Skip to main content
SpringerLink
Log in

Topological Derivatives of Shape Functionals for Elasticity Systems

  • Conference paper
Optimal Control of Complex Structures

Part of the book series: ISNM International Series of Numerical Mathematics ((ISNM,volume 139))

Abstract

The exact form of topological derivative (TD) and the computational procedure for its calculation is derived for a class of shape functionals in 3D elasticity. The derivation is based on asymptotic expansion of solutions. TD is used numerical shape optimization and for solving shape inverse problems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H.A. Eschenauer, V.V. Kobelev, A. Schumacher, Bubble method for topology and shape optimization of structures Struct. Optimiz 8(1994), pp. 42–51.

    Article  Google Scholar 

  2. S. Garreau, PH. Guillaume, M. Masmoudi, The topological asymptotic for PDE systems: the elasticity case (2000) to appear.

    Google Scholar 

  3. D. Görde, Singuläre Störung von Randwertproblemen durch ein kleines Loch im Gebiet, Zeitschrift für Analysis und ihre Anwendungen Vol.4(5)(1985), pp. 467–477.

    Google Scholar 

  4. H. G. Hahn, Elastizitätstheorie, B.G. Teubner Stuttgart, 1985.

    MATH  Google Scholar 

  5. A. Herwig, Elliptische Randwertprobleme zweiter Ordnung in Gebieten mit einer Fehlstelle, Zeitschrift für Analysis und ihre Anwendungen No.8(2)(1989), pp. 153–161.

    MathSciNet  MATH  Google Scholar 

  6. A. M. Il’In, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Translations of Mathematical Monographs, Vol. 102, AMS 1992.

    MATH  Google Scholar 

  7. L. Jackowska-StrumiŁŁo, J. SokoŁowski, A. Żochowski, The topological derivative method and artificial neural networks for numerical solution of shape inverse problems, Rapport de Recherche No. 3739, 1999, INRIA-Lorraine.

    Google Scholar 

  8. M. Kachanov, I. Tsukrov, B. Shafiro, Effective moduli of solids with cavities of various shapes Appl. Mech. Rev. Volume 47, Number 1, Part 2, pp. S151–S174.

    Google Scholar 

  9. T. Lewinski, J. Sokolowski, Optimal Shells Formed on a Sphere. The Topological Derivative Method. Rapport de Recherche No. 3495, 1998, INRIA-Lorraine

    Google Scholar 

  10. T. Lewinski, J. Sokolowski, Topological derivative for nucleation of non-circular voids Contemporary Mathematics, American Math. Soc. vol 268, 2000, pp. 341–361.

    Article  MathSciNet  Google Scholar 

  11. T. Lewinski, J. Sokolowski, Energy change due to appearing of cavities in elastic solids in preparation.

    Google Scholar 

  12. A. Shumacher, Topologieoptimierung von Bauteilstrukturen unter Verwendung von Lochpositionierungkriterien, Ph.D. Thesis, Universität-Gesamthochschule-Siegen, Siegen, 1995.

    Google Scholar 

  13. J. SokoŁowski, J-P. Zolesio, Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer Verlag, 1992.

    Book  MATH  Google Scholar 

  14. J. SokoŁowsei, A. Żochowski, On topological derivative in shape optimization, SIAM Journal on Control and Optimization. Volume 37, Number 4, 1999, pp. 1251–1272.

    Article  MathSciNet  Google Scholar 

  15. J. SoxoŁowsxl, A. Żochowski Topological derivative for optimal control problems, Proceedings of Fifth International Symposium on Methods and Models in Automation and Robotics, Miçdzyzdroje, Poland, August, 1998, pp. 111–116; special issue of Control and Cybernetics Recent Advances in Control of PDEs 28(1999) pp. 611–626.

    Google Scholar 

  16. J. SokoŁowski, A. Żochowski, Topological derivatives for elliptic problems, Inverse Problems, Volume 15, Number 1, 1999, pp. 123–134.

    Article  MathSciNet  MATH  Google Scholar 

  17. J. SokoŁowski, A. Żochowski, Topological derivatives of shape functionals for elasticity systems Les prépublications de l’Institut Elie Cartan 35/99.

    Google Scholar 

  18. J. SoxoŁowsxi, A. Żochowski, Optimality conditions for simultaneous topology and shape optimization Les prépublications de l’Institut Elie Cartan 7/2001

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Elie Cartan, Laboratoire de Mathématiques, Université Henri Poincaré Nancy I, B.P. 239, 54506, Vandoeuvre lès Nancy Cedex, France

    Jan Sokołowski

  2. Systems Research Institute of the Polish Academy of Sciences, ul. Newelska 6, 01-447, Warszawa, Poland

    Antoni Żochowski

Authors
  1. Jan Sokołowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antoni Żochowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Fachbereich Mathematik, Arbeitsgruppe 10, Technische Universität Darmstadt, Schlossgartenstrasse 7, 64289, Darmstadt, Germany

    Günter Leugering

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer Basel AG

About this paper

Cite this paper

Sokołowski, J., Żochowski, A. (2001). Topological Derivatives of Shape Functionals for Elasticity Systems. In: Hoffmann, KH., Lasiecka, I., Leugering, G., Sprekels, J., Tröltzsch, F. (eds) Optimal Control of Complex Structures. ISNM International Series of Numerical Mathematics, vol 139. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8148-7_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8148-7_19

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9456-2

  • Online ISBN: 978-3-0348-8148-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation