Published in: Proceedings of the Twenty Second Nordic Seminar on Computational Mechanics

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1 Downloadd from vbn.aau.dk on: aprl 09, 019 Aalborg Unvrs Implmnaon of Moldng Consrans n Topology Opmzaon Marx, S.; Krsnsn, Andrs Schmd Publshd n: Procdngs of h Twny Scond Nordc Smnar on Compuaonal Mchancs Publcaon da: 009 Documn Vrson Publshr's PDF, also known as Vrson of rcord Lnk o publcaon from Aalborg Unvrsy Caon for publshd vrson (APA): Marx, S., & Krsnsn, A. S. (009). Implmnaon of Moldng Consrans n Topology Opmzaon. In L. Damkld, L. Andrsn, A. S. Krsnsn, & E. Lund (Eds.), Procdngs of h Twny Scond Nordc Smnar on Compuaonal Mchancs (pp. 39-4). Dparmn of Cvl Engnrng, Aalborg Unvrsy. DCE Tchncal Mmorandum, No. 11 Gnral rghs Copyrgh and moral rghs for h publcaons mad accssbl n h publc poral ar rand by h auhors and/or ohr copyrgh ownrs and s a condon of accssng publcaons ha usrs rcogns and abd by h lgal rqurmns assocad wh hs rghs.? Usrs may download and prn on copy of any publcaon from h publc poral for h purpos of prva sudy or rsarch.? You may no furhr dsrbu h maral or us for any prof-makng acvy or commrcal gan? You may frly dsrbu h URL dnfyng h publcaon n h publc poral? Tak down polcy If you blv ha hs documn brachs copyrgh plas conac us a vbn@aub.aau.dk provdng dals, and w wll rmov accss o h work mmdaly and nvsga your clam.

2 Procdngs of h Twny Scond Nordc Smnar on Compuaonal Mchancs Aalborg Unvrsy 009 ISSN DCE Tchncal Mmorandum No. 11 Implmnaon of moldng consrans n opology opmzaon S. Marx and A. Krsnsn* Esbjrg nsu of Tchnology Aalborg Unvrsy, Aalborg, Dnmark mal: sm1076@sudn.aau.dk, ask@cvl.aau.dk Ky words: Topology opmzaon, njcon moldng, manufacurng consrans Summary In many cass h opology opmzaon mhod yld nadmssbl soluons n rspc o a parcular manufacurng procss,.g. njcon moldng. In h prsn work s chosn o focus on h mos common njcon moldng paramrs/facors drmnng h qualy of h mold gomry,.. unform hcknss, fllng of h d and jcon of h moldd m,.. xruson. Th mnond njcon mold paramrs/facors ar nroducd n h opology opmzaon by dfnng a cnrln of h nal doman and hn pnalz lmns n rspc o h dsanc o h dfnd cnrln of h doman. Inroducon On of h wdly usd procsss for manufacurng s njcon moldng. In hs papr an ffor s mad o mplmn consrans gvn by h procss of njcon moldng no h procss of opology opmzaon. Inroducon of manufacurng consrans hav bn consdrd by Ish al [3] whch proposs a modfd fram basd un cll approach provdng symmrcal cross scons. Also commrcal opology opmzaon sofwar nclud mhods o mpos manufacurng consrans as dvsd by Schramm al. [4], [5] suggsng a coupld opologysz approach. In h prsn work h opology opmzaon mhod dvlopd by Bndsø al. [1] s usd as bass. Algorhm o mpos njcon moldng consrans n opology opmzaon An algorhm mplmnng moldng consrans n h procss of opology opmzaon s dscrbd shorly n h followng. Th approach s o dfn h srucurs cnr lns on h background of opology opmzaon and usng hos lns o wakn h scons, whch would mak h njcon moldng of h soluon dffcul or mpossbl. Th us of lns By dfnng h cnr lns of h srucur h followng consrans of h njcon moldng procss can b asly mplmnd: Fllng of h d: I hr ar a unnrrupd lns from h usr dfnd placmn of h njcon nozzl o all pars of h srucur, han h lqud maral can fly n all pars of h srucur n h moldng procss. Drcon of draw: If h lns ar lmd n a way ha rspcs h drcon of draw, h possbl jcon of h m s guarand. Unform hcknss and no nrmda dnss: If all lmns, whch cnr pon s closr o h ln han half h hcknss, ar assgnd wh h dnsy 1 and all ohrs wh h dnsy 0 (or a vry small valu o avod sngulars), hn h rsulng m s of unform hcknss and dos no conan nrmda dnss. 39

