Skip to main content
SpringerLink
Log in

Topological Derivative in Shape Optimization

  • Reference work entry
Book cover Encyclopedia of Optimization

Article Outline

Introduction

Definitions

Formulation

  Three-Dimensional Anisotropic Elastic Body with a Small Cavity

  Contact Problem for Plane Elasticity

  Solution of the Elasticity System in the Ring

Cases

  Plane Isotropic Elasticity System

  Three-Dimensional Isotropic Elasticity Systems

Conclusions

References

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 2,500.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 2,499.99
Price excludes VAT (USA)
  • Durable hardcover 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

References

  1. Allaire G, de Gournay F, Jouve F, Toader A-M (2005) Structural optimization using topological and shape sensitivity via a level set method. Control Cybern 34:59–80

    MATH  Google Scholar 

  2. Allaire G, Jouve F (2007) Minimum stress optimal design with the level set method, to appear in Int J Boundary Elem Appl

    Google Scholar 

  3. Amstutz S, Horchani I, Masmoudi M (2005) Crack detection by the topological gradient method. Control Cybern 34:119–138

    MathSciNet  Google Scholar 

  4. Amstutz S, Andrä H (2006) A new algorithm for topology optimization using a level-set method. J Comput Phys 216(2):573–588

    Article  MATH  MathSciNet  Google Scholar 

  5. Auroux D, Masmoudi M Image processing by topological asymptotic expansion. J Math Imag Vis, to appear

    Google Scholar 

  6. Bonnet M, Guzina BB (2004) Sounding of finite solid bodies by way of topological derivative. Int J Numer Methods Eng 61:2344–2373

    Article  MATH  MathSciNet  Google Scholar 

  7. Burger M, Hackl B, Ring W (2004) Incorporating Topological Derivatives into Level Set Methods. J Comput Phys 194:344–362

    Article  MATH  MathSciNet  Google Scholar 

  8. Eschenauer HA, Kobelev VV, Schumacher A (1994) Bubble method for topology and shape optimization of structures. Struct Optim 8:42–51

    Article  Google Scholar 

  9. Fulmanski P, Laurain A, Scheid J-F, Sokolowski J (2007) A levelset method in shape and topology optimization for variational inequalities. Int J Appl Math Comput Sci 17(3):413–430

    Article  MATH  MathSciNet  Google Scholar 

  10. Fulmanski P, Laurain A, Scheid J-F, Sokolowski J (2007) Level set method with topological derivatives in shape optimization Les Prépublications de l'Institut Elie Cartan

    Google Scholar 

  11. Garreau S, Guillaume P, Masmoudi M (2001) The topological asymptotic for PDE systems: the elasticity case. SIAM J Control Optim 39(6):1756–1778

    Article  MATH  MathSciNet  Google Scholar 

  12. Hahn HG (1985) Elastizitätstheorie. Teubner, Stuttgart

    MATH  Google Scholar 

  13. Hintermüller M (2005) Fast level-set based algorithms using shape and topological sensitivity information. Control Cybern 34(1):305–324

    MATH  Google Scholar 

  14. Hintermüller M (2006) A combined shape Newton and topology optimization technique in real-time image segmentation. In: Biegler L, Ghattas O, Heinkenschloss M, Keyes D, van Bloemen Waanders B (eds) Real-Time PDE-Constrained Optimization, SIAM

    Google Scholar 

  15. Il'in AM (1992) Matching of Asymptotic Expansions of Solutions of Boundary Value Problems. AMS, p 281

    Google Scholar 

  16. Jackowska-Strumiłło L, Sokołowski J, Żochowski A, Henrot A (2002) On numerical solution of shape inverse problems. Comput Optim Appl 23:231–255

    Article  MATH  MathSciNet  Google Scholar 

  17. Kamotski IV, Nazarov SA (1998) Spectral problems in singular perturbed domains and selfadjoint extensions of differential operators. Trudy St.-Petersburg Mat Obshch 6:151–212 (Engl. transl. in: Proceedings of the St. Petersburg Mathematical Society (2000), Am Math Soc Transl Ser 2 6:127–181, 199, Am Math Soc, Providence, RI)

    Google Scholar 

  18. Laurain A (2006) Domaines singulierements perturbes en optimisation de formes. Ph.D. dissertation, Institut Elie Cartan, Nancy, http://tel.archives-ouvertes.fr/tel-00139595

  19. Leugering G, Sokolowski J Topological derivatives for ellipticv problems on graphs. In: Novotny A, Tarocco E, de Souza Neto E (eds) The book for Raul Feijoo, to be published by International Center for Numerical, Methods in Engineering (CIMNE), Barcelona, in press

    Google Scholar 

  20. Lewinski T, Sokolowski J (1998) Optimal shells formed on a sphere. The topological derivative method. RR-3495, INRIA-Lorraine

