.3. Fnte element analyss s performed to obtan the Stress Influence Coeffcent (SIC) of the roadarm usng ANSYS by applyng 18 quas-statc loads. To compute the multaxal crack ntaton lfe of the roadarm, the equvalent von Mses stran approach [20] s employed. The fatgue lfe contour n Fg. 6 shows crtcal nodes and the shortest lfe s lsted n Table 4. The computaton for fatgue lfe predcton and for desgn senstvty requre, respectvely, 6950 and 6496 CPU seconds (for desgn parameters) on an HP 9000/782 workstaton. Torson Bar Intersecton 1 b1, b2 Intersecton 2 b3, b4 Intersecton 3 b5, b6 x' 3 x' 2 Intersecton 4 b7, b8 Center of the Roadwheel x' 1 ( b ) Fnte Element Model of Roadarm Model Fgure 5. Geometry and Fnte Element Model for Roadarm Model Table 4. Crtcal Nodes for Crack Intaton Fatgue Lfe Node ID Lfe [Load Cycle] Lfe [Year] E E E E Fgure 6. Contour for Crack Intaton Fatgue Lfe Roadarm Fgure 4. Mltary Tracked Vehcle x' 3 20 n. Intersecton 1 Intersecton 3 Intersecton 2 Intersecton 4 x' 1 x' 2 x' 2 ( a ) Geometry of Roadarm Model The random varables and ther statstcal propertes for the crack ntaton lfe predcton are lsted n Table 5. Eght tolerance random parameters characterze four cross sectonal shapes of the roadarm. The contour of a cross sectonal shape conssts of four straght lnes and four cubc curves, as shown n Fg. 7. Sde varatons ( x 1 -drecton) of the cross sectonal shapes are defned as the random parameters b1, b3, b5, and b7 for ntersectons 1 to 4, respectvely, and vertcal varatons ( x 3 -drecton) of the cross sectonal shapes are defned usng the remanng four random varables. For relablty analyss, a falure functon s defned as L( X) G( X ) = 1 (17) Lt where L( X) s the number of servce blocks to ntate crack at node 885 and L t s the number of target servce blocks to ntate crack n the structural component. The number of blocks at node 885 for the current desgn s 9.998E+6 (20 seconds per block), whch consttutes the shortest lfe of the component. The target crack 6 Copyrght 2001 by ASME

7 ntaton fatgue lfe s set as 0.1 years (.e., 1.577E+5 cycles) to llustrate the concave performance functon. Table 5. Defnton of Random Varables for Crack Intaton Fatgue Lfe Predcton Random Dstrbuton Mean Value Std. Dev. Varables Type Tolerance b Normal Tolerance b Normal Tolerance b Normal Tolerance b Normal Tolerance b Normal Tolerance b Normal Tolerance b Normal Tolerance b Normal the effcency n relablty analyss. Rather, t s found that PMA wth the sphercal equalty constrant s easer to solve than RIA wth a complcated constrant. In other words, t s easer to mnmze a complex cost functon subject to a smple constrant functon than to mnmze a smple cost functon subject to a complcate constrant functon. Cubc Curves b, = 1,3,5,7 x' 3 Straght Lnes b, = 2,4,6,8 ( a ) MPFP Search Space n RIA x' 1 Cross Sectonal Shape Desgn Parameters: b1, b3, b5, b7 Desgn Parameters: b2, b4, b6, b8 Fgure 7. Defnton of Random Parameters n Roadarm Model The conventonal AMV and proposed HMV method are used to calculate the relablty of the crack ntaton lfe. Begnnng at the mean pont, the HMV method has converged to MPP at x = [1.872, 3.093, 1.708, 2.830, 2.218, 2.755, 4.758, 2.836] T wth a target relablty ndex β t = 3.325, as obtaned from RIA. In contrast, the AMV method has dverged due to oscllaton. Consstent wth the prevous concave functon examples, the HMV method has converged whle the AMV method has dverged. Table 6. MPP Search Hstory n Roadarm Durablty Model AMV HMV G(X) β G(X) β ς N.A N.A N.A Dverged RIA VS. PMA IN RELIABILITY ANALYSIS It has been reported [3] that the sze of the search space n a relablty analyss