JCAMECH Vol. XX, No. XX, XX 20XX, pp 307-318 DOI: 10.22059/jcamch.2017.237353.160 Multi-objctiv Optimization of wb profil of railway whl using Bi-dirctional Evolutionary Structural Optimization Ali Asghar Atai 1,, Ehsan Azarlu 2 1 School of Mchanical Enginring, Collg of Enginring, Univrsity of Thran, Iran. 2 Mchanical Enginring Dpartmnt, Islamic Azad Univrsity-Karaj Branch, Karaj, Iran Rcivd: 9 July. 2017, Accptd: 20 Sp. 2017 Abstract In this papr, multi-objctiv optimization of railway whl wb profil using bidirctional volutionary structural optimization (BESO) algorithm is invstigatd. Using a finit lmnt softwar, static analysis of th whl basd on a standard load cas, and its modal analysis for finding th fundamntal natural frquncy is prformd. Th von Miss strss and critical frquncy as th problm objctivs ar combind using diffrnt wight factors in ordr to find th snsitivity numbr in th mthod, which spcifis which lmnts to b omittd and which to b addd. Th itrativ procss is continud until convrgnc to an a priori spcifid matrial volum. Th rsultd wb profils show that whn th strss is important, matrial rmoval is from th middl part of th wb, whil for frquncy as th important objctiv, th rmoval is from nar th rim part of th wb. Th suggstd profil, corrsponding to qual wight factor for th objctivs, has a bttr volum and strss stat compard to a standard wb profil, and has a mor uniform strss distribution. Howvr, highr natural frquncy, compard to that of th standard profil, ar obtaind for largr frquncy wight factors, although with a biggr volum. In th nd, considring manufacturability of th whl, th jaggd profil rsultd from BESO is rplacd with a fittd smooth curv and prforming th finit lmnt analysis on it. It is sn that thr is an improvmnt in th obtaind objctivs for th smoothnd profil, with no significant chang in volum. Kywords: BESO, Topology optimization, Railway whl, Multi-objctiv optimization. Corrsponding Author. Tl.: +98 21 61119951; Fax: +98 21 88013029 Email Addrss aata@ut.ac.ir 307
Atai and Azarlu 1. Introduction Topology optimization is on of intrsting and important filds of structural nginring, looking for th bst possibl placmnt of matrial in th structur so that whil th loads ar carrid safly, minimum amount of matrial is usd. For continuous structurs, such as 2D and 3D bams, columns, vn micro-structurs, diffrnt mthods of topology optimization, such as SIMP (Solid Isotropic Matrial with Pnalization), ESO (Evolutionary Structural Optimization), and BESO (Bidirctional Evolutionary Structural Optimization) ar dvlopd. Ths mthods start with an initial matrial domain and gradually, add and/or dlt matrial to/from rgions which has grat/littl positiv ffct on structural prformanc, which might b masurd by stiffnss, strain nrgy, strss distribution, tc. Sinc arly dvlopmnt of ESO by Xi and Stvn [1], it was mployd for optimizing structural prformanc, such as buckling [2]. In ESO, only matrial limination was prformd, starting from a sufficintly larg continuum, and gradual infficint matrial rmoval. It was thn changd to AESO (Adaptiv Evolutionary Structural Optimization), in which matrial addition in high gradint placs was prformd. BESO, combind th advantags of both, considring simultanous addition/limination of matrial. Huang and Xi improvd th prformanc of BESO by liminating msh dpndncy and facilitating convrgnc [3], and applid it to svral structural problms, and from thr svral authors usd th mthod for various static and dynamic optimization of structurs, including solid-fluid intraction applications [4-5]. In ordr to prsnt an ovrall comparison of th mthods statd