Design Tolerance Optimization using LS-OPT

Transcription

1 Design Tolerance Opimizaion using LS-OPT Anirban Basudhar, Nielen Sander, Imiaz Gandikoa, Åke Svedin 2, Kaharina Wiowski 3 Livermore Sofware Technology Corporaion, Livermore, CA, USA 2 DYNAmore Nordic, Sweden 3 DYNAmore GmBH, Germany Inroducion In design opimizaion, i is ypical o minimize an objecive funcion, such as he cos of he design, while consraining oher responses. As a resul, he final design ofen lies a he limi of permissible behavior. However, a sligh perurbaion of he design or he loading condiions can lead o an inadmissible response. In addiion, he objecive funcion may be highly sensiive o perurbaions leading o a much worse value han he epeced opimum values. To avoid hese issues, reliabiliybased design opimizaion (RBDO) and robus opimizaion are ofen performed [,2]. Tolerance values play an imporan role in deermining he effec of uncerainies on he opimal design [3-5]. The olerance inervals define he variable probabiliy densiy funcion (PDF) bounds and, herefore, affec he failure probabiliy. In he absence of complee probabiliy densiy informaion, hey faciliae he search for a wors case design; an opimal design wih zero failure wihin he olerance inervals can be deermined. In his work, LS-OPT is used o perform olerance-based design opimizaion using a muli-level scheme [6]. The ouer level consiss of an opimizaion problem ha maimizes he olerance value such ha here is no failure wihin he olerance inerval (i.e. neligible probabiliy of failure). The nominal design parameers and heir olerance values are he opimizaion variables for his level; hus, each sample in he ouer level uniquely defines he nominal values and bounds of he variable probabiliy densiy funcions (PDFs). The probabiliy of failure for each sample is deermined using an inner level Mone Carlo analysis. This value is hen eraced as an ouer level response and is consrained o be close o zero during he opimizaion. The ouer level opimizaion is formulaed such ha a balance beween robusness and performance is acheved. For problems wih epensive analysis, e.g. crash anaysis, a wo sep approach is presened. For such problems, he firs sep involves he consrucion of high fideliy global meamodels for he sysem responses and saving heir formulae o a file. In he second sep, he previously consruced meamodels are used o replace he acual epensive funcion analysis. In oher words, once he global meamodels are consruced, he muli-level olerance scheme does no need any epensive funcion evaluaion. The olerance opimizaion mehodology is applied o opimize he hickness design parameers for a Chevrole ruck impac problem as well as he associaed olerances. The deails of he LS-OPT seup are illusraed hrough his design problem. The newly developed feaures and eniies of LS-OPT used in his work include muli-level opimizaion using LS-OPT sage, ransfer variable usage, eporing meamodel formulae o a file and paramerizaion of LS-OPT aribues such as disrbuion properies. A muli-objecive olerance-based opimizaion is performed o obain a radeoff. In he following secions, he olerance opimizaion mehod is eplained in greaer deail along wih some resuls. In Secion 2, he olerance opimizaion formulaion is provided. Secion 3 involves he he applicaion of he proposed mehodology o a Chevrole pickup ruck along wih he seps required for seing up he problem in LS-OPT. In Secion 4, resuls for he ruck opimizaion problem are presened. Finally, Secion 5 provides a summary of he work and some fuure work ha will help in furher sreamlining he seup in LS-OPT. 204 Copyrigh by DYNAmore GmbH

