11 th World Congress on Structural and Multidisciplinary Optimisation. Po Ting Lin 1

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1 11 th World Congress on Structural and Multdscplnary Optmsaton 07 th -12 th, June 2015, Sydney Australa Utlzaton of Gaussan Kernel Relablty Analyses n the Gradent-based Transformed Space for Desgn Optmzaton wth Arbtrarly Dstrbuted Desgn Uncertantes Po Tng Ln 1 1 Department of Mechancal Engneerng, Chung Yuan Chrstan Unversty, Chungl, Taoyuan, Tawan, potngln@cycu.edu.tw 1. Abstract Several Relablty-Based Desgn Optmzaton (RBDO) algorthms have been developed to solve engneerng optmzaton problems under desgn uncertantes. Some exstng methods transform the random desgn space to standard normal desgn space to estmate the relablty assessment for the evaluaton of the falure probablty. When the random varable s arbtrarly dstrbuted and cannot be properly ftted to any known form of probablty densty functon, the exstng RBDO methods, however, cannot perform relablty analyss ether n the orgnal desgn space or n the standard normal space. Ths paper proposes a novel method, Ensemble of Gradent-based Transformed Relablty Analyses (EGTRA), to evaluate the falure probablty of arbtrarly dstrbuted random varables n the orgnal desgn space. The arbtrary dstrbuton of the random varable s approxmated by a merger of multple Gaussan kernel functons. Each Gaussan kernel functon s transformed to a sngle-varate coordnate that s drected toward the gradent of the constrant functon. The falure probablty s then estmated by the ensemble of each kernel relablty analyss. Ths paper further derves a lnearly approxmated probablstc constrant at the desgn pont wth allowable relablty level n the orgnal desgn space usng the aforementoned fundamentals and technques. Numercal examples wth generated random dstrbutons show EGTRA s capable of solvng the RBDO problems wth arbtrarly dstrbuted uncertantes n the orgnal desgn space. 2. Keywords: gradent-based transformaton; Gaussan kernel densty estmaton; relablty-based desgn optmzaton; arbtrarly dstrbuted desgn uncertanty. 3. Introducton In engneerng desgn, tradtonal determnstc optmzaton has been successfully appled to mprove qualty, processes and reduced costs. However, uncertantes have to be consdered to make desgns more confdent and relable. These tradtonal determnstc approaches for optmzaton of components, products and systems are slowly beng replaced n the past decades of approaches that ntegrate probablstc consderatons. These probablstc consderatons were nvestgated and have been known to be materal property devaton [1], allowable falure probabltes from standards [2], producton condton [3], reported ncdences of falures or satsfactons [4] and operatng condtons [5], to menton a few. The basc concept behnd Relablty-Based Desgn Optmzaton (RBDO) methods s to ntegrate and consder these probablstc factors n the optmzaton process and many approaches have been developed n the past. Relablty Index Approach (RIA) [6-9] formulated the probablstc constrants based on the evaluatons of relablty ndces. Ln et al. [10] resolved the convergence problems of RIA [11] by modfyng the relablty ndex to correctly evaluate the falure probablty for both feasble and nfeasble desgn ponts. Performance Measure Approach (PMA) [11-15], on the other hand, mplements an nverse relablty analyss, whch determnes the performance measure of a target desgn pont. Probablstc constrants are then formulated from these performance measures. Observatons of the strengths and weaknesses of both RIA and PMA has led to the development of Hybrd Relablty Approach (HRA) [16, 17]. HRA uses a selecton factor to determne whether PMA or MRIA would be effcent to use. Dervatons of a Unfed Relablty Formulaton (URF) [18] had revealed how varous RBDO algorthms can be reformulated nto one general equaton based on the relablty analyses n the standard normal desgn space and the fundamental aspect of lnear expansons wth allowable relabltes. Recently, the desgn optmzaton problems wth arbtrarly dstrbuted uncertantes have drawn hgh attentons [19-21] because they cannot be properly solved by RBDO algorthms that requre transformaton of the orgnal desgn space to the standard normal desgn space. Therefore, the stuatons such as unknown random dstrbuton type, undetermnable transformaton to the standard normal desgn space, and nsuffcent nformaton about the random dstrbuton, are dffcult for most RBDO algorthms. Thus, a new method that s capable of effcently solvng the RBDO problem wth arbtrarly dstrbuted uncertantes n the orgnal desgn space s desrable. In ths paper, a novel method called Ensemble of Gradent-based Transformed Relablty Analyses (EGTRA) s derved based on estmaton of arbtrary dstrbuton n the orgnal desgn space usng Kernel Densty Estmaton 1

