This work explores the design of piezoelectric transducers based on functional material gradation, here named functionally graded piezoelectric transducer (FGPT). Depending on the applications, FGPTs must achieve several goals, which are essentially related to the transducer resonance frequency, vibration modes, and excitation strength at specific resonance frequencies. Several approaches can be used to achieve these goals; however, this work focuses on finding the optimal material gradation of FGPTs by means of topology optimization. Three objective functions are proposed: (i) to obtain the FGPT optimal material gradation for maximizing specified resonance frequencies; (ii) to design piezoelectric resonators, thus, the optimal material gradation is found for achieving desirable eigenvalues and eigenmodes; and (iii) to find the optimal material distribution of FGPTs, which maximizes specified excitation strength. To track the desirable vibration mode, a mode-tracking method utilizing the ‘modal assurance criterion’ is applied. The continuous change of piezoelectric, dielectric, and elastic properties is achieved by using the graded finite element concept. The optimization algorithm is constructed based on sequential linear programming, and the concept of continuum approximation of material distribution. To illustrate the method, 2D FGPTs are designed for each objective function. In addition, the FGPT performance is compared with the non-FGPT one.
Almajid, A., Taya, M. and Hudnut, S.2001. ‘‘Analysis of Out-of-plane Displacement and Stress Field in a Piezocomposite Plate with Functionally Graded Microstructure ,’’ International Journal of Solids and Structures , 38(19):3377-3391.
2.
Almeida, S.M.R., Paulino, G.H. and Silva, E.C.N.2008. ‘‘A Simple and Effective Inverse Projection Scheme for Void Distribution Control in Topology Optimization,’’Structural and Multidisciplinary Optimization, DOI: 10.1007/s00158-008-0332-6 (in press).
3.
Banks-Sills, L., Eliasi, R. and Berlin, Y.2002. ‘‘Modeling of Functionally Graded Materials in Dynamic Analyses,’’ Composites B, 33(1):7-15.
4.
Bendsøe, M.P. and Sigmund, O.2003. Topology Optimization: Theory, Methods and Applications , Springer, Berlin.
5.
Birman, V. and Byrd, W.L.2007. ‘‘Modeling and Analysis of Functionally Graded Materials and Structures,’’ Applied Mechanics Review, 60(5):195-216.
6.
Carbonari, R.C., Silva, E.C.N. and Paulino, G.H. 2009. ‘‘Multi-actuated Functionally GradedPiezoelectric Micro-tools Design: A Multiphysics Topology Optimization Approach,’’ International Journal for Numerical Methods in Engineering, 77(3):301-336.
7.
Carbonari, R.C., Silva, E.C.N. and Paulino, G.H.2007. ‘‘Topology Optimization Design of Functionally Graded Bimorph-type Piezoelectric Actuators,’’ Smart Materials and Structures, 16(6):2607-2620.
8.
Carbonari, R.C., Silva, E.C.N. and Nishiwaki, S.2005. ‘‘Design of Piezoelectric Multi-actuated Microtools Using Topology Optimization,’’ Smart Materials and Structures, 14(6):1431-1447.
9.
Carrrillo-Heian, E.M., Carpenter, D., Paulino, G.H., Gibeling, J.C. and Munir, Z.A.2001. ‘‘Dense Layered Molybdenum Disilicide-Silicon Carbide Functionally Graded Composites Formed by Field-activated Synthesis ,’’ Journal of the American Ceramic Society , 84(5):962-968.
10.
Chakraborty, A., Gopalakrishnan, S. and Kausel, E.2005. ‘‘Wave Propagation Analysis in Inhomogeneous Piezo-composite Layer by The Thin Layer Method,’’ International Journal for Numerical Methods in Engineering, 64(5):567-598.
11.
Diaz, A. and Kikuchi, N.1992. ‘‘Solution to Shape and Topology Eigenvalue Optimization Problems Using a Homogenization Method,’’ International Journal for Numerical Methods in Engineering, 35(7):1487-1502.
12.
Ewins, D.J.1988. Modal Testing: Theory and Practice, Research Studies Press, England.
13.
Guest, J.M., Prevost, J.H. and Belytschko, T.2004. ‘‘Achieving Minimum Length Scale in Topology Optimization Using Nodal Design Variables and Projection Functions , ’’ International Journal for Numerical Methods in Engineering , 61(2):238-254.
14.