abov, ESO has th shortcoming just matrial limination, and thus th initial continuum must b larg nough to ncountr th xpctd final optimum configuration. Morovr, onc a matrial is liminatd during th itrativ procss of this mthod, it is not possibl to bring it back if it would b ndd for th optimum topology. SIMP is quit similar to BESO, xcpt that it considrs a continuous variabl for matrial xistnc, which starts at an initial valu and is gradually moving towards 1 (prsnc of matrial) or zro (limination of matrial), rsulting in th final topology. Th first author s xprinc with th mthod has shown that th succss and convrgnc of th mthod is much dpndnt on how th matrial xistnc paramtr is updatd in ach itration, and it may divrg. Whil BESO suffrs much lss from this problm, by considring a binary matrial xistnc variabl. Thr ar rsarchrs that support ithr mthod, but BESO has gaind a popularity of its own and som bliv that BESO has grat potntial, spcially considring th latst nhancmnts, spcially whn combind with othr tchniqus lik gntic algorithms [6]. In trms of th application considrd in this work, a railway wagon whl is considrd hr. Whls function both as carrying wagon load (statically and dynamically), and conducting th wagon in its railway. Thr ar diffrnt typs of whls basd on th typ of wagon and loading on thm, som of which ar shown in Fig. 1 [7]. Thr ar also profils claimd to b of lssr strss undr loading, and of standard profils in diffrnt parts of th world [8] (s Fig. 2). Thr ar fw rsarchs rgarding th optimization of prformanc of wagon whls. Hirakawa and Sakamoto studid th variation of ffctiv paramtrs on fractur of whls [9]. Nilsn and Frdö usd th mthod dsign of xprimnts for optimization of whls. This justifis our motivation for a nw rsarch in this fild, spcially for applying BESO to a 3D round structur, for multi-objctiv optimization. Fig. 1 Diffrnt typs of whl wb profils [6] (a) (b) Fig. 2 (a) Low-strss whl, (b) EN standard whl [7] In this work, multi-objctiv optimization of th wagon whl, considring uniformity of strss distribution and natural frquncy of th whl is studid. In th nxt sction, th mthod of BESO, tailor-mad for multi-objctiv optimization of th problm at hand is xplaind in dtail. In sction 3, finit lmnt modling and static and dynamic analysis of th whl, ndd to xtract BESO paramtrs ar xplaind, and rsultd 308
Vol. 48, No. 2, Dcmbr 2017 optimizd whl wb profils, considring diffrnt wight factors (prioritis) for th objctiv ar prsntd. Th rcommndd compromis profil is furthr analyzd, by smoothning th jaggd wb profil rsultd from FE-BESO, in ordr to tak manufacturability into account. It is shown that th modifid optimizd profil considrably outprforms th standard whl profil, at last in th fram of th two objctivs considrd. 2. Mthod of BESO for multi-objctiv optimization In This sction, th mthod of BESO, which is modifid to b suitd for th currnt multi-objctiv optimization problm at hand, is xplaind in dtail. BESO is an volutionary structural optimization mthod with th proprty of both liminating th lss fficint matrial lmnts from th structur, and adding a prviously liminatd matrial lmnt, which now provs to b fficint, in ach itration of an itrativ procdur. Dcision on liminating or adding lmnts is basd on an lmnt-wis paramtr, calld snsitivity numbr, which shows th snsitivity of th objctiv(s) to th dsign variabl (hr, prsnc or absnc of lmnts). Formally, BESO tris to minimiz a charactristic proprty of structur volum constraint for th structur, as follows Minimiz f = N i=1 P i N i=1 (1) Subjct to: V V i x i = 0 x i = 0 or 1 in which f is th objctiv function, N is