A NEW STOCHASTIC PROGRAMMING METHOD FOR VEHICLE STRUCTURES

Transcription

1 A EW STOCHASTIC PROGRAMMIG METHOD FOR VEHICLE STRUCTURES Yan Fu and Steve Wang Ford Research Laboratory, MD, SRL, Ford Motor Company, Dearborn, MI, Emal: Urmla M. Dwekar and Kemal H. Sahn Center for Uncertanty Systems: Tools for Optmzaton & Management, Carnege Mellon Unversty, Pttsburgh, PA ABSTRACT Ths paper presents an applcaton of a new stochastc programmng method for vehcle sde mpact desgn. onlnear response surface models are employed as the 'real' models for the sde mpact related performance functons to conduct ths study. The man goal s to enhance vehcle sde mpact crash performance whle mnmzng vehcle weght under varous uncertantes. The new algorthm allevates the computatonal burden of ecessve model evaluatons by estmatng the obectve and constrant functons durng the optmzaton process through a reweghtng method. The effcency and accuracy of ths algorthm are demonstrated by solvng a real-world vehcle safety desgn problem. Keywords: Stochastc Programmng, Optmzaton Under Uncertanty, onlnear Programmng, Sde Impact ITRODUCTIO The computatonal analyss of crashworthness for vehcle mpact has become a powerful and effcent tool n reducng the cost and development tme for a new product. Today, nonlnear fnte element-based crash analyss codes are commonly used to smulate many laboratory vehcle crash events, e.g. frontal mpact, sde mpact, nteror head mpact, and rear mpact. umercal optmzaton s a useful and systematc tool for automatcally selectng approprate desgn parameters. It has been wdely used and acheved sgnfcant results n the automotve ndustry. Sobesk et al. ( conducted the multdscplnary desgn optmzaton for a car body structure under constrants of ose, Vbraton, and Harshness (VH and roof crush. Kodyalam et al. ( reported a multdscplnary optmzaton of a vehcle system n a scalable hgh-performance computng envronment. For vehcle safety optmzaton, Yang et al. ( conducted a feasblty study of vehcle safety CAE optmzaton. Stander ( nvestgated the crashworthness problem usng both the response surface method and massvely parallel programmng. Yang et al. ( compared appromate methods for safety optmzaton of large systems n terms of accuracy and effectveness. To ensure product qualty and the consstency of product performance, uncertantes n modelng, smulaton, manufacturng, and desgn varables, must be consdered. Relablty based desgn optmzaton methods formulated the constrants that nclude uncertanty wth approprate probabltes. Mavrs et al. ( and Gu et al. ( employed relablty based desgn optmzaton for vehcle crashworthness usng a global response surface model. Cho and Youn ( developed robust and effcent relablty based desgn optmzaton methods for both conve and concave functons. Du and Chen ( proposed a smlar approach. Youn et al. ( appled a relablty based desgn optmzaton method for vehcle sde mpact desgn. Koch et al. ( utlzed a robust optmzaton method for solvng a desgn for s-sgma problem for vehcle crashworthness. Yang et al. ( conducted relabltybased multdscplnary desgn optmzaton for a full vehcle desgn. Ths paper employs a new stochastc programmng method called better optmzaton of nonlnear uncertan systems, (Sahn and Dwekar, for vehcle sde mpact desgn. uses samplng to estmate obectve and constrant functons wth uncertantes. It reduces the computatonal burden of ecessve model smulatons durng the optmzaton process through a reweghtng method. Snce crash analyss s computatonally ntensve, global response surface models, whch are generated usng stepwse regresson coupled wth the optmal Latn hypercube samplng for desgn of eperment, are treated as the real models for ths study (Gu et al.. The followng sectons are organzed as follows: Frst, the detals of s ntroduced, and then a vehcle sde mpact desgn problem amng at mnmzng vehcle weght whle mantanng or enhancng vehcle sde mpact performance s presented. It follows by the robustness assessment of the baselne and determnstc optmal desgns as well as obtanng stochastc optmal desgns wth the consderaton of varous uncertantes usng both the tradtonal approach and. Results of both methods are then compared. Fnally, the concluson s summarzed at the end. BETTER OPTIMIZATIO OF OLIEAR UCERTAI SYSTEMS ( A general stochastc optmzaton approach nvolves two computatonally ntensve recursve loops: ( the nner samplng loop, and ( the outer optmzaton loop (Fgure. The commonly used method for the nner samplng loop s Monte Carlo samplng (MCS technque, whch uses random samples selected from assumed nput dstrbuton to obtan estmates of the output dstrbutons and assocated statstcal characterstcs, such as mean, varance, or percentles. A numercal method for solvng the outer optmzaton loop of nonlnear programmng problems (LPs s sequental quadratc programmng (SQP. As optmzaton progresses and new decson varables are determned, shfts n uncertan

