AN OPTIMAL DESIGN OF UNDERWATER PIEZOELECTRIC TRANSDUCERS OF NEW GENERATION Maria Shevtsova Rhein-Main University of Applied Sciences, 6597, Wiesbaden, Germany; Southern Scientific Center of Russian Academy of Sciences, 34402, Rostov-on-Don, Russia email: maria.shevtsova@hs-rm.de Andrey Nasedkin Southern Federal University, Institute for Mathematics, Mechanics, and Computer Science in the name of I.I. Vorovich, 344090 Rostov-on-Don, Russia Sergey Shevtsov, Igor Zhilyaev Southern Scientific Center of Russian Academy of Sciences, 34402, Rostov-on-Don, Russia Shun-Hsyung Chang National Kaohsiung Marine University, 857, Kaohsiung City, Taiwan (R.O.C.) This paper presents the method to improve the performance of underwater transducers based on the porous piezoelectric ceramics. The Pareto-based approach has been used for the optimization of two types of devices: multilayered projector with an active layer made of porous piezoelectric ceramics and the membrane-type hydrophone with a perforated active piezoelectric diaphragm (pmut). The simulations have been performed by means of the live-link of MATLAB and finite-element (FE) software Comsol Multiphysics. The numerical results are obtained and presented for both types of devices.. Introduction With the rapid development in piezoelectric materials science the micro-electromechanical systems (MEMS) technology for creating PZT-based nanoscale devices for receiving and radiation of directed sound made a great progress. The design of powerful MEMS-based projectors and highsensitive hydrophones is one of the research interests in underwater acoustics [-3] which is very little discovered so far [4-6]. Such devices demonstrate performances exceeding those of their macro-scale counterparts and low production costs per device [7, 8]. Piezoelectric micro-machined ultrasonic transducers (pmuts) present a new approach to sound detection and generation that can overcome the shortcomings of conventional transducers due to their high efficiency and accomplishment in fabrication technology [9]. The pmut s sound sensoring element is a micromachined multi-layered membrane with a piezoactive layer, typically a thin PZT film [, 0-]. The use of porous and perforated piezoelectric elements for sonar transducers gives such advantages as an impedance matching between the active element and the acoustic medium, a small size compared with the wavelength of sound, an ability to create both sharp and uniform directivity patterns [2-4]. In the network of presented research a multilayered projector with an active element of porous piezoelectric ceramics and a pmut-hydrophone, based on the nanoscale perforated membrane, have been simulated and optimized.
In the first part of our investigation a finite-element model and the coupled problem of acoustics and piezoelectricity for the multilayered projector based on the porous PZT-4 ceramics has been implemented. Next section describes the Pareto-based approach to the optimization of such type of transducers. The second part of our work is devoted to the similar research with regard to a hydrophone device based on the perforated nanoscale PZT-membrane. For the coupled problems formulation the dependencies [5] of the full sets of effective modules on porosity or the degree of perforation of the PZT-elements have been used. These dependencies allowed to significantly decrease the number of degrees of freedom. The final part contains the optimization results and their discussion. 2. Coupled Problem for the Multilayered Projector Based on Porous PZT-Ceramics This section describes a coupled problem of acoustics and electric elasticity for an axialsymmetric model of an ultrasonic 4-layered transducer with an active piezoelectric layer made of porous piezoelectric ceramics (Figure ). The lateral and bottom surfaces were covered with a protective layer. A perfectly matched layer (PML) has been placed on the outer boundary of the acoustic layer. This layer simulates an antireflection boundary condition. Thicknesses of the substrate (4), piezoelectric (3) and protective (5) layers has been taken equal to a half longitudinal wavelength at the thickness oscillation mode; the thickness of the matching layer (2) corresponds to the ¼ wave length, and the thickness of the acoustic () layer ¾ wavelength of the respective wave at an operating frequency of 300kHz. A coupled problem has been performed by means of FE package COMSOL Multiphysics by the joint use of two modes: Piezoelectrics and Pressure Acoustics (Helmholtz equation) Figure : The scheme of multilayered transducer placed in acoustic medium 2 p p 0, 2 0 0 c s () 2f rad/s f Hz - vibration p p x, (hereinafter an implicit dependency on where angular frequency, frequency, 3 is meant), 0 000 kg/m density of water, c s 500 m/s speed of sound in water. In the 2-D axial symmetric case, a radial coordinate r and axial coordinate z are independent; only an angular coordinate is dependent, while im