Adaptive multi-point sequential sampling methodology for highly nonlinear automotive crashworthiness design problems

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1 11 th World Congress on Structural and Multdscplnary Optmsaton 07 th -12 th, June 2015, Sydney Australa Adaptve mult-pont sequental samplng methodology or hghly nonlnear automotve crashworthness desgn problems Slang Zhang*, Zhengchao Song, Guohong Sh, Rongyng Qu Pan Asa Techncal Automotve Center, Shangha, P.R. Chna, 1. Abstract Automotve crashworthness desgn s a hghly expensve and non-lnear problem. In metamodel-based crash desgn problem, the predcton error o the metamodel may nduce a local or a wrong optmum. In the past ew years, the mult-pont objectve-orented sequental samplng methods have been demonstrated an ecent way to mprove the ttng uracy and nd the true optmum. However exstng nllng crtera are restrcted to specy the number o the sequental samples obtaned n each teraton. It s not practcal or complex engneerng desgn problems. In ths paper, a new adaptve mult-pont sequental samplng method s developed. The sequental sample sze s determned by the predcton states o the ttng metamodels. To demonstrate the benets, the new proposed method s appled to a hghly nonlnear crashworthness desgn problem. Results show that the proposed method can mtgate the eect o the predcton error, and more ecently denty the crashworthness desgn soluton compared to the conventonal approach. 2. Keywords: Metamodel-based optmzaton, objectve-orented sequental samplng method, adaptve mult-pont strategy, crashworthness desgn. 3. Introducton Fnte element (FE) smulatons have been a useul tool or replacng the physcal tests n crashworthness desgn. However, hgh delty FE models are oten computatonally ntensve, takng hours and even days to complete one computaton cycle. A common approach to address ths challenge s to employ metamodelng method predctng the smulaton responses. The metamodel provdes a cheap-to-run surrogate model to approxmate the complex smulatons [1]. The eectveness o derent metamodelng technques vary based on the derent modelng crtera, amount o avalable samples, and the behavor o the smulaton responses [2]. However n complex engneerng optmzatons, the prmal challenge s how to determne the number o samples requred and how to allocate samples. Comparng to tradtonal one-stage DOE methods (Orthogonal expermental desgn, Unorm Desgn, Latn Hypercube Desgn et. al), sequental samplng methods have been dented as a more ecent strategy. In prevous nvestgatons, the sequental samplng crtera can be classed nto two categores: model-orented and objectve-orented. The model-orented methods ocus on the goal o creatng a globally urate metamodel, whle the objectve-orented sequental samplng strateges have been demonstrated to have a hgher ecency o ndng the global optmum [3]. The most wdely used objectve-orented sequental samplng crtera, Ecent Global Optmzaton (EGO) algorthm, s rst developed by Jones [4]. The EGO method only nds one pont n one teraton, resultng n many sequental cycles beore reachng convergence. To take advantage o the parallel computaton capablty and save the total amount o teratons, a mult-pont samplng strategy s needed. Schonlau [5] dened the concept o mult-pont sequental samplng method. Vana [6] extended the Probablty o Improvement uncton to nclude multple ponts at the same tme. Zhu and Zhang [7] developed a new double-loop strategy to nd q samples va Krgng Belever method. However exstng nllng crtera are restrcted to specy the number o the sequental samples obtaned n each teraton. It s not practcal or complex engneerng desgn problems. In ths paper, a new adaptve mult-pont sequental samplng method s developed. The ollowng secton revews the concept o mult-pont sequental samplng methods, and ntroduces the proposed adaptve strategy. A new nllng crteron s developed to determne whether there s a need to nd one more sample. In Secton 5, to demonstrate the eectveness, the proposed adaptve mult-pont sequental samplng method s appled to an automotve crashworthness desgn problem. Fnally, the dscussons and concluson are summarzed n Secton 6. 4: Adaptve mult-pont sequental samplng methodology or complex engneerng optmzaton 4.1. Mult-pont Sequental samplng method or constraned optmzaton problem For a constraned engneerng optmzaton problem, the mathematcal ormulaton can be dened as: mn : y( s.t : g ( β, 1, 2,..., k (1) 1

