The optimal design support system for shell components of vehicles using the methods of artificial intelligence

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1 IOP Confrnc Sris: Matrials Scinc and Enginring PAPER OPEN ACCESS Th optimal dsign support systm for shll componnts of vhicls using th mthods of artificial intllignc To cit this articl: M Szczpanik and A Potralski 216 IOP Conf. Sr.: Matr. Sci. Eng Rlatd contnt - Dsign Support Systm for Opn Distanc Larning Studnt Tamwork A Putranto and Y D Pradipto - Form finding analysis basd on variational mthod for multistabl structur utilizing snap-through bhaviour T Kurokawa, M Shigmatsu and M Kohiyama Viw th articl onlin for updats and nhancmnts. This contnt was downloadd from IP addrss on 4/9/218 at 18:14

2 2th Innovativ Manufacturing Enginring and Enrgy Confrnc (IManEE 216) IOP Conf. Sris: Matrials Scinc and Enginring 161 (216) 1219 doi:1.188/ x/161/1/1219 Th optimal dsign support systm for shll componnts of vhicls using th mthods of artificial intllignc M Szczpanik and A Potralski Institut of Computational and Mchanical Enginring, Faculty of Mchanical Enginring, Silsian Univrsity of Tchnology, Poland miroslaw.szczpanik@polsl.pl Abstract. Th papr is dvotd to an application of th volutionary mthods and th finit lmnt mthod to th optimization of shll structurs. Optimization of thicknss of a car whl (shll) by minimization of strss functional is considrd. A car whl gomtry is built from thr surfacs of rvolution: th cntral surfac with th hols dstind for th fastning bolts, th surfac of th ring of th whl and th surfac conncting th two mntiond arlir. Th last on is subjctd to th optimization procss. Th structurs ar discrtizd by triangular finit lmnts and subjctd to th volum constraints. Using proposd mthod, matrial proprtis or thicknss of finit lmnts ar changing volutionally and som of thm ar liminatd. As a rsult th optimal shap, topology and matrial or thicknss of th structurs ar obtaind. Th numrical xampls dmonstrat that th mthod basd on volutionary computation is an ffctiv tchniqu for solving computr aidd optimal dsign. 1. Introduction Optimal proprtis of shll componnts of vhicls can b providd by using th computr aidd optimization tchniqus. For xampl, an appropriat strngth of structurs can b stablishd by changing thir shap, topology and matrial proprtis. Th choic of optimal shap and topology ar th ky lmnts dfining th ffctivnss of structurs, and thus finding thm is th problm of grat practical intrst. Shap and topology structural optimization is a vry activ rsarch ara. Svral compting approachs for topology optimization xist [1][2][]. Mor rcntly, intllignt optimal dsign tchniqus basd on bio-inspird mthods lik th volutionary algorithm (EA), th particl swarm optimizr (PSO) and artificial immun systm (AIS) hav found applications to structural optimization problms. Th main fatur of volutionary mthod is to simulat of biological procsss basd on hrdity principls (gntics) and th natural slction (th thory of volution) to crating of optimal individuals (solutions) prsntd by singl chromosoms. Evolutionary algorithms ar usually applid in th situations whn th optimization problms ar too complicatd for th traditional gradint optimization mthods. Th task considrd in th prsnt work is rlatd to such a problm. This task consists in crating an ffctiv optimization algorithm for shll-structurs in rspct of topology [4], shap and matrial or thicknss arrangmnt. Th main advantag of th volutionary algorithm is th fact that this approach dos not nd any information about th gradint of th fitnss function and givs a strong probability of finding th global optimum. Th fitnss function is calculatd for ach chromosom in ach gnration by solving th boundary-valu problm by mans of th finit lmnt mthod (FEM). In ordr to solv th optimization problm th fitnss function, Contnt from this work may b usd undr th trms of th Crativ Commons Attribution. licnc. Any furthr distribution of this work must maintain attribution to th author(s) and th titl of th work, journal citation and DOI. Publishd undr licnc by Ltd 1