could affect the effcency of the MP(F)P search. However, based on numercal examples n ths paper, t has been found that szes of the MP(F)P search spaces may not be crucal to ( b ) MPP Search Space n PMA Fgure 8. MP(F)P Search Spaces Fgure 8(a) llustrates the MPFP search space n RIA over the desgn parameter space, where the frst-order safety relablty k ndces n Eq. (4) are βs,form = β ( ) j j d = Tx ( G j = 0), j=1, 2. Relablty analyss n RIA s carred out by determnng the mnmum dstance between the mean value desgn pont and MPFP on the falure surface G j ( X ) = 0, j = 1, 2. The MPP search space n PMA s llustrated n Fg. 8(b), where the probablstc performance measures n Eq. (5) are Gp, ( ) j FORM = Gj x β j = β, j = 1, 2. t Relablty analyss n PMA s performed by determnng the mnmum performance value on the explct sphere of the target relablty β j ( d ) = βt, j = 1, 2. Comparng Fgs. 8(a) and (b), the MPP search space n PMA s smaller than the MPFP search space n RIA f the constrant at the mean value desgn pont s largely nactve or largely volated wth the large negatve relablty ndex, such as the frst probablstc constrant. Thus, the MPP search n PMA, wth the easer 7 Copyrght 2001 by ASME

8 optmzaton problem n Eq. (7), mght be better than RIA n terms of effcency and robustness. On the other hand, the MPFP search space n RIA s smaller f the constrant at the mean value desgn pont s near actves or lghtly volated, such as the second probablstc constrant n Fg. 8. In ths case, although RIA has a smaller MPFP search space, the optmzaton problem n Eq. (6) s not easer to solve than that of PMA. As a result, a comparson of the effcency n RIA and PMA s not clear regardng effcency and, as such, wll be examned closely n ths secton. In ths study, the HL-RF method s used for RIA and both the proposed HMV and conventonal AMV methods are used for PMA. For RIA vs. PMA n the relablty analyss, the roadarm durablty model used n Example 5 wll be demonstrated n Example 6 and 7. Example 6: RIA wth a Larger Search Space than PMA (Such as G 1 n Fg. 8) In Table 7, a target crack ntaton fatgue lfe of L t = 300 year s specfed so that the MPFP search space n RIA becomes larger than the MPP search space n PMA on the nfeasble regon, as represented by the largely volate determnstc constrant at the mean value desgn pont. At the second teraton, RIA has dverged - the lfe at the frst MPFP becomes nfnte (1.0E+20 load cycle or 6.34E+13 years) and all desgn senstvtes become zero, whch lead to falure of RIA. In contrast, PMA does not have numercal dffculty n relablty analyss wthn the prescrbed MPP search space. Ths example shows that PMA usng the HMV method s better than RIA n terms of stablty. Table 7. Relablty Analyss for L t = 300 year Iteraton RIA (HL-RF) PMA (AMV, HMV) G β G β G β E Dverged Example 7: RIA wth a Smaller Search Space than PMA (Such as G 2 n Fg. 8) In Table 8, the target crack ntaton fatgue lfe L t s specfed as 10 years so that the MPFP search space n RIA s smaller than the MPP search space n PMA on the nfeasble regon, as represented by the slghtly volate determnstc constrant at the mean value desgn pont. In ths case, RIA searches for the MPFP n the smaller search space than PMA wth β t = 3.0. However, PMA s more effcent than RIA, snce the PMA optmzaton problem n Eq. (7) s easer to solve. Note that the HMV method demonstrates superorty over the AMV method, whch has dverged, as shown n Table 8. Based on the examples presented n ths secton, t can be concluded that PMA s superor to RIA, regardless of szes of the MP(F)P search spaces. Consequently, t s recommended to use PMA wth the sphercal equalty