BOTTOM STRUCTURE FOR DutchEVO CAR: FORMULATION OF THE PROBLEM AND THE ADJUSTMENT OF THE OPTIMIZATION SYSTEM

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1 INTERNATIONAL DESIGN CONFERENCE - DESIGN 2002 Dubrovn, May 14-17, BOTTOM STRUCTURE FOR DutchEVO CAR: FORMULATION OF THE PROBLEM AND THE ADJUSTMENT OF THE OPTIMIZATION SYSTEM Natala S. Ermolaeva, Krll G. Kavelne and Jan L. Spoormaer Keywords: lghtweght desgn, optmal desgn, mechanstc approxmaton 1. Introducton The DutchEVO project has been ntated as a part of Delft Interfaculty Research Cooperaton (DIOC) program Smart Product Systems n order to stmulate nnovatve, multdscplnary research. The purpose of the project s to develop a nowledge base for sustanable product desgn. A sustanable car, n accordance to DutchEVO project tas, means envronmentally frendly, affordable, lghtweght car satsfyng all current and/or future legslatons on safety, emssons, recyclablty etc. The methodology of lghtweght desgn s expected to be the most valuable approach to reach the goals of the project. The applcaton of a structural optmzaton technque wth respect to the mnmum mass drectly supports ths methodology. One of the desgn possbltes for the DutchEVO car, whch s consdered as advantageous for the car structure, helpng to solve the safety lghtweght desgn conflct, s the rased floor concept [Kanter de, Vlot, Kandachar and Kavelne 2001]. Ths concept was developed later nto the type of a platform structure for the car body, when the bottom part of the car carres the man loads. Accordng to automotve standards [Fenton 1998], a car body desgn s based, ntally, on bendng and torsonal stffness as the man requrements concernng statcal loads (crashworthness s outsde the scope of the current sub-project). The optmzaton problem s formulated here for szng optmzaton of the DutchEVO car bottom structure wth respect for mnmum mass. The structural optmzaton system developed earler [Ermolaeva and Spoormaer 2001], based on the Multpont Approxmaton method wth Response Surface fttng (MARS) [Toropov 1989] and MSC.Marc FEA code, s adjusted here to be appled to the desgn of the consdered structure. Problem dependent nterfaces need to be programmed n order to connect structural optmzaton and FE analyss programs. In current paper, bendng stffness, strength and structural stablty (non-buclng behavour) constrants were consdered as the frst step of system applcaton. Torsonal stffness requrement wll be ncluded later, after the proper adjustment of the system. The possblty to use dfferent materals for the bottom structure s evaluated. Here we consder conventonal structural materals and two compostes wth synthetc and natural fbres. 2. Structural and fnte element model The concept desgn of the DutchEVO car suggests the bottom structure to be constructed of panels and stffeners, latter formng some nd of a frame on whch panels are fxed (Fg. 1). The frame conssts of cross-members between two renforced sde rocer beams and two bacbone profles. It s supposed to be made of a wrought alumnum alloy. The panels can be of an alumnum alloy too, but also there INDUSTRIAL SOLUTIONS 627

Structure of the bottom part of DutchEVO car: 1 vertcal and 2, 3, 4 horzontal components of the sde rocer beams; 5, 6 components of the lower and 7, 8 upper panels of the floor; 9, 13 longtudnal and

2 s a possblty to apply sandwch or composte panels nstead. The compostes renforced by natural or synthetc fbres can reduce weght essentally. We also consder steel as a conventonal materal for automotve body structures for comparson and for the adjustment of the optmzaton system. Fgure 1. Structure of the bottom part of DutchEVO car: 1 vertcal and 2, 3, 4 horzontal components of the sde rocer beams; 5, 6 components of the lower and 7, 8 upper panels of the floor; 9, 13 longtudnal and 10, 11, 12 transverse stffeners; 14 bacbone profles The FE model s presented n Fg. 2. It was tested for the appled desgn load n the mddle of the span between hnges used as the supports. In ths model, we use the four-node blnear thc-shell element from MSC.Marc standard element lbrary [MSC.Marc Volume B: Element Lbrary, Verson 2000]. A lnear mechancal analyss s performed. Stresses and strans are calculated n each of the fve layers of the element and n four ntegraton ponts, wthn the plane of each layer. We also use the buclng opton (va egenvalue analyss) to obtan the frst fve buclng modes and the correspondng crtcal load. 3. Optmzaton problem 3.1 Formulaton A structural optmzaton problem for the gven bottom structure of the DutchEVO car s formulated here as follows: mnmze the mass of the structure under the constrants of bendng stffness, strength and buclng; the structural model along wth the supports and the desgn load s presented n Fg. 1 and Fg. 2; (the requrement for the torsonal stffness wll be ncluded later, frst the optmzaton system should be adjusted to the gven model) desgn varables are the thcnesses of all plan structural elements,.e. the components of the lower and upper panels and flats formng the cross-members (see Fg. 1); sde constrants are lmted from below based on senstvty analyss, and from above based on manufacturablty of polymers. 628 INDUSTRIAL SOLUTIONS

