Topology optimization considering the requirements of deep-drawn sheet metals

Transcription

1 th World Congress on Structural and Multdscplnary Optmsaton 7 th - th, June 5, Sydney Australa Topology optmzaton consderng the requrements of deep-drawn sheet metals Robert Denemann, Axel Schumacher, Serk Febg, Unversty of Wuppertal, Faculty D Mechancal Engneerng, Char for Optmzaton of Mechancal Structures, Wuppertal, Germany denemann@un-wuppertal.de schumacher@un-wuppertal.de Volkswagen AG, Braunschweg, Germany, serk.febg@volkswagen.de. Abstract Topology-optmzed desgns for mnmum complance or mnmum stress at mnmum mass are often framework structures due to ther homogeneous stress dstrbuton over the cross secton and therefore the best possble materal utlzaton. From the manufacturng s pont of vew complex framework structures, whch often develops durng topology optmzaton, are dffcult to manufacture because of possble undercuts. Manufacturng of these desgns s often only possble by jonng of numerous components or by D prntng. For mass producton sheet metal parts manufactured by deep drawng are often more effcent concernng the costs n relaton to ther performance. Therefore we mplemented a manufacturng constrant to the D topology optmzaton based on the densty method ensurng that thn walled structure results. Thereby more flexblty for the md surface desgn and also for cut-outs s reached compared to the optmzaton based on CAD-parameters. Also a varable thckness dstrbuton for talored blanks can be acheved. Results for deep drawng structures wth optmzed topologes wll be compared wth optmzed structures wthout manufacturng restrcton due to ther performance.. Keywords: topology optmzaton, sheet metals, deep drawng, manufacturng constrant, thn walled structures. Introducton The optmzaton of shell structures s mportant n the feld of mechancal engneerng, but also n cvl engneerng and archtecture (roof structures). In these felds a strengthened research has taken place n recent years. Ansola et. al [] propose a combnaton of CAD-parameters for the md surface descrpton and the SIMP-algorthm for the dentfcaton of optmal cut-outs. Thereby the optmzaton algorthm runs serally through the shape optmzaton of the md surface and afterwards the topology optmzaton. Ths approach was taken up by Hassan et al. [] and a smultaneous shape- and topology-optmzaton was ntroduced. The shape optmzaton takes place n the Fnte Element Model, whch shape can be modfed by control ponts of splnes. Both methods hghly depend on the parametrzaton of the md surface. Zenkewcz and Campbell [] use the de coordnates as desgn varables nstead of the CAD-parameters. Thereby a larger freedom of desgn s acheved. However by usng senstvtes of coordnates of boundary des the fnte element mesh becomes rregular. Yonekura et al. [] keep the mesh regularty for small shape modfcatons. In lterature there are few attempts for the optmzaton of shells based on sold elements. Lochner-Aldnger and Schumacher [5] use the densty method and extract sosurfaces of the element denstes as md surfaces.. The new approach for topology optmzaton for deep-drawn sheet metals Our new approach uses the homogensaton method [6] on a lnear voxel mesh. The derved method Sold Isotropc Materal wth Penalsaton (SIMP) ntroduces materal wth the artfcal densty < ρ and Young s modulus E n element (see equaton ). E s the Young s modulus of the basc materal. By ncreasng the penalty exponent s over. ntermedate denstes are penalzed and thereby the optmzed desgn rather converges to a black&whte desgn. E s ρ E = () Because of the use of senstvtes our approach s sutable for lnear statc load cases. All knds of objectve functons or constrants can be used, f ther senstvtes are kwn.