2 where x s the desgn varables; y and g represent the objectve response and constrant responses; β s the th constrant threshold. When the objectve and constrant responses are replaced by metamodels ( yˆ ( and g ˆ( ), consderng the metamodelng mperecton, the predcton error aects the optmzaton uracy and constrant easblty, especally n hgh-dmensonal and hghly-nonlnear engneerng problems. The objectve-orented strategy can spread new samples to balance the optmzaton exploraton and uracy mprovement. Evaluatng the eects o predcton error on the objectve responses y ˆ( and the constrants g ˆ (, the generalzed expected mprovement uncton (GEI) o a constraned optmzaton problem can be dened as [7]. max : GEI( REI( EV( (2) ymn yˆ( ) ymn yˆ( ) where : REI( ) ( ymn yˆ( )) x x x x Φ + σ y φ y ( ) y ( ) σ x σ x k gˆ ( ) EV( ) β x x Φ 1 g( ) σ x where k s the number o constrant responses; y mn s the mnmal objectve response o the sampled ponts; yˆ ( and gˆ ( ndcate the predcted value o the objectve and constrant response respectvely; φ( and Φ ( represent the probablty densty uncton and cumulatve densty uncton o a standard normal dstrbuton. It s an ecent way to choose the global and quas-local optmums o the GEI uncton as the sequental [8]. It should be noted that exstng mult-ponts methods are developed to obtan a constant number q o sequental samples. But n real engneerng problems, t s dcult to guess how many samples are needed n each cycle. A complex problem wth a small q stll needs many teratons, whle wth an over large q may nduce ntensve smulatons. The ollowng secton wll ntroduce a new adaptve mult-pont sequental samplng method. The number q n each teraton s decded by the predcton states o the optmzaton problems adaptvely Adaptve mult-pont sequental samplng method or complex engneerng optmzaton problem The nllng crteron s the most mportant actor n sequental samplng process. In order to mprove the sequental samplng ecency, the weghted contrbuton o a new pont s developed to replace the conventonal generalzed Expected Improvement uncton. The moded qgei uncton s ormulated as: GEI, q 1 m (3) qgei GEI, q > 1 qgei1 where q s the number o the sequental samples obtaned n each teraton; qgei 1 represents the qgei value o the 1 st sequental sample (q 1); m s the power number. Ater the rst pont s ound, the power uncton m o GEI downplays the relatve contrbutons o the new ponts. As shown n Fgure 1, when m 1, the qgei uncton represents the relatve GEI value. As the m value ncreases, the regons wth small GEI wll be dmnshed. I m s set to 2, the pont where the GEI value s less than 10% o the qgei 1 wll be neglected. I m s set to 4, the pont where the GEI value s less than 35% o the qgei 1 wll be neglected. Usng the qgei uncton n the sequental samplng process, more eorts wll be made n the regons wth hgher contrbuton. 1/ k Fgure 1: The nluence o the m value Fgure 2: The lowchart o the proposed adaptve n the qgei uncton mult-pont sequental samplng process The true soluton s lkely to be near the neror optmum o the nllng crteron, rather than the global optmum wth the largest EI unctonerror! Reerence source not ound.. In ths paper, the concept o the Krgng Belever strategy s adopted n the adaptve sequental samplng method. Derent rom any strateges n prevous studes, the Krgng Belever strategy treats the predcted response as the true response durng the sequental samplng process. The sequental samples obtaned n each teraton are allocated to the global and quas-local optmums. 2

3 The lowchart o the proposed method s shown n Fgure 2. Step 1: Generate a set o samples, and extract the smulated responses o these N tranng samples. Step 2: Update the DOE matrx o the n samples (N ntal DOE samples and all sequental samples). Step 3: Based on the true observatons at x n and predcted responses at x q, the Krgng models o the objectve responses y ˆ( and constrant responses c ˆ ( are constructed. x q s the sequental samples n the q th teraton. Step 4: Maxmze the nllng crteron qgei and nd the next sequental sample x q. Step 5: Check the convergence. I the 1 st stoppng crteron s satsed, go to Step 8. Step 6: Evaluate the predcted response y ˆ( and c ˆ ( o the newly added pont x q, and set q q+1. Step 7: Add the sample x q nto the tranng DOE samples. Step 8: Check the convergence. I the 2 nd stoppng crteron s satsed, the sequental samplng process s converged and goes to Step 10. Step 9: Smulate the obtaned samples x q by FE models, and add these ponts nto the tranng samples n. Step 10: Ater the sequental samplng process s termnated, the nal desgn soluton wll be ound. 5. Engneerng applcaton n a crashworthness