Computer Algebra for the Formation of Structural Matrices of Piezoceramic Finite Elements

  • Algimantas Čepulkauskas
  • Regina Kulvietienė
  • Genadijus Kulvietis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3992)


This paper deals with the description of a theoretical background of systematic computer algebra methods for the formation of structural matrices of piezoceramic finite elements. The efficiency of computer algebra application was compared here with the numerical integration methods of forming the structural matrices of the finite elements. To this end, the computer algebra system VIBRAN was used. Two popular finite elements for modelling piezoceramic actuators of the sector-type and the triangular one are discussed. All structural matrices of the elements were derived, using the computer algebra technique with the following automatic program code generation.


Computer Algebra Piezoelectric Actuator Structural Matrice Ultrasonic Motor Numerical Integration Scheme 
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  1. 1.
    Bansevicius, R., Barauskas, R., Kulvietis, G., Ragulskis, K.: Vibromotors for Precision Microrobots. Hemisphere Publishing Corp., USA (1988)Google Scholar
  2. 2.
    Bansevicius, R., Čepulkauskas, A., Kulvietienė, R., Kulvietis, G.: Computer algebra for real-time dynamics of robots with large numbers of joints. In: Bubak, M., van Albada, G.D., Sloot, P.M.A., Dongarra, J. (eds.) ICCS 2004. LNCS, vol. 3039, pp. 278–285. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  3. 3.
    Bansevicius, R., Parkin, R., Jebb, A., Knight, J.: Piezomechanics as a Sub-System of Mechatronics: Present State of the Art, Problems, Future Developments. IEEE Transactions on Industrial Electronics 43(1), 23–30 (1996)CrossRefGoogle Scholar
  4. 4.
    Bansevicius, R., Kulvietis, G., Mazeika, D.: Optimal Electrodes Configuration of Piezodrives. In: Solid Mechanics and Its Applications, vol. N 73, pp. 77–83. Kluwer Academic Publ., Dordrecht (2000)Google Scholar
  5. 5.
    Belevicius, R., Pedersen, P.: Analysis and Sensitivity Analysis by Computer Algebra for a Third Order Plate Finite Element. Comp. & Struct. 49, 243–252 (1993)MATHCrossRefGoogle Scholar
  6. 6.
    Besseling, J.F.: Finite Element Properties, Based upon Elastic Potential Interpolation. In: Hybrid and Mixed Finite Element Meth. Int. Symp. Atlanta, 1981, pp. 253–266. John Wiley, Chichester (1983)Google Scholar
  7. 7.
    Han, J.M., Adriaens, T.A., Willem, L., de Koning, R.B.: Modelling Piezoelectric Actuators. IEEE/ASME Transactions on Mechatronics 5(4), 331–337 (2000)CrossRefGoogle Scholar
  8. 8.
    Laouafa, F., Royis, P.: An Iterative Algorithm for Finite Element Analysis. Journal of Computational and Applied Mathematics, 164–165, 469–491 (2004)Google Scholar
  9. 9.
    Liew, K.M., He, X.Q., Kitipornchai, S.: Finite Element Method for the Feedback Control of FGM Shells in the Frequency Domain via Piezoelectric Sensors and Actuators. Computer Methods in Applied Mechanics and Engineering 193(3), 257–273 (2004)MATHCrossRefGoogle Scholar
  10. 10.
    Marczak, R.J.: Object-Oriented Numerical Integration – a Template Scheme for FEM and BEM Applications. Advances in Engineering Software, Corrected Proof, Available online (July 11 , 2005) (in Press)Google Scholar
  11. 11.
    Oliveira, A., Sousa, P., Costa Branco, P.J.: Surface Deformation by Piezoelectric Actuator: from Park and Agrawal Models to a Simplified Model Formulation Sensors and Actuators. A: Physical 115(2–3), 235–244 (2004)Google Scholar
  12. 12.
    Parashar, S., Das Gupta, A., von Wagner, U., Hagedorn, P.: Non-linear Shear Vibrations of Piezoceramic Actuators International. Journal of Non-Linear Mechanics 40, 429–443 (2005)MATHCrossRefGoogle Scholar
  13. 13.
    Samal, M.K., oths: A Finite Element Model for Nonlinear Behaviour of Piezoceramics under Weak. Electric Fields Finite Elements in Analysis and Design 41(15), 1464–1480 (2005)CrossRefGoogle Scholar
  14. 14.
    Storck, Heiner, Wallaschek, Jörg: The Effect of Tangential Elasticity of the Contact Layer between Stator and Rotor in Travelling Wave Ultrasonic Motors. International Journal of Non-Linear Mechanics 38, 143–159 (2003)MATHCrossRefGoogle Scholar
  15. 15.
    Tenchev, R.T.: A Study of the Accuracy of some FEM Stress Recovery Schemes for 2D Stress Concentration Problems. Finite Elements in Analysis and Design 29(2), 105–119 (1998)MATHCrossRefGoogle Scholar
  16. 16.
    Tzou, H.S.: Piezoelectric Shells (Distributed Sensing and Control of Continua). Kluwer Academic Publishers, Dordrecht (1993)Google Scholar
  17. 17.
    Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method. McGraw-Hill, New York (1991)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Algimantas Čepulkauskas
    • 1
  • Regina Kulvietienė
    • 1
  • Genadijus Kulvietis
    • 1
  1. 1.Vilnius Gediminas Technical UniversityVilniusLithuania

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