2 (KDE) [22, 23] and relablty analyses along the gradent-based transformed drecton [24]. A lnearly approxmated probablstc constrant s formulated at the desgn pont determned at the allowable relablty level. The aforementoned dervatons and technologes are conducted n the orgnal space; therefore, EGTRA s expected to be capable of solvng RBDO problems wth arbtrarly dstrbuted uncertantes. Two numercal examples are solved n order to nvestgate the numercal performances of the proposed method. 4. Dervaton of Ensemble of Gradent-based Transformed Relablty Analyses (EGTRA) A typcal RBDO problem s formulated as follows: Mn f d!" # $ P f, =1...M (1) ( ) s.t. P ( X) > 0 where X s the randomly dstrbuted desgn varable; d s the mean of X and s commonly used as the desgn varable; f s the objectve (or cost) functon; s the th constrant functon and 0 represents the th safe regon; P f, s th allowable falure probablty. There are M constrants and L varables. Several RBDO algorthms have been nvestgate the falure probablty n Eq. (1) durng the optmzaton process to reach the optmalty, feasblty and relablty smultaneously. However, some exstng methods cannot perform the relablty analyss properly when the followng condtons occur: (1) Dstrbuton type of X s unknown. (2) Transformaton of X to the th standard normal desgn space U s dffcult to determne or does not exst. (3) Data about the random dstrbuton s nsuffcent. Ths paper assumes the relatve postons between the samplng ponts and the desgn pont reman constant and are ndependent from the locaton of the desgn pont. The followng gradent-based transformaton [24] s frst consdered to transform the orgnal coordnate to the sngle varate desgn space y toward the drecton of the th constrant gradent v = x (d) x (d) 1 : y,p = (s p d) v (2) where the desgn pont d s the orgn of the th gradent-based transformed desgn space, denoted by Ω ; the value of y,p s the projecton of the p th samplng pont n Ω. Fgure 1 (a) llustrates the transformaton to the constrant gradent drecton and the mappng of the samplng ponts to Ω. A Most Probable Target Pont (MPTP) n Ω, denoted by y #, s defned such that the summaton of the cumulatve probablty of each kernel functon from the MPTP to nfnty s equal to the allowable falure probablty, as llustrated n Fgure 1 (b). Therefore, y # s determned by solvng the followng equaton: N 1 N Φ p (y # ) = P f, (3) In Eq. (3), Φ p s the p th p=1 Gaussan Cumulatve Dstrbuton Functon (CDF) and s defned as below: Φ p (y 0 ) = σ 1 ( 2π ) 1 exp[ 0.5(y y,p ) 2 σ 2 ]dy (4) y 0 (a) (b) (c) Fgure 1. Illustraton of the proposed method n Ω : (a) Transformaton of samplng ponts to Ω ; (b) Determnaton of Most Probable Target Pont; (c) Determnaton of Allowable Relablty Pont. where σ s a shape parameter n KDE [22, 23]. The sze of σ s crtcal for the accuracy of the estmaton of the 2

3 arbtrarly dstrbuted PDF. The locaton of MPTP s essental for the estmaton of the desgn pont wth allowable relablty and the further formulaton of probablstc constrant. The RBDO process n Ω s expected to move the MPTP to the lmt state; therefore, the value of y # also represents the allowable tolerance near the lmt state, as llustrated n Fgure 1 (c). The th Allowable Relablty Pont (ARP) y A s then determned by y A = y * y # = (d) x (d) 1 # y (5) where y * s the Most Probable Falure Pont (MPFP) n Ω. In the end, the RBDO procedure s expected to move the desgn pont to the locaton of the ARP. Thus, the falure probablty of the new desgn pont s expected to reach the allowable level. Usng URF [18], the lnearly approxmated probablstc constrant s formulated below: (d x A ) v 0 (6) where x A s the th Allowable Relablty Pont (ARP) n the orgnal desgn space. Therefore, a fnal formulaton of the th probablstc constrant usng the EGTRA s gven as follows: {d d (k ) +[ (d (k ) ) x (d (k ) ) 1 +y # (k ]v ) (k } v ) 0 (7) From the derved Eq. (7), t s noted that y # s not only a most probable allowable tolerance but also a newly defned relablty assessment n the orgnal desgn space for the proposed EGTRA method. 5. Numercal Examples Ths secton frst ntroduces the procedure of random dstrbuton generaton that s used n ths paper. Two mathematcal problems are then solved by the proposed EGTRA. 