r, z pr, ze p,, (2) where m is the wave number. An equation for the acoustic pressure has the form: 2 2 r p p m rp r 0 (3) r 0 r z 0 z 0 cs r Piezo Axial Symmetry mode is governed by the equations in the stress-charge form E σ c ε e* E, (4) S D e ε ε E where ε is the strain tensor, is the stress tensor, E is the electric field vector, D is the electric E displacement vector; c is the elastic stiffness tensor at the constant electric field; e is the tensor of S stress coefficients; ε is the tensor of dielectric permittivity at a constant mechanical stress. Coupling between solid end acoustic media has been provided by means of the boundary conditions. The top of the transducer undergoes simultaneously an acoustic pressure -p in the z- 2 ICSV23, Athens (Greece), 0-4 July 206
direction and the inward accelerations the same direction. An acoustic pressure and the inward acceleration act also on the lateral surface but in the radial direction. The bottom face is fixed. An axially symmetric boundary condition was defined on the left boundary; when the acoustic medium was in an ideal contact with the PML.The boundary between an acoustic medium and the PML is a rigid wall. An acoustic medium was excited by a harmonic electric potential with amplitude of 00V, applied onto the upper bound of the piezoelectric layer, when its lower bound was grounded. The considered range of 00-400 khz comprises the first frequency of the thickness vibration mode. 3. Optimization Problem for a Multilayered Piezoelectric Projector In this section the formulation of an optimization problem for the multilayered projector based on the porous piezoelectric ceramics is presented. As a substrate material we propose tungsten due to its heightened acoustic impedance and mechanical rigidity [2, 7]. Mechanical properties of matching layer, acoustic window and the value of piezoceramic layer s porosity have been accepted in the capacity of the design variables during the optimizing the following objective functionals: -maximum sound pressure level (SPL) in the direct beam, measured at a meter distance from the sound source: SPL f2 20lg p p ref, f 2 f where p sound pressure at the measurement point, pressure, f2 f2 f ; -the maximum Transmitting Current response (TCR): p p( f ) d f (5) f f f 5 p ref 2 0 Pa threshold of the sound 2 p TCR d f, (6) f I 2 f where I f amplitude of the electric current through the piezoelectric layer at a constant potential difference; - minimum unevenness of SPL in the test frequency range of 00-400kHz f 2 2 p p p f df (7) f 2 f During the solving of the considered problem, several objective functionals are simultaneously to be optimized. Thereby there is a set of feasible solutions. The optimization problem can be formulated as follows min x,..., px. (8) xx The most effective approach to its solution is the construction of the Pareto frontier. A feasible solution x~ X is considered to be Pareto efficient if there is no such a solution x X, that x ~ x, for all k,..., p x ~ x for at least one i,..., p, where k 2 k k, and i i the number of objective functions. The set of all Pareto optimal solutions X P is usually called the Pareto frontier. 7 The space of design variables was taken as 7-dimensional, and the following coordinates in were considered: porosity of the piezoelectric layer, Young s modulus E, Poisson s rations and densities of both matching and acoustic window layers. The ranges of the design variables have been accepted according to the referenced data on the on the characteristics of water absorption and the acoustic impedance; porosity was limited to the value of 0.5. Solution of the optimization problem is given by a live-link of MATLAB and Comsol Multiphysics. During the solution process the ICSV23, Athens (Greece), 0-4 July 206 3
transducer s design parameters have been varied in MATLAB, then the FE Comsol Multiphysics model has been called by MATLAB; the objective functions have been iteratively calculated at selected values of design variables. Obtained data have been analyzed and illustrated by means of MATLAB and Mathcad. 4. FE Model and the Coupled Problem Formulation for a Hydrophone Device with an Active Perforated Piezoelectric Membrane Figure 2: Figure 3: Scheme of the micromachined hydrophone is shown in the Figure 2. Diameter of the simulated device is 7 mm, the thickness of active membrane is 0.5 mm. Hydrophone operates in the frequency range of 40-200 khz. The cylindrical region occupied by the liquid has 5 cm diameter and 5 cm height. The device contains a polymeric layer, a silicon bottom layer and several perforated membranes: SiO 2, piezoelectric ceramics with glued platinum electrodes and one more SiO 2 layer. The active piezoelectric membrane is perpendicularly polarized and distantly located from the mid-plane of the sandwich-device. Because of this feature the membrane undergoes the biaxial stretching deformations under the action of high hydrostatic pressures. The holes