Optimal shape and location of piezoelectric materials for topology optimization of flextensional actuators

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1 Optmal shape and loaton of pezoeletr materals for topology optmzaton of flextensonal atuators ng L 1 Xueme Xn 2 Noboru Kkuh 1 Kazuhro Satou 1 1 Department of Mehanal Engneerng, Unversty of Mhgan, Ann Arbor, MI Department of Cvl and Envronmental Engneerng, Unversty of Mhgan, Ann Arbor, MI Abstrat Pezoeletr atuator has been nreasngly used n MEMS system due to ts advantage of generalty and flexblty. A flextensonal atuator onsst of a pezoeram deve, whh an onvert eletral energy nto mehanal energy and ve versa, and a flexble mehanal struture, whh an onvert and amplfy the output pezoeram dsplaement n the desred dreton and magntude. A reent researh n ths area s optmzng the topology of the mehanal part whle fxng the eletral part. In ths researh, the loaton and shape of the pezoeletral omponent s optmzaton as an dsrete problem. The mxed optmzaton problem has been solved by two-layer optmzaton proedure ombned wth SLP and GA. Optmal result s presented and dsussed. 1 INTRODUCTION Pezoeletr atuators are beng nreasngly used n varous novel applatons [1]. A pezoeletr atuator usually onssts of two man omponents [2]: a mehanal part, whh s a flexble struture, and an eletral part, whh s the pezoeletr materal blok. One of the mportant ssues of usng pezoeletr atuator s to mprove ther performane for a ertan amount of pezoeletr materal, whh s the goal of pezoeletr atuator desgn. Desgn of pezoeletr atuators has been greatly advaned durng the past ten years. Researhers are fousng on every aspet and every omponent n order to aheve the best performanes, from pezo-eram omposte desgn, optmal szng and loatng to topology optmzaton. Topology optmzaton wth homogenzaton method was proposed by Bensdφe and Kkuh to desgn the stffest struture. Ths method s then appled to desgn omplant mehansms[4][5] and omposte materals. Sne the mehanal part of flextensonal atuators s atually a omplant mehansm, pezoeletr transduer [6] and thermal atuators [7][8] have been also desgned usng topology optmzaton tehnque. In ths prevous work of pezoeletr atuator optmzaton, topology and shape of the mehanal part of the atuator was desgned, however, the loaton and shape of the pezoeletr materal are fxed. Ths topologal desgn optmzaton was able to generate effetve mehanal struture that greatly mproved the performanes of the pezoeletr atuator. There have also been desgn optmzaton tehnques developed for the eletral part of pezoeletr atuator. The plaement and sze of pezo-materal was optmzed [10]. In a reent researh, the dstrbuton of pezoeletr materal n the optal MEMS was optmzed [11]. However the desgn of the flexble struture have not been taken nto aount. Sne the effetveness of the atuator s desvely dependent on both mehanal and eletral part, t s desrable to desgn both parts. The work presented here s based on the topology optmzaton tehnques and extends the desgn varables for shape and loaton of pezoeletr materal n the extended desgn doman. Dsrete optmzaton problem has been formulated n order to make the eletral part desgn onsstent wth the fnte element model n topology optmzaton. Two layered optmzaton tehnque has been developed. An example s presented here to support the tehnque. 2 OPTIMIZATION TECHNIQUES The use of fxed grd s the key pont n topology optmzaton tehnque. Thus the fnte element model does not hange durng the optmzaton proess and exessve dstorton to the fnte elements an be avoded. In order to aommodate the two desgn parts n the same fnte element model, the shape and loaton parameters are easly hosen as dsrete varables, whle the topology desgn parameters are stll ontnuous. Ths requres the problem to be dealt as a mxed varable optmzaton problem. In order to utlze the exstng topology

