PARTITION HOLE DESIGN FOR MAXIMIZING OR MINIMIZING THE FUNDAMENTAL EIGENFREQUENCY OF A DOUBLE CAVITY BY TOPOLOGY OPTIMIZATION

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1 ICSV4 Cns Australia 9- July, 007 PARTITION HOLE DESIGN FOR MAXIMIZING OR MINIMIZING THE FUNDAMENTAL EIGENFREQUENCY OF A DOUBLE CAVITY BY TOPOLOGY OPTIMIZATION Jin Woo L and Yoon Young Kim National Crativ Rsarch Initiativs Cntr for Multiscal Dsign, School of Mchanical and Arosac Enginring, Soul National Univrsity San 56- Shilim-9-dong, Kwanak-gu, Soul, 5-74, Rublic of Kora jw06nam@yahoo.co.kr Abstract A toology otimization mthod is dvlod to minimiz or maximiz th fundamntal ignfrquncy of a doubl cavity. Th acoustic modl consists of two rctangular cavitis and a holy artition. Bcaus th numbr and locations of hols in th artition affct th ignfrquncis of th doubl cavity significantly, th undrlying acoustical charactristics of th doubl cavity can b controlld by adjusting thm. In this work, th ignfrquncy control roblm is formulatd as an acoustical toology otimization roblm whr th fundamntal acoustic ignfrquncy is minimizd or maximizd for an allowd hol volum. For th formulation, th artition is dividd into sub-artitions, ach of which has variabl matrial rortis. Whn a sub-artition has acoustical rortis of, it is rgardd as a hol. Intrmdiat stats btwn and a rigid body ar introlatd by a carfully-slctd nalizd function in ordr to roduc a clar hol distribution at th convrgd itration.. INTRODUCTION An acoustical toology otimization mthod is alid to th artition hol dsign of a hol-artitiond doubl cavity. Th doubl cavity is a simlifid cavity modl suggstd in th rvious ar [], whr th acoustical charactristics of a assngr vhicl comartmnt with a trunk wr invstigatd. Rcntly, a toology otimization mthod has bn mloyd to acoustical dvic dsign sinc it was suggstd by Bndsø and Kikuchi []. Jnsn and Sigmund [3] formulatd an acoustical toology otimization roblm for systmatic dsign of acoustical dvics. L t al. [4] otimizd th toology of thin-body for th radiation and scattring of sound using gntic algorithms. Wadbro and Brggrn [5] rsntd an acoustic horn dsign mthod by toology otimization. Dühring [6] alid toology otimization to obtain otimal matrial distribution in th ciling. Howvr, an acoustical toology otimization roblm for artition hol dsign has not bn rortd. A doubl cavity consists of two cavitis of th sam cross-sction and a holy artition btwn thm. Th total cross-sctional ara and th osition of hols strongly affct - -

2 ignfrquncis of longitudinal acoustic mods of th doubl cavity. Th goal of this work is to find an otimal hol distribution for th minimum and maximum fundamntal ignfrquncis of th doubl cavity for a givn constraint on th total hol volum. In this ar, th artition hol dsign roblm is formulatd as an acoustical toology otimization roblm. To this nd, th artition btwn two cavitis is dividd into sub-artitions, whos acoustical rortis can vary during otimization rocss. Carfully-slctd introlation functions ar usd to nsur that ach sub-artition bcoms a hol or a rigid sub-artition at th final otimizd rsults. To carry out ignfrquncy analysis, finit lmnt modl is mloyd and a gradint-basd otimization algorithm is mloyd to udat dsign variabls.. ACOUSTIC TOPOLOGY FORMULATION As shown in Figur, a doubl cavity consists of two rctangular cavitis and a artition. Cavity and Cavity ar filld with, and th artition btwn th two cavitis is dividd into svral sub-artitions, ach of which is filld with an artificial matrial. Assuming that th width and th hight of th cavity ar smallr than its lngth, only longitudinal acoustic mods ar considrd. Acoustic rssur of th acoustic systm is govrnd by th Hlmholtz quation: ω + ρ K = 0, () whr ρ is th dnsity of acoustic mdium, and th angular frquncy ω is dfind by ω = π f, whr f is a frquncy. Th bulk modulus K is dfind by th sound sd in acoustic mdium. K = ρc, whr c is Figur. A doubl cavity with a holy artition Using a finit lmnt mthod, quation () is convrtd to [ M] P = 0 K ω () whr P is th acoustic rssur vctor. Th global stiffnss matrix K and th global mass matrix M ar xrssd by using an assmbly orator A as follows: n= N - -