2 varables are observed, resultng n a new nput dstrbuton. The model s re-evaluated for another sample set whch s generated from the new dstrbuton. Durng the optmzaton teratons, even for small sample szes, the repeated evaluaton of the model s a sgnfcant bottleneck. Stochastc Optmal Desgn Probablstc Obectve & Constrant Functons Optmzer Stochastc Modeler Intal Desgn Varables Desgn Varables calculatng the obectve and constrant functons. The tradtonal approach, shown n the center, reles on developng a samplng loop and evaluatng ths loop for every sample that s generated usng the nput dstrbuton. The new approach follows the bg arrows n Fgure. Instead of runnng the model for every sample pont, the output dstrbutons are estmated based on the base case nput and output dstrbutons. An ntal set of samples,, L, followng unformly dstrbuted base case dstrbuton s generated and the model s run to determne the base case output dstrbutons. Let's assume s multvarate wth K-dmenson ( K f s unvarate and ts densty can be calculated for one varable at a tme accordng to the kernel densty approach as: Output Dstrbuton Input Dstrbuton g (, k h π e, k h, k ( MODEL where h s the wdth for the Gaussan kernel, whch depends on the sample sze and varance of the samples. Fgure : A General Approach for Stochastc Optmzaton A new stochastc programmng method called developed by Sahn and Dwekar ( s employed for stochastc optmzaton. The man dea of shown n Fgure can be summarzed as an effcent appromaton for Cumulatve Dstrbuton Base Case Output For ndependent varables, the ont probablty densty for each pont s calculated as: Probablty Dstrbuton Base Case Input g ( K, k k ( probablty Stochastc Modeler probablty Changed Input Probablty Dstrbuton Estmated Output Cumulatve Dstrbuton SAMPLIG LOOP MODEL probablty probablty Reweghtng Scheme Fgure : Densty Estmaton Approach for Stochastc Optmzaton After determnng the model output,, L, for each nput sample,, L,, the optmzaton algorthm generates new desgn varables. Ths s followed by the generaton of a new set of samples (,, L, for the new nput dstrbuton. Let's assume s multvarate wth K-dmenson ( K f s unvarate, and the probablty densty of each,, L, n the new nput dstrbuton s determned for one varable at a tme through the kernel densty appromaton as: f (, k h e π, k, k h For ndependent varables, the ont probablty densty for each pont,, L, n the new nput dstrbuton s calculated as: f ( K f (, k k ( (

3 At ths stage, the model s not re-run; nstead, a reweghtng approach s appled to appromate the probablstc behavors of the new output dstrbutons. The mean ( µ rato and varance ( σ rato of the output dstrbutons are calculated by Equaton and, and then the obectve and constrant functons can be obtaned. µ rato f ( ω f ( ω f ( where ω σ rato s a ω [ ω weght E[ ] ( E[ ] ] ( µ The detaled eplanaton of can be found n Sahn and Dwekar (. The advantage of ths technque s bypassng of the model evaluatons for the new sample set durng each teraton, whch s computatonally ntensve for stochastc optmzaton. Ths s partcularly crtcal for optmzaton algorthms that rely on evaluatng the gradents numercally, by perturbng every desgn varable by a small ncrement and calculatng the change n the obectve and constrant functons. Evaluatng the model for an entre sample set (due to uncertantes over and over agan durng the optmzaton process s neffcent for comple models. VEHICLE SIDE IMPACT DESIG PROBLEM A large-scale applcaton of the proposed algorthm n the desgn of vehcle sde mpact s llustrated n Fgure. The system model ncludes a full-vehcle fnte element (FE structural model, a FE sde mpact dummy model, and a FE deformable sde mpact barrer model. The