Guo, H., Cannata, J., Zhou, Q. and Shung, K.2005. ‘‘Design and Fabrication of Broadband Graded Ultrasonic Transducers with Rectangular Kerfs,’’ IEEE Transaction on Ultrasonics, Ferroelectrics, and frequency Control, 52(11):2096-2102.
15.
Guo, N. and Cawley, P.1992. ‘‘Measurement and Prediction of the Frequency Spectrum of Piezoelectric Disks by Modal Analysis, ’’ Journal of the Acoustical Society of America, 92(6):3379-3388.
16.
Ha, H. and Cho, S.2006. ‘‘Design Sensitivity Analysis and Topology Optimization of Eigenvalue Problems for Piezoelectric Resonators, ’’ Smart Materials and Structures, 15(6):1513-1524.
17.
Haftka, R.T., Gürdal, Z. and Kamat, M.P.1990. Elements of Structural Optimization, Kluwer Academic Publishers, Dordrecht.
18.
Hauke, T., Kouvatov, A., Steinhausen, R., Seifert, W., Beige, H., Langhammer, H.T. and Abicht, H.2000. ‘‘Bending Behavior of Functionally Graded Materials,’’ Ferroelectrics, 238(1):195-202.
19.
Kim, T.S. and Kim, Y.Y.2000. ‘‘MAC-based Mode-tracking in Structural Topology Optimization,’’ Computers and Structures , 74(3):375-383.
20.
Kim, J.K. and Paulino, G.H.2002. ‘‘Isoparametric Graded Finite Elements for Nonhomogeneous Isotropic and Orthotropic Materials,’’ ASME Journal of Applied Mechanics, 69(4):502-514.
21.
Kino, G.2000. Acoustic Waves: Devices, Imaging, and Analog Signal Processing, Corrected edn, Prentice-Hall, Inc. New Jersey.
22.
Kögl, M. and Silva, E.C.N.2005. ‘‘Topology Optimization of Smart Structures: Design of Piezoelectric Plate and Shell Actuators,’’ Smart Materials and Structures, 14(2):387-399.
23.
Kruusing, A.2000. ‘‘Analysis and Optimization of Loaded Cantilever Beam Microactuators,’’ Smart Materials and Structures , 9(2):186-196.
24.
Le, C.H.2006. Achieving Minimum Scale and Design Constraints in Topology Optimization: A New Approach, University of Illinois at Urbana-Champaign, Master thesis.
25.
Lee, H.-J.2005. ‘‘Layerwise Laminate Analysis of Functionally Graded Piezoelectric Bimorph Beams,’’ Journal of Intelligent Material Systems and Structures, 16(4):365-371.
26.
Lerch, R.1990. ‘‘Simulation of Piezoelectric Devices by Two- and Three-dimensional Finite Elements,’’ IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 37(2):233-247.
27.
Ma, Z.D., Kikuchi, N. and Cheng, H.C.1995. ‘‘Topology Design for Vibration Structures,’’Computer Methods in Applied Mechanics and Engineering, 121(1-4):259-280.
28.
Matsui, K. and Terada, K.2004. ‘‘Continuous Approximation of Material Distribution for Topology Optimization,’’ International Journal for Numerical Methods in Engineering, 59(14):1925-1944.
29.
Miyamoto, Y., Kaysser, W.A., Rabin, B.H., Kawasaki, A. and Ford, R.G.1999. Functionally Graded Materials: Design, Processing and Applications, Kluwer Academic Publishers, Dordrecht.
30.
Naillon, M., Coursant, R.H. and Besnier, F.1983. ‘‘Analysis of Piezoelectric Structures by a Finite Element Method,’’ Acta Electronica , 25(4):341-362.
31.
Paulino, G.H., Jin, Z.H. and Dodds, R.H.2003. ‘‘Failure of Functionally Graded Materials,’’ In: Karihaloo, B. and Knauss, W.G. (eds), Comprehensive Structural Integrity, Vol. 2, Chap. 13, pp. 607-644, Elsevier Science, New York.
32.
Qui, J., Tani, J., Ueno, T., Morita, T., Takahashi, H. and Du, H.2003. ‘‘Fabrication and High Durability of Functionally Graded Piezoelectric Bending Actuators,’’ Smart Materials and Structures, 12(1):115-121.
Rubio, W.M., Buiochi, F., Adamowski, J.C. and Silva, E.C.N.2009. ‘‘Modelling of Functionally Graded Piezoelectric Ultrasonic Transducers,’’Ultrasonics , 49(4-5):484-494.