2 2 Tolerance Opimizaion Mehodology This secion presens he formulaion of he olerance opimizaion [3-5] problem solved in his work. The mehodology used in his work is summarized using Figure. In general, olerance opimizaion involves a large number of funcion evaluaions. Therefore, o avoid high compuaion cos for problems wih epensive analysis, he mehodology used in his work consiss of wo seps. Meamodels for he responses are consruced in he firs sep and hese compuaionally inepensive approimaions are hen used for he olerance opimizaion, which is performed in he second sep. Firs, in sep, a zero olerance problem is solved as a deerminisic single ieraion opimizaion using LS-OPT. The opimizaion problem can be formulaed as: min ( ).. ( ) 0, =,2,3 () h where he opimizaion variables are nominal variables. I should be noed ha he soluion of he above opimizaion formulaion may no be robus. However, he above problem is solved o achieve wo objecives - o obain a arge value of for sep 2, where a consrain is applied on, and o obain high fideliy meamodel approimaions of he responses ha are used o replace he epensive analysis in sep 2 olerance opimizaion. I is imporan o obain he response approimaions as replacemens of he epensive analysis, as olerance opimizaion involves a high number of funcion evaluaions. In sep 2, he olerance opimizaion is solved as a single or muli-objecive problem. This work consiss of a muli-objecive opimizaion wih wo objecive funcions. One of he wo objecives is a olerance scale facor, which is he raio of he olerance and nominal values, and he second objecive is he nominal value of he sep objecive funcion. I should be noed ha can be a vecor of muliple objecive funcions, bu is reaed as a scalar here for simpliciy. The opimizaion in sep 2 is se up as a wo-level problem in LS-OPT. The nominal design variables and he olerance value are opimized in he ouer level while he inner level compues he probabiliy of failure a any design alernaive. The failure probabiliies in he inner level are calculaed based on he previously consruced meamodel approimaions. The ouer level opimizaion is: ma, s.. {, f } g( ) 0 f ( ) ηf * [ δ, + δ ] [ δ, + δ ] where is he vecor of relaive olerance values or he olerance scale facors, δ is he vecor of absolue olerance values, is he opimal objecive funcion value for sep soluion and η is a facor greaer han. I should be noed ha he absence of failure wihin a large olerance inerval implies high robusness of he soluion, bu a very large olerance will also lead o loss of performance. Therefore, i is imporan o include he original objecive funcion as a consrain while maimizing he olerance. A design obained in his manner srikes a balance beween robusness and nominal performance. Figure shows he general overview of he mulilevel olerance-based muliobejcive opimizaion. Equaion (2) can also be wrien as: {, f } ma, s.. Pf (, ) Parge where ( g (, ) > 0 ),, P g (, ) * { P ( p > 0 ), P( f ( ) > ηf )} P f = L and P arge 0., (2) (3) 204 Copyrigh by DYNAmore GmbH

3 Sep : Deerminisic objecive funcion f minimizaion wih consrain(s) g < 0 min f s.. g ( ) ( ) 0 f and g meamodels Opimal f * f Sep 2: Mulilevel (wo-level) olerance opimizaion P f Level : Ouer level Opimizaion Conrol Variables:, { P( g( ma, s.. Level 2: Inner level Reliabiliy Assessmen Noise Variables:,Transfer Variables:, =, consrains : g,, f P f (, ) ( ) P arge, ~ 0 * ( ) ( f (, ) > ηf )} ) > 0), L, P g p (, ) > 0, P * ( ) 0, f (, ) ηf FIGURE Tolerance Opimizaion Summary 3 Tolerance Opimizaion Using LS-OPT - Design Problem and Seup The design problem discussed in his paper consiss of a muli-objecive design opimizaion formulaed o opimize he finie elemen (FE) model of he Chevrole C2500 pickup ruck developed by Naional Crash Analysis Cener (NCAC) [7]. The goal of he opimizaion process was o obain a ligher and robus design wihou compromising he crashworhiness characerisics of he vehicle model given by he inrusion disance and crash pulse responses. The hickness of a few pars seleced based on heir conribuionn o he overall crash energy absorpion were reaed as he design variables. A oal of nine pars were seleced and due o symmery in he design, he number of design variables was reduced o si. Figure 2 shows he FE model of he Chevrole C2500 pickup ruck and he design pars seleced for he opimizaion. FIGURE 2 FE model of Chevrole C2500 pickup ruck (lef) and he pars seleced for opimizaion (righ). The olerance opimizaion mehodology eplained in secion 2 was applied o he Chevrole C2500 pickup ruck. The following discussion perains o he seup of he olerance opimizaion of he vehicle model using LS-OPT. The opimizaion is performed in wo seps. The firs sep consiss of a deerminisic opimizaion and he olerance opimizaion is performed in he second sep. The second sep furher consiss of a wo-level (muli-level) seup as eplained below.