th numbr of lmnts, P i is som charactristic proprty of th ith lmnt (such as stiffnss, von-miss strss, contribution to natural frquncy, tc.), V i is th volum of th i th lmnt, V* is th targt volum for th structur, and x i shows th xistnc (1) or absnc (0) of th i th lmnt. But instad of dirct minimization of objctiv, its snsitivity to th xistnc/absnc of lmnts is takn into account, which causs som liv lmnts to di, and som dad lmnts to bcom liv, subjct to a gradual volution of th volum of th structur that approachs th targt volum V* upon convrgnc. In this work, th whl is considrd both from static and from dynamic point of viw. Usually for static prformanc in BESO, th stiffnss or strain nrgy of th structur is th objctiv. But thr ar altrnativs minimizing th avrag von-miss strss in th structur [10]. It can b shown that ths two schms ar almost quivalnt [11]. In th work, th scond schm is adoptd bcaus th von-miss strss is radily availabl from finit lmnt analysis. In this rgard, th snsitivity paramtr of this objctiv for th i th lmnt is takn as α i,static = σ Vonmiss (2) in which, σ Vonmiss is th von-miss strss for th i th lmnt. As for th dynamic (natural frquncy) analysis, th snsitivity paramtr is takn to b [12] α i,dynamic = 1 { m i } T (ω 2 i [M ] [K ]){ i } i (3) in which m i, { 1 }, [M ], and [K ] ar th mass, lmnt displacmnt vctor basd on th first mod shap, mass matrix and stiffnss matrix of th i th lmnt, rspctivly, and ω 1 is th first natural frquncy. Topological optimization looks for a mor uniformly strssd structur with th last amount of matrial. Thrfor, th optimization problm is statd in its simplst form as follows with a volum constraint. Bfor combining th snsitivity numbrs to a singl on for a combination of objctivs, thy ar normalizd in th intrval [0, 1] as follows in ordr to mak th combination maningful α i,ns α i,nd = α i,static α max static = α i,dynamic max α dynamic max α static α static (4) min (5) max α dynamic min α dynamic in which α i,ns and α i,nd ar th normalizd snsitivity numbr for static and dynamic analysis, rspctivly, and th suprscripts min and max indicat th minimum and maximum corrsponding snsitivity numbrs among all lmnts, rspctivly. Nxt, analogous to th combination of multi-objctivs into a singl on using th wightd sum, th multiobjctiv snsitivity numbr for ach lmnt of th structur is dfind by α i,multi = λ s (α i,ns ) +λ m (α i,nd ) (6) in which λ s and λ m ar th wight factors for th static and dynamic objctivs, rspctivly, varying in th rang [0, 1], showing th importanc of ach objctiv. In th numrical part, ths wight factors ar rlatd by λ m 1, and ar varid in th rang to gt topologis with diffrnt priority for th objctivs, as it will b sn in th nxt sction. On of th main issus in ESO is that sinc th structur is discrtizd into lmnts, th snsitivity numbrs, which dtrmin which lmnts ar liminatd or addd, bcom discontinuous and thrfor rsult a chckrboard pattrn (s Fig. 3) in th optimizd structur, which is unaccptabl from manufacturability point of viw. To ovrcom this problm in BESO, th lmnt snsitivity numbrs ar rdfind as follows in ordr to smoothn thir distribution. This also hlps th 309