3 Th dfnon of lns A ach sp of raon h lns ar dfnd subsqunly sarng a h usr dfnd placmn of h njcon nozzl by addng h nghborng nod, whch boh rspcs h angl of draw and s assgnd wh h hghs dnsy. Th nods ar assgnd wh h avrag dnsy of h lmns, whch cnr pons ar whn h crcl wh h damr of h hcknss. Th nod s no oband n h ln, f h assgnd dnsy s lowr han a cran valu, n h usd cass 0.. If wo nghborng nods hav h sam assgnd dnsy, h crcl s damr s ncrasd. Adjusng h complanc snsvy Th complanc snsvy of an lmn s adjusd n rspc o h dsanc bwn s cnr pon and h ln: f ds (1) max 1;n -sady 30 f ds ds In hs quaon h xprsson s h adjusd snsvy for h lmn, s h unadjusd snsvy for h lmn, s h usr dfnd hcknss, s h dsanc form lmn o h ln, n sady s h numbrs of sps of raon whr h ln was kp unchangd. Dfnon of h volum fracon Th volum of h srucur s dfnd by h sum of all lmns volum, whch cnr pon s closr o h ln han half of h hcknss and a nh or a housandh of h ohr lmns volum, f h ln lngh s ncrasng or sady rspcvly. v fr N l v N l v 1, wh h funcon fn sady 30 f ds 0001f n d 30 sady 1 f ds ds () Advanags of hs algorhm Th soluon from hs algorhm s always an m, whch asly can b manufacurd by h procss of njcon moldng, has unform hcknss and h jcon s possbl. I s opmzd by h procss of opology opmzaon whn h lms gvn by h manufacurng consrans. 40

4 Bcaus h combnaon of ln and hcknss dfns, whch lmns ar kp and whch lmns ar lf ou of h soluon, h mhod of SIMP s dspnsabl. Dsadvanags of hs algorhm Th algorhm ncrass h calculaon m, bcaus h ln has o b bul up squnally n vry sp of raon and hs ncluds assgnd nods wh dnsy. Afrwards h complanc snsvy of all lmns s adjusd n rspc o h dsanc bwn h lmn s cnr pon and h ln. Th calculaon m s no ncras havly compard o ohr flrng mhods. Vrfcaon of h algorhm Th algorhm s vrfd by a sandard cas, a quadrac canlvr bam wh h forc appld n h cnr of h fr sd. Th algorhm s rsul s compard o h rsul found by h opology opmzaon procss wh Esbjrg flrng and h SIMP-pnaly of 3.5. Esbjrg flrng Nw algorhm SIMP-pnaly p=3.5 Afr 83 sps: Ovrall chang of dnsy < 1/10,000 Complanc: Afr 1 sps: Ovrall chang of dnsy < 1/10,000 Complanc:

5 Th soluon of h Esbjrg algorhm has a complanc of ca. 0, whl h nw algorhm s a ll wakr wh complanc of 0.7. In h Esbjrg flrng h hghr valu of h complanc s h valu shown afr opmsaon, sll undr h SIMP pnaly. Th lowr valu s cland for hs nflunc. Th ll dffrnc shows, ha a fw lmns hav nrmda dnss. Th lowr valu n h nw algorhm s h complanc of all h lmns assgnd wh h dnsy 1, whl h hghr valu s h complanc for all lmns. In rspc o h lmaons o h nw algorhm hs sms rasonabl. Th dffrnc n h complanc can b xpland wh h aras abov and blow h cnr of h rgh dg. Hr h Esbjrg flrng has maral, whch volas h lmaon of unform hcknss. Thrfor hs aras ar no a par of h soluon of h nw algorhm. A h uppr rgh cornr hr s an ara blow h dark ara, whch s dark n h soluon of h opology opmsaon usng h Esbjrg flrng. Ths ara and h smlar and h lowr lf cornr vola h drcon of draw and ar hrfor no a par of h nw algorhm s soluon. Concludng rmarks Th nw algorhm mgh b a usful ool for dvlopng and opmzng ms, f s ngrad n commrcal sofwar. Insad of srnghnng or waknng ndvdual lmns, h procss works on scons. Rfrncs [1] Bndsø, M.P. and Sgmund, O., Topology Opmzaon, Thory, Mhods and Applcaons, nd don, Sprngr Vrlag, 00, p 1-68, [] Gorgsn, L.S., Mornsn, L., Topology Opmzaon, Implmnaon of a Dynamc cv, Aalborg Unvrs Esbjrg, 007 [3] Ish, K. and Aomura, S., Topology Opmzaon for h Exrudd Thr Dmnsonal Srucur wh Consan Cross Scon, JSME Inrnaonal Journal, Srs A, Vol. 47, No., 004 [4] Schramm, U., Thomas, H.L., Zhou, M. and Voh, B., Topology opmzaon wh Alar OpSruc, n Procdngs of h Opmzaon n Indusry II Confrnc, Banff, CAN (1999). [5] Alar OpSruc, Usr s Manual v7.0, Alar Engnrng Inc., Troy, MI (003). 4