system model conssts of, shell elements and, nodes. In the FE smulaton of the sde mpact event, the barrer has an ntal velocty of. kph ( mph and mpacts the vehcle structure. The CPU tme for one nonlnear FE smulaton usng the RADIOSS software s appromately hours on a SGI Orgn machne. The desgn goal s to mantan or enhance sde mpact test performance whle mnmzng the vehcle weght. Fgure : Vehcle Sde Impact Model For sde mpact protecton, the vehcle desgn must meet or eceed regulated sde mpact requrements specfed by the rato ( ( vehcle market. Two prmary sde mpact protecton regulatons are Federal Motor Vehcle Safety Standard o. n the Unted States and ECE Unted atons Economc Commsson For Europe Regulaton o. n Europe. In ths study, the ECE sde mpact test confguraton s used. The dummy's responses are the man metrc n sde mpact studes. The crash dummy crtera specfed n the ECE sde mpact regulaton nclude abdomen load, vscous crtera (upper, mddle, and lower, rb deflectons (upper, mddle, and lower, and pubc symphyss force. The dummy's responses must meet or eceed ECE crtera. Other concerns n sde mpact desgn are the velocty of the B-pllar at the mddle pont (V B- Pllar and the velocty of the front door at the B-pllar (V Door. Table. Regulatons and Desgn Targets ECE Desgn Performance Symbol Regulaton Target Abdomen Load (k Ab.Load.. VC.. Vscous Crtera Mddle VC Mddle.. (m/s VC.. RbDefl. Rb Mddle RbDefl. Mddle Deflecton RbDefl. Pubc Symphyss Force (k PSF. Table. Propertes of Desgn and Uncertan Varables ame of Varable Thckness of B-Pllar nner. Thckness of B-Pllar renforcement. Thckness of floor sde nner. Thckness of cross member # & #. Thckness of door beam. Thckness of door belt lne renforcement. Thckness of roof ral. Desgn Varable Bound Baselne Bound Uncertan Varable Dstr. Std. Type Dev. ormal. ormal. ormal. ormal. ormal. ormal. ormal. Materal of B-Pllar nner MST MST HSST ormal. Materal of floor sde nner MST MST HSST ormal. Barrer heght... ormal. Barrer httng poston... ormal. ote: s the baselne value for desgn varable. There are desgn varables used n the desgn optmzaton of vehcle sde mpact. The desgn varables are the thckness ( - and materal property of crtcal parts (,, as shown n Table. All thckness desgn varables (unt, mm are contnuous whch are allowed to vary over the range from. to., where s the baselne value for desgn varable. The two materal desgn varables (unt, GPa are dscrete, and can be ether mld steel (MST or hgh strength steel (HSST. In ths sde mpact CAE model, t s assumed that there are normally dstrbuted uncertantes around ther nomnal value for all the desgn varables.

4 Addtonal two nose factors (uncertan varables are barrer heght and httng poston, whch also follow normal dstrbutons. The standard devatons (Std. Dev. for all uncertan varables are lsted n Table. A response surface method (RSM developed by Gu et al. ( s employed to demonstrate for vehcle crashworthness as numercal computaton s out of scope of ths paper. The RSM used the quadratc backward-stepwse regresson method coupled wth the optmal Latn hypercube samplng to generate a global response surface model for the dummy test performance n vehcle sde mpact. The models can be summarzed as Equaton. Weght V V VC B-Pllar Door Ab.Load VC VC PSF Mddle RbDefl. RbDefl. RbDefl. Mddle The stochastc optmzaton problem for crashworthness of vehcle sde mpact s formulated as Equatons. It s noted that f a response (output dstrbuton follows normal dstrbuton, µ+.σ wth respect to desgn specfcaton lmt s the equvalent of desgnng for a relablty level of % (or probablty of falure of %. Even for a nonnormal response dstrbuton, µ+.σ stll provdes a good ndcator for a relable desgn under uncertanty. It s also noted that the proposed method can be easly mplemented for robust desgn by changng the obectve to mnmze the varance of a response, such as σ [ Weght]. ( Mnmze µ [Weght] Subectve µ [V µ [V µ [Ab. µ [VC µ [VC µ [VC µ [RbDefl. µ [RbDefl. µ [RbDefl. µ [PSF] where B- Pllar Door RESULTS : to ] +. ] +. Load] Mddle +. µ [ ] s the σ [ ] s the +. : +. ] +. ] +. ] +. Mddle σ [V ] +. ] +. ] +. σ [PSF] mean σ [V varance σ [Weght] Door σ [Ab. σ [VC σ [VC σ [VC of B- Pllar ]. σ [RbDefl.. response of ]. Mddle Load] ]. σ [RbDefl. σ [RbDefl. response. ]. ]., Mddle. ] ] ] Table. Comparson of the Baselne and Determnstc Optmal Desgns Baselne Desgn Desgn Varable Thckness of B-Pllar nner Thckness of B-Pllar renforcement Thckness of floor sde nner Thckness of cross member # & # ( Determnstc Optmal Desgn.... Thckness of door beam. Thckness of door belt lne renforcement. Thckness of roof ral. Materal of B-Pllar nner MST HSST Materal of floor sde nner MST MST Robustness Assessment µ[] +.