35.
Santare, M.H., Thamburaj, P. and Gazonas, G.A.2003. ‘‘The Use of Graded Finite Elements in the Study of Elastic Wave Propagation in Continuously Nonhomogeneous Materials ,’’ International Journal of Solids and Structures , 40(21):5621-5634.
36.
Santare, M.H. and Lambros, J.2000. ‘‘Use of Graded Finite Elements to Model the Behavior of Nonhomogeneous Materials,’’ Journal of Applied Mechanics, 67(4):819-822.
37.
Shindo, Y., Narita, F. and Nakagawa, J.2009. ‘‘Dynamic Electromechanical Response and Self-Sensing of Functionally Graded Piezoelectric Cantilever Transducers,’’Journal of Intelligent Material Systems and Structures , 20(1):119-126.
38.
Sigmund, O. and Torquato, S.1999. ‘‘Design of Smart Composite Materials Using Topology Optimization,’’ Smart Materials and Structures , 8(3):365-379.
39.
Silva, E.C.N., Carbonari, R.C. and Paulino, G.H.2007. ‘‘On Graded Elements for Multiphysics Applications ,’’ Smart Materials and Structures , 16(6):2408-2428.
40.
Silva, E.C.N., Nishiwaki, S. and Kikuchi, N.2000. ‘‘Topology Optimization Design of Flextensional Actuators,’’ IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 47(3):657-671.
41.
Silva, E.C.N. and Kikuchi, N.1999. ‘‘Design of Piezoelectric Transducers Using Topology Optimization,’’ Smart Materials and Structures , 8(3):350-364.
42.
Silva, E.C.N., Fonseca, O.J.S., Montero de Espinosa, F., Crumm, A.T., Brady, G.A., Halloran, J.W. and Kikuchi, N. 1999. ‘‘Design of Piezocomposite Materials and Piezoelectric Transducers Using Topology Optimization - Part I,’’ Archives of Computational Methods in Engineering, 6(2):117-182.
43.
Soto, C.A. and Diaz, A.R.1993. ‘‘Layout of Plate Structures for Improved Dynamic Response Using a Homogenization Method,’’Proceedings of the 1993 ASME DETC, pp. 667-674, Advances in Design Automation, Albuquerque, NM.
44.
Steinhausen, R., Kouvatov, A., Beige, H., Langhammer, H.T. and Abicht, H.-P.2004. ‘‘Poling and Bending Behavior of Piezoelectric Multilayers Based on Ba(TiSn)O3 Ceramics ,’’Journal of the European Ceramic Society, 24(6):1677-1680.
45.
Sutter, R.T., Camarda, J.C., Walsh, L.J. and Adelman, M.H.1988. ‘‘Comparison of Several Methods for Calculating Vibration Mode Shape Derivates,’’ AIAA Journal , 26(12):1506-1511.
46.
Taya, M., Almajid, A.A., Dunn, M. and Takahashi, H.2003. ‘‘Design of Bimorph Piezocomposite Actuators with Functionally Graded Microstructure,’’ Sensors and Actuators A-Physical, 107(3): 248-260.
47.
Wang, B.P.1990. ‘‘Improved Approximate Methods for Computing Eigenvector Derivates in Structural Dynamics,’’ AIAA Journal, 29(6):1018-1020.
48.
Wang, B.L. and Noda, N.2001. ‘‘Design of a Smart Functionally Graded Thermopiezoelectric Composite Structure,’’ Smart Material Structure , 10(2):189-193.
49.
Yang, J. and Xiang, H.J.2007. ‘‘Thermo-electro-mechanical Characteristics of Functionally Graded Piezoelectric Actuators,’’ Smart Materials and Structures, 16(3):784-797.
50.
Yong, Y.K. and Cho, Y.1996. ‘‘Numerical Algorithms for Solutions of Large Eigenvalue Problems in Piezoelectric Resonators, ’’ International Journal for Numerical Methods in Engineering , 39(6):909-922.
51.
Zhang, Z. and Paulino, G.H.2007. ‘‘Wave Propagation and Dynamic Analysis of Smoothly Graded Heterogeneous Continua Using Graded Finite Elements, ’’ International Journal of Solids and Structures, 44(11):3601-3626.
52.
Zhong, Z. and Yu, T.2006. ‘‘Vibration of a Simply Supported Functionally Graded Piezoelectric Rectangular Plate,’’ Smart materials and Structures, 15(5):1404-1412.
