Domain sensitivity for singular elliptic problems: Harmonic transformation approach

  • A. Zochowski
Contributed Papers
  • 52 Downloads

Abstract

A method for computing the sensitivities of functionals depending on the solutions of elliptic equations defined over variable domains is presented. It is based on the material derivative approach and allows the uniform treatment of both singular and nonsingular cases. The novelty consists in defining the vector field connected with the domain transformation as the solution of an auxiliary elliptic equation. Such a choice does not restrict the range of admissible goal functionals and has many advantages from the numerical point of view. It allows one also to consider singular domain variations.

Key Words

Shape optimization elliptic problems material derivatives shape derivatives singularities 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cea, J., andHaug, E. J., Editors,Optimization of Distributed Parameter Structures, Sijthoff and Noordhoff, Amsterdam, Holland, 1981.Google Scholar
  2. 2.
    Choi, K. K., Haug, E. J., andKomkow, V.,Design Sensitivity Analysis of Structural Systems, Academic Press, New York, New York, 1986.Google Scholar
  3. 3.
    Choi, K. K., andSeong, H. G.,A Domain Method for Shape Design Sensitivity Analysis of Built-up Structures, Computer Methods in Applied Mechanics and Engineering, Vol. 57, No. 1, pp. 1–15, 1986.Google Scholar
  4. 4.
    Choi, K. K., andSeong, H. G.,Boundary Layer Approach to Shape Design Sensitivity Analysis, Mechanics of Structures and Machines, Vol. 15, No. 2, pp. 241–263, 1987.Google Scholar
  5. 5.
    Zolesio, J. P.,The Material Derivative (or Speed) Method for Shape Optimization, Optimization of Distributed Parameter Structures, Edited by J. Cea and E. J. Haug, Sijthoff and Noordhoff, Amsterdam, Holland, pp. 1089–1150, 1981.Google Scholar
  6. 6.
    Chenais, D.,On the Existence of a Solution in a Domain Identification Problem, Journal of Mathematical Analysis and Applications, Vol. 52, pp. 189–219, 1975.Google Scholar
  7. 7.
    Grisvard, P.,Elliptic Problems in Nonsmooth Domains, Pitman, London, Great Britain, 1985.Google Scholar
  8. 8.
    Vasilopoulos, D.,On the Determination of Higher-Order Terms of Singular Elastic Stress Fields Near Corners, Numerische Mathematik, Vol. 53, pp. 51–95, 1988.Google Scholar
  9. 9.
    Zochowski, A.,The Design of a Two-Dimensional Domain, Analysis and Algorithms of Optimization Problems, Edited by K. Malanowski and K. Mizukami, Springer-Verlag, Berlin, Germany, pp. 111–134, 1986.Google Scholar
  10. 10.
    Zochowski, A., andMizukami, K.,A Stress Versus Compliance Constraint in a Minimum Weight Design, Computers and Structures, Vol. 18, No. 1, pp. 9–13, 1984.Google Scholar
  11. 11.
    Zochowski, A., andMizukami, K.,Identification of the Shape of Elastic Body, Proceedings of the 9th IFAC World Congress, Budapest, Hungary, 1984.Google Scholar
  12. 12.
    Zochowski, A., andHolnicki, P.,Interactive Method of Variational Grid Generation, Journal of Computational and Applied Mathematics, Vol. 26, pp. 281–287, 1989.Google Scholar
  13. 13.
    Zolesio, J. P.,Multiplication dans les Espaces de Besov, Proceedings of the Royal Society of Edinburgh, Vol. 78A, pp. 113–117, 1977.Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • A. Zochowski
    • 1
  1. 1.Systems Research InstitutePolish Academy of SciencesWarsawPoland

Personalised recommendations