3 Fgure 2. FE model of the bottom structure of DutchEVO car. P - desgn load; H hnges The mathematcal formulaton of ths optmzaton problem n general form can be wrtten as follows: Mnmze subject to 0 N mn, x R F (1) ( ) 1, j = 1,..., M and A x B, = 1 N Fj x,..., (2) where x s a vector of desgn varables, F 0 s an objectve functon, F j (j = 1,,M) are constrant functons and A, B (I = 1,,N) are sde lmts. In gven formulaton M = 3 and N = 14. Soluton of the optmzaton problem s an teratve process that nvolves repettve evaluatons of the objectve and constrant functons. The objectve functon s expressed va the volume of the fnte elements. Ths volume s calculated by means of a user subroutne based on nodes coordnates. Constrants are specfed as follows: F ( ) δ f max δ 1 x = (3) δ 0 where d max (x) s the maxmum node deflecton n the drecton of the desgn load P; f d s the factor of safety for bendng stffness; d 0 s the allowed deflecton; F σ f eq,max σ 2 = (4) σ f where s eq,max (x) s the maxmum equvalent stress under the desgn load chosen from all ntegraton ponts and layers of all fnte elements n the model; f s s the factor of safety for bendng strength; s f s the stress at falure (meanng of stress at falure depends on the type of materal [Ashby 1999]); F = Pf buc (5) 3 Pcr where f buc s the factor of safety for buclng mode; P cr (x) s the lowest value of crtcal load. The values of d max (x), s eq,max (x) and P cr (x) result from the mechancal behavour analyss n each desgn pont. INDUSTRIAL SOLUTIONS 629

4 3.2 Method The Multpont Approxmaton method based on Response Surface analyss (MARS) [Toropov 1989] was used as the structural optmzaton tool. Ths optmzaton technque s based on approxmatons of objectve and constrant functons n specally or randomly dstrbuted ponts of a desgn space. The random plan of numercal experments gves an advantage to perform a contnuous procedure even f there s no soluton n one or several ponts of the desgn space. Ths stuaton often occurs durng the FEA when structure bucles and FE program termnates wth wrong ext number. The MARS technque sps ths pont and searches for the nearest pont where the soluton exsts. Accordng to the approxmaton concept of the method the orgnal functons (1) and (2) are replaced by approxmate ones that consderably reduce computatonal tme. Instead of the orgnal optmzaton problem a successon of smpler approxmated sub-problems, smlar to the orgnal one, s formulated: mnmze subject to N ( x, a ) mn, x R ~ F (6) 0 ~ F j (, a ) 1, j = 1,..., M and A x B, A A, B B, = 1,..., N x (7) The current sub-optmal pont x s consdered as a startng pont of the next ( + 1) teraton. Each functon F ~ n (6) and (7) s an explct approxmaton; a s a vector of tunng parameters; A and B are move lmts defnng the range of applcablty of the approxmatons. The weght least-squares mnmzaton problem s to be solved to determne the components of vector a. The teraton process stops when one of the termnaton condtons occurs, for example, f the approxmatons are suffcently good, none of the move lmts s actve, and the search sub-regon s small enough. The MARS programmng code contans the common type of bult-n approxmatons: lnear, multplcatve and recprocal. In order to reduce the computatonal tme, the mechanstc approach can be mplemented to develop smplfed approxmatons of the constrant functons. It was shown earler [Ermolaeva and Spoormaer 2001] that f the mechancal behavour of an optmzed structure, or the parts of ths structure, can be even approxmately descrbed by engneerng equatons then mechanstc approxmatons can be easly derved and appled to correspondng mechancal constrants. In ths case, the convergence of the optmzaton procedure needs less number of teratons and, hence less structural analyss calculatons. The mechanstc approxmatons derved for the sde rocer of the DutchEVO car, represented by hollow rectangular beam, reduced the number of FEA runs by 3-4 tmes [Ermolaeva and Spoormaer 2001]. Mechanstc approxmatons for the consdered structure wll be shown n the followng secton. 4. Optmzaton system adjustment 4.1 MARS optmzaton system The optmzaton system usually combnes three followng parts: optmzaton programs (optmzer), problem-dependent analyzer and nterface programs that connect the optmzer wth the analyss software. These nterfaces depend on the structure of the analyss code and have to be created for each specfc applcaton. We showed earler [Ermolaeva and Spoormaer 2001] how the MARS optmzaton code [Toropov 1989] could be lned to MSC.Marc FEA pacage. The MSC.Marc FEA pacage provdes a user subroutnes feature, whch allows users to substtute ther own subroutnes for those exstng n the program [MSC.Marc Volume D: Users Subroutnes and Specal Routnes, Verson 2000]. It suffcently smplfes nterface programmng and maes the MSC.Marc FEA code attractve to user wllng to nterfere n the calculaton process. In addton to subroutnes used n 630 INDUSTRIAL SOLUTIONS

