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Topology optimization of piezoelectric smart structures for minimum energy consumption under active control

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Abstract

This paper investigates topology optimization of the electrode coverage over piezoelectric patches attached to a thin-shell structure to reduce the energy consumption of active vibration control under harmonic excitations. The constant gain velocity feedback control method is employed, and the structural frequency response under control is analyzed with the finite element method. In the mathematical formulation of the proposed topology optimization model, the total energy consumption of the control system is taken as the objective function, and a constraint of the maximum allowable dynamic compliance is considered. The pseudo-densities indicating the distribution of surface electrode coverage over the piezoelectric layers are chosen as the design variables, and a penalized model is employed to relate the active damping effect and these design variables. The sensitivity analysis scheme of the control energy consumption with respect to the design variables is derived with the adjoint-variable method. Numerical examples demonstrate that the proposed optimization model is able to generate optimal topologies of electrode coverage over the piezoelectric layers, which can effectively reduce the energy consumption of the control system. Also, numerical comparisons with a minimum-volume optimization model show the advantage of the proposed method with respect to energy consumption. The proposed method may provide useful guidance to the layout optimization of piezoelectric smart structures where the energy supply is limited, such as miniature vibration control systems.

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References

  • Aridogan U, Basdogan I (2015) A review of active vibration and noise suppression of plate-like structures with piezoelectric transducers. J Intell Mater Syst Struct 26(12):1455–1476

    Article  Google Scholar 

  • Bazin L, Mitra S, Taberna PL, Poizot P, Gressier M, Menu MJ, Barnabé A, Simon P, Tarascon J-M (2009) High rate capability pure Sn-based nano-architectured electrode assembly for rechargeable lithium batteries. J Power Sources 188(2):578–582

    Article  Google Scholar 

  • Beckert W, Kreher WS (2003) Modelling piezoelectric modules with interdigitated electrode structures. Comput Mater Sci 26:36–45

    Article  Google Scholar 

  • Brennan MC, McGowan A-MR (1997) Piezoelectric power requirements for active vibration control. Proc. SPIE 3039, Smart Structures and Materials 1997: Mathematics and Control in Smart Structures. https://doi.org/10.1117/12.276584

  • Bruant I, Gallimard L, Nikoukar S (2010) Optimal piezoelectric actuator and sensor location for active vibration control, using genetic algorithm. J Sound Vib 329(10):1615–1635

    Article  Google Scholar 

  • Carbonari RC, Silva EC, Paulino GH (2009) Multi-actuated functionally graded piezoelectric micro-tools design: a multiphysics topology optimization approach. Int J Numer Methods Eng 77(3):301–336

    Article  MATH  Google Scholar 

  • Chevva K, Sun F, Blanc A, Mendoza J (2015) Active vibration control using minimum actuation power. J Sound Vib 340:1–21

    Article  Google Scholar 

  • Darivandi N, Morris K, Khajepour A (2013) An algorithm for LQ optimal actuator location. Smart Mater Struct 22(3):035001

    Article  Google Scholar 

  • Devasia S, Meressi T, Paden B, Bayo E (1993) Piezoelectric actuator design for vibration suppression-placement and sizing. J Guid Control Dyn 16(5):859–864

    Article  Google Scholar 

  • van Dijk NP, Maute K, Langelaar M, Van Keulen F (2013) Level-set methods for structural topology optimization: a review. Struct Multidiscip Optim 48(3):437–472

    Article  MathSciNet  Google Scholar 

  • Donoso A, Sigmund O (2009) Optimization of piezoelectric bimorph actuators with active damping for static and dynamic loads. Struct Multidiscip Optim 38(2):171–183

    Article  Google Scholar 

  • Dou S, Jensen JS (2015) Optimization of nonlinear structural resonance using the incremental harmonic balance method. J Sound Vib 334:239–254

    Article  Google Scholar 

  • Du J, Olhoff N (2007a) Minimization of sound radiation from vibrating bi-material structures using topology optimization. Struct Multidiscip Optim 33(4–5):305–321

    Article  Google Scholar 

  • Du J, Olhoff N (2007b) Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps. Struct Multidiscip Optim 34(2):91–110

    Article  MathSciNet  MATH  Google Scholar 

  • Dühring MB, Jensen JS, Sigmund O (2008) Acoustic design by topology optimization. J Sound Vib 317(3):557–575

    Article  Google Scholar 

  • Ghasemi H, Park HS, Rabczuk T (2017) A level-set based IGA formulation for topology optimization of flexoelectric materials. Comput Methods Appl Mech Eng 313:239–258

