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J ournal o f A pp Published Onli n http://dx.doi.or g Open Access Mul t Proj 3 Depar t ABSTRA C This paper c o layer. The de p tained and all Pareto-frontie MATLAB. Keywords: P o U 1. Introdu c Porous p iezo c cal imaging, other p iezoel e tions are refe r tures for aco u zoelectric ma t tivity, extend e ing to the aco u the developi n structures of t On the base o ous materials tion problem structure hav e tors and com b types of mod e transmission. rating sound a In order to e acoustic tran s including: lo w with water, d high mechani c widely used d vices based o n their effectiv e p lied Mathemat i n e November 2 0 g /10.4236/jamp . t iobjec t ecto r w A. V. N 1 Mathem a 2 Departmen t t ments of Micro e C T o ncerns the o p p endences of t owe d to decr e r calculation h o rous Piezoce r nderwate r Ap p c tion c eramic mater i ultrasonic tr a e ctric devices. r red to the de u stic transduc e t erials due to t e d frequency u stic medium. n g of methods t he transducer s o f developed n effective mo statement a n e been imple m b ined devices — e rn devices u s A projector i s a nd transmitti n e ffectively fu l s ducers have w impedance d urability; hi g c al load and h d evices in un d n the p iezoele e applications c i cs and Ph y sics , 0 13 (http://ww w .2013.16017 t ive Op t w ith Po r N asedkin 1 , M a tical Modeling t of Electric al E n e lectronics & M p timization p r o t he effective m e ase the numb e h as been s o lve d r amics; Multi o p lications i als are widel y a nsducers, h y A large num b velopmen t of e rs based on t hei r high p ie z bandwidth an The aim of pr for synthesi z s for underwa t n umerical met h duli determin n d solving fo r m ented. Hydr o — transceivers s ed for sonar s a transduce r n g it to the a c l fill its p urp o to meet man y for better ac o g h sensitivity; h ydrostatic p re d erwate r acou s ctric effect. T h c aused by b ot h , 2013, 1, 89-94 w .scirp.org/jour n t imal D r ous Pi e . S. Shevtso v Department, S o n gineer ing, Nat i M arine Engineer i Email: m Receiv o ble m for mu l m oduli for p or o er of design v d using the li v o bjective Opti m y used in me d y drophones a n b e r of investig a effective stru the porous pi z oelectric sen s d better matc h r esented work ing the opti m t e r applicatio n h od for the po e the optimiz a r whole devi c o phones, proje are the gene r and underwat r used for gen c oustic mediu m o se, underwat e y requiremen t o ustic matchi n ability to ta k ssur e. The m o s tics are the d h e expansion o h the success o 4 n al / jamp ) D esign o e zocom v a 1 , J.-C. Li u o uthern Federal U i onal Taiwan O c i ng, National K a m ariamarcs@bk . e d October 201 3 l tilayered ultr a o us p iezoelec t v ariables. The v e-link of finit e m ization; Pare d i- nd a- c- e- s i- h - is m al n s. o r- a- c e c- r al e r e- m . e r t s, n g k e o s t e- o f o f the p ie z the mi c ogies t active h for rec e the p r o ance m overco m eleme n of laye r of the w over t h ramics. diate l a transd u theoret i lectric p roper t lectric charac t lectric thickn e for the energy . lectric u other pi In pr p roper t of Und e posite A u 2 , S.-H. Ch a U niversity, Ros c ean Universi ty , a oshiung Marin e . ru 3 a sonic transdu t ric material o n multiobjectiv e e -element (FE ) to Frontier; M z oelectric ma t c ro-electrome c t o create dev i h ydrophones a e iving and ra d o blems at real m atching. The u m ing this dif f nt made of de n r s are needed; w hole structu r h is difficulty i s This entails a yers and b et t u ce r and medi u i cal [4,5] inv e ceramics ma y t ies of transd u materials. P o t erized by lo w modulus 31 d e ss p iezoelect r efficient con v . Therefore, t h u ltrasonic tra n i ezoelectric d e r evious work s t ies of p iezoe l e rwate r A ctive E a ng 3 , J.