4 Sep : The deerminiic zero olerance opimizaion problem required for sep was formulaed as: min.. _ ( ) _ _ ( ) 0, =,2 _ ( ) 0 (4) h where _ is he oal mass of he seleced pars normalized by is baseline value, _ _ are he normalized design consrains given by he crash pulse responses and _ is he normalized design consrains given by he he inruion disance. These are required o be less han or a leas equal o heir baseline values. are he par hickness design variables ha need o be beween heir respecive lower and upper bounds. The opimizaion problem given by Equaion 4 is solved using a meamodel-based single ieraion ask available in LS-OPT. A relaively high number of samples (000) were used in his sep o obain high fideliy global meamodel approimaions for he responses. This, however, allows one o avoid many more simulaions in he poenially epensive second sep. The following figure shows he seup required in he LS-OPT ool for solving Sep. FIGURE 3 LS-OPT seup for Sep. Radial basis funcion (RBF) neworks were used o formulae he approimae equaions for he design responses using a oal of 000 sample poins seleced using he space filling sampling echnique. Through opimizaion, he oal scaled mass of he seleced pars was reduced o compared o he baseline value of. The single ieraion deerminisic opimizaion in sep serves wo purposes. Firs, he deerminisic opimal oal mass value, afer some relaaion, is used as a arge o consrain he mass during he olerance-based opimizaion in he second sep. This ensures robusness of he mass in he final design. In addiion, he mahemaical epressions of he RBF meamodels consruced in his sep are auomaically saved ino he DesignFuncions. file. These approimaions are used o replace he epensive LS-DYNA FE crash analysis during he olerance opimizaion in he second sep. I should be noed ha LS-OPT saves he approimaion formulae in Malab compaible forma. In his work, however, he formulae in he DesignFuncions file were manually convered o perl compaible forma using some simple modificaions. Sep 2: This sep involves muli-objecive opimizaion wih oal scaled mass of he design pars and he olerance scale facor as he objecives. The opimizaion in sep 2 is se up as a wo-level (ouer and inner) problem in LS-OPT. The ouer level consiss of a direc simulaion-based muliobjecive opimizaion wih a olerance scale facor and he nominal hickness of he design pars as he design 204 Copyrigh by DYNAmore GmbH

5 variables. The analysis sage of he ouer level consiss of an LS-OPT sage ha represens he inner level Mone Carlo analysis. Thus, for each ouer level sample corresponding o a unique nominal variable value and olerance value, an inner level Mone Carlo analysis is conduced o compue he corresponding probabiliies of failure. The ouer level LS-OPT sage iself represens an inner level LS-OPT seup ha conains a userdefined sage wih a perl scrip as he inpu file. The perl scrip compues he response values for he ruck impac and is based on he sep meamodel approimaions. An eample of a simplified perl scrip obained by modifying he DesignFuncions file of he sep is shown in Figure 4. The scrip conains 9 parameers ha are auomaically deeced in he inner level seup. However, hree of hese are dependen on he oher variables. The remaining 6 parameers are defined as noise variables. FIGURE 4 Simplified sep 2 inner level perl inpu file obained by modifying he DesignFuncions daabase. The acual inpu file conains muliple response approimaions. The probabiliy densiy funcions (PDF) of hese variables needs o be parameerized wih respec o he ouer level variables, as each ouer level sample represens a unique variable PDF. Therefore, in addiion o he variables auomaically deeced in he perl scrip, he nominal design variables and he olerance scale facor are manually added o he inner level seup as ransfer variables. These ransfer variables added o he inner level.lsop seup file, which is also he ouer level LS-OPT sage inpu file, are auomaically deeced as parameers in he ouer level. These are se as opimizaion variables in he ouer level, bu are reaed as consan parameers in he inner level. For each ouer level sample, he corresponding ransfer variable values are subsiued in he inner level seup file. The noise variable PDFs in he inner level are parameerized using he ransfer variables or heir dependens using he "&" operaor. The design variables for boh inner and ouer level LS-OPT analysis are shown in Figure 5 and Figure 6. For each sample of he ouer level, he inner level LS-OPT analysis consiss of a Mone Carlo analysis of he Chevrole C2500 pickup ruck o deermine he probabiliy of eceeding he crash response bounds. However, i is noeworhy ha he epensive ransie dynamic FE crash analysis is replaced by meamodel approimaions wihin he perl inpu file. This allows significan savings in he compuaional ime. The inner level reliabiliy analysis for probabiliy of failure calculaion is se up as a meamodel-based Mone Carlo analysis wih 00 space filling sample poins (Figure 7). 204 Copyrigh by DYNAmore GmbH

6 FIGURE 5 Inner level variable seup for sep 2 (lef). Parameerizaion of PDF using & operaor (righ). FIGURE 6 Ouer level sage seup (lef) and variable seup (righ) FIGURE 7 LS-OPT seup for he inner loop meamodel-based Mone Carlo analysis.