    Article  MathSciNet  Google Scholar 

  • Gonçalves JF, De Leon DM, Perondi EA (2017) Topology optimization of embedded piezoelectric actuators considering control spillover effects. J Sound Vib 388:20–41

    Article  Google Scholar 

  • Hu J, Zhang X, Kang Z (2017) Layout design of piezoelectric patches in structural LQR optimal control using topology optimization. Submitted

  • Huang X, Xie Y-M (2010) A further review of ESO type methods for topology optimization. Struct Multidiscip Optim 41(5):671–683

    Article  Google Scholar 

  • Kang Z, Tong L (2008) Integrated optimization of material layout and control voltage for piezoelectric laminated plates. J Intell Mater Syst Struct 19:889–904

  • Kang Z, Wang R, Tong L (2011) Combined optimization of bi-material structural layout and voltage distribution for in-plane piezoelectric actuation. Comput Methods Appl Mech Eng 200(13):1467–1478

    Article  MathSciNet  MATH  Google Scholar 

  • Kögl M, Silva EC (2005) Topology optimization of smart structures: design of piezoelectric plate and shell actuators. Smart Mater Struct 14(2):387

    Article  Google Scholar 

  • Liu H, Zhang W, Gao T (2015) A comparative study of dynamic analysis methods for structural topology optimization under harmonic force excitations. Struct Multidiscip Optim 51(6):1321–1333

    Article  MathSciNet  Google Scholar 

  • Ma Z-D, Kikuchi N, Cheng H-C (1995) Topological design for vibrating structures. Comput Methods Appl Mech Eng 121(1):259–280

    Article  MathSciNet  MATH  Google Scholar 

  • Mello LAM, Kiyono CY, Nakasone PH, Silva ECN (2014) Design of quasi-static piezoelectric plate based transducers by using topology optimization. Smart Mater Struct 23(2):025035

    Article  Google Scholar 

  • Mukherjee A, Joshi S (2001) Design of actuator profiles for minimum power consumption. Smart Mater Struct 10(2):305

    Article  Google Scholar 

  • Nanthakumar S, Lahmer T, Zhuang X, Park HS, Rabczuk T (2016) Topology optimization of piezoelectric nanostructures. J Mech Phys Solids 94:316–335

    Article  MathSciNet  Google Scholar 

  • Nguyen M, Nazeer H, Dekkers M, Blank D, Rijnders G (2013) Optimized electrode coverage of membrane actuators based on epitaxial PZT thin films. Smart Mater Struct 22(8):085013

    Article  Google Scholar 

  • Noh JY, Yoon GH (2012) Topology optimization of piezoelectric energy harvesting devices considering static and harmonic dynamic loads. Adv Eng Softw 53:45–60

    Article  Google Scholar 

  • Olhoff N, Du J (2014) Topological design for minimum sound emission from structures under forced vibration. In: Topology Optimization in Structural and Continuum Mechanics. Springer, Vienna 341–357

  • Olhoff N, Du J (2016) Generalized incremental frequency method for topological designof continuum structures for minimum dynamic compliance subject to forced vibration at a prescribed low or high value of the excitation frequency. Struct Multidiscip Optim 54(5):1113–1141

    Article  MathSciNet  Google Scholar 

  • Peng F, Ng A, Hu Y-R (2005) Actuator placement optimization and adaptive vibration control of plate smart structures. J Intell Mater Syst Struct 16(3):263–271

    Article  Google Scholar 

  • Ruiz D, Bellido J, Donoso A, Sánchez-Rojas J (2013) Design of in-plane piezoelectric sensors for static response by simultaneously optimizing the host structure and the electrode profile. Struct Multidiscip Optim 48(5):1023–1026

    Article  Google Scholar 

  • Shu L, Wang MY, Fang Z, Ma Z, Wei P (2011) Level set based structural topology optimization for minimizing frequency response. J Sound Vib 330(24):5820–5834

    Article  Google Scholar 

  • Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidiscip Optim 48(6):1031–1055

    Article  MathSciNet  Google Scholar 

  • Sigmund O, Torquato S (1997) Design of materials with extreme thermal expansion using a three-phase topology optimization method. Journal of the Mechanics and Physics of Solids 45:1037–1067

  • Silva ECN, Kikuchi N (1999) Design of piezoelectric transducers using topology optimization. Smart Mater Struct 8(3):350

    Article  Google Scholar 

  • Sun D, Tong L (2005) Design optimization of piezoelectric actuator patterns for static shape control of smart plates. Smart Mater Struct 14(6):1353

    Article  Google Scholar 

  • Svanberg K (2002) A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J Optim 12(2):555–573