-K. W t ov-on-Don, R u , Zhongzheng, T e University, K a ce r with acti v n porosity ha v e optimization ) package Co m M ultilayered A c erials science c hanical syste m i ces based on a nd p rojectors , d iation of dire c izing such st r u se of interm e f iculty. How e n se p iezoelect r and this inv o r e. The effect i s the use of po both decreas e t e r acoustic a g u m. Various e e stigations sho w y significantl y u cers and exp a o rous p iezoco m w e r values of at almost th r ic modulus v ersion of ele c h e use of p oro u n sducers, hyd r e vices is very p s [6,7] it was l ectric materi a Acous t E leme n W u 3 u ssia T aiwan a oshiung, Taiw a v e porous p iez v e been p revi o problem b as e m sol Multiphy s c oustic Projec t and the devel m s (MEMS), nanoscale m , arrays of tra n c te d sound [1] r uctures is th e e diate layers a e ve r in case o r ic ceramics a o lves big ener g i ve method fo r o rous p iezoel e ed number of g reemen t b et w e xperimental [ w , that p orou s y improve th e a n d the use o f m posite mate r the transvers e e same value 33 d [6,7] re s c trical and m e u s materials i n r ophone struct u p romising tod a investigated a ls depend on JAMP t ic n ts a n oelectric o usl y ob- e d on the s ics with t o r ; oping of technol- m ultilayer n sducers . One of e imped- a llows to o f active a number g y losses r getting e ctric ce- interme- w een the 2,3] and s piezoe- e desired f piezoe- r ials are e piezoe- e s of the s ponsible e chanical n piezoe- u res and a y. how the porosity Open Access 90 for the cera m tric hardness. the full set of The aim o f tion of multi l lectric active expressed thr o obtained dep e ciently decre a underwater a c such layers a s tric, backing p 1) we formu l electroelastici optimized ob j sure level (S and the mea n frequency ra n riables are: Y layer, p rotect i damping p ar a layers; p oros i investigation to-frontier ca l the 6D space are feasible. The coupl e the live-link with MATL A about 30% - 4 Figure 1. The acoustic medi u layer, 3—acti v 5—protective f m ics of differe n Such depend e material cons t f presented res e l ayere d transd u element whic h o ugh the valu e e ndencies of e f a se the numbe r c oustic p roje c s : an acoustic w p late and p rot e l ate the coupl ty in axial-s y j ectives we in t PL), transmit t n -square valu e n ge from 100 Y oung’s mod u i ve foam lay e a mete r and sti i ty of an acti v we use the a l culation, i.e. of design vari e d problem is of the FE pa A B. The best r 4 0% of p oros i scheme of mu l u m: 1—acoust v e porous piezo e f oam layer. nt connectivit y e ncies have be t ants. e arch is a s tru c u ce r with the h all material’ e of porosity. f fective modu l r of design v a c to r which st r w indow, matc h e ctive foam la y e d problem o y mmetric for m t roduce avera g t ing current r e of the SP L to 400 kHz. u les of an a c e r, and match i ffness dampi n v e p iezoelectr i a pproach b as e building the s ables where a l numerically i a ckage Coms o r esults have b i ty for p iezoel l tilayered tran s ic window lay e lectric layer, 4 A. V. NA S y and ferroele c e en obtained fo c tural optimiz a porous piezo s parameter a r The p revious l l i allow to suf f a riables. For t h r ucture contai n h ing, piezoele c y ers (see Fi g u r f acoustics a n m ulation. As t h g e d sound p r e r esponse (TC R L irregularity i The design v a c oustic windo i ng layer; ma n g parameter o i c layer. In t h e d on the Par s e t of points i l l the objectiv e mplemented b o l Multiphysi c een obtained ectric layer a n s ducer placed er, 2—matchi n 4 —backing pla t S EDKIN ET c- for a- e- r e ly f i- h e n s c- r e n d h e e s- R ) in a- w ss o f h is e- in e s by c s at nd in n g t e, mecha n ers tha t terials. p aram e than fo metho d timizat i 2. De p Po r For th e approa c effecti v ative v o use of t obtain compo s but an model o cells— pores [ 2 a skele come t h tion th e ters ca n from pi ten-Sa n p resen t ten-Sa n The with t h dium f p resen t thod, m by tria n into ac c corres p calcula t unifor m experi m Fi gu 33 () S r with th e the de p Sander of p ol a obtain e also ca n and n u data fr o tained i and gi v and th e AL. n ical and dam p t may charact e Electro-acou s e ter, is signif i r the dense c e d s might be e f i on of enough p endencies r osit y e effective p r c h presented i v e moduli met h o lumes for po t he FE techno l the structure s ite material. A adequate mi c o f the p iezoce r cube s, some o 2 ]. However, ton at a large h is problem t h e ory can be u s n be built fr o i ezoelectric m a n der’s metho d t e d work both n der’s method experimental h e obtained re s f erroelectric h a t e d below, th e m arked by cir c n gles. The de p c oun t the inh o p ond to dotte d t ions p erfor m m p olarization m ental data. u re 2 shows 33 33 ()/ ( 0 SS p e experimenta l p endencies are methods. Th e a rization field ’ ed without thi s n be seen that