2 optmzaton software, a spef two-layered optmzaton method s proposed to separate the ontnuous and dsrete desgn varables n two optmzaton proedures, one over the other. That s: mn f( x) = mn[mn f( x)] ab, a b Thus the optmzaton problem has been deomposed nto two layers of optmzaton problems and the nner and outer optmzatons are performed by homogenzaton desgn method and genet algorthm respetvely. 2.1 HOMOGENIZATION DESIGN METHOD Fgure 1: Homogenzaton Desgn Method The topology optmzaton problems s formulated as a problem of fndng the optmal dstrbuton of materal propertes n an extended fxed doman. Where some struture ost funton s maxmzed. Therefore the fnte element model does not hange durng the optmzaton proess. Ths tehnque s appled based on the homogenzaton method. To relax the optmzaton problem, a mrostruture proposed by Bendsφe and Kkuh[1] s defned n eah pont of the doman whh s a unt ell wth a retangular hole nsde (Fgure 1). The use of mrostruture allows the ntermedate materals rather than only vod or full materal n the fnal soluton. The desgn varables are the dmensons α, β and the orentaton θ of the mro-hole. In ths sense the problem s to optmze the materal dstrbuton n a perforate doman wth nfnte mrosope vods. The effetve propertes of the porous materal, are alulated usng the homogenzaton methods. 2.2 GENETIC ALGORITHM Genet algorthms are searh algorthms based on the mehans of natural seleton and natural genets [12]. It s a very effent and robust method of dsrete optmzaton. The reason of hoosng GA n ths researh s that GA searhes a very large spae and t explots hstoral nformaton to speulate on new searh ponts. These make GA a speedy and effent algorthm. A genet algorthm reles on the proess of reproduton, rossover and mutaton of notons to reah the global or near global optmum. Reproduton s a proess by whh the ndvduals are oped aordng to ther objetve funton values. Crossover nvolves random matng of newly reprodued ndvduals n the matng pool. Mutaton s the oasonal random alteraton of a strng poston. Mutaton s neessary beause although reproduton and rossover effently searh and mx exstng, oasonally they may result n loss of some nfeasble solutons. Hgh performane notons are repeatedly tested and exhanged n the searh for better and better performane. GA s haraterzed by parameters p ( rossover probablty ) and p ( mutaton probablty). m 2.3 OPTIMIZATION PROCEDURE The two layered optmzaton proedure s shown n Fgure 2. Topology optmzaton, the nner layer, ontans a Sequental Lnear Programmng optmzer and fnte element analyss and alulatons as evalutaton. A tolerane for desgn varables are spefed as termnatng rtera. If the value of objetve s gettng worse and also all of the varables have smaller than 10% hange from ther prevous value, optmzaton s termnated and the urrent desgn s returned as the optmal. In the outer layer, the genet alogrthm optmzaton s onduted by omeral optmzaton software SIGHT5.5. GA parameters are automatally generated and updated nternally durng the proess. It s observed durng several experments from dfferent ntal desgn that after 150 teratons, GA does not produe a sgnfantly better desgn, but wll osllate wthn a small range. Due to the harasterst a maxmum teraton number of 200 s hosen to get the best result wthn a short alulaton tme. The topology optmzaton part and dsrete genet algorthm optmzaton are onneted through a program whh generates topology optmzaton nputs (fem.p and opt.p) wth the dsrete varables generated by genet algorthm optmzaton. 3 PROBLEM FORMULATION The problem formulaton s smlar to that of the topology optmzaton of flextensonal atuator as n[1]. In ths work, for smplty, the eletral part s fxed to be onepee retangular blok algn n horzontal dreton wth the dmensons and loaton beng dsrete desgn varables. The desgn problem and extended desgn doman s show n Fgure DESIGN VARIABLES Contnuous desgn varables are those of topology optmzaton: α, β (0,1]: Dmensons of the mrosop vods n homogenzaton desgn method; θ [0, π] : Orentaton of the mrosop vods;