Atai and Azarlu mthod to bcom msh-indpndnt (diffrnt msh sizs giv ris to mor or lss th sam optimizd structur) [10]. subdomain Ω i with its cntr at th cntr of th lmnt and with a radius of r min is considrd (Fig. 4). Th paramtr r min plays th rol of idntifying nods that influnc th snsitivity of i th lmnt. It should b larg nough so that at last on lmnt surrounding th i th lmnt is includd in th domain. On th othr hand, in this schm, th siz of th subdomain is kpt constant, irrlvant of th msh siz. Having th snsitivity numbrs of all nods in th domain, th snsitivity numbr of i th lmnt is rdfind as Fig. 3 An xampl of chckrboard pattrn in th ESO mthod [10] Th tchniqu, which is calld filtration, gos as follows. First, for vry nod j in th domain, th nodal snsitivity numbr is calculatd by Fig. 4 Circular subdomain of th i th lmnt for smoothning th snsitivity numbrs M α j n = w i α i (7) i=1 in which M is th total numbr of lmnts common at nod j, and w i is th wight factor of th i th lmnt givn by w i = 1 M 1 (1 with th proprty M w i = 1 i=1 i th lmnt Ωi r ij M i=1 r ij + r min ) (8) (9) Hr, rij is th distanc from th cntr of i th lmnt to nod j. Th abov dfinition of wight factors implis that lmnt snsitivity numbr has gratr ffct on closr nods. Nxt, ths nodal snsitivity numbrs ar usd to smoothn th lmnt snsitivity numbr throughout th msh. For vry lmnt i, a circular α i = k j=1 k j=1 w(r ij)α j n (10) w(r ij ) in which k is th total numbr of nods in th subdomain and w(r ij) is th linar wight factor dfind by w(r ij ) = r min r ij (j = 1,2,., k) This filtration schm hlps to liminat both chckrboard pattrn and msh dpndncy. Anothr problm is th instability in th procss, which may show oscillations in th volving topology during th procss. To rmdy this issu, th history of volutions of snsitivity numbrs is takn into account by dfining α i k = α i k + α i k 1 2 (12) in which k is th currnt itration numbr and th updatd snsitivity numbr is usd for th nxt itration. In ordr to rach th targt volum V upon convrgnc, in th itrativ procss, a targt volum V k+1 is considrd for th nxt itration, which is changd stp by stp until th final targt volum V is rachd. It is this itrativ targt volum which govrns how many lmnts should b addd to or dltd from th currnt topology. This volum is dfind by V k+1 = V k (1 ± ER) (k = 1,2,3, ) (13) in which ER is th volutionary volum ratio (Th plus/minus sign is for th cas whr th starting volum is lss/gratr than th final targt volum). This paramtr should b st at a rasonabl valu so that thr is a smooth gradual chang of topology from ach itration to nxt. Onc th targt volum V is rachd in an itration, it is kpt constant for th rmaining itrations, i.. V k+1 =V*. (11) 310
Vol. 48, No. 2, Dcmbr 2017 To dcid which lmnts to b addd or dltd in ach itration, th snsitivity numbr of ach lmnt for th whol domain is calculatd as xplaind abov, and snsitivity numbrs ar sortd (it should b notd that void lmnts also gt a nonzro snsitivity numbr basd on th filtration schm). Thrshold snsitivity th th numbrs α dl and α dl ar considrd for dlting and adding lmnts rspctivly. For th ith lmnt, if α i th α dl, thn that lmnt is dltd (x i is st to 0), and if α i α th add, th lmnt is addd (x i is st to 1). Th thrsholds ar st in ach itration so that th targt volum V k+1 for th nxt itration is mt. Furthrmor, a paramtr AR max (maximum volum addition ration, i.. maximum of th ratio of addd lmnts to th total numbr of lmnts in th domain) controls how many lmnts ar addd in ach itration, in ordr to avoid loss of intgrity of topology in cas to many lmnts ar to b addd. As a convrgnc critrion, both th targt volum and th combind objctiv function corrsponding to th combind snsitivity numbr (i.. λ s maximum von-miss strss + λ m ngativ of th first natural frquncy) ar trackd and if thir chang is lss than a crtain tolranc, thn th procdur stops. 3. In summary, th flowchart for th multi-objctiv BESO is prsntd in Fig. 5. 