5.1 Generatons of Arbtrarly Dstrbuted Random Varables Ths paper consders the desgn pont at the mean value of the generated random dstrbuton. All problems are solved n the two-dmensonal desgn space,.e. L = 2, for better llustraton of the results. The followng dstrbutons are artfcally generated by specfed procedures for research purpose and may not be seen n real world. Ths paper ntentonally generates some dstrbutons that are concavely ranged and are dffcult for conventonal relablty analyses. Fgure 2 (a) shows the generated heat-shaped dstrbuton wth ts mean pont at the orgn of the coordnate. If t s mproperly consdered as a Gaussan dstrbuton, as shown by the dashed contour, the standard devatons along the x and y drectons wll be computed as [0.3896, ], respectvely. Fgure 2 (b) to (d) show the generated lke -shaped, star-shaped and corona-shaped dstrbutons, respectvely. If these dstrbutons are mproperly consdered as Gaussan dstrbutons, the standard devatons along the x and y drectons are computed as [0.2493, ], [0.3596, ] and [0.4329, ] for the lke -shaped, star-shaped and corona-shaped dstrbutons, respectvely. The heart-shaped dstrbuton shows a concave dstrbuton supported n the entre desgn space. The lke -shaped dstrbuton shows a combned dstrbuton that s partally supported n n the entre doman and partally supported n a sem-nterval. The star-shaped dstrbuton shows a unform dstrbuton n a concave regon. The corona-shaped dstrbuton shows a dstrbuton supported n the entre desgn space wth a vod regon at the center. (a) (b) (c) (d) Fgure 2. Generated (a) heart-shaped, (b) lke -shaped, (c) star-shaped and (d) corona-shaped dstrbutons. 5.2 Example 1: A Lnear Math Problem The frst example s a lnear mathematcal problem [10, 11], whch s shown n Eq. (8). Mn d d 1 + d 2 s.t. P[g 1 = X 1 2X > 0] P f ; P[g 2 = 2X 1 X > 0] P f ; 0.1 d 1, d 2 10 In ths paper, two varous levels of allowable falure probabltes are nvestgated: P f =1% and 30%. The ntal (8) 3

4 desgn pont s located at [5, 5]. Fgure 3 shows the optmal solutons usng EGTRA wth N = to solve the problem n Eq. (8) n the generated random dstrbutons. The subfgures (a) to (d) are the results for Pf = 1% whle the subfgures (e) to (h) are for Pf = 30%. The red lnes represent the lmt states of the orgnal constrants and the black lnes are the lnearly approxmated probablstc constrants, whch are determned usng the Eq. (7). Because each kernel relablty analyss n EGTRA s completed along the same gradent drecton, the requred functon evaluaton of each constrant per teraton s only 3. Monte Carlo Smulatons (MCS) wth 10 5 samplng ponts were used to evaluate the true falure probabltes. The numercal results showed the proposed method s capable of solvng the problems wth the generated desgn uncertantes wth accurate estmatons of relabltes n the orgnal desgn space,.e. errors were less than 1%. Fgure 3. Solutons of example 1 n varous dstrbutons and allowable falure probabltes: (a) heart, 1% ; (b) lke, 1% ; (c) star, 1% ; (d) corona, 1% ; (e) heart, 30% ; (f) lke, 30% ; (g) star, 30% ; (h) corona, 30%. 5.3 Example 2: A Nonlnear Benchmark Problem The second example s a well-known benchmark mathematcal problem [10, 11, 25, 26] that contans three nonlnear constrants. Because the thrd constrant s nactve, t s removed and the followng problem formulaton s consdered n ths paper. Mn d1 + d2 d (9) s.t. P[g1 = 1 ( X 12 X 2 ) 20 > 0] Pf ; P[g 2 = 1 ( X 1 + X 2 5)2 30 ( X 1 X 2 12)2 120 > 0] Pf ; 0.1 d1, d2 10 In ths problem, two varous levels of allowable falure probabltes,.e. Pf = 1% and 30%, are studed. EGTRA s used to solve the problem wth N = The ntal desgn pont s located at [5, 5]. The rest of the problem settns the same as the frst example. Fgure 4 (a) to (d) show the optmal solutons for Pf = 1% whle the subfgures (e) to (h) show the ones for Pf = 30%. EGTRA was capable of solvng the gven problems wth only 3M functon evaluatons per teraton. However, the accuracy of EGTRA slghtly dropped because lnear approxmatons were utlzed for the nonlnear constrants n Eq. (9),.e. error ncreased up to around 6%. Fgure 4. Solutons of example 2 n varous dstrbutons and allowable falure probabltes: (a) heart, 1% ; (b) lke, 1% ; (c) star, 1% ; (d) corona, 1% ; (e) heart, 30% ; (f) lke, 30% ; (g) star, 30% ; (h) corona, 30%. 6. Conclusons 4