constrant n relablty analyss and not RIA wth the complcate constrant for all cases. RBDO USING PMA WITH HMV METHOD As descrbed n Secton 2.1, the probablstc constrants n the RBDO model can be evaluated by two dfferent relablty analyses: RIA and PMA. Based on the results of prevous sectons n relablty analyss, the comparatve study between the conventonal RIA and the enhanced PMA wth the HMV method s extended to RBDO of a bracket problem n ths secton. Table 8. Relablty Analyss for L t = 10 years RIA (HL-RF) PMA (AMV) PMA (HMV) G β G β G β G β Dverged Example 8: Bracket Problem n RBDO Model Fgure 9 shows desgn parameterzaton and stress analyss result of a bracket at the ntal desgn. A total of 12 desgn parameters are selected to defne the nner and outer boundary shapes of the bracket model whle mantanng symmetry. Desgn parameterzaton s performed by selectng the control ponts of the parametrc curves. The bracket s modeled as a plane stress problem usng 769 nodes, 214 elements, and 552 DOF wth the thckness of 1.0 cm. The boundary condton s mposed to fx two lower holes. Usng FEM, stress analyss requred sec., whle DSA requred 35.44/12=2.95 sec. per desgn varable. The bracket s made of steel wth E = 207 GPa, ν = 0.3, and the yeld stress of σ=400 MPa. Probablstc constrants are defned on two crtcal regons usng the von Mses stress as shown n Fg. 9(b). Random parameters are defned n Table 9 and SQP optmzer s used wth a target relablty ndex of β=3.0 n the RBDO model N d 2 d 3 d 12 d 7 d 11 d 9 d 8 d1 d6 d 5 d 10 d4 G p3, G p4 G p1,g p2 ( a ) Desgn Parameterzaton ( b ) Stress Contour at Intal Desgn Fgure 9. Intal Bracket Desgn Fgure 10 shows several desgn teratons throughout the RBDO process. At the optmum desgn, the overall area s substantally reduced at the nner boundary and slghtly at the outer boundary. Fgure 11 (a) shows the stress contour at the MPP of the ntal desgn where all probablstc constrants are largely nactve. Fgure 11 (b) shows the stress contour at the MPP of the optmum desgn. 8 Copyrght 2001 by ASME

9 Random Varable Table 9. Random Varables n Bracket Model Lower Mean Upper Std. Dstrb. Desgn Value Desgn Dev. Type Bound (Desgn) Bound Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal model, the RIA fals to converge n relablty analyss, whereas PMA successfully obtans an optmal desgn for the bracket model. In addton, PMA wth the HMV method performs better than wth the conventonal AMV method n terms of numercal effcency (195 analyses vs. 295 analyses). ( a ) Stress At Intal Desgn ( b ) Stress At Optmum Desgn Fgure 11. Analyss Results Comparson ( a ) Intal Desgn ( b ) 1 ST RBDO Iteraton 0.9 Cost ( a ) Volume Hstory ( c ) 4 TH RBDO Iteraton ( d ) 7 TH RBDO Iteraton Gp1 Gp2-0.8 Gp3 Gp ( b ) Probablstc Constrant Hstory ( e ) Optmum Desgn Fgure 10. Shape Desgn Hstory n RBDO Process Desgn hstores are shown n Fg. 12. The area of the relablty-based optmum desgn s reduced by 47% of the orgnal area. The frst probablstc constrant becomes actve whle other probablstc constrants nactve at the optmum desgn wth 99.9% relablty as shown n Fg.12 (b). The sgnfcantly changed shape desgn parameters are 12 th, 1 st, and 2 nd parameters. In Table 10, the PMA wth both HMV and AMV methods s compared to RIA n terms of computatonal effcency and robustness. As n the roadarm D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D ( c ) Desgn Parameter Hstory Fgure 12. Desgn Optmzaton wth 99.9 % Relablty 9 Copyrght 2001 by ASME