u merical resul t om [10]. The i n [11] strong l v e about 35% d e results f rom [ p ing p aramete e rize a wide r a s tic efficiency i cantly highe r e ramics. Obtai n f fectivel y use d wide ra nge o f of the Effe c operties dete r i n details in [ 6 h ods [2,6-8], m o rous p iezoele c l ogies. There a of a two- p h a At low p erce n c rostructure o f r amic cubic l a o f which are r a this model m a number of po h e algorithms s ed. In case o f om pores, wh i a terial. One o f d [9] has b ee n methods: ran d were used. data [2,4,1 0 s ults for a p o r a rdness PZT- 4 e curves, relat e c les, and the W p endences, o b o mogeneity o f d lines; dash- d m ed at taking i ; and the das h the relative 0 ) on p orosi t l data [10,11]. linear for b o t results obtai n ’ s inhomogen e s hypothesis l all four depen t s are in b ett e values of el e l y decrease w i d iscrepancy w [ 9]. rs of interme d a nge of p oly m , defined by t r for porous n e d data and p d for the struc t f transducers. c tive Modu l r mination we 6 ,7] and b ase d m odeling of r e c tric material s a re several m e a se cubic p iez nt of porosity a f porous mat e a ttice consistin a ndomly decla r a y lose conne c o res. In order based on the f low porosity t i le at high po f such method n analyzed i n d o m method a 0 -12] were c o r ous material w 4 . For all the e d to the ran d W itten-Sander b taine d witho u f the p olarizati d o t lines corre s i nto account t h e d lines ind i electric p er m t y p in co m One can conc t h random an d n e d with the h y e ity differ fr o l ess than 1% dencies are v e er agreement w e ctric p ermitt i i th increasing w ith the calcul a JAMP d iate lay- m eric ma- t he TCR material p ropose d t ural op- l i on used an d on the e present- s and the e thods to oelectric a simple e rial is a g of unit r ed to be c tivity of to over- percola- t he clus- o rosity— s is Wit- n [8]. In a n d Wit- o mpare d w ith me- figures, d o m me- method, ut taking on field, s pond to t he non- i cate the m ittivity m parison lude that d Witten- y pothesis om those - 2%. It e ry close, w ith the i vity ob- porosity a ted data Open Access Figure 2. The d porosity. After the c a () E cp , ( i e p characteristic s coefficients d where effE s a trix). The co m the experime n longitudinal ( piezoelectric m As can be related to th e creases with i n method. It is o assuming of better agree m 3(a)). At the s lectric modul u rosity for b ot h that the max i approximatel y about 3% - 5 % At the se c during the si m ture, the obt a polarization f Witten-Sande r properties for 3. Coupled Electroe l Piezoele c In presented ultrasound p i e cations in its surface and t h protective la y d ependency of a lculation the f ) p , () S ii p , s of p iezoelec t 31 d and 33 d a re the comp o m parison of th e n tal data [2, 4 Fi g ure 3(a)) a m oduli. seen the dep e Witten-San d n creasing p or o o bvious that t h polarization f m en t with th e s ame time, th e u s 33 d are e s h methods (Fi g i mu m error is y 10% - 15% % for the data [ c on d stage of m ulation of a m a ined depende n f ield’s inhom o r method ha v the active ele m Problem o l asticit y fo r c tric Trans d investigation, e zoelectric tra n axial-symm e h e bottom o f a y e r made of f relative electri c f ull set of the e we obtained t ric ceramics ( () ii ddp o nents of the c e obtained de p 4 ,13] are p erf o a nd transvers a endency for d e r method, s i o sity compare d h e relationshi p f ield’s inhom o e experiment a e dependencie s s sentially ind e g ure 3(b)). It still quite si g for the data [ 2,4]. the p resente d m ultilaye r tra n n cies at the a o geneity and v e been acce p m ent. f Acoustics r a Multila y d ucer we consider n sduce r for u n e tric formulat i transdu cer ar e f oam. Transd u A. V. NA S c permittivity o e ffective mod u such importa n as p iezoelect r () ( ) E i eps p c ompliance m a p endencies wi t o rmed for b o t a l (Figure 3( b 31 d coefficie n i gnificantly d d to the rando m p obtained at t h o geneity is in a l data (Fi g u r s for the pi ezo e penden t of p o should be no t e g nifican t and from [13], a n d investigatio n n sducer’s stru a ssuming of t h related to t h p ted as materi and y ered a model of a n derwate r app l i on. Cylindric e covered wit h u ce r consists o S EDKIN ET o n u li nt r ic ) , a- t h t h b )) n t, e- m h e a r e e- o - ed is nd n, c- h