reduce the residual stresses in the diaphragm and give an additional degree of freedom, which allows the adjusting of the acoustic impedance and electrical properties of a piezoelectric active layer. Such perforation in the two-dimensional structure is similar to the porosity in the 3-D structures, however the technology of their preparation [8, 3, 4] allows to change the step and diameter of the perforated holes. The use of perforated membranes allows to provide a high perceived effectiveness in the broad frequency band at the minimal unevenness of the sensitivity frequency response as well. This effect is achieved as a result of a regular perforation by the holes with about 5 µm diameters (Fig. 3), when the dissipative properties of the protective layer are varied controllably. A coupled problem of piezoelectricity and hydroacoustics was formulated and implemented in the form of an axial symmetric FE model of the hydrophone in Comsol-Multiphysics package. Figure 4 demonstrates geometry, properties of the layers and boundary conditions. Back side of the active membrane is faced into the vacuum cavity, therefore the back side is accepted to be free. Sound pressure is applied onto the outer horizontal surface of the liquid. In order to exclude the reflections, considered system was surrounded by the perfectly matching layer PML, when the impedance of the lower boundary was assumed to be equal to the impedance of water. At the steady-state vibration mode, a solution of the Helmholtz equation has been accepted as x, t px exp it Sketch of pmut hydrophone design. The scheme of perforation. p for the liquid medium and for the PML. Equations of electroelasticity described the areas occupied by the elements of the device. All the material constants of the perforated piezoelectric membrane and the intermediate SiO 2 membrane are given as empirical dependencies on the degree of perforation provided in [4]. The conductive plane of the piezoelectric membrane was grounded, when the top is connected to an active electrical resistance of MOhm. 4 ICSV23, Athens (Greece), 0-4 July 206
Figure 4: The geometry and boundary conditions of the hydrophone s model. An electric potential V generated on the input resistance is determined at each frequency by means of the membrane surface integrating of the current density J and the subsequent solving of the algebraic equation V R J nd 0. The hydrophone sensitivity was estimated by the ratio of the electric potential amplitude to the sound pressure amplitude, as well as by the quantity ~ ~ SR 0logV V ref, where V ~ ~ ~ ref V and the values of the potentials V, V correspond to the ref acoustic pressure amplitude of Pa. Frequency response of the hydrophone sensitivity was calculated in Comsol Multiphysics by means of FRF-analysis (Piezoelectric mode) and time-harmonic analysis (Pressure acoustics mode). The frequency range, which contains only the first natural frequency of the receiving membrane, was used for the following solution of the optimization problem. 5. Optimization of a pmut-hydrophone In order to implement the structural optimization of the hydrophone device as the coordinates of a 6-D space the next design variables have been taken: relative area of perforation S, the Young's modulus Y w, Poisson ratio w, density w, mass damping dm and stiffness damping dk Rayleigh parameters of the outer polymeric layer. The ranges of the mechanical parameters variation have been chosen according to the passport data for available polymeric materials with low water absorption Yw 8;5 GPa, w 0.2;0.48, 3 w 000;600kg/ m, 7 6 dm, dk 0 ; 0 ; the degree of perforation was limited by the maximal allowable deflection requirement of 0.25 mm at a depth of S 0;0.5. For each design space point 3 different quality criteria were used: the sensitivity 200 m p response averaged over the frequency range 40-200 khz S S f S and maximum R R R max f f, f2 S R S R (in db/pa), an averaged variation of sensitivity inside the studied frequency band f, f. The optimization algorithm has been implemented in MATLAB. Each call of 2 the FE model produces one point in the criterion space. Then, the 2-D projections of the design 6 space have been analyzed to select the subsequent target 6-dimensional intervals for the values of the design variables. Temporary design variables were introduced because during the first numerical experiments it was clarified that the sensitivity of the objective functions to the taken design variables was not enough. These variables are the first bending frequency mode of the outer polymeric membrane, the Rayleigh damping factor and the degree of perforation. Variation of these parameters has showed a significant impact on the first natural frequency and figure of merit of the whole membrane. All the obtained numerical optimization results are illustrated and discussed in the next section. p ICSV23, Athens (Greece), 0-4 July 206 5