10 Table 10. Computatonal Effcency and Robustness n RIA and PMA PMA RIA Opt. HMV AMV HL-RF Lne Search Anal yss Lne Search Anal yss Lne Search Anal yss N.A Opt mum Falure to Converge CONCLUSIONS Advances n the RBDO are made by developng the HMV method for the PMA n ths paper. It has been shown that PMA wth a sphercal equalty constrant s easer to solve than RIA wth a complcate constrant n relablty analyss. However, t has been found that the conventonal MPP search algorthm, the AMV method, exhbts numercal nstablty and neffcency for the concave performance functon. Therefore, the HMV method s proposed for effectve evaluaton of probablstc constrants n the RBDO process n order to take advantages of PMA. Based on numercal effcency and robustness n relablty analyss, the HMV method s very effectve numercal tool for estmatng probablstc constrants n the RBDO process. The comparson study between RIA and PMA has been extended to the RBDO problem, demonstratng that the PMA usng HMV method provdes the best result n the RBDO process. ACKNOWLEDGMENTS Research s partally supported by the Automotve Research Center sponsored by the U.S. Army TARDEC. REFERENCES 1. Madsen, H.O., Krenk, S., and Lnd, N.C., 1986, Methods of Structural Safety, Prentce-Hall, Englewood Clffs, NJ. 2. Palle, T.C. and Mchael J. B., 1982, Structural Relablty Theory and Its Applcatons, Sprnger-Verlag, Berln, Hedelberg. 3. Tu, J. and Cho, K.K., 1999, A New Study on Relablty-Based Desgn Optmzaton, Journal of Mechancal Desgn, ASME, Vol. 121, No. 4, 1999, pp Tu, J., Cho, K.K., and Park, Y.H., 2001, Desgn Potental Method for Robust System Parameter Desgn, to appear n AIAA Journal. 5. Wu, Y.T., Mllwater, H.R., and Cruse, T.A., 1990, Advanced Probablstc Structural Analyss Method for Implct Performance Functons, AIAA Journal, Vol. 28, No. 9, pp Wu Y.T., 1994, Computatonal Methods for Effcent Structural Relablty and Relablty Senstvty Analyss, AIAA Journal, Vol. 32, No. 8, pp Hasofer, A.M. and Lnd, N.C., 1974, Exact and Invarant Second-Moment Code Format, Journal of Engneerng Mechancs Dvson ASCE, 100(EMI), pp Lu, P.L. and Kureghan, A.D., 1991, Optmzaton Algorthms For Structural Relablty, Structural Safety, Vol. 9, pp Wang, L.P. and Grandh, R.V., 1994, Effcent Safety Index Calculaton For Structural Relablty Analyss, Computers & Structures, Vol. 52, No. 1, pp Wang, L.P. and Grandh, R.V., 1996, Safety Index Calculaton Usng Intervenng Varables For Structural Relablty, Computers & Structures, Vol. 59, No. 6, pp Enevoldsen, I. and Sorensen, J.D., 1994, Relablty-Based Optmzaton In Structural Engneerng, Structural Safety, Vol. 15, pp Wu, Y.-T. and Wang, W., 1996, A New Method for Effcent Relablty-Based Desgn Optmzaton, Probablstc Mechancs & Structural Relablty: Proceedngs of the 7 th Specal Conference, pp Yu, X., Cho, K.K., and Chang, K.H., 1997, A Mxed Desgn Approach for Probablstc Structural Durablty, Journal of Structural Optmzaton, Vol. 14, No. 2-3, pp Grandh, R.V. and Wang, L.P., 1998, Relablty-Based Structural Optmzaton Usng Improved Two-Pont Adaptve Nonlnear Approxmatons, Fnte Elements n Analyss and Desgn, Vol. 29, pp Rackwtz, R. and Fessler, B., 1978, Structural Relablty Under Combned Random Load Sequences, Computers & Structures, Vol. 9, pp Hohenbchler, M. and Rackwtz, R., 1981, Nonnormal Dependent Vectors n Structural Relablty, Journal of the Engneerng Mechancs Dvson ASCE, 107(6), Arora, J.S., 1989, Introducton to Optmum Desgn, McGraw- Hll, New York, NY. 18. Yu, X., Chang, K.H. and Cho, K.K., 1998, Probablstc Structural Durablty Predcton, AIAA Journal, Vol. 36, No. 4, pp CADSI Inc., DADS User s Manual, Rev. 7.5, Oakdale, IA, DRAW, Durablty and Relablty Analyss Workspace, Center for Computer-Aded Desgn, College of Engneerng, The Unversty of Iowa, Iowa Cty, IA, Copyrght 2001 by ASME