2 .. Calculaton of the md surface To allow the manufacturng by deep drawng n a sngle formng step, the optmzed structure must have - undercuts. - a constant wall thckness. Thereby the thnnng durng the formng process s neglected. By t consderng the formng process also the materal hardenng and resdual stresses are gred. These two manufacturng constrants can be acheved by modfyng the senstvtes of the objectve functon. An ncrease of the element denstes s only allowed near the current md surface. Thus the md surface can move accordng to the senstvtes. The md surface can be found by calculatng the average of the element coordnates n the punch drecton weghted wth the element denstes. Fgure shows the procedure of dervng the md surface from the volume mesh. Only a sngle cross secton s dsplayed. Intally the user has to defne the global punch drecton. The mesh s dvded nto columns wth the same wdth w, whch s the element edge length. The mdpont of each column s calculated by equaton. element densty ξ element mdpont punch drecton column boundary ground lne element poston ξ pont of md surface Fgure : Calculaton of md surface ξ = ρ ξ m () ρ ξ are the dstances between the element mdponts from a ground lne. For one exemplary column these dstances are marked as grey arrows. The mdpont of each element decdes to whch column the element belongs. The connecton of all mdponts wth dstance ξ m present the md surface.. Penalzaton of senstvtes In order to get a shell structure senstvtes far away from the md surface are penalzed. The penalzaton factor P for the senstvtes of each element s calculated by equaton. = a P atan b d π b () d s the mnmum dstances between the mdpont of element and the md surface. b s the user defned desred wall thckness, a / b descrbes the dscreteness of the penalty functon (see Fgure ). A larger quotent a / b ensures that the shell thckness does t exceed b, but slows down the convergence rate. The penalsaton factor s P,. rmalzed ] [

3 P b / d Fgure : Graph of penalsaton functon for element senstvtes as a functon of the dstance from the md surface.. Convergence The movement of the md surface can stagnate, f the penalsaton of the senstvtes s stronger than the mprovement of the objectve functon. Ths problem s solved by alternatng the desred wall thckness b. By ncreasng the desred wall thckness, elements are accumulated at the sde of the shell, where the senstvtes are larger, by decreasng the desred wall thckness the shell s mdface has moved to an mproved desgn. Also the penalsaton of ntermedate denstes has an nfluence on the convergence. Fgure shows the movement of the md surface. Only a sngle cross secton s dsplayed. Even f a lower located md surface would be better for the possble objectve functon complance, the stffness of the structure would temporarly decrease due to the lower stffness of elements wth penalzed ntermedate densty (mage at the rght). That s the reason why the optmzaton starts wthout penalsaton of ntermedate denstes (penalty exponent s = ) untl a convergence crteron s reached. Thereby at least the tensle/compressve stffness remans the same between the mages on the left and the rght. After the ncrease of the penalty exponent ths convergence problem s also solved by alternatng the desred wall thckness. element densty md surface Fgure : Movement of md surface through change of elements denstes. Optmzaton procedure Fgure shows the optmzaton algorthm. Convergence crteron can be the change of the objectve functon from one teraton to the next one or the maxmum change of element denstes. Convergence crteron s the mprovement of the objectve functon after the alternaton of the desred wall thckness. Durng the alternaton of the desred wall thckness and at the start of the optmzaton, the current desred wall thckness bcurr s larger than the desred wall thckness b. 5. Examples In the followng example topology optmzatons of a cantlever beam (see Fgure 5a) wth and wthout manufacturng constrant are performed. The complance s mnmzed consderng a volume fracton constrant of 6.5 %. The desgn space s dscretsed by 8 8 elements. One end of the structure s fxed, at one edge a lne load of N/mm s appled. The elements at the lne load are defned as n-desgn space. A senstvty flter wth the radus of r = (.7 element edge lengths) and a penalty exponent s = are used. The materal s steel wth Young s Modulus E = MPa and Posson s Rato ν =..

4 Start desgn, penalty exponent, desred wall thckness, current desred wall thckness Calculaton of senstvtes Flterng & penalzaton of senstvtes Update of element denstes Fnte element analyss Decrease current desred wall thckness Convergence crteron reached? Desred wall thckness b = current desred wall? thckness Increase current desred wall thckness Convergence crteron reached? Penalty exponent Penalty exponent? Optmzed deep drawable sheet metal found Fgure : Optmzaton procedure mm z Mses Stress [MPa] 8 6 y x a) b) Complance [Nm] 5. Cantlever Beam wthout manufacturng constrant Wthout the manufacturng constrant a complance of 8.7 Nmm s acheved (see Fgure 5b/c). 6 teratons were necessary followed by a fnal converson to a black&whte desgn. The convergence crteron s the mprovement of the objectve functon per teraton of less than. % per teraton. 8 Iteraton c) Fgure 5: Cantlever Beam: a) FE-Model wth loads and boundary condton, b) Stresses of fnal desgn (converted to black&whte desgn) wthout manufacturng constrant, c) Complance hstory

