Advertisement

Journal of Mathematical Sciences

, Volume 174, Issue 2, pp 229–242 | Cite as

A posteriori error estimators of finite-element approximations for problems of forced harmonic vibrations of piezoelectrics

  • F. V. Chaban
  • H. A. Shynkarenko
Article
  • 19 Downloads

A variational problem of determination of the amplitude of the velocity vector of elastic displacements and the electric potential of a piezoelectric with instantaneous memory that executes steady-state vibrations under the influence of harmonic loads of given circular frequency is formulated. We establish conditions of well-posedness of this class of problems and find a priori estimates of convergence of approximations of the finite-element method to their solution. A posteriori error estimators of approximations of the finite-element method, which enable us to determine the distribution of energy norms of errors by solving local problems of the residual of approximation at each finite element of triangulation, are constructed. An algorithm for calculating estimators of this type is described in detail for two-dimensional problems for both triangular and tetragonal finite elements, which may be extended to the three-dimensional case. The efficiency and reliability of the estimator of piecewise bilinear approximations of the finite-element method is illustrated by numerical solutions of a model problem.

Keywords

Variational Problem Harmonic Load Posteriori Error Estimator Galerkin Scheme Galerkin Discretization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    O. Yu. Zharii and A. F. Ulitko, Introduction to Mechanics of Nonstationary Vibrations and Waves [in Russian], Vyshcha Shkola, Kiev (1989).Google Scholar
  2. 2.
    H. Kvasnytsya and H. A. Shynkarenko, “Comparison of simple a posteriori estimators of errors of the finite-element method for problems of elastostatics,” Visn. L’viv. Univ. Ser. Prykl. Mat. Inform., Issue 7, 162–174 (2003).Google Scholar
  3. 3.
    W. Nowacki, Efekty Elektromagnetyczne w Stalych Cialach Odksztalcalnych, PWN, Warsaw (1993).Google Scholar
  4. 4.
    V. Z. Parton and B. A. Kudryavtsev, Electromagnetoelasticity of Piezoelectric and Electroconducting Bodies [in Russian], Nauka, Moscow (1988).Google Scholar
  5. 5.
    H. A. Shynkarenko, “Statement and solvability of initial boundary-value problems of electroviscoelasticity,” Visn. L’viv. Univ. Ser. Mekh.-Mat., Issue 33, 10–16 (1990).Google Scholar
  6. 6.
    G. A. Shinkarenko, “Projection-mesh approximations for variational problems of pyroelectricity. I. Statement of problems and analysis of steady-state forced vibrations,” Differ. Uravn., 29, No. 7, 1252–1260 (1993).MathSciNetGoogle Scholar
  7. 7.
    N. A. Shul’ga and A. M. Bolkisev, Vibrations of Piezoelectric Bodies [in Russian], Naukova Dumka, Kiev (1990).Google Scholar
  8. 8.
    F. V. Chaban and H. A. Shynkarenko, “Constructing of h-adaptive finite-element method for piezoelectricity problem,” Zh. Obchysl. Prykl. Mat. Ser. Obchysl. Mat., 97, Issue 1 (97), 1–9 (2009).Google Scholar
  9. 9.
    F. Chaban and H. Shynkarenko, “The construction and analysis of a posteriori error estimators for piezoelectricity stationary problems,” Oper. Theory: Adv. Appl., 191, 291–304 (2009).MathSciNetGoogle Scholar
  10. 10.
    A. Premount, Mechatronics: Dynamics of Electromechanical and Piezoelectric Systems, Springer, Berlin (2006).Google Scholar
  11. 11.
    J. Yang, An Introduction to the Theory of Piezoelectricity, Springer, Berlin (2005).MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • F. V. Chaban
    • 1
  • H. A. Shynkarenko
    • 1
    • 2
  1. 1.Franko Lviv National UniversityLvivUkraine
  2. 2.Opole University of TechnologyOpolePoland

Personalised recommendations