desgn problem In ths secton, the benets o the proposed adaptve sequental samplng method are demonstrated n a complex crashworthness desgn example. Two derent strateges are consdered n ths secton: l Conventonal mult-pont sequental samplng method wth a constant q (GEI_cq): the sequental samplng method ound q samples n each teraton. The sequental nllng crteron s dened by Eq. (2). l Proposed adaptve mult-pont sequental samplng method (GEI_aq): the sequental samples ound n each teraton are determned by the predcton states, and the nllng crteron s dened by Eq. (3) Crashworthness desgn applcaton In the automotve crashworthness desgn, FE smulatons are used to predct crash perormances. Snce ull sze automotve smulaton models are computatonally expensve, metamodelng technques are wdely utlzed to buld surrogate models. In ths secton, a rontal mpact desgn problem s utlzed to demonstrate the eectveness o the proposed adaptve mult-pont sequental samplng method n real engneerng desgn. The FE model s shown n Fgure 3. The average mesh sze s 5 mm. For the rontal mpact nvestgaton, the regulatons and test conguratons n the Chna Natonal Crash Legslaton o rontal mpact (GB ) are ollowed. Consderng the stran rate senstvty o the sheets n hgh speed mpact, stress versus plastc stran curves under derent stran rates are dened n a load table. These curves are obtaned rom physcal tenson experments. Fgure 3: Full-sze nte element model o rontal mpact smulaton The rontal sde ral s the crtcal part n absorbng rontal mpact, as shown n Fgure 4. The sheet gauge and the component shape are mportant or absorbng the mpact energy. Consderng the symmetry o the ral structure, 11 sheet gauges and 16 shape varables are chosen as the desgn varables, as shown n Table 1. In ths crashworthness desgn problem, the Eectve Acceleraton s dened as the objectve response, whle ten crash perormances (Ecency g e, structural Intrusons g nt1 ~ g nt9 ) and mass g M, are treated as the constrants. (a) Gauge varables (b) Shape varables Fgure 4: Desgn varables o the crashworthness desgn 3

4 Table 1: Desgn varables o the automotve crashworthness desgn Gauge DVs Shape DVs Component Varables DV Orgnal/mm LB/mm UB/mm Upper Ren. 1 Dv1 x Upper Ren. 2 Dv2 x Upper Ren. 3 Dv3 x Upper Ren. 4 Dv4 x Frontal sde ral outer Dv5 x Lower Ren. 1 Dv6 x Lower Ren. 2 Dv7 x Lower Ren. 3 Dv8 x Lower Ren. 4 Dv9 x Frontal sde ral nner 1 Dv10 x Frontal sde ral nner 2 Dv11 x Upper Ren. 1 SP1 Dv12 x Upper Ren. 1 SP2 Dv13 x Upper Ren. 2 SP1 Dv14 x Upper Ren. 2 SP2 Dv15 x Upper Ren. 3 SP1 Dv16 x Upper Ren. 3 SP2 Dv17 x Upper Ren. 4 SP1 Dv18 x Upper Ren. 4 SP2 Dv19 x Lower Ren. 1 SP1 Dv20 x Lower Ren. 1 SP2 Dv21 x Lower Ren. 2 SP1 Dv22 x Lower Ren. 2 SP2 Dv23 x Lower Ren. 3 SP1 Dv24 x Lower Ren. 3 SP2 Dv25 x Lower Ren. 4 SP1 Dv26 x Lower Ren. 4 SP2 Dv27 x Sequental mprovement and optmzaton results All structural perormances are nterpolated by Krgng method. The optmzaton ormulaton s dened as: mn : s. t. : c c g / β c11 gm / β11 where β represents the th constrant target. Based on 180 samples generated by the Latn Hypercube method, the c metamodel-based optmzaton results are shown n Table 2. But when the optmzaton soluton s conrmed by the FE smulaton model, there has a large dscrepancy between predcted and smulated objectve response. And two constrant responses (c 10, c 11 ) volate the desgn lmts. The predcton error msleads to nd an neasble soluton. In order to mtgate the predcton error, the mult-pont sequental samplng method s used. Table 2: The optmzaton results based on ntal DOE samples Opt. Result Target Krgng-based Smulaton conrmaton Objectve mn c c c Constrants c c c c c g g nt1 nt9 / β 1 / β e 9 10 (4) 4

5 c c c To demonstrate the benets o the proposed method, the conventonal mult-pont sequental samplng strategy GEI_cq wth a constant q s also adopted n ths example, ormulated as: max : GEI_cq( where : REI( REI( EV( ( ˆ ( ) Φ 1/11 mn ˆ ( ) x + σ ( ) φ σ x mn ˆ ( ) x σ ( ) x 11 1 cˆ ( ) x EV( Φ 1 ( ) σ c x In ths nllng crteron, mn represents the mnmal objectve response value o the sampled ponts, and q 5 samples are newly added n each teraton. The lmt crteron s utlzed n ths crashworthness desgn problem: when GEI_cq 1 s less than 1%, the sequental samplng process wll be termnated. The GEI_cq method s converged ater 5 teratons. The optmzaton soluton s obtaned based on ntal tranng samples and the newly added samples. The soluton s conrmed by FE smulaton. Fgure 5 llustrates the convergence hstory o the GEI_cq method. The objectve response