5 prevous verson of the MARS system [Ermolaeva and Spoormaer 2001], that allow one to get necessary nformaton for partcular nodes and elements, for current problem we employ the subroutne ufconn.f, n order to obtan a connectvty matrx used further for the volume of elements estmaton. All programs: optmzer, analyzer and nterfaces, are combned n MARS optmzaton system va start_optmzaton.bat fle to start the optmzaton procedure. A scheme of the developed optmzaton system one can fnd n [Ermolaeva, Wllemen and Spoormaer, 2001]. 4.2 Model senstvty analyss In order to evaluate the senstvty of the constrants and objectve functon to the change n varables, the followng analyss was done. The FEA of the modelled structure was performed for one of the varables changed whle the other varables beng constant. It resulted n the followng (see geometry components thcness of whch correspond to the desgn varables n Fg. 1): geometry 1 (walls of the rocer beams) and geometry 5 (outer panels of the lower part of the floor) nfluence the crtcal load sgn (negatve values of a crtcal load cannot be processed by the optmzaton code); geometres 2, 3 and 4 (horzontal components of the rocer beams) affect mostly the bendng stffness constrant and stress-stran behavour of the structure; geometres 6 (central panel of the lower part of the floor), 7, 8, (components of the floor upper panel), 9, 13 (longtudnal stffeners) and 14 (a component for engne support at the front part) nfluence generally the crtcal load. the buclng constrant appeared to be the most senstve functon. Accordng to ths analyss the lower lmts has been adjusted to the current optmzaton problem. 4.3 Mechanstc approxmaton Approxmatons chosen for the current optmzaton problem are the followng: lnear for the objectve functon, recprocal for bendng stffness constrant, multplcatve for strength constrant and, ntally, for buclng constrant. Tral calculatons showed that all approxmatons except for buclng constrant are suffcently good. For the buclng constrant, we tred also the recprocal approxmaton from the set of the bult-n functons and proposed to use mechanstc approach to derve an approxmaton that s close to the buclng behavour of the gven structure. Wthn the mechanstc approach, the smplfed model s employed to buld the approxmatons (ndces and j are omtted here for smplcty): ~ ~ F =, (8) ( x, a) F( f, a) where f(x) s the functon presentng the structural response usng the smplfed model. The approxmaton based on the smplfed model satsfes all basc requrements. It reflects the man propertes of the orgnal complex model; t s computatonally less expensve and relatvely noseless [Toropov and Marne 1996]. The gven structure s a complex bult-up constructon consstng of many plane elements. The structural nstablty of ths structure s defned manly by the local buclng modes of ts components. It s reasonable to tae nto account the local modes of each structural element, the crtcal load for whch s proportonal to the thcness of the component n power three. Then the approxmaton functon for buclng constrant could be wrtten n the followng form: ~ F N (, a) 3 = a0 + = 1 a x x. (9) 3 Analyss of dfferent types of approxmatons for buclng constrant (see Fg. 3) gves the followng result. Applcaton of multplcatve approxmaton shows the tendency to non-convergence of the optmzaton problem. Comparson of recprocal and mechanstc approxmatons results n less varatons of analysed functon for the latter approxmaton type. INDUSTRIAL SOLUTIONS 631