3 N N = k n M mn n = An = K A = (3) From th condition to obtain a nontrivial solution from quation (), th following charactristic quation of th doubl cavity is obtaind: dt[ K ω M] = 0 (4) Th dsign domain is th artition which is dividd into sub-artitions. Th width of a sub-artition is qual to that of th artition. Each sub-artition is suosd to bcom a hol or a rigid sub-artition at th nd of otimization itration sts. On dsign variabl χ is assignd to ach sub-artition and it has valus of btwn 0 and. Th two limiting valus corrsond to ( χ = 0 ) or a rigid body ( χ = ). Thrfor, it changs acoustic rortis of th sub-artition and is udatd by th MMA (Mthod of Moving Asymtots [7]) during th otimization rocss. It rquirs th snsitivity of th r th ignfrquncy with rsct to a dsign variabl χ, which is givn as r χ P ω T K = T r Pr r Pr MPr χ ω P T r M Pr χ (5). Dsign roblm I: ignfrquncy minimization To minimiz th ignfrquncy of th fundamntal acoustic mod of th doubl cavity, th following objctiv function and th constraint ar mloyd. Min L = 0 χ f (6-a) Subjct to N = χ / N V (6-b) whr f is th fundamntal ignfrquncy and Vo is th ratio of th total hol volum to th artition volum. Th symbol of N is th numbr of total sub-artitions. Introlation functions of dnsity ρ and bulk modulus K of th th sub-artition affct significantly final otimizd rsults; without ror choics, it would b difficult to idntify hol distributions rcisly. For this minimization roblm, w mloyd ( χ ) = / ρ + χ ( / ρ ) / ρ ρ o rigid / (7-a) ( ) = / Κ + χ ( / Κ ) / Κ χ Κ (7-b) rigid / whr th subscrits and rigid stand for and a rigid body, and is th nalization aramtr

4 . Dsign roblm II: ignfrquncy maximization To obtain th maximum ignfrquncy of th fundamntal acoustic mod of th doubl cavity, th following acoustical toology otimization roblm is st u: Max L = log f 0 χ (8-a) Subjct to N = χ / N V (8-b) In this cas, th following introlation functions ar shown to b aroriat: o ρ Κ ( χ ) = ρ + χ ( ρ ρ ) (9-a) rigid ( ) = Κ + χ ( Κ Κ ) χ (9-b) rigid 3. NUMERICAL RESULTS Sinc a hol distribution in th z dirction was considrd, a two-dimnsional finit lmnt modl in x-z lan was mloyd for th convninc of analysis. Th numrical data usd for all dsign roblms wr as follows: l =.50 m, l = 0.50 m, l = 0.0 m h = 0.3 m, N = 0, V = 0. o 3 ρ =. kg/m, = 343 m/s 5 c, ρ rigid = 0 ρ, crigid = 0 c Considring th hol volum ratio V o = 0., two sub-artitions wr xctd to bcom hols. Th valus of th nalty xonnts wr = for th minimization roblm and = 3 for th maximization roblm. To minimiz th risk of obtaining a local minimum or maximum, tn initial gusss wr usd. It was xctd that this stratgy could yild th global minimum or maximum ignfrquncy. 3. Dsign roblm I: numrical rsults Th hol artition dsign roblm formulatd in quations (6) and (7) was solvd to obtain otimal hol distribution for th minimum ignfrquncy of th first ignmod. Figur lots itration historis of th first ignfrquncy for tn initial dsign variabls. Among 0 cass, two cass yildd 57.3 Hz as a minimum ignfrquncy. Figur 3 shows svral initial dsign variabls and otimizd dsign variabls. Black rrsnts a rigid body and whit rrsnts a hol in an otimizd toology. Nithr filtring nor ost-rocssing was not usd to obtain this otimal hol distribution. Cas 4 is takn as th solution to this toology otimization roblm. Two hols ar locatd conscutivly at th dg of th artition. Figur 4 shows acoustic mod of th doubl cavity at th minimum ignfrquncy