e h e i al a n l i- al h a o f Figure cients o four la y layer, m constr u the bo match e dent u p reflect p iezoe l have be the lo n acousti matchi n same f couple d the FE p licati o monic quenc y the in h p ressu r where is a s p freque n b ound a b ound a and ax i 1). AL. 3. The depen d n porosity. y ers, such as: m atching laye u ction is surro u o undar y of a n e d layer (PM L p on the PML at the interfac l ectric layer, p e en chosen t o n gitudinal wa v c window t h n g layer—1/ 4 f requency of a d problem of a package CO M o n modes: P r analysis) and y response ana l h omogeneous H r e p : 0 = 1000 kg/ m p ee d of soun d n cy with f a ry conditions a ry between a i al symmetry d encies of rela t backing p lat r and acousti c u nde d with a n n acoustic m L ) is placed s o from a non- P e (Figure 1). p rotective an d o be approxi m v e at thickne s h ickness—ap p 4 of the long i a bou t 300 k H a coustics and M SOL Multip h r essure Acou s Piezo Axial l ysis). Sound w H elmholtz eq u 2 00 1p c m 3 is a fluid d e d , 2 f Hz denotin g are following a n acoustic m (on the left b t ive piezoelect r e, p iezoelectr i c window. T h n acoustic me d m ediu m the p o that the wa v P ML mediu m The thickness e d backing p la t m ately a half l e s s vibration m p roximately 3 i tudinal wave H z. We form u electroelastic i h ysics and its s tics mode (t Symmetry m o w aves are gov e u ation for the 2 20 s p c , e nsity, 1 5 s c rad/s is the g the freque n : sound hard w m ediu m and t h oundary), (se e JAMP 91 r ic coeffi- i c active h e whole d ium. On p erfectly v es, inci- m , do not e s of the t e layers ength of m ode; an 3 /4, and e s at the u late the i ty using two ap- ime-har- o de (fre- e rned by acoustic (1) 5 00 m/s angular n cy. The w all (the h e PML) e Figure A. V. NASEDKIN ET AL. Open Access JAMP 92 The constitutive relations for the piezoelectric active layer are taken in the stress-charge form as follows: * E S σcεeE Deεε E, (2) where ε is the strain tensor, σ is the stress tensor, E is the electric field vector, D is the electric displace- ment vector; E c is the tensor of elastic stiffness moduli at constant electric field; e is the tensor of piezoelectric moduli (stress coefficients); S ε is the tensor of electric permittivity moduli at con s tant mechanical stress. Coupling between solid end acoustic media is pro- vided by the boundary conditions: the top and the sides of a transducer undergo both an acoustic pressure and the inward accelerations. The bottom of a transducer is fixed; on the left boundary we consider an axial symmetry con- dition; on the top of piezoelectric layer the constant elec- tric potential with amplitude 100 V in whole studied fre- quency band is applied, when the bottom is grounded. Since the dimensions of the investigated transducer are quite small, this type of projector cannot be used to gen- erate directional sound and therefore we will consid er th e sound pressure level only in a direct ray. When the transducer is placed into acoustic medium the thickness vibration mode is excited at frequencies from approx- imately 100 to 400 kHz. This frequency range was used during the following optimization of transducer’s para- meters. 4. Multiobjective Optimization of the Piezoelectric Transducer Ther e is a wide ran ge of mat erial s that c an be used as the constituent layers of a transducer; this proves the possi- bility to vary their mechanical properties within the wide scope. It should be noted that we chose tungsten as a backing plate material to generate the thickness v ibration mode of a PZT layer because of its large mechanical stiffness and high acoustic impedance. In order to for- mulate the optimization problem let us introduce six de- sign variables: porosity o f an active layer (por), Young’s modules of an acoustic window layer (aw E), matching layer (m E), and protective foam layer ( f E ); mass damping parameter (1 R) and stiffness damping parame- ter (2 R) of layers. In our investigation we considered three objectives: sound pressure level (SPL) in direct ray measured at the 1m distance from the sound source and transmitting current response (TCR) to be maximized; the deviation of SPL is to be minimized. SPL is repre- sented in decibels as follows 1 20lg ref ppp, (3) where 1 p is the sound pressure at the measurement point; and 5 210 ref p Pa is the threshold of sound pressure. TCR