3 2th Innovativ Manufacturing Enginring and Enrgy Confrnc (IManEE 216) IOP Conf. Sris: Matrials Scinc and Enginring 161 (216) 1219 doi:1.188/ x/161/1/1219 dsign variabls and constraints should b formulatd. Coupling th finit lmnt mthod and th volutionary algorithm givs an ffctiv and fficint altrnativ optimization tool, which nabls solving a larg class of th optimization problms of mchanical structurs for xampl structur of car whl. Th main fatur of th first proposd optimization mthod is th volutionary distribution of th matrial, changing its matrial proprtis or by changing its thicknss in th construction of car whl. This procss lads to th limination of th part of matrial from th whl construction and as a rsult th nw shap and th topology of th car whl mrgs. Using intrpolation surfacs (hypr surfac) rducs th numbr of th dsign variabls and shortns th tim of th computation. Th application of th profssional program of th finit lmnt mthod MSC NASTRAN in this mthod nabls th optimization of th complx mchanical systms. Th numrical xampls confirm th fficincy of th proposd optimization mthod and dmonstrat that th mthod basd on volutionary computation is an ffctiv tchniqu for solving computr aidd optimal dsign problms. 2. Formulation of volutionary optimization problm Evolutionary algorithms ar algorithms sarching th spac of solutions and thy oprat on populations of individuals. Each individual which consists of chromosoms in population rprsnts a singl solution. All chromosoms consist of gns which ar dsign variabls in optimization problms. Nxt th fitnss function is computd in ordr to adaptation ach individual in population. All gns of an individual dcid about th fitnss function valu. Th algorithm of volutionary optimization has a svral stags. First an initial population of individuals is cratd. Th dsign variabls (gns) in randomly way ar gnratd. In th nxt stp, th fitnss function valu for ach chromosom is computd. In th nxt stp gns ar changd using volutionary oprators and working on th parnt population individuals. Th offspring population is cratd on th bass of modifid individuals with prvious stp. Th algorithm is continuing itrativly till th nd of th computation. Th stop condition of th computation can b formulatd as th maximum numbr of itrations. Two typs of oprators ar usd: mutation and crossovr. Mutation is rprsntd by uniform mutation, mutation with Gaussian distribution and boundary mutation. Th last oprator is simpl and arithmtical crossovr. Considr a 2D structur which, at th bginning of an optimization procss, occupis a domain Ω in E, boundd by a boundary Γ. Th domain of this structur Ω is filld by a homognous and ( ) isotropic matrial (paramtrs: Young s modulus E and a Poisson ratio ν ). Th thicknss of th structur g is also constant at th bginning of th volutionary procss. Th shlls ar considrd in th framwork of thory of lasticity. During th optimization procss th domain Ω t, its boundary Γ t and th fild of Young s modulus E( x) = E t, x Ω t or th thicknss g ( x ) = gt can chang for ach itaration t (for t=, E =const, g =const). Th optimization procss procds in an nvironmnt in which th structur fitnss is dscribing by th objctiv is to minimiz th strss functional ψ ( σ ) d (1) J = Ω whr ψ is an arbitrary function of strss tnsor σ, with a constraint imposd on th volum of th structur Ω V Ω V max (2) In ordr to solv th formulatd problm Finit Elmnt modls of th structurs ar considrd [5]. Th structur is dividd into finit lmnts Ω, = 1, 2,..., R, and nod displacmnts ar calculatd by solving a systm of linar algbraic quations 2