3 Dsrete desgn varables are those of pezoeletr materal blok: fem.p & opt.p template Outer loop: Genet Algorthm Inner Loop: SLP Start Intalzaton Buld FEM model fem.p & opt.p FEM analyss N Converge? Optmal Topology Max ter #? Optmal Result End N New values for dsret desgn varables Senstvty Analyss Mutaton / Seleton rossover Fgure 2: Flow har of the two- layered optmsaton 2.Volume onstrant of the mehanal part: m = 1 (1 αβ ) V V 0 e Where V s the volume of a full element wth no vods. e 3. The total area of the pezoeletr blok s assumed to satsfy: ab A 0 4. The pezoeletr blok must reman nsde of the desgn doman: x x x l u y y y l u sup where for dfferent desgn doman, the boundary values are dfferent. In some engneerng ase, they are defned by onsderng also the engneerng feasblty. 5. Coupled equlbrum equatons for three dfferent loadng ases: a Nb, N : Dmensons of pezoeletr blok; x N, y N : Coordnates of the lower-left orner of pezoeletr blo V { 1,1} : Dreton of the appled voltage, whh determnes the dsplaement dreton of the pezoeletr blok for ths gvng polarzaton; V (X,) a desred dsplanement b H K K uuu U F =, k 1,2,3 ( ) = T k K K φφ Q where ( k ) represents three dfferent loadng ondton onsdered for objetve funton, shown s Fgure 4. U and Ö are dsplaements (mehanal degrees of freedom) and voltage (eletral degree of freedom) respetvely n k-th loadng ase. K H s the global uu stffness matrx, alulated through homogenzed elast tensor. H 1 = ( x, y)( I å( χ)) d 3.2 CONSTRAINTS Fgure 3: desgn problem 1.Sde onstrants of the mrosop vods: where χ s the haraterst deformaton, and represents the unt ell mrostruture. (Detals refer to [4]). Also, K s the deletr matrx and K s the φφ pezoeletr (oupled) matrx. F α sup 0 α α < 1 sup 0 β β < 1 β sup sup where and are upper lmts for dmenson of the vods. These prevent the exstene of the zero stffness, whh may ause the ll posed stffness matrx. Fgure 4: Loadng ases for mult-objetve funton

4 The loadng ases are as shown n Fgure 4: 1) only voltage s appled at the pezoeletr materal; 2) only a dummy load s appled at the pont of desred dsplaement and 3) the voltage s fxed and dummy load s appled. Case (1) and (2) formulate the mutual mean omplane to meet the knemats requrement and Case (3) formulate the mean omplane to meet the stffness requrement. 3.3 OBJECTIVE FUNCTION Ths mxed varable problem has the same objetve funton as the topology optmzaton problem [4]. For ompleton, we repeat the formulaton of the multobjetve problem. Consderng the three loadng ases n Fgure 4. The objetves are: 1. Maxmze the mutual mean omplane: A spef flextensonal atuator desgn problem s hosen as a follow-up to a prevous desgn problem. The struture layout are hosen to be both the mehanal part and the eletral part nsde of a spefed desgn doman as llustrated n Fgure 5, are optmzed Nshwak, et al[9] gave an example desgn problem by fxng the pezoeletr part at the entral bottom of the desgn doman. The topology optmzaton result s shown n Fgure 6 and values of nputs and outputs are shown n Table 1. MMC = whh s the knemats requrement or flexblty requrement. 2. Mnmze the mean omplane: MC = whh s the strutural requrement or stffness requrement. The mult-objetve funton s formulated as: 4 EXAMPLES T (1) H (2) K K uuu (1) T (2) K K T (3) H (3) K K uuu (3) T (3) K K max Fgure 5: Desgn doman of the example φφ φφ MMC f = MC In ths two-layered optmzaton problem, the loaton and dmensons of pezoeletr part are optmzed wth genet algorthm, whle the mehanal part s optmzed wth prevous topology method. A fnte element method s used to alulate the objetve funton The followng parameters are used n ths problem: However, t should be noted that t s not neessary to assgn unts to the parameters and varables, sne the topology optmzaton s appled on lnear elast fttous materal. Assgnng unts to the parameters and varables does not have any physal meanng. Optmal result A B 2 5 x, y 11,8 5 V -1 (-) 1 (+) Objetve Dsplaement at desred pont Fgure 6: Prevous result L = 40, H = 20; A = 100; 0 x = 0, y = 0; Prevous result Table 1: Optmal result ompared wth the prevous

