On Shape Optimization for an Evolution Coupled System

  • G. Leugering
  • A. A. Novotny
  • G. Perla Menzala
  • J. SokołowskiEmail author
Open Access


A shape optimization problem in three spatial dimensions for an elasto-dynamic piezoelectric body coupled to an acoustic chamber is introduced. Well-posedness of the problem is established and first order necessary optimality conditions are derived in the framework of the boundary variation technique. In particular, the existence of the shape gradient for an integral shape functional is obtained, as well as its regularity, sufficient for applications e.g. in modern loudspeaker technologies. The shape gradients are given by functions supported on the moving boundaries. The paper extends results obtained by the authors in (Math. Methods Appl. Sci. 33(17):2118–2131, 2010) where a similar problem was treated without acoustic coupling.


Piezoelectricity Electromechanical interaction Shape sensitivity analysis 


  1. 1.
    Cagnol, J., Zolésio, J.P.: Shape derivative in the wave equation with Dirichlet boundary conditions. J. Differ. Equ. 334(3), 175–210 (1999) CrossRefGoogle Scholar
  2. 2.
    Cardone, G., Nazarov, S.A., Sokołowski, J.: Asymptotic analysis, polarization matrices, and topological derivatives for piezoelectric materials with small voids. SIAM J. Control Optim. 48(6), 3925–3961 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Delfour, M.C., Zolésio, J.P.: Shapes and Geometries. Advances in Design and Control. SIAM, Philadelphia (2001) Google Scholar
  4. 4.
    Eshelby, J.D.: The elastic energy-momentum tensor. J. Elast. 5(3–4), 321–335 (1975) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Geis, W., Mishuris, G., Sändig, A.-M.: Piezoelectricity in multi-layer actuators modelling and analysis in two and three dimensions. Berichte aus dem Institut für Angewandte Analysis und Numerische Simulation 23, Universität Stuttgart (2003) Google Scholar
  6. 6.
    Gurtin, M.E.: An Introduction to Continuum Mechanics. Mathematics in Science and Engineering, vol. 158. Academic Press, New York (1981) zbMATHGoogle Scholar
  7. 7.
    Gurtin, M.E.: Configurational Forces as Basic Concept of Continuum Physics. Applied Mathematical Sciences, vol. 137. Springer, New York (2000) Google Scholar
  8. 8.
    Lasiecka, I.: Mathematical Control Theory of Coupled PDEs. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 75. SIAM, Philadelphia (2002) zbMATHCrossRefGoogle Scholar
  9. 9.
    Leugering, G., Novotny, A.A., Menzala, G. Perla, Sokołowski, J.: Shape sensitivity analysis of a quasi-electrostatic piezoelectric system in multilayered media. Math. Methods Appl. Sci. 33(17), 2118–2131 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Melnik, V.N.: Existence and uniqueness theorems of the generalized solution for a class of non-stationary problem of coupled electroelasticity. Sov. Math., Izv. VUZ, Math. 35(4), 24–32 (1991) MathSciNetGoogle Scholar
  11. 11.
    Mercier, D., Nicaise, S.: Existence, uniqueness and regularity results for piezoelectric systems. SIAM J. Math. Anal. 37, 651–672 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Sokołowski, J., Zolésio, J.P.: Introduction to Shape Optimization—Shape Sensitivity Analysis. Springer, New York (1992) zbMATHCrossRefGoogle Scholar
  13. 13.
    Wein, F., Kaltenbacher, M., Schury, F., Leugering, G., Bänsch, E.: Topology optimization of piezoelectric actuators using the simp method. In: 10th Workshop on Optimization and Inverse Problems in Electromagnetism—OIPE, Ilmenau, Germany (2008) Google Scholar
  14. 14.
    Wein, F., Kaltenbacher, M., Schury, F., Leugering, G., Bänsch, E.: Topology optimization of a piezoelectric loudspeaker coupled with the acoustic domain. In: Proceedings WCSMO-08, Lisbon, Portugal (2009) Google Scholar
  15. 15.
    Wein, F., Kaltenbacher, M., Schury, F., Leugering, G., Bänsch, E.: Topology optimization of a piezoelectric-mechanical actuator withsingle- and multiple-frequency excitation. Int. J. Appl. Electromagn. Mech. 30(3–4), 201–221 (2009) Google Scholar
  16. 16.
    Wein, F., Kaltenbacher, M., Kaltenbacher, B., Leugering, G., Bänsch, E., Schury, F.: On the effect of self-penalization of piezoelectric composites in topology optimization. Struct. Multidiscip. Optim. 43, 405–417 (2011) MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  • G. Leugering
    • 1
  • A. A. Novotny
    • 2
  • G. Perla Menzala
    • 2
  • J. Sokołowski
    • 3
    • 4
    Email author
  1. 1.Department of MathematicsUniversity of Erlangen NurembergErlangenGermany
  2. 2.Laboratório Nacional de Computação Científica LNCC/MCTCoordenação de Matemática Aplicada e ComputacionalPetrópolisBrasil
  3. 3.Institut Elie Cartan, UMR 7502 (Nancy Université, CNRS, INRIA), Laboratoire de Mathématiques, Université Henri Poincaré Nancy IVandoeuvre-lès-Nancy CedexFrance
  4. 4.Systems Research Institute of the Polish Academy of SciencesWarszawaPoland

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