4 2th Innovativ Manufacturing Enginring and Enrgy Confrnc (IManEE 216) IOP Conf. Sris: Matrials Scinc and Enginring 161 (216) 1219 doi:1.188/ x/161/1/1219 KU=F () whr U is a column matrix of unknown displacmnts, F is a known column matrix of acting forcs and K is a known global stiffnss matrix of th structur whos lmnts ar givn as follows: K =B T DBV (4) whr D and B ar th known lasticity and gomtrical matrics, rspctivly, V rprsnts th volum of th finit lmnt.. Th ida of volutionary optimization Th distribution of th Young s modulus or thicknss E( x), x Ωt in th shll is dscribd by a hiprsurfac W ( x), x H. Th hiprsurfac W ( x ) is strtchd undr H E and th domain Ω t is includd in H, i.. ( Ωt H ) [6]. Th shap of th hiprsurfac W ( x ) is controlld by gns h j, j=1,,n, which crat a chromosom min max ch = h, h,..., h,..., h, h h h 1 2 j N (5) j min max whr h - th minimum valu of th gn and h - th maximum valu of th gn. Gns ar valus of th function ( ) h W x, j=1,2,,n. W x in intrpolation nods x j, i.. j = ( j ) Th assignation of Young s moduli or thicknss to ach finit lmnt Ω, = 1,2,..., R is adquatly prformd by th mapping: E = W ( x ), x Ω, = 1,2,..., R (6) g = W ( x ), x Ω, = 1,2,..., R (7) It mans that ach finit lmnt can hav diffrnt matrial. Whn th valu of th Young s modulus or thicknss for th -th finit lmnt is includd in: th intrval E < Emin (or g < gmin ), th finit lmnt is liminatd and th void is cratd, th intrval Emin E < Emax (or gmin g < gmax ), th finit lmnt rmains having th valu of th Young s modulus from this matrial (figur 1). <,..., E,..., E min) Elimination of FE Emin < h min, hmax> Dsign variabl < Emin,..., E,..., Emax> Existnc of FE Emax Figur 1. Rquirmnts for limination and xistnc of th finit lmnts Two typs of intrpolations which aim at th appropriat slction of mass dnsitis ar usd. Th first of thm is th multinomial intrpolation, th othr th intrpolation bass on th nighborhood of lmnts [1].On th basis of obtaind rsults th intrpolation bass on th nighbourhood of

5 2th Innovativ Manufacturing Enginring and Enrgy Confrnc (IManEE 216) IOP Conf. Sris: Matrials Scinc and Enginring 161 (216) 1219 doi:1.188/ x/161/1/1219 lmnts is bttr than multinomial intrpolation. In addition to scond intrpolation bass on th nighbourhood of lmnts th optional numbr of control points can b loadd and all working spac always is usd. It is following advantag of this mthod. In addition to incras numbr of control points prmit to obtain mor accurat distribution of hypr surfac but for biggr numbr of control points, numbr of individuals in ach gnration must b incrasd. In ordr to improv optimization rsults two diffrnt additional hav bn introducd: th additional procdur aiding th topology optimization [9][1], th assignation of matrials [1]. Using th first mthod (th additional procdur aiding th topology optimization), on can chang matrial proprtis of finit lmnts during th immun optimization procss and som lmnts ar liminatd. As a rsult, th optimal shap, topology and matrial of structur ar obtaind [9][1]. Using scond mthod th Young s modulus valu should b dvidd into th nforcd numbr of subintrvals (qual to th numbr of matrials). Each subintrval rprsnts anothr matrial whos th Young s modulus blongs to this subintrval. Th subintrvals should hav th sam lngth and thir cntrs corrspond with Young s modulus valus of suitabl matrials [1]. START Discrtization of th domain Gnration of initial population Intrpolation of th surfac of Young s modulus (adquatly thicknss) Assignation Young s moduli for FEs Chck th constraint imposd on volum NO n=n+1 If ys Solution of boundary valu problm by FEM If no Additional procdur for th aid of volutionary optimization Evaluation of th fitnss function Evolutionary algorithm Th stop condition END YES Figur 2. Evolutionary optimization of shlls 4