6. Optimization Results and Analysis 6. Numerical Results for the Optimized Multilayered Projector The feature of the Pareto-frontier approach lie in the assignment of the admissible values of the objective functions. The valid values have been taken to satisfy the following inequalities: p 50 db, p 2 db, TCR 400 db/a. For an illustration of the structure of the Pareto set using the level lines of each functional, the Pareto set was projected onto the subspaces the two design parameters: porosity and an impedance of the matching layer (Fig.5), the impedances of the PZT layer and matching layer or impedances of the acoustic window and matching layer. It is obvious (Fig. 5) that the TCR which characterizes the energetic efficiency of the transducer, reaches maximal values when the impedance of a matching layer is lower than 8 MRayl and porosity of the piezoelectric layer is greater than 0.25. The maximum sound pressure level is achieved when the values of the matching layer s impedance change in the range of 9-2 GPa for the piezoelectric layer made of dense PZT-ceramics, and 3-6 GPa with increasing porosity from about 0.3 to 0.4. In the Figure 6 the frequency-response characteristics of the sound pressure level (a) and the transmitting current response (b) for the optimal parameters, which are taken from the optimized region (blue star), outside the optimum area (red star), as well as for the projector with a dense active layer Figure 5: Projections of the criteria set presented as the contour lines of objectives (black star), levels on the subspaces of design variables. It is obvious are presented. that the maximum sound pressure level of the transducer with a dense active layer and a projector with an optimal set of parameters are very close to each other, when the smallest unevenness of the SPL corresponds to the optimal set of parameters. It can be seen (Fig. 6, b) that the maximum values of TCR are observed for the projector with a set of optimal parameters, and the minimum values for the piezoelectric transducer with the dense active layer. 6.2 Optimization Results for the pmut-hydrophone The projection of the design variables set and the corresponding contour lines for the averaged sensitivity and its maximum deviation over the desired frequency range are presented below. Figure 7 demonstrates the feasible area set and Pareto frontiers area assuming the feasible design have the averaged sensitivity no less than.25 μv/pa, and the maximum deviation of sensitivity no more than 25 db. All the designs in the feasible area have the relative perforation value 0.5 0.35. The choice of the values for each feasible objective depended on qualifying standards to the optimized design, requested by user. The multi-objective optimum area is filled with dark-green colour in the a) b) Figure 6: The frequency responses of SPL(a) and TCR(b) 6 ICSV23, Athens (Greece), 0-4 July 206
Figure 7: The contour lines plot for the averaged sensitivity and maximum deviation of the sensitivity, obtained for the designs with admissible diaphragm deflection. choice of feasible values for the objectives is rather arbitrary and due to customer s preferences, but presented multi-objective optimization approach allows to find the feasible designs taking into account all constraints caused by the operating conditions, technological and structural limitations. Figure 6.4; the green and yellow areas are the Pareto frontier. These areas were reconstructed by means of more than 50,000 virtual designs, when only ~20,000 were feasible. Despite some obvious local distortions of the contour lines, caused by the finite number of the simulated designs, the best sensitivity of designs with less flexural stiffness of multilayered hydrophone sandwich diaphragm can be confirmed. The flexural stiffness is decreased with the decreasing of Young s module of protective layer and increasing of perforated area. For three different designs, which are labelled in Figure 7 by four-, five-, and six-pointed stars, the FRF for sensitivities are presented in Figure 8. It is obvious that the Figure 8: Three FRFs of the receiving sensitivity for hydrophone s designs, labelled by the stars in Figure 6.3. 7. Conclusions The method of multiobjective optimization with regard to the underwater transducers based on the porous piezoelectric ceramics has been proposed. The coupled axially-symmetric problems of piezoelectricity and acoustics have been formulated and solved by means of MATLAB and Comsol Multiphysics software. An optimization problem for the multilayered piezoelectric projector with the three objective functions was formulated and numerically solved. The key independent functions: sound pressure level (SPL), transmitting current response (TCR) and the standard deviation of the sound pressure level in the frequency range from 00 to 400 khz have been optimized using the Pareto-based approach in the 7-D space of design variables, such as porosity of PZT-layer, Young s modules, densities and