5 Fgure 7 shows the optmal onfguraton after 200 runs. Table 1 ompares the values of varables and objetves of the optmal desgn and the orgnal desgn. In the orgnal desgn, the dmensons of the pezoeletr blok s 20*5 and s loated at (11,1), whle n the optmzed desgn, the sze of the pezoeletr blok s redued to 22*2, and the loaton s moved up to (11,8). Wth the optmzed desgn, the objetve funton value s nreased by 175%, and the dsplaement s nreased by 33%. It s obvous the optmal result s an aeptable mprovement to the prevous result. Also, by allowng more freedom to the desgn, we an expet less stress onentraton n the sense that dfferene of stresses at dfferent pont s smaller. Genet algorthm has shown ts advantage of handlng dsrete desgn varables. If pure mutaton s used to try dfferent desgn, t needs a total of more than 576,000 topology optmzaton teratons, whh take more than 400 days to fnsh. By genet algorthm a feasble loal optmal soluton an be obtaned wthn 2 days. If more omputatonal tme are allowed, a better global optmal an be possbly obtaned. (a) (b) Fgure 7: Optmal onfguraton (a) materal dstrbuton (b) threshold result To verfy the result, an mage proessng tehnology s appled to the optmal materal dstrbuton to obtan a fnte element model of a sold struture. Then the fnte element analyss s onduted wth ABAQUS. The struture s deformed n the desrable manner as shown n Fgure 8. 5 CONCLUSIONS In ths researh, optmal pezoeletr atuator desgn was aheved by gvng more desgn freedom than the topology optmzaton. The two layered optmzaton proedure has suessfully ollaborate two optmzaton tehnques and provded onsstently mproved desgn. Genet Algorthm has worked n an effent manner. The tehnque presented here an be also appled to three dmensonal desgn optmzaton problems. The same methodology an be utlzed n desgns of other atuator and strutures, whh ontans two or more dfferent materals. For nstane, the future desgn of thermal atuators, b-materal omplant mehansms and MEMS an be possbly benefted from ths method. Furthermore, more dsrete desgn varables an be added, suh as materal types and atuaton type, and more objetve funton an be onsdered for the desgn of eonom and envronmental onsous mehansms and deves. Aknowledgment Fgure 8: Deformaton verfaton from ABAQUS The authors thank Engneous In for generously provdng SIGHT lene. Sne ths work s based on ourse projet of ME558 (Dsrete optmzaton methods) n Fall 2000 at the Unversty of Mhgan, the authors are grateful for the opportunty from Unversty of Mhgan. Supports and onerns from omputatonal mehans lab are also appreated.

6 Referenes [1] C. Near (1996) Pezoeletr atuator tehnology. Proeedng of SPIE [2] K. Ontsuka, A. Dogan, J. Tressler, Q. Xu, S. oshkawa and R. Nownham (1995) Metaleram omposte transduer, The Moone. Journal of Intellgent Materal Systems and Strutures [3] M. Bendsφe and N. Kkuh (1988) Generatng optmal topologes n strutural desgn usng a homogenzaton method. Computer Methods n Appled Mehans and Engneerng [4] O. Sgmund (1997) On the desgn of omplant mehansms usng topology optmzaton Mehans of Strutures and Mahnes [5] S. Nshwak, M. Freker, S. Mn and N. Kkuh (1998) Topology optmzaton of omplant mehansms usng homogenzaton method. Internatonal Journal for Numeral Methods n Engneerng [8]. L, B. Chen and N. Kkuh (1999) Topology optmzaton for mehansms of thermal atuaton wth Eo-materals. Proeedng of the th Internatonal Conferene on Eomaterals [9] S. Nshwak, E. Slva,. L and N. Kkuh (1998) Topology optmzaton for flextensonal autators AIAA onferene St. Louse [10] J. Man, E. Gara and D. Howard (1994) Optmal plaement and szng of pared pezoatuators n beams and plates. Smart Materal and Struture [11] Gabbert, U. and Weber, C-T (1999) Optmzaton of pezoeletr materal dstrbuton n smart strutures. Proeedngs of SPIE- The Internatonal Soety for Optal Engneerng [12] D. Goldberg (1989) Genet algorthms n searh, optmzaton and mahne learnng Addson-Wesley Publshng [6] E. Slva, N. Kkuh (1999) Desgn of pezoeletr transduer usng topology optmzaton. Smart Materals and Strutures [7] J. Jonsmann, O. Sgmund, S. Bouwstra(1999) Complant eletro-thermal mroatuators. Proeedng of the th IEEE Internatonal Conferene on MEMS