4. Modling, analysis, and optimization Finit lmnt analysis lis at th cor of BESO. Thrfor, th usual squnc of introducing gomtry, matrial, mshing, loading and boundary conditions, and analysis usd in commrcial FE softwar is xplaind hr. Fig. 6 shows th cross sction of a so-calld sshapd whl. It is usd in this rsarch for its rim, hub, and ovrall dimnsions. Ths dimnsions ar takn from standard no. UIC 515-1 [13]. In ordr to hav a mor ralistic whl-rail intraction during loading, th rim is modld using th dtaild gomtry from this standard, shown in Fig. 7. Th gomtry of th wb is what w try to optimiz using BESO. But th gomtry shown in Fig. 6 is usd latr on for comparison with th optimizd whl. For rail modling, th gomtry is takn from standard UIC-60, shown in Fig. 8 [15]. Nxt, th matrial is dfind, which is stl in this cas with proprtis E=200 GPa, θ = 0.3, and ρ=7800 kg/m 3. In th mshing part, a 3D modl of th whl is gnratd. First, th cross sction of th whl with a block for th wb (which will b mptid latr on using BESO) is considrd (Fig. 9) as an ara and is mshd. Nxt, th mshd ara is rvolvd about th whl axis to gnrat th 3D whl, with 20 sctors and 8 divisions in ach sctor, mad of SOLID185 lmnts (Fig. 10). To gt th propr msh siz, a msh snsitivity analysis was don with a typical run. Th whl cross sction was divid using diffrnt msh sizs and th maximum von- Miss strss obtaind was takn as th convrg paramtr. Th hardwar was an Intl(R) Cori5 systm with 8GB of RAM. Th rsults ar shown in tabl 1. For th finst msh siz in th tabl, th hardwar was not abl to giv a rsult. Basd on th von-miss strss valus and hardwar limitations, a msh siz of 5 by 5 was considrd as appropriat for all th analyss. For boundary conditions, sinc th whl hub is connctd to th axl almost rigidly, all th displacmnts and rotations for th innr surfac of th whl hol wr st to zro. As for th loading, th BS EN_13979 standard considrs diffrnt load cass with point loads on th whl rim [16]. On of ths cass is for th whl moving in straight lin (Fig. 11). In this cas, a vrtical load of F z with a magnitud of 1.25 tims th wagon wight pr whl must b applid at th point indicatd. In ordr to hav a bttr grasp of whl-rail intraction for this loading, part of th rail was modld and mshd and was brought into contact with th whl at th point of F z. Th xtrnal loads on th rail wr st in an quivalnt mannr so that th standard load Fz would b on th whl (Fig. 12). For th cas study considrd hr, a forc of F z=98.8 kn was applid on th whl. Onc th modl is prpard, two typs of analyss ar don as follows in ordr to calculat th snsitivity numbrs for BESO: 1- A static analysis is prformd with th dfind loading to gt th von-miss strss as th static snsitivity numbr (Eq. 4). 2- To gt th frquncy snsitivity numbr, a modal analysis is don and basd on th mass and stiffnss matrics of lmnts, and th first natural frquncy, th snsitivity numbr is calculatd using Eq. (5). Ths two analyss ar prformd using a macro for th FE softwar in ach itration of BESO. 311
Atai and Azarlu Fig. 5 Flowchart of multi-objctiv BESO 312
Vol. 48, No. 2, Dcmbr 2017 Msh siz Tabl 1. Msh snsitivity analysis Total numbr of nods Total numbr of lmnts Maximum von-miss strss (Pa) 10mm 10mm 83858 73452 0.779 10 9 7.5mm 7.5mm 123538 111852 0.861 10 9 5mm 5mm 230578 215852 0.884 10 9 2.5mm 2.5mm 782526 758384 NA Fig. 6 Cross-sction of EN standard whl [14] (dimnsions in mm) Fig. 7 Gomtry of standard whl rim with 100-720 mm diamtr [13] Fig. 9 Cration of whl cross sction Fig. 8 Gomtry of UIC-60 rail [15] (dimnsions in mm) Fig. 10 (a) A whl with 20 sctors, (b) 3D mshing of sctors Exprimnting with th mthod, th following BESO paramtrs provd to b promising for th problm at hand: 313