6 7 6 Approxmatons: multplcatve recprocal mechanstc buclng constrant number of teratons Fgure 3. Behavour of the buclng constrant of dfferent approxmaton types 4.4 Senstvty to the ntal desgn In order to be sure that the optmal soluton s global, the senstvty of the constrants and objectve functon to the ntal desgn change was analysed. The startng pont n the space of desgn varables was taen n three varants: model Bend_09_1 x for = 5,, 9, 13, 14 were taen at ther lower sde lmts; all other varables were taen equal to 1 mm; model Bend_09_2 4 mm for all varables; model Bend_09_3 8 mm for all varables. The values of the constrants and the objectve functon for the optmal solutons under dfferent ntal desgns are presented n Table 1. Model Number of response analyses Table 1. Senstvty to the ntal desgn Bendng stffness constrant Strength constrant Buclng constrant Optmal mass, g Error, % Bend_09_ Bend_09_ Bend_09_ As one can see the dfference n optmal mass s wthn the acceptable range of engneerng error. Buclng mode for these optmal solutons s the same of local type: upper panel between longtudnal and transverse stffeners (Fg. 4). 5. Results of the optmzaton Results of the structural optmzaton for the bottom structure made of steel are presented n the prevous Secton. Accordng to the results n Table 1, the ntal desgn for model Bend_09_2 can be consdered as preferable, because of the mnmal value of the objectve functon and lowest number of FEA runs (consequently, the lowest number of teraton steps and computatonal tme). The bendng stffness constrant appears to be actve n ths soluton. Optmsaton of the gven structure made of other materals needs addtonal adjustment through the analyses of the optmsaton process and acheved solutons. Approxmaton functons and ntal desgn were taen as n the prevous model for steel structure. Lower bound lmts were changed. They were set up durng the system adjustment. Ths maes dffcult to compare the optmsaton results, because due to these changes the ntal formulaton of the optmsaton problem cannot be consdered completely the same for the structures made of dfferent materals. Nevertheless, optmsaton solutons for a wrought alumnum alloy structure (model Bend_10_b) and the structure composed of 632 INDUSTRIAL SOLUTIONS

7 an alumnum alloy frame covered wth composte plates of two dfferent compostons are gven n Table 2. Appled composte materals were polypropylene (PP) matrx renforced wth randomly dstrbuted glass (model Bend_11_b) or hemp fbres (model Bend_12_a); the latter s the representatve of natural fbre compostes. Fgure 4. Buclng of the bottom structure As one can see n Table 2, under the gven formulaton of optmzaton problem the bottom structure made of alumnum alloy has the lowest weght 51.8 g, whch s about 46% lghter than the same steel structure. The vector of desgn varables (thcnesses of structural components n mm) correspondng to ths optmal soluton s the followng x = (1.0, 1.1, 3.0, 3.6, 1.3, 2.3, 1.5, 2.2, 1.8, 1.7, 4.0, 1.0, 3.4, 1.8). The bendng stffness and buclng are the actve constrants. The structures made of alumnum alloy frame covered wth the PP composte floor panels are 13% heaver than the same structure from only alumnum alloy. The current result cannot be consdered as drect recommendaton for fnal desgn. It s true only for gven formulaton of the consdered optmzaton problem, and can be far from the real optmum. Let us call that no technologcal aspects of assembly, as jont materal (weld or adhesve), or producton, except of the lmtaton of thcnesses due to manufacturablty of polymer plates, are appled here. On the other hand, t s possble that the weght of the bottom structure can be further decreased. Consdered here composte floor panels are comparatvely wea due to the random dstrbuton of short fbres. If the mechancal propertes of the appled compostes can be mproved, due for example to orented fbres, or specal treatment of fbres and/or matrx, then the alumnum frame structure wth the PP/hemp fbres composte floor panels mght be lghter and can compete wth the alumnum alloy structure. The torson stffness should be ncluded nto consderaton to mae the optmal problem formulaton complete for statc load resstance of the DutchEVO car bottom structure. model Bendng stffness constrant Table 2. Optmzaton results Strength constrant Buclng constrant Optmal mass, g Optmal mass relatve to steel Bend_10_b Bend_11_b Bend_12_a Concluson In current paper, the szng optmzaton problem of the DutchEVO car bottom structure wth respect to mnmum mass under bendng load case requrements was formulated. FE model of the gven structure was bult and tested. Problem dependent nterfaces were programmed n order to connect structural optmzaton and FE analyss programs. The MARS optmzaton system based on the Multpont Approxmaton method wth Response Surface fttng and MSC.Marc FEA code was INDUSTRIAL SOLUTIONS 633