5 Figur. Itration historis of th st ignfrquncy in th Dsign roblm I. Figur 3. Changs in otimal artition layouts for initial dsign variabls in th Dsign roblm I. Figur 4. Acoustic mod at a minimum ignfrquncy of th st ignmod (57.3Hz) - 5 -

6 3. Dsign roblm II: numrical rsults Th hol artition dsign roblm st u in quations (8) and (9) was solvd for th first ignfrquncy. As in Dsign roblm I, a maximum valu among tn cass is takn as th max solution to this dsign roblm ( f = 76. Hz ). Figur 5 comars itration historis of th first ignfrquncy. Figur 6 shows a hol distribution in th artition of th otimizd rsult, whr two hols ar locatd symmtrically around th y-axis on th artition. Figur 7 dislays th acoustic mod of th doubl cavity at th maximum fundamntal ignfrquncy. As in Figur 4, it is rovd that hols distribution strongly affcts acoustic rssur distribution as wll as ignfrquncy of th hol-artitiond doubl cavity. W calculatd th st ignfrquncy of th doubl cavity for svral symmtrically-locatd hols as shown in Figur 8. On intrsting oint is that th ignfrquncy for two hols locatd at th cntr of artition is qual to that for two hols locatd at both nds of th artition ( f = 68.4 Hz ). That is to say, a maximum ignfrquncy could b xctd whn two saratd hols ar locatd som lacs btwn cntr and dg, rsctivly. This brif invstigation shows th validity of th rsult obtaind in this maximization roblm. Figur 5. Itration historis of th st ignfrquncy in Dsign roblm II. Figur 6. Hol distribution in th artition of th otimizd rsult

7 Figur 7. Acoustic mod at a maximum ignfrquncy of th st ignmod (76. Hz). Figur 8. Comarison of ignfrquncis for svral symmtric hol distributions. 4. CONCLUSIONS W formulatd two acoustical toology otimization roblms to ffctivly dsign a holy artition so that th hol-artitiond doubl cavity could hav th dsird acoustical charactristics. Diffrnt objctiv functions, constraints and introlation functions wr mloyd to obtain th minimum and maximum ignfrquncy of th fundamntal ignmod of th doubl cavity. W succssfully obtaind th rasonabl hol distribution on th artition for minimum or maximum ignfrquncis, whr acoustic mods wr comard focusing on th acoustic rssur distribution around th artition. Th hol distribution in th artition affcts not only an ignfrquncy of th doubl cavity but also its ignmod strongly. Th numrical rsults showd th ossibility that our dsign stratgy basd on toology otimization could b alid to a two-dimnsional holy artition dsign roblm. Although w dividd th artition into only 0 sub-artitions, th formulation dvlod in this work can b asily alid to th hol-artitiond doubl cavity having mor than 0 sub-artitions. ACKNOWLEDGMENT This rsarch was suortd by th National Crativ Rsarch Initiativs Program (Kora Scinc and Tchnology Foundation grant No ) contractd through th Institut of Advancd Machinry and Dsign at Soul National Univrsity

8 REFERENCES [] J.W. L, J.M. L and S.H. Kim, Acoustical analysis of multil cavitis connctd by ncks in sris with a considration of vanscnt wavs, Journal of Sound and Vibration 73, (004). [] M.P. Bndsø and N. Kikuchi, Gnrating otimal toologis in structural dsign using a homognization mthod, Comutr Mthods in Alid Mchanics and Enginring 7, 97-4 (988). [3] J. S. Jnsn and O. Sigmund, Systmatic dsign of acoustic dvics by toology otimization, Procdings of th Twlfth Intrnational Congrss on Sound and Vibration (ICSV), -4 July 005, Lisbon, Portugal [4] J. L, S. Wang and A. Dikc, Toology otimization for th radiation and scattring of sound from thin-body using gntic algorithms, Journal of Sound and Vibration 76, (004). [5] E. Wadbro and M. Brggrn, Toology otimization of an acoustic horn, Comutr Mthods in Alid Mchanics and Enginring 96, (006). [6] M.B. Dühring, Toological dsign otimization of structurs, Machins and Matrials, Sringr 37, (006). [7] K. Svanbrg, Th mthod of moving asymtots a nw mthod for structural otimization, Intrnational Journal for Numrical Mthods in Enginring 4, (987)