σ[] Robustness Assessment µ[] +.σ[] Deter. Deter. Constrant Response Response Response Value Obectve functon, Weght (Kg Mnmze..... V B-Pllar. (m/s. (.%.. V Door. (m/s.... (.% Ab.Load. (K.... VC. (m/s.... VC Mddle. (m/s.... VC. (m/s.... RbDefl. (mm.... RbDefl. Mddle (mm RbDefl. (mm (.% (.%. (.% PSF. (K. (.%. (.%.. (.% ote: s the baselne value for desgn varable. In order to compare the performances of the determnstc and stochastc optmal desgns, the determnstc optmzaton problem s frst solved n ths secton, and the robustness

5 assessment usng MCS wth a very large number of samples (, are conducted for both the baselne and determnstc optmal desgns. The results are shown n Table. The determnstc optmal desgn can mprove the test performance n vehcle sde mpact as well as reducng vehcle weght wthout consderng uncertantes. However, the results also show that by the consderaton of the uncertantes, some of the constrants are sgnfcantly volated by the baselne desgn (e.g..% volaton on the lower rb deflecton and by the determnstc optmal desgn (e.g..% on the pubc symphyss force and total of three volatons. satsfes the constrants better than the others, but results n slghtly hgher vehcle weght. It s noted that all stochastc optmal desgns found by have less weght and better test performance n vehcle sde mpact than those of the baselne and determnstc optmal desgns (shown n Table under the uncertantes. Table presents the relatve errors of estmatons compared to, Monte Carlo smulatons for all the responses of the stochastc optmal desgns (Table found by. The relatve error s calculated as follows: The results obtaned by both the tradtonal approach and wth dfferent numbers of HSS samples are shown n Table. It s shown that wth mnmum smulatons yelds a smlar stochastc optmal desgn compared to that found by the tradtonal approach. Table also shows that the frst four stochastc optmal desgns found by (wth the number of HSS ponts less than, save slghtly more vehcle weght than that found by the tradtonal approach, however, they slghtly volate (by less than.% two of the ten constrants. The stochastc optmal desgn wth, HSS ponts satsfes the constrant of the lower rb deflecton better than the frst four, but wth slght volaton (by.% on the constrant of the publc symphyss force and provdes less vehcle weght savng. The stochastc optmal desgn wth, HSS ponts ( µ +.σ ε MCS, % ( ( µ +.σ MCS, where represents response Table. Comparson of Stochastc Optmal Desgns ( µ +.σ In general, the estmaton technque can provde good predctons of the actual response values under the uncertantes. The mamum relatve error reduces as the number of HSS samples ncreases. There are a few relatve errors are larger than %. However, they do not affect the results n negatve manners, as the constrants wth relatvely larger errors are mostly nactve. The accuraces of the most crtcal constrants, such as lower rb deflecton and publc symphyss force, have greater mpacts on the results than the others. Methods Mean values of varables SQP + MCS (, Thckness of B-Pllar nner (mm. Thckness of B-Pllar renforcement (mm. Thckness of floor sde nner (mm. Thckness of cross member # & # (mm. Thckness of door beam (mm. Thckness of door belt lne renforcement (mm. Thckness of roof ral (mm Materal of B-Pllar nner HSST HSST HSST HSST HSST HSST HSST Materal of floor sde nner MST MST MST MST MST MST MST Barrer heght (mm Barrer httng poston (mm Responses µ[weght] +.σ[weght] (Kg µ[v B-Pllar ] +.σ[v B-Pllar ]. (m/s µ[v Door ] +. σ[v Door ]. (m/s µ [Ab.Load] +.σ [Ab.Load]. (K µ [VC ] +.σ [VC ]. (m/s µ [VC Mddle ] +.σ [VC Mddle ]. (m/s µ [VC ] +.σ [VC ]. (m/s µ [RbDefl. ] +.σ [RbDefl. ] (mm µ [RbDefl. Mddle ] +.σ [RbDefl. Mddle ] (mm µ [RbDefl. ] +.σ [RbDefl. ] (mm. (.% (.% (.% (.%.. µ [PSF] +.σ [PSF]. (K.. (.%. (.%. (.%. (.%. (.%. ote: s the baselne value for desgn varable. All response values are confrmed by MCS wth, samples.