8 adjusted here to be appled to the desgn of the consdered structure. The mechanstc approach was mplemented to defne the buclng constrant approxmaton functon. The possblty to use dfferent materals for the bottom structure was evaluated. In current paper, we consdered conventonal structural materals, steel and wrought alumnum alloy, and two compostes wth synthetc and natural fbres. It was shown that under gven formulaton of the optmzaton problem the bottom structure made of a wrought alumnum alloy has a lowest weght, whch s about 46% less than for steel structure. The other consdered materals also showed the potental to be used n gven structural applcaton. In current paper, bendng stffness, strength and structural stablty (non-buclng behavour) constrants were consdered as the frst step of system applcaton. In order to mae an optmal desgn complete wth respect to statcal loads resstance, the torsonal stffness requrement should be consdered n addton. Ths wll be done as the next step of the development and applcaton of the optmzaton system. Acnowledgement The authors are very grateful to Prof. V.V. Toropov (Unversty of Bradford, UK) for provdng, as a matter of courtesy, the MARS optmzer code. References Ashby, M.F., Materals selecton n mechancal desgn, Butterworth-Henemann, Oxford, UK, Ermolaeva, N.S., Spoormaer, J.L., Applcaton of MARS optmzaton system to the optmal desgn of DutchEVO car sde rocer beam, Proceedngs of the Fourth World Congress of Structural and Multdscplnary Optmzaton, WCSMO-4, June 4-8, 2001, Dalan, Chna, Cheng, G., Gu, Y., Lu, S. and Wang, Y., 2001, publcaton 9. Ermolaeva, N.S., Wllemen, R., Spoormaer, J.L., Optmal desgn of corrugated-core sandwch panel Proceedngs of the 3 rd ASMO UK/ISSMO conference, Harrogate, North Yorshre, UK, 9 th -10 th July 2001, ISBN: , Quern, O.M., Unversty of Leeds, UK, pp Fenton, J., Handboo of Automotve Body Constructon and Desgn Analyss, Professonal Engneerng Publshng, London and Bury St Edmunds, UK, Kanter de, J.L.C.G., Vlot, A., Kandachar, P., Kavelne, K,. Toward a new paradgm n car desgn, Proceedngs of Internatonal Conference on Materals for Lean Weght Vehcles 4 (LWV4), October 2001, Hertage Motor Centre, Gaydon, UK, 2001 (to be publshed). MSC.Marc Volume B: Element Lbrary, Verson 2000, MSC.Software Corporaton, MSC.Marc Volume D: Users Subroutnes and Specal Routnes, Verson 2000, MSC.Software Corporaton, Toropov, V.V., Smulaton approach to structural optmzaton, Structural Optmzaton, Vol. 1, 1989, pp Toropov, V.V., Marne, V.L., The Use of Smplfed Numercal Models as Md-Range Approxmatons, Proceedngs of the 6-th AIAA/USAF/NASA/ISSMO Symposum on Multdscplnary Analyss and Optmzaton, Bellevue WA, September 4-6, 1996, Part 2, 1996, pp Ermolaeva N.S., Dr., Delft Unversty of Technology, Faculty of Desgn Engneerng and Producton, Industral Desgn Engneerng, Landbergstraat 15, 2628 CE, Delft, the Netherlands Tel: Fax: Emal: N.Ermolaeva@o.tudelft.nl 634 INDUSTRIAL SOLUTIONS

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