7 The four design consrains defined in he Mone Carlo analysis are he hree crash responses (inrusion disance and crash pulses) and a consrain for he oal mass of he design pars. The crash responses are required o be less han or a leas equal o he baseline values. Since he opimum value of he oal mass in sep was , a relaed value of 0.9 was given as he upper bound for he oal mass. I is imporan o inroduce his addiional mass consrain in order o ensure a minimum level of mass reducion wihin he enire olerance inerval. Figure 8 shows he consrain definiion of he inner level Mone Carlo analysis. FIGURE 8 Consrain definiions for inner loop Mone Carlo analysis. The resul of he Mone Carlo analysis is a lis of probabiliies of failure of he crash responses given by he probabiliies of eceeding he consrain bounds. Since each inner level Mone Carlo analysis corresponds o one sample of he ouer level, he probabiliies of failure of he crash responses are defined as he responses of he ouer level LS-OPT sage. The values of he probabiliies can be obained from he lsop_repor file generaed during he inner level Mone Carlo analysis. Since lsop_repor is a e file, GenE response eracion ool of LS-OPT was used o erac he values of probabiliy of failure. Apar from he probabiliies of failure, he mean oal mass of he pars was also eraced from he inner level. Once he responses of ineres were defined for he ouer level, a muliobjecive opimizaion o maimize he design olerance and minimize he mean oal mass was formulaed as: ma, s.. {, scaled _ mass( ) } P( scaled _ mass P( scaled _ disp ( ) P( scaled _ sage_ pulse P( scaled _ sage2 _ pulse ( ) > ) > 0.9) P P ( ) ( ) arge arge > ) P > ) P arge arge where is he olerance scale faco scaled _ mass is he nominal mass a. The negaive sign indicaes ha he value of he scaled mass is minimized during he opimizaion process. is he probabiliy of failure of he crash responses, evaluaed using he inner loop Mone Carlo analysis. A robus soluion is desired boh wih respec o he objecive and he consrains, such ha any variaion wihin he olerance limis does no lead o failure. To alleviae he effec of uncerainies, he olerance scale facor is maimized. All he hicknesses are assumed o have he same olerance scale facor or percenage olerance. is, herefore, a scalar. The nominal mass is simulaneously minimized, and he problem is solved as a muli-objecive opimizaion o obain a rade-off beween nominal mass and he olerance. Figure 9 shows he LS-OPT ouer level objecivee funcion and consrain definiions. The seup of he ouer level ask, which is se as direc opimizaion, is shown in Figure 0. or and ( ) (5)

8 FIGURE 9 LS-OPT seup for ouer loop objecives (lef) and design consrains (righ). FIGURE 0 LS-OPT seup for ouer loop design opimizaion. 4 Resuls and Discussion The muli-objecive opimizaion problem given in equaion 5 and shown in Figure 0 was solved using he direc simulaion based opimizaion ask of he LS-OPT wih geneic algorihm as he opimizaion echnique. A populaion size of 00 designs per ieraion was considered wih a maimum of 50 ieraions. Thus, 00 ouer loop samples were analyzed during each ouer level ieraion, wih each sample consising of is own inner level Mone Carlo analysis. A Pareo opimal fron obained using he NSGAII mehod available in LS-OPT was used o deermine he radeoff beween he objecive funcions, i.e. olerance scale facor vs. he nominal mass as shown in Figure and Figure 3. FIGURE Tradeoff beween he objecive funcions over ieraions.