is the ratio of an absolute value of sound pressure 1 p, to the amplitude of electric current I through the active element: 1I SpI. (4) There is a wide range of approaches to structural opti- mization. In the framework of multi-criteria optimization problem (MOO) when several objective functions exist, there is no unique solution, but a number of optimum solutions exist. In this situation the most suitable way to optimization is a calculation of a so-called Pareto opti- mum or Pareto-frontier. Using the Pareto approach we suppose the assignment of a set of choices for all objec- tive components that are Pareto efficient. By confining the set of choices to only the Pareto-efficients instead of considering the full range for each parameter, it is possi- ble to make trade-offs within this set. During the solving of considered optimization problem the three integrals were assumed to be optimized: 2 1 21 f f p pfdff f , (5) 2 1 2 21 f f ppfdf pff , (6) 2 1 21 f f pf TCRdff f If , (7) where p , p and TCR represent an averaged SPL, deviation of SPL and TCR, respectively; 1 f and 2 f are the boundaries of the frequency range. Obviously the construction of a Pareto-frontier was complicated for the three-dimensional space of objective functions. In order to overcome this difficulty the illu- stration of a Pareto-frontier has been represented using the level lines. At the numerical problem solving MAT- LAB varies design parameters for the transducer, calls the FE model simulated by Comsol Multiphysics, and is carry out the multiple computations of the objectives. Then obtained data are being analyzed and illustrated using the set of complimentary procedures, written in MATLAB (see Figures 4(a), (b)). 5. Numerical Results and Discussion At the analysis of the simulation results it was founded that the influence of both damping designs variables on the objectives are negligible. So only two obtained pro- jections of the criteria points set on the spaces of other design variables are presented in Figure 4. Figure 4(a) corresponds to the projection on subspace of two design variables: porosity of active layer and Young’s modulus Open Access Figure 4. Pro j contour lines o variables. of matching l side p rotecti v window. For ing bounds fo 1500 dB/A of SPL. On t h and dashed li n the 2D subs p represent the seen from Fi g racterized by porosity of a c maximum va l variation of p SPL value sh i matching lay e porosity; on t corresponds t quency respo n j ections of th e o f objectives le v ayer; Figure 4 v e layer and Y the studied o b r the feasible v – for TCR, a n h e presented f i n es are the p r o p aces of the d intersections g ure 4(a) that TCR, reach e c tive layer is g l ues of SPL a orosity. One c i fts with the g r er from about t he other han d t o lower p or o n se of SPL ar e e criteria set p v els on the su b 4 (b) for You n Y oung’s mod u b jectives we u v alues: 150 nd 2.5 dB f i gures the ar e a o jections of P a d esign variabl e of optimum a the energetic e s desirable v a g reate r than 0. 3 a re reached a t c an observe t h r owth of Y ou n 1.6 GPa till 2. d the lower Y o o sity. The mo e reached whe n A. V. NA S p resented as t h b spaces of desi g n g’s modulus o u lus of acous t u sed the follo w dB for the S P for the deviati o a s between sol i a reto frontier o e s. Green are a a reas. It can b efficiency, ch a lues when t h 3 . However, t h t a considerab h a t the optimu m n g’s modulus o 2 GPa at high o ung’s modul u s t uniform fr n the percent o S EDKIN ET h e g n o f t ic w - P L, o n i d o n a s b e a- h e h e le m o f e r u s e- o f p orosit y that th e cantly quenc y the val u time t h windo w areas. For c cy res p design For from t h tains p last se t made o (Youn g side th e the Par e For ponses the Fi g The g optima l uneve n cases o p orosit y level i n (1), (2 ) The s o range i base o f on 10 % ure 5( b utmost are tak e device taining the w h TCR d respon s 6. Co n Ten de p Table 1 lated p r St u de s Insi d optim u Outs i optim u Den s AL. y is greater t h e Young’s m o influences o n y response; th e u es greater t h h e Young’s m w layers doe s c larity, we p r e p onses obtaine variables that the first set a h e obtained o p arameters b ei n t corresponds t o f dense p iezo e g ’s modulus o e optimum ar e e to frontier. each set of d for SPL and ures 5(a) and g raphs shown l set of desig n n ness of soun d o f active