Poisson ratios of the acoustic window and matching layer. Projections of the Pareto frontier have been visualized through an introduction of the "auxiliary" design parameters - acoustic impedances of the layers, which allowed to significantly reduce the number of degrees of freedom and the dimension of the space of design variables. According to the series of numerical experiments transducer with porous piezoelectric ceramics showed significantly higher electroacoustic efficiency compared to the device with dense active layer. By analogy the problem of a multi-criteria optimization for the hydrophone with perforated nanoscale piezoelectric membrane was solved. The hydrophone sensitivity and the frequency response unevenness in a given frequency band have been optimized in 6-D space of design parameters: degree of perforation, Young's modulus, Poisson ratio, density, and two Rayleigh damping parameters of the outer polymer layer. In much the same way the interim design variables were introduced in order to increase the sensitivity of the target parameters: Rayleigh damping factor of the ICSV23, Athens (Greece), 0-4 July 206 7
polymeric membrane, degree of perforation and frequency of the first bending mode. Performed numerical experiments confirmed the advantage of a perforated active piezoelectric diaphragm, that allows tuning of the electromechanical coupling and operated bandwidth of device. The proposed optimization methodology allows to purposefully choose materials for structural elements of the acoustic multilayered piezoelectric transducers. A functional methodology for the developers of new piezo-active materials was proposed; it allowing to predict the properties of produced ceramics with specified type of connectivity and porosity, as well as providing an efficient optimization of the characteristics of the piezoelectric transducers and arrays due to the substantial reduction in the number of degrees of freedom of the optimized design; and takes into account operational, design and technological constraints. REFERENCES Akasheh, F., Myers, T., Fraser, J. D., Bose, S., and Bandyopadhyay, A. Development of piezoelectric micromachined ultrasonic transducers, Sensors and Actuators A: Physics, (2 3), 275-287, (2004). 2 Sherman, C. H. and Butler, J. L. Transducers and Arrays for Underwater Sound, Springer Verlag, 60 pp., (2007). 3 Sathishkumar, R., Vimalajuliet, A., Prasath, J. S., Selvakumar, K., and Veer Reddy, V. H. S. Micro size ultrasonic transducer for marine applications, Indian Journal of Science and Technology, 4 (), 8-, (20). 4 Cheng, Y. T., Liou, J. C., Hou, K. C., Chang, S. H., Chu, D. R. C., Parinov, I. A., et al. The Fabrication of Hydrophone Based on Epitaxial PZT Film for Acoustic Device Applications, Physics and Mechanics of New Materials and Their Applications, Nova Science Publishers, 373-382, (203). 5 Li, Z., Huang, A., Luan, G., and Zhang, J. Finite Element Analyzing of Underwater Receiving Sensitivity of PMN-0.33PT Single Crystal Cymbal Hydrophone, Ultrasonics, 44, 759-762, (2006). 6 Shevtsov, S. N., Parinov, I. A., Zhilyaev, I. V., Chang, S. H., Lee, J. C. Y., Wu, P. C., Lin, C. F., and Wuu D. S. Proceedings of the 3rd International Conference on Theoretical and Applied Mechanics TAM-2, 88-93, (202). 7 Nicolaides, K., Nortman, L., and Tapson, J. The effect of backing material on the transmitting response level and bandwidth of a wideband underwater transmitting transducer using -3 piezocomposite material, Physics Procedia, 3 (), 04-045, (2009). 8 Yaacob, M. I. H., Arshad, M. R., and Manaf, A. A. Response estimation of micro-acoustic transducer for underwater applications using finite element method, Indian Journal of Geo Marine Science, 40 (2), 76-80, (20). 9 Yaacob, M. I. H., Arshad, M. R., and Manaf, A. A. Theoretical characterization of square piezoelectric micro ultrasonic transducer for underwater applications, Proceeding of the7th International Symposium on Mechatronics and its Applications- ISMA0, (200). 0 Muralt, P. Ferroelectric thin films for micro-sensors and actuators: A review, Journal of Micromechanics and Microengineering, 0 (2), 36-46 (2000). Ito, M., Okada, N., Takabe, M., Otonari, M., Akai, D., Sawada, K., and Ishida, M. High sensitivity ultrasonic sensor for hydrophone applications, using an epitaxial Pb(Zr,Ti)O 3 film grown on SrRuO 3 /Pt/γ- Al 2 O 3 /Si, Sensors and Actuators A: Physics, 45-46, 278-282 (2008). 2 Au, W. W. L., Hastings, M. C. Principles of Marine Bioacoustics, Springer, 27-56, (2008). 3 Kara, H., Ramesh, R., Stevens, R., Bowen, C. R. Porous PZT ceramics for receiving transducers, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 50 (3), 289-296, (2003). 4 Nasedkin, A. V., Shevtsova, M. S., Liu, J.-C., Chang, S.-H., Wu, J.-K. Multiobjective optimal design of underwater acoustic projector with porous piezocomposite active elements, Journal of Applied Mathematics and Physics, (6), 89 94, (203). 8 ICSV23, Athens (Greece), 0-4 July 206