9 The opimizaion process reached convergence a 50 ieraions as shown by he change in hyper volume and spread over ieraions in Figure 2. The final Pareo Opimal Fron consising of he nondominaed soluions is shown in Figure 3. Since he evaluaion of crash responses in he inner loop is based on he approimae analyical equaions obained from sep, all he non-dominaed design poins of he Pareo Opimal Fron obained from he muli-objecive opimizaion were analyzed hrough LS-DYNA FE analysis o deermine he rue nominal crash responses. Figure 3 shows boh he meamodel-based and simulaions-based Pareo Opimal Fron. The simulaion-based Pareo Opimal Fron shows ha he mos Pareo Opimal poins of he muli-objecive opimizaion have saisfied he crash response design crieria. A few infeasible design poins in he simulaion-based Pareo fron are due o he error in predicion accuracy of he approimae analyical equaions obained in sep ha could arise due o some inheren noise in he problem. An ineresing hing o noe, however, is ha he infeasibiliy of nominal design is observed only for some of hose Pareo Opimal poins ha have low olerance values. This is an epeced resul as a higher olerance miigaes some of he effecs of approimaion/soluion process inaccuracies also. This furher underscores he imporance of olerance-based design opimizaion compared o deerminisic opimizaion. FIGURE 2 Hyper volume change (lef) and change in spread (righ) of he muli-objecive opimizaion showing convergence. FIGURE 3 Meamodel-based (lef) and simulaion-based (righ) rade-off beween scaled mass. olerance and The Pareo Opimal Fron (Figure 3) shows ha a minimum mass soluion of approimaely scaled mass is obained, being he baseline mass of he seleced pars. This corresponds o a vehicle mass reducion of 22.8 kg. Among he Pareo Opimal poins obained, he mass reducion varies beween 7.54 kg and 22.8 kg. A maimum olerance soluion of abou 0.03 or 3% olerance is obained. A knee is observed in he radeoff plo and i is seen ha approimaely 2% olerance can be achieved wihou significan increase in he mass; he corresponding increase in masss is a mere 0.8 kg.

10 5 Summary and Fuure Work A mehodology for applicaion of LS-OPT o mulilevel olerance-based opimizaion is presened. I has been sucessfully applied o opimize he FE model of Chevrole C2500 pickup ruck. A se of Pareo Opimal soluions is obained wih a rade-off beween he olerance and he nominal mass of he vehicle. Significanly higher robusness is achieved as a resul of considering olerances. A maimum olerance of 3% has been obained using he opimizaion echnique. A he same ime, a mass reducion varying beween 7.54 kg and 22.8 kg has also been achieved compared o he baseline. In order o reduce he overall compuaional cos, he epensive ransien dynamic finie elemen analysis of he ruck was replaced by simpler analyical equaions obained using he meamodeling echniques. The usabiliy of LS-OPT can be improved using wo enhancemens o he curren sofware. Firs, he user-defined solver ha uses analyical formula previously generaed using LS-OPT will be replaced by a meamodel impor feaure in he near fuure. In addiion, i is also possible o inroduce a new ask for olerance opimizaion ha will eliminae he need for a mulilevel seup, replacing i wih a simpler single level seup. 6 References [] Youn BD, Choi KK. "Selecing probabilisic approaches for reliabiliy-based design opimizaion". AIAA Journal 2004;42():24 3. [2] Sanchez, S.M. "Robus design: seeking he bes of all possible worlds". In Proceedings of he 2000 Winer Simulaion Conference, eds Joines JA, Baron RR, Kan K, and Fishwick PA , Insiue of Elecrical and Elecronic Engineers. Piscaaway, NJ. [3] Schjaer-Jacobsen, H., Madsen, K., "Algorihms for wors-case olerance opimizaion," IEEE Transacions on Circuis and Sysems, 979; 26(9): [4] Arenbeck H., Missoum S., Basudhar A. and Nikravesh P. "Reliabiliy-Based Opimal Design and Tolerancing for Mulibody Sysems Using Eplici Design Space Decomposiion". J. Mech. Des. 200, 32(2). [5] Jordaan J. P., Ungerer C. P. "Opimizaion of Design Tolerances hrough Response Surface Approimaions". J. Manuf. Sci. Eng. 2002; 24: [6] Sander N., Basudhar A., Gandikoa I., Svedin Å., Wiowski K. "LSOPT: Saus and Oulook". LS- DYNA Users Forum, Bamberg, Germany, Ocober 6-8, 204 [7] Naional Crash Analysis Cener. Finie elemen model archive, hp:// Nov Copyrigh by DYNAmore GmbH