    Article  MathSciNet  MATH  Google Scholar 

  • Takezawa A, Makihara K, Kogiso N, Kitamura M (2014) Layout optimization methodology of piezoelectric transducers in energy-recycling semi-active vibration control systems. J Sound Vib 333(2):327–344

    Article  Google Scholar 

  • Takezawa A, Daifuku M, Nakano Y, Nakagawa K, Yamamoto T, Kitamura M (2016) Topology optimization of damping material for reducing resonance response based on complex dynamic compliance. J Sound Vib 365:230–243

    Article  Google Scholar 

  • Vicente W, Picelli R, Pavanello R, Xie Y (2015) Topology optimization of frequency responses of fluid–structure interaction systems. Finite Elem Anal Des 98:1–13

    Article  Google Scholar 

  • Wallenhauer C, Kappel A, Gottlieb B, Schwebel T, Lüth T (2009) Efficient class-B analog amplifier for a piezoelectric actuator drive. Mechatronics 19(1):56–64

    Article  Google Scholar 

  • Wang S, Tai K, Quek S (2006) Topology optimization of piezoelectric sensors/actuators for torsional vibration control of composite plates. Smart Mater Struct 15(2):253

    Article  Google Scholar 

  • Xu B, Jiang J, Ou J (2007) Integrated optimization of structural topology and control for piezoelectric smart trusses using genetic algorithm. J Sound Vib 307(3):393–427

    Article  Google Scholar 

  • Yamamoto T, Yamada T, Izui K, Nishiwaki S (2015) Topology optimization of free-layer damping material on a thin panel for maximizing modal loss factors expressed by only real eigenvalues. J Sound Vib 358:84–96

    Article  Google Scholar 

  • Yan K, Cheng G, Wang BP (2016) Topology optimization of plate structures subject to initial excitations for minimum dynamic performance index. Struct Multidiscip Optim 53(3):623–633

    Article  MathSciNet  Google Scholar 

  • Yoon GH (2010) Structural topology optimization for frequency response problem using model reduction schemes. Comput Methods Appl Mech Eng 199(25):1744–1763

    Article  MathSciNet  MATH  Google Scholar 

  • Yoon GH (2013) Acoustic topology optimization of fibrous material with Delany–Bazley empirical material formulation. J Sound Vib 332(5):1172–1187

    Article  Google Scholar 

  • Zhang X, Kang Z (2013) Topology optimization of damping layers for minimizing sound radiation of shell structures. J Sound Vib 332(10):2500–2519

    Article  Google Scholar 

  • Zhang X, Kang Z (2014a) Dynamic topology optimization of piezoelectric structures with active control for reducing transient response. Comput Methods Appl Mech Eng 281:200–219

    Article  MathSciNet  Google Scholar 

  • Zhang X, Kang Z (2014b) Topology optimization of piezoelectric layers in plates with active vibration control. J Intell Mater Syst Struct 25(6):697–712

    Article  Google Scholar 

  • Zhang X, Kang Z, Li M (2014) Topology optimization of electrode coverage of piezoelectric thin-walled structures with CGVF control for minimizing sound radiation. Struct Multidiscip Optim 50(5):799–814

    Article  MathSciNet  Google Scholar 

  • Zhang W, Liu H, Gao T (2015) Topology optimization of large-scale structures subjected to stationary random excitation: an efficient optimization procedure integrating pseudo excitation method and mode acceleration method. Comput Struct 158:61–70

    Article  Google Scholar 

  • Zorić ND, Simonović AM, Mitrović ZS, Stupar SN (2012) Optimal vibration control of smart composite beams with optimal size and location of piezoelectric sensing and actuation. J Intell Mater Syst Struct 24:499–526

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Acknowledgements

The authors would like to thank Prof. Krister Svanberg for providing the source code of the GCMMA algorithm. The authors also acknowledge the support of the Natural Science Foundation of China (11602049, 11425207, and U1608256), China Postdoctoral Science Foundation (2015 M581328), and Research Project of State Key Laboratory of Mechanical System and Vibration (MSV201603).

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Authors and Affiliations

  1. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, China

    Xiaopeng Zhang & Zhan Kang

  2. Department of Transportation and Environmental Systems, Graduate School of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima, Hiroshima, Japan

    Xiaopeng Zhang & Akihiro Takezawa

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  1. Xiaopeng Zhang
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  2. Akihiro Takezawa
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Correspondence to Zhan Kang.

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Zhang, X., Takezawa, A. & Kang, Z. Topology optimization of piezoelectric smart structures for minimum energy consumption under active control. Struct Multidisc Optim 58, 185–199 (2018). https://doi.org/10.1007/s00158-017-1886-y

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  • DOI: https://doi.org/10.1007/s00158-017-1886-y

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