Atai and Azarlu Fig. 11 Straight path and loading point on th whl for this cas [16] Fig. 12 Equivalnt rail loading for standard forc on th whl V = 0.1V total, r min =12.5mm, AR max =0.5, ER = 0.03 Th multi-objctiv optimization was prformd using wightd sum mthod and considring wight factors λ s and λ m, for static and dynamic analysis snsitivity numbrs, rspctivly. Fig. 13 shows th rsultd wb profil basd on diffrnt wight factor combinations. It is sn that whn th dynamic rspons is lss important (lowr valus of λ m ), th wb is thinnr in th middl, but onc th critical frquncy bcoms of gratr importanc (highr valus of λ m ), th wb bcoms thinnr at th rim junction. This could b justifid by considring th wb lik a bam. Whn lowr maximum strsss ar ndd, th bam bcoms thickr at both nds, which ar closr to loads, to diminish th strss. On th othr hand, whn high natural frquncis ar ndd, th bam bcoms thickr clos to th rigid nd (hub) to incras th stiffnss and thus th frquncy. Anothr intrsting point is that whn static rspons bcoms important ( 1), th strss tnds to b mor uniform throughout th wb, which is a good sign that BESO is working. Figs. 14 to 16 ar a proof of convrgnc of BESO, for diffrnt wight factors. Thy show that both th volum fraction and von-miss strss rach asymptotically to a final valu as th itration numbr is incrass. Thy ar comparabl to Fig. 3.3 of Rf. 17. In ordr to mak a comparison btwn th optimizd whl and th standard on, an FE modl of th standard whl is gnratd and valuatd statically and dynamically (Fig. 17). It is sn that th strss distribution in th standard whl is lss uniform compard to that of th optimizd whl ( 1) in Fig. 15. Th prformanc of th standard whl and thos of BESO with diffrnt wight factors ar prsntd in tabl 2. It is sn that all th BESO whls hav a somwhat lssr volum than that of th standard whl. Th BESO whls hav lowr maximum von-miss strsss (for highr valus of λ s ) and highr natural frquncis (for lowr valus of λ s ), than thos of th standard whl, which is xpctd. To introduc a compromis optimum solution, th profil of ( 0.5, λ m = 0.5), is suggstd, with a volum of about 3 prcnt lss than that of th standard whl, a lowr maximum von-miss strss, and a comparabl natural frquncy. In ordr to addrss th manufacturability of th optimizd whls, th suggstd compromis wb profil with jaggd dgs is considrd and th dgs ar smoothnd with a polynomial curv (Fig. 18). Th modifid, smoothnd profil is r-analyzd to s how th prformanc is changd. Fig. 19 shows th von-miss strss distribution for this profil, and th prformanc valus ar givn in tabl 3. It is sn that th volum incras is ngligibl compard to th original BESO profil. But mor importantly, th maximum von-miss strss is dcrasd 17.5 prcnt, and th natural frquncy is incrasd by 11.5 prcnt, compard to th standard whl. This shows th ability of BESO in making lightr structurs with high prformanc. 5. Conclusion. In this work, multi-objctiv optimization of railway whl wb profil was invstigatd using BESO. Static and dynamic prformanc of th whl whr analyzd using finit lmnt. Basd on th combind snsitivity numbr and with diffrnt static and dynamic objctiv wight factors, diffrnt wb profils wr obtaind, with a bttr static and/or dynamic prformanc compard to th standard whl. Th compromis suggstd solution with smoothnd profil dgs outprformd th standard on significantly. As a final not, it should b statd that du to lack of nough information and quipmnt, it was not possibl for th authors to do a vrification of th findings. It is suggstd as a futur work to do xprimnts on th suggstd optimal whl, to vrify that indd is optimum or clos to it. 314