and two crtcal constrant responses c 10 /c 11 are montored. The objectve response reduced rom 1.00 to 0.92, achevng 8% mprovement, whle two crtcal constrant response c 10 and c 11 are successvely approachng to the desgn target 1. It demonstrates that the mult-pont sequental samplng method GEI_cq can mtgate the predcton error n both objectve response and all constrant responses, and ensure the uracy and easblty o the desgn soluton. (5) (a) Objectve response (b) Constrant responses c 10 and c 11 Fgure 5: Convergence hstory o the GEI_cq method The newly proposed adaptve mult-pont sequental samplng method GEI_aq do not need to dene a number q, and can nd a proper amount o sequental samples based on the predcton states o the ttng models. The nllng crteron o the proposed method s dened as: GEI _ cq, 1 max : GEI _ aq (6) 2 aq 1, > 1 ( GEI _ cq / GEI _ ) Smlar to the GEI_cq method, when the GEI_aq 1 value s less than 1%, the sequental mprovement process termnates. The convergence hstores o the proposed GEI_aq method are shown n Fgure 6. The conrmed objectve response reduced rom 1.00 to 0.91, whle two crtcal constrants satsy the desgn requrements. The FE smulated results shows that the proposed GEI_aq can mprove the objectve response, and ensure the easblty o two crtcal constrants c 11 and c 11. (a) Objectve response (b) Constrant responses c 10 and c 11 Fgure 6: Convergence hstory o the proposed GEI_aq method The sequental samples obtaned by these two methods are compared n Fgure 7. In the 1 st sequental teraton, the 5

6 GEI_cq wth a constant q explored and ound 5 new samples to mprove the ttng states. But when the proposed adaptve method GEI_aq s used, 8 samples are allocated n the desgn space. It demonstrates that based on ntal 180 tranng samples, the ttng responses o the crashworthness desgn problem has large predcton error, and the number q used n the GEI_cq method s not enough. Ater the 1 st teraton, the nterpolaton uracy o the crash responses has been mproved. Fewer ponts are needed n the next teratons. The conventonal GEI_cq method wth a constant q allocated more and more samples on the ponts wth lower contrbuton. In summary, the proposed method s converged n the 4 th teraton and 16 sequental samples are newly added. Comparng to the conventonal GEI_cq method, the proposed strategy converge to the true crashworthness soluton aster. It demonstrates a hgher ecency n the complex engneerng desgn problem. Fgure 7: The samples obtaned by two derent sequental samplng methods 6. Dscussons and conclusons A ew observatons are made: l The proposed adaptve mult-pont sequental samplng method can decde the sample sze by the predcton states o the desgn responses. It s benecal or the problems where smulaton models are computatonally expensve and the parallel computng ablty can be utlzed to calculate many smulatons at the same tme. l The crashworthness desgn s a hghly nonlnear problem. It s ound that comparng to conventonal sequental samplng method, the proposed adaptve strategy not only can mprove the objectve response (Eectve Acceleraton ) and ensure the easblty o ten crash constrant responses, but also can converge to the true soluton n ewer teratons. It demonstrates the eectveness and the ecency o the newly proposed method. 7. Reerences [1] G.G. Wang and S. Shan, Revew o metamodelng technques n support o engneerng desgn optmzaton, Internatonal Journal o Mechancal Desgn, 129(4), , [2] R. Jn, W. Chen and T.W. Smpson, Comparatve studes o metamodelng technques under multple modellng crtera. Structural and Multdscplnary Optmzaton, 23(1), 1-13, [3] S.K, Chen, Y. Xong and W. Chen, Multresponse and multstage metamodelng approach or desgn optmzaton. AIAA Journal, 47(1), , [4] D.R. Jones, M. Schonlau and W.J. Welch, Ecent global optmzaton o expensve black-box unctons. Journal o Global Optmzaton, 13(4), , [5] M. Schonlau, Computer experments and global optmzaton. Ph.D. dssertaton, Unversty o Waterloo, Waterloo, [6] F.A.C. Vana and R.T. Hatka, Surrogate-based optmzaton wth parallel smulatons usng the probablty o Improvement, 13th AIAA/ISSMO Multdscplnary Analyss and Optmzaton Conerence, Fort Worth, USA, September 13-15, [7] P. Zhu, S.L. Zhang and W. Chen, Mult-pont objectve-orented sequental samplng strategy or constraned robust desgn, Engneerng Optmzaton, 2014, DOI: / X [8] W. Ponweser, T. Wagner and M. Vncze, Clustered multple generalzed expected mprovement: A novel nll samplng crteron or surrogate models, IEEE Congress on Evolutonary Computaton, Hong Kong, P.R. Chna,