6 2th Innovativ Manufacturing Enginring and Enrgy Confrnc (IManEE 216) IOP Conf. Sris: Matrials Scinc and Enginring 161 (216) 1219 doi:1.188/ x/161/1/ Evolutionary optimization algorithm of topology, shap and matrial or thicknss Th minimization of th fitnss functions (1) or (2) with rspct to th chromosom (5) is prformd by mans of an volutionary algorithm [7][8] with th floating point rprsntation (figur 2). Aftr th FEM discrtisation th starting population of chromosoms is randomly gnratd. At th nxt stp th main loop of th optimization algorithm is prformd. Oprations includd in th main loop lad to th calculation of th fitnss function. It rquirs that th boundary-valu problm should b solvd by th FEM. Aftr th calculation of th fitnss function for all of th chromosoms in th population th volutionary algorithm is applid. Th volutionary algorithm contains th following oprators: th ranking slction, th simpl and arithmtical crossovrs, th uniform and boundary mutations and th cloning. As th rsult a nw offspring population is cratd. Th nd of th algorithm s work, i.. a brak in th main loop activity, occurs aftr th dclard gnration numbr. Th algorithm can b also stoppd, whn aftr th spcifid itration numbr th chang of th fitnss function is vry small [9]. 5. Numrical xampl car whl optimization Th problm is to find optimal shap, topology and thicknss of a car whl by th minimization of th strss functional and by th volum constraint. Th gomtry of a car whl with charactristic dimnsions (Tabl 1) is prsntd. Thr surfacs of rvolution built th structur (figur ): th cntral surfac with th hols dstind for th clamping scrws, th surfac of th ring of th whl and th surfac conncting th two mntiond arlir. Th conncting surfac is optimizd. Th shllstructur is loadd with a prssur c and th tangnt forc s (figur 4b). Th support is applid on th cntral surfac in th dirction of th rotation axis of th whl. Th nods in th aras nar th hols dstind for th clamping scrws ar fixd (figur 4b). Th symmtry of th car whl is imposd (rvolution of th 1/5 part of th structur). Figur 4a prsnts distribution of th control points of th intrpolation hiprsurfac. In this way th numbr of dsign variabls (gns) ar dcrasd and th symmtrical rsults ar obtaind. Th input data and th paramtrs of volutionary algorithm ar prsntd in th tabls 2 and, rspctivly. Th maps of thicknss prsnts th rsults of th optimization aftr 1th gnrations (figur 5). All th prsntd solutions ar includd in th sarch spac and ar optimal th scond optimization critrion is th sthtic aspct dpndnt on th car usr. Th rsults corrspond vry wll with th car whls producd by diffrnt automotiv companis (figur 5). dimnsions of a whl LK LP LW Figur. Gomtry and charactristic dimnsions of a car whl Tabl 1. Charactristic dimnsions of a car whl diamtr of th whl LW 55.6 mm width of a tyr LF 175 mm diamtr of th whls spacing LK 11 mm diamtr of a whl hub LP 6 mm thicknss of th whl hub mm thicknss of a tyr 8 mm LF 5

7 2th Innovativ Manufacturing Enginring and Enrgy Confrnc (IManEE 216) IOP Conf. Sris: Matrials Scinc and Enginring 161 (216) 1219 doi:1.188/ x/161/1/1219 Tabl 2. Input data to th optimization task tangnt forc s [N] prssur c [Mpa] numbr of dsign variabls numbr of control points numbr of th chromosoms matrial rang of th chang xisting of an limination of of th gns [mm] lmnt an lmnt V max [cm ] aluminium g < 1 1 g Tabl. Paramtrs of volutionary algorithm boundary simpl cloning uniform mutation arithmtical crossovr mutation crossovr 2% 5% 5% 1% 1% (a) (b) Figur 4. Car whl: (a) distributing of th control points of th intrpolation hiprsurfac; (b) boundary conditions (a) (b) (c) (d) 6