5 Some exstng RBDO algorthms transformed the orgnal random desgn space to the standard normal desgn space n order to perform the relablty analyses for the evaluaton of falure probabltes. However, these relablty analyses cannot be properly executed when the transformaton to the standard normal desgn space cannot be determned. A new RBDO algorthm, Ensemble of Gradent-based Transformed Relablty Analyses (EGTRA), was developed to solve desgn optmzaton problems wth arbtrarly dstrbuted uncertantes n the orgnal desgn space. The arbtrarly dstrbuted PDF was approxmated by KDE and then transformed to a sngle-varate coordnate toward the constrant gradent drecton. The entre relablty analyss s decomposed to multple kernel relablty analyses n the gradent-based transformed desgn space and the results are merged together for the formulaton of a lnearly approxmated probablstc constrant functon. Because each kernel relablty analyss s completed along the same gradent drecton, the requred functon evaluaton of each constrant per teraton s only 3. The numercal results showed the proposed method s capable of solvng the problems wth the generated desgn uncertantes wth accurate estmatons of relabltes n the orgnal desgn space. The newly developed method does not need transformaton to the standard normal desgn space and relablty analyses that requre addtonal functon evaluatons. EGTRA s able to perform very accurate relablty analyss for lnear RBDO problems and the accuracy reduces when the constrants are nonlnear. EGTRA s a method that requres nformaton about the samplng ponts of the arbtrarly dstrbuted random varables. Insuffcent samplng ponts may lead to naccurate estmatons of PDF and falure probabltes. The performance of EGTRA wll reduce when the samplng at the tal of the dstrbuton s nsuffcent. Ths may happen when the amount of samplng ponts and the level of allowable falure probablty are both very low. 7. Acknowledgments The supports from Mnstry of Scence and Technology, Tawan (grant number MOST E ), Research and Development Center for Mcrosystem Relablty, and Center for Robotcs Research at Chung Yuan Chrstan Unversty, Tawan are greatly apprecated. 8. References [1] H.-S. Jung and S. Cho, "Relablty-based topology optmzaton of geometrcally nonlnear structures wth loadng and materal uncertantes," Fnte elements n analyss and desgn, vol. 41, pp , [2] B. Ellngwood, Development of a probablty based load crteron for Amercan Natonal Standard A58: Buldng code requrements for mnmum desgn loads n buldngs and other structures, 577, US Department of Commerce, Natonal Bureau of Standards, [3] R. D. Pope and R. A. Kramer, "Producton uncertanty and factor demands for the compettve frm," Southern Economc Journal, pp , [4] S. S. Hasan, J. M. Leth, B. Campbell, K. L. Smth, and F. A. Matsen III, "Characterstcs of unsatsfactory shoulder arthroplastes," Journal of shoulder and elbow surgery, vol. 11, pp , [5] M. Karamouz and H. V. Vaslads, "Bayesan stochastc optmzaton of reservor operaton usng uncertan forecasts," Water Resources Research, vol. 28, pp , [6] E. Nkolads and R. Burdsso, "Relablty Based Optmzaton: A Safety Index Approach," Computers & Structures, vol. 28, pp , [7] I. Enevoldsen, "Relablty-Based Optmzaton as an Informaton Tool," Mechancs of Structures and Machnes, vol. 22, pp , [8] I. Enevoldsen and J. D. Sorensen, "Relablty-Based Optmzaton n Structural Engneerng," Structural Safety, vol. 15, pp , [9] S. V. L. Chandu and R. V. Grandh, "General Purpose Procedure for Relablty Based Structural Optmzaton under Parametrc Uncertantes," Advances n Engneerng Software, vol. 23, pp. 7-14, [10] P. T. Ln, H. C. Gea, and Y. Jalura, "A Modfed Relablty Index Approach for Relablty-Based Desgn Optmzaton," Journal of Mechancal Desgn, vol. 133, , [11] J. Tu, K. K. Cho, and Y. H. Park, "A New Study on Relablty Based Desgn Optmzaton," Journal of Mechancal Desgn, vol. 121, pp , [12] B. D. Youn and K. K. Cho, "An Investgaton of Nonlnearty of Relablty-Based Desgn Optmzaton Approaches," Journal of Mechancal Desgn, vol. 126, pp , [13] B. D. Youn and K. K. Cho, "Selectng Probablstc Approaches for Relablty-Based Desgn Optmzaton," AIAA Journal, vol. 42, pp , [14] B. D. Youn, K. K. Cho, and L. Du, "Adaptve Probablty Analyss Usng an Enhanced Hybrd Mean Value Method," Structural and Multdscplnary Optmzaton, vol. 29, pp , [15] B. D. Youn, K. K. Cho, and L. Du, "Enrched Performance Measure Approach for Relablty-Based Desgn Optmzaton," AIAA Journal, vol. 43, pp , [16] P. T. Ln, Y. Jalura, and H. C. Gea, "A Hybrd Relablty Approach for Relablty-Based Desgn 5

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