Vol. 48, No. 2, Dcmbr 2017 Initial wb block 1 λ m = 0 0.9 λ m = 0.1 0.7 λ m = 0.3 0.5 λ m = 0.5 0.3 λ m = 0.7 0.1 λ m = 0.9 Fig. 13 Rsultd profils from multi-objctiv optimization with diffrnt wight factors for th objctivs 315
Atai and Azarlu Fig. 14 History of man von Miss strss and volum fraction for λ m = 0, 1 Fig. 15 History of man von Miss strss and volum fraction for λ m = 0.5, 0.5 Tabl 2. Comparison of volum, maximum von-miss strss and first natural frquncy of standard and optimizd whl wb profil whl wb profil (Figs. 8 & 15) Standard λ m = 0 1 λ m = 0.1 0.9 λ m = 0.3 0.7 λ m = 0.5 0.5 λ m = 0.7 0.3 λ m = 0.9 0.1 volum (10-3 m 3 ) 0.1902 0.1899 0.1895 0.1875 0.1850 0.1890 0.1887 Maximum von- Miss strss (10 8 Pa) 0.360 0.355 0.351 0.357 0.354 0.597 0.764 First natural frquncy (HZ) 118.6695 101.4532 103.0177 102.60 105.0534 121.834 124.1121 Fig. 16 History of man von Miss strss and volum fraction for λ m = 0.9, 0.1 Fig. 18 Smoothning of th discontinuous optimizd profil, λ m = 0.5, 0.5 Fig. 17 von Miss strss distribution for EN standard whl 316
Vol. 48, No. 2, Dcmbr 2017 Fig. 19 Distribution von Miss strss for smoothnd profil. Tabl 3. Volum, maximum von Miss strss and first natural frquncy for th optimizd smoothnd profil Volum 10-3 (m 3 ) 0.18757 Maximum von- Miss strss 10 8 (Pa) 0.297 Natural frquncy (HZ) 132.367 Rfrncs [1] Xi, Y.M. and G.P. Stvn, A simpl volutionary procdur for structural optimization. Computrs & structurs, 1993. 49(5): p. 885-896. [2] Manickarajah, D., Y. Xi, and G. Stvn, Optimisation of columns and frams against buckling. Computrs & Structurs, 2000. 75(1): p. 45-54. [3] Huang, X. and Y. Xi, Convrgnt and mshindpndnt solutions for th bi-dirctional volutionary structural optimization mthod. Finit Elmnts in Analysis and Dsign, 2007. 43(14): p. 1039-1049. [4] Shabani Nodhi.S, S. R. Falahatgar, R. Ansari, Studying th ffcts of dsign paramtrs on th final topology of planar structurs by improvd bidirctional volutionary Structural optimization mthod, Modars Mchanical Enginring, Vol. 16, No. 5, pp. 29-38, 2016. (in Prsian) [5] Shobiri, V., Th topology optimization dsign for crackd structurs. Enginring Analysis with Boundary Elmnts, 2015. 58: p. 26-38. [6] Cazacu R, Grama L. Ovrviw of structural topology optimization mthods for plan and solid structurs. Annals of th Univrsity of Orada, Fascicl of Managmnt and Tchnological Enginring. 2014 Dc. [7] Farzangan.M,A.Ohadi Esfhani, S. H. Hosini, Invstigating th ffct of wb curvatur of wagon whl on itsphysical prformanc using, FEM,15th annual intrnational ISME confrnc, 2007. (in Prsian) [8] Frmér, M., Optimization of a railway fright car whl by us of a fractional factorial dsign mthod. Procdings of th Institution of Mchanical Enginrs, Part F: Journal of Rail and Rapid Transit, 1994. 208(2): p. 97-107. [9] Hirakawa, K. and H. Sakamoto, Effct of dsign variation on railroad whl fractur. ASME papr, 1981. [10] Huang, X. and M. Xi, Evolutionary topology optimization of continuum structurs: mthods and applications. 2010: John Wily & Sons. [11] Liang, Q.Q., Y.M. Xi, and G.P. Stvn, Optimal topology slction of continuum structurs with displacmnt constraints. Computrs & Structurs, 2000. 77(6): p. 635-644. [12] Cho, K.-H., J.Y. Park, S.P. Ryu, S.Y. Han., Rliability-basd topology optimization basd on bidirctional volutionary structural optimization using multi-objctiv snsitivity numbrs. Intrnational Journal of Automotiv Tchnology, 2011. 12(6): p. 849-856. [13] Cod, U., 510-2, 2004. Trailing stock: whls and whlsts. Conditions concrning th us of whls of various diamtrs. 317
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