8 2th Innovativ Manufacturing Enginring and Enrgy Confrnc (IManEE 216) IOP Conf. Sris: Matrials Scinc and Enginring 161 (216) 1219 doi:1.188/ x/161/1/1219 () (f) (g) (h) Figur 5. Th rsults of th car whl optimization: (a), (c), (), (g), th bst obtaind solution - maps of thicknss, (b), (d), (f), (h) ral car whls which ar in good coincidnc with obtain solutions 6. Conclusions An ffctiv tool of volutionary optimization of shlls has bn prsntd. Using this approach shap, topology and matrial or thicknss optimization ar prformd simultanously. Th important fatur of this approach is a strong probability of finding th global optimal solutions rcivd by implmntation of th volutionary algorithm and its univrsality consisting in application of this approach to th optimization of th 2-D structurs in plan strss/strain [1]. Evolutionary mthod blong to mthods basd on population of solutions and thy hav som intrsting faturs which can b considrd as altrnativ to artificial immun systm [11][12][1][14] or particl swarm optimizrs [1][15][16]. Comparison btwn AIS, PSO and EA in th papr [6][17][18][19] is prsntd. Rfrncs [1] Bndsø M P 1989 Optimal shap dsign as a matrial distribution problm Struct. Optim. [2] Burczyński T and Osyczka A 2 Evolutionary Mthods in Mchanics Proc. IUTAM Symposium Kluwr Dordrcht [] Xi Y M and Stvn G P 1997 Evolutionary Structural Optimization London Springr [4] Yamada T t al. 21 A topology optimization mthod basd on th lvl st mthod incorporating a fictitious intrfac nrgy Comput. Mthods Appl. Mch. Engrg. [5] Zinkiwicz O C and Taylor R L 2 Th Finit Elmnt Mthod Buttrworth Hinmann Oxford [6] Burczyński T Potralski A and Szczpanik M 215 Immun and Swarm Optimization of Structurs Advancs in Evolutionary and Dtrministic Mthods for Dsign Optimization and Control in Enginring and Scincs Chaptr no 19 [7] Arabas J 21 Lcturs on Evolutionary Algorithms WNT Warszawa (in polish) [8] Michalwicz Z 1996 Gntic algorithms + data structurs = volutionary algorithms Springr- Vrlag Brlin [9] Burczyński T and Szczpanik M 21 Intllignt optimal dsign of spatial structurs Computr 7

9 2th Innovativ Manufacturing Enginring and Enrgy Confrnc (IManEE 216) IOP Conf. Sris: Matrials Scinc and Enginring 161 (216) 1219 doi:1.188/ x/161/1/1219 & Structurs Elsvir [1] Szczpanik M and Burczyński T 212 Swarm optimization of stiffnrs locations in 2-D structurs Bulltin of th Polish Acadmy of Scincs Tchnical Scincs vol 6 no 2 pp [11] Potralski A Szczpanik M Ptaszny J Kuś W and Burczyński T 21 Hybrid artificial immun systm in idntification of room acoustic proprtis Invrs Problms in Scinc and Enginring Taylor & Francis [12] Potralski A 214 Optimization of mchanical structurs using artificial immun algorithm Byond Databass Architcturs and Structurs Communications in Computr and Information Scinc vol 424 pp [1] Potralski A Szczpanik M and Burczyński T 215 Immun Optimal Dsign of 2-D and -D Structurs Intrnational Confrnc on Artificial Intllignc and Soft Computing (ICAISC) Lctur Nots on Artificial Intllignc vol 912 Springr pp [14] Potralski A 215 Data procssing in immun optimization of th structur, Byond Databass Architcturs and Structurs (BDAS) Communications in Computr and Information Scinc vol 521 Springr pp 9-19 [15] Potralski A Szczpanik M and Burczyński T 215 Swarm and immun computing of dynamically loadd rinforcd structurs Intrnational Confrnc on Artificial Intllignc and Soft Computing (ICAISC) Lctur Nots on Artificial Intllignc vol 912, Springr pp [16] Szczpanik M Potralski A Ptaszny J and Burczyński T 212 Hybrid Particl Swarm Optimizr and Its Application in Idntification of Room Acoustic Proprtis Swarm and Evolutionary Computation - Intrnational Symposia SIDE 212 and EC 212 Hld in Conjunction with ICAISC 212 Zakopan Poland Lctur Nots in Computr Scinc vol 7269 Springr [17] Potralski A Szczpanik M Dziatkiwicz G and Kuś W 21 Comparison btwn PSO and AIS on th basis of idntification of matrial constants in pizolctrics Springr-Vrlag Brlin Hidlbrg ICAISC 21 Part II LNAI vol 7895 pp [18] Potralski A Szczpanik M Bluch W and Burczyński T 214 Optimization of composit structurs using bio-inspird mthods Artificial intllignc and soft computing ICAISC 214 vol 8468 pp [19] Szczpanik M Potralski A Długosz A Kuś W and Burczyński T 21 Bio-inspird optimization of thrmomchanical structurs Springr-Vrlag Brlin Hidlbrg ICAISC 21 Part II LNAI vol 7895 pp