6 Table. Relatve Errors of Estmatons Compared to, MCS Relatve Error % Methods Responses µ [Weght] +.σ [Weght] (Kg µ [V B-Pllar ] +.σ [V B-Pllar ] (m/s µ [V Door ] +. σ [V Door ] (m/s µ [Ab.Load] +.σ [Ab.Load] (K µ [VC ] +.σ [VC ] (m/s µ [VC Mddle ] +.σ [VC Mddle ] (m/s µ [VC ] +.σ [VC ] (m/s µ [RbDefl. ] +.σ [RbDefl. ] (mm µ [RbDefl. Mddle ] +.σ [RbDefl. Mddle ] (mm µ [RbDefl. ] +.σ [RbDefl. ] (mm µ [PSF] +.σ [PSF] (K Mamum Relatve Error % COCLUSIO A new stochastc programmng method ( s successfully appled to a vehcle sde mpact problem for obtanng stochastc optmal desgns. The results demonstrate that can sgnfcantly reduce the computatonal resources by usng an appromaton technque for appromatng performance functons nstead of runnng model smulatons durng the optmzaton loops. The results of ths study are prmarly based on the assumptons that both of the global response surface and fnte element models are vald, and the dstrbutons of random (uncertan varables are also accurate. However, the valdatons of these models and the statstcal analyss of these random varables are beyond the scope of ths paper. REFERECES Youn, B. D., Cho, K. K., Yang, R. J., and Gu, L., "Relablty-Based Desgn Optmzaton for Crashworthness of Vehcle Sde Impact", to appear n Structural and Multdscplnary Optmzaton Journal,. Cho, K. K., and Youn, B. D., "Hybrd Analyss Method for Relablty-Based Desgn Optmzaton", n Proceedngs of ASME Desgn Engneerng Techncal Conferences, Pttsburgh, Pennsylvana, September. Du,., and Chen, W., "A Most Probable Pont-Based Method for Effcent Uncertanty Analyss", Desgn Manufacturng, Vol., o., pp. -,. Gu, L., Yang, R. J., Cho, C. H., Makowsk, M., Faruque, M., and L, Y., "Optmzaton and Robustness for Crashworthness," Internatonal Journal of Vehcle Desgn, Vol., o., pp. -,. Hesterberg, T., "Weghted Average Importance Samplng Defensve Mture Dstrbutons", Technometrcs, Vol., pp. -,. Kalagnanam, J. R., and Dwekar, U. M., "An Effcent Samplng Technque for Offlne Qualty Control", Technometrcs, Vol., pp. -,. Koch, P.., Yang, R. J., and Gu, L., "Desgn for S Sgma through Robust Optmzaton", to appear n Structural and Multdscplnary Optmzaton Journal,. Kodyalam, S., Yang, R. J., Gu, L., and Tho, C. H., "Large-Scale, Multdscplnary Optmzaton of a Vehcle System n a Scalable, Hgh Performance Computng Envronment", DETC/DAC-, n Proceedngs of ASME Desgn Engneerng Techncal Conferences, Pttsburgh, Pennsylvana, September. Mavrs, D. and Bandte, O., "A Probablstc Approach to Multvarate Constraned Robust Desgn Smulaton," SAE -,. Sahn, K. H. and Dwekar, U. M., "Better Optmzaton of onlnear Uncertan Systems (: A ew Algorthm for Stochastc Programmng usng Reweghtng through Kernel Densty Estmaton", to appear n Annual of Operatonal Research,. Slverman, B. W., Densty Estmaton for Statstcs and Data Analyss, Chapman & Hall, Boca Raton, USA,. Sobesk, J. S., Kodyalam, S., and Yang, R. J., "Optmzaton of Car Body under Constrants of ose, Vbraton, and Harshness (VH and Crash", Structural and Multdscplnary Optmzaton Journal, Vol., o., pp. -,. Stander,., "Crashworthness Technology Usng Response Surface Methodology and Massvely Parallel Programmng," Poster paper at Optmzaton n Industry-II Banff, Canada,. Yang, R. J., Tseng, L., agy, L., and Cheng, J., "Feasblty Study of Crash Optmzaton," ASME, Vol. -, pp. -,. Yang, R. J., Gu, L., Law, L., Gearhart, C., Tho, C. H., Lu,., and Wang, B. P., "Appromatons for Safety Optmzaton of Large Systems," Proceedngs of ASME Desgn Engneerng Techncal Conferences, September, Baltmore, Maryland,. Yang, R. J., Gu, L., Tho, C. H., Cho, K. K., and Youn B. D., "Relablty-Based Multdscplnary Desgn Optmzaton of a Full Vehcle Desgn", AIAA--, AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamcs, and Materals Conference, Denver, Colorado, Aprl -,.