5 5. Cantlever Beam wth manufacturng constrant The same optmzaton task as n chapter 5. s performed by usng the optmzaton procedure for thn walled structures descrbed n chapter.. The desred wall thckness s b = ( element edge lengths). Ths s the thnnest possble structure that ensures that a bendng stress state can be represented wth lnear volume elements. The punch drecton was chosen as z. The convergence crtera and were the mprovement of the objectve functon per teraton of less than. %. The penalsaton parameter for the manufacturng restrcton s chosen as a = 5. Fgure 6a shows the complance hstory of the topology optmzaton process. In Fgure 6b) the success of the alternaton of the desred wall thckness between ntermedate result and can be seen. In Fgure 7 the desgn changes durng the optmzaton process are shown. 6 Iteraton Current desred wall thckness [multple of element edge length] Complance [Nm] SIMP penalty exponent s Complance [Nm] 8 5 Iteraton a) 5 b) Fgure 6: Complance hstory: a) whole Optmzaton (logarthmc scale) wth change of penalty exponent s, b) detal of convergence hstory wth change of desred wall thckness Element densty Iteraton Iteraton 9 Iteraton 68 5 Iteraton 76 Iteraton 77 6 Iteraton 5 Fgure 7: Element denstes of ntermedate results durng the optmzaton (elements wth densty x >. ) 5

6 In Fgure 8a the fnal black&whte desgn of the shell structure s shown. Ths structure reaches a complance of. Nmm at a bucklng safety of In comparson to the optmzaton wthout manufacturng constrant the complance s.7 % worse, whereby the manufacturng s much easer. Mses Stress [MPa] 8 6 a) b) Fgure 8: Stress a) of fnal desgn wth manufacturng constrant (converted to black&whte desgn), b) converted to surface model wth shell elements In order to check the qualty of the fnte element model wth sold elements, a surface model of the optmzed desgn wth the same volume has been created. Thereby the complance ncreases by.6 %. As to be seen n Fgure 8, also the stresses are very smlar, although the sold model s calculated wth only three lnear voxels across the sheet metal thckness. 6. Fnal remarks Besdes the shown applcaton examples, the manufacturng constrant for the topology optmzaton of deep drawable sheet metals has been tested for several structures wth multple load cases. The results are promsng, but we have to te that the gradent based optmzaton method wll fnd most probably only local optma. Compared to gradent based topology optmzatons wthout manufacturng constrant the presented method needs more teratons and the objectve functon of the optmzed desgns s usually worse, but t can be guaranteed that the structures can be manufactured easly. Further research actvtes wll focus on the mprovement of the computatonal effcency, multshell structures, stress- and bucklng constrants, the mplementaton of the deep drawng smulaton n the optmzaton and automatzaton of the converson to a surface model n order to perform a followng shape optmzaton. 7. References [] R. Ansola, J. Canales, J.A. Tarrago and J. Rasmussen, On smultaneous shape and materal layout optmzaton of shell structures, Structural and Multdscplnary Optmzaton,, 75-8,. [] B. Hassan, S.M. Tavakkol and H. Ghasemnejad, Smultaneous shape and topology optmzaton of shell structures, Structural and Multdscplnary Optmzaton, 8, -,. [] O.C. Zenkewcz and J.S. Campbell, Shape Optmzaton and sequental lnear programmng, In: R.H. Gallagher and O.C. Zenkewcz, Optmum Structural Desgn, John Wley & Sons, Chchester, 97. [] M. Yonekura, M. Shmoda and Y. Lu, Optmal Free-form Desgn of Shell Structure for Stress Mnmzaton, th World Congress on Structural and Multdscplnary Optmzaton, Orlando, USA,. [5] I. Lochner-Aldnger and A. Schumacher, Homogenzaton method, In: S. Adraenssens, P. Block, D. Veenendaal, C. Wllams, Shell Structures for Archtecture Form Fndng and Optmzaton, Routledge, New York, [6] M.P. Bendsøe, Optmal shape desgn as a materal dstrbuton problem, Structural Optmzaton,, 9-, 989 6