elem e y . The maxi m n the investig a ) , and (3) are o und pressure s the best. It e f dense ceram i % . The peak o b b )) shows th a resonance pr e n from the o p has a sufficie n a constant a m h ole frequenc y d oes not wors e s e for the sou n n clusion p endencies o f 1 . Design vari a r ojectors. u die d s ign aw E , GPa d e the u m area2.5 i de the u m area0.5 s e PZT 0.5 h an 0.25. It’s o o dulus of ac o n the uniform e optimum q u h an or equal t o m oduli of a pr s not impact e sen t below th r d for the thre e is contained i n a ll the design p timu m areas . n g outside th e t o a transduc e e lectric ceram i o f acoustic wi n e a, the other f o d esign variabl TCR are cal c 5(b), respecti v in Figure 5, a n parameters d pressure lev e e n t with dense m u m deviation a te d frequenc y 8.5 dB/21 d B level inside e xceeds the S P i cs on 5%, an d b serve d in th e at a transduce r r operties whe n p timal area. If nt p erforman c m plitude of t h y range, the r e n the unifor m nd pressure. f material cons a bles for the t h f E , MPa m E , GPa 30 16 5 20 30 16 o bvious (Fi gu o ustic windo w ity of the S P u antities corre s o 2 GPa. At t r otective and on TCR in o r ee groups of e sets of valu e n Table 1. variables we . The second e Pareto fron t er with an act i i cs; the first pa n dow) was ta k o u r variables b es the freque n c ulated and pl v ely. a clearly show s provides a m e l as compari n ceramics (3) a of the sound y range for th e B /12 dB, resp the whole f r P L of p roject o d SPL for desi g e TCR graph ( r (1) demonst r n the design v the electronic c e that allows h e applied p o t r esonance fe a m ity of the f r tants on p oro s h ree examples 17 10 R 2 7 10 R 1.5 0.6 2.3 1.2 1.5 0.6 JAMP 93 u re 4(b)) w signifi- P L’s fre- s pond to t he same acoustic o ptimu m frequen- e s of the re taken set con- t ier. The i ve layer a ramete r k en out- b elong to n cy res- l otted on s that the m uch less n g to the a nd 20% pressure e designs ectively. r equenc y or on the gn (2)— ( see Fig- r ates the v ariables exciting to main- t ential in a tures of r equenc y s ity were of simu- 7 por 0.4 0.2 0 A. V. NASEDKIN ET AL. Open Access JAMP 94 Figure 5. The frequency responses of SPL (a) and TCR (b). successfully obtained for the porous piezocomposite ma- terials of different connectivity in order to optimize the hydroacoustic performance of multilayered projector based on the active PZT layer with varied porosity. These effective modules were calculated using the FE method at the assumption of homogeneous and inhomo- geneous polarization field. The last dependencies were used at the statement and solving the optimization prob- le m d ue to t he be st agreement with the experi mental d at a. Obtained dependencies allowed to reduce the number of design variables to six (porosity of an active layer; Young’s modules of an acoustic window layer, protec- tive and matching layers; mass and stiffness damping parameters of layers). On the base of the Pareto optimal- ity the set of feasible designs in a six dimensional design space was reconstructed using three objectives: averaged sound pressure level, transmitting current response and the standard deviation of the SPL in a frequency range from 100 to 400 kHz. A comparative analysis of three examples of the simulated designs has been performed. It showed the best performance of a projector with porosity near 40% and elastic modules of intermediate layers tuned to achieve the best acoustic impedances matching between the structure and acoustic medium. 7. Acknowledgements This work is partially supported by the Russian Founda- tion for the Basic Researches (Grant 12-08-31350) and by National Science Council of Taiwan (Project NSC99- 2923-E-022-001-MY3). REFERENCES [1] R. Sathishkumar, “Micro Size Ultrasonic Transducer for Marine Applications,” Indian Journal of Science and Technology, Vol. 4, No. 1, 2010, pp. 8-13. [2] I. Getman and S. Lopatin, “Theoretical and Experimental Investigation of the Porous PZT Ceramics,” Ferroelec- trics, Vol. 186, 1996, pp. 301-304. http://dx.doi.org/10.1080/00150199608218088 [3] R. Ramesh, H. Kara and C. R. Bowen, “Finite Element Modelling of Dense and Porous Piezoceramic Disc Hy- drophones,” Ultrasonics, Vol. 43, No. 3, 2005, pp. 173- 181. http://dx.doi.org/10.1016/j.ultras.2004.05.001 [4] A. N. 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