    Google Scholar 

  21. Lewinski T, Sokolowski J (2000) Topological derivative for nucleation of non-circular voids Contemporary Mathematics. Am Math Soc 268:341–361

    MathSciNet  Google Scholar 

  22. Lewinski T, Sokolowski J (2003) Energy change due to appearing of cavities in elastic solids. Int J Solids Struct 40:1765–1803

    Article  MATH  Google Scholar 

  23. Lewinski T, Sokolowski J, Żochowski A (1999) Justification of the bubble method for the compliance minimization problems of plates and spherical shells. CD-Rom, 3rd World Congress of Structural and Multidisciplinary Optimization (WCSMO-3) Buffalo/Niagara Falls, NY, May 17–21

    Google Scholar 

  24. Mazja WG, Nazarov SA, Plamenevskii BA (1991) Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten. 1, 2. Akademie-Verlag, Berlin. (English transl.: Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, vol 1, 2. Birkhäuser, Basel (2000))

    Google Scholar 

  25. Muskhelishvili NI (1953) Some Basic Problems on the Mathematical Theory of Elasticity. Noordhoff, Groningen

    Google Scholar 

  26. Nazarov SA, Plamenevsky BA (1994) Elliptic Problems in Domains with Piecewise Smooth Boundaries. De Gruyter Exposition in Mathematics 13. Walter de Gruyter, Berlin

    MATH  Google Scholar 

  27. Nazarov SA, Sokołowski J (2003) Asymptotic analysis of shape functionals. J Math Pures Appl 82(2):125–196

    MATH  MathSciNet  Google Scholar 

  28. Nazarov SA, Sokołowski J (2004) Topological derivative of the Dirichlet integral due to formation of a thin ligaments. Siberian Math J 45:341–355

    Article  MathSciNet  Google Scholar 

  29. Nazarov SA, Sokołowski J (2006) Self adjoint extensons for the Neumann laplacian in application to shape optimization. Act Math Sin Engl Ser 22:879–906

    Article  MATH  Google Scholar 

  30. Nazarov SA, Slutskij AS, Sokołowski J (2005) Topological derivative of the energy functional due to formation of a thin ligament on a spatial body. Folia Math 12:39–72

    MATH  MathSciNet  Google Scholar 

  31. Novotny AA (2003) Anàlise de Sensibilidade Topológica. Ph.D. Thesis. LNCC/MCT, Petrópolis

    Google Scholar 

  32. Novotny AA, Feijo RA, Padra C, Taroco E (2005) Topological-Shape Sensitivity Analysis Applied to Topology Design of Kirchhoff's Plate Bending Problem. Control Cybern 34(1):339–361

    MATH  Google Scholar 

  33. Rao M, Sokołowski J (2000) Tangent sets in Banach spaces and applications to variational inequalities. Les prépublications de l'Institut Élie Cartan 42

    Google Scholar 

  34. Rocha de Faria J, Novotny AA, Feijo RA, Taroco E, Padra C (2007) Second Order Topological Sensitivity Analysis. Int J Solids Struct 44(14–15):4958–4977

    Article  MATH  Google Scholar 

  35. Sokołowski J, Żochowski A (1999) On topological derivative in shape optimization. SIAM J Control Optim 37(4):1251–1272

    Article  MATH  MathSciNet  Google Scholar 

  36. Sokołowski J, Żochowski A (1999) Topological derivative for optimal control problems. Control Cybern 28(3):611–626

    MATH  Google Scholar 

  37. Sokołowski J, Żochowski A (1999) Topological derivatives for elliptic problems. Inverse Problems 15(1):123–134

    Article  MATH  MathSciNet  Google Scholar 

  38. Sokołowski J, Żochowski A (2001) Topological derivatives of shape functionals for elasticity systems. Mech Struct Mach 29(3):333–351

    Google Scholar 

  39. Sokołowski J, Żochowski A (2003) Optimality conditions for simultaneous topology and shape optimization. SIAM J Control Optim 42:1198–1221

    Article  MATH  MathSciNet  Google Scholar 

  40. Sokołowski J, Żochowski A (2005) Modelling of topological derivatives for contact problems. Num Math 102:145–179

    Article  MATH  Google Scholar 

  41. Sokołowski J, Żochowski A (2007) Asymptotic analysis and topological derivatives for shape and topology optimization of elasticity problems in two spatial dimensions. Prépublication IECN, 16

    Google Scholar 

  42. Sokołowski J, Zolesio J-P (1992) Introduction to Shape Optimization. Shape Sensitivity Analysis. Springer, Berlin

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques, Université Henri Poincaré, Nancy, France

    Jan Sokolowski

  2. Systems Research Institute of the Polish Academy of Sciences, Warsaw, Poland

    Antoni Zochowski

Authors
  1. Jan Sokolowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antoni Zochowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Chemical Engineering, Princeton University, Princeton, NJ, 08544-5263, USA

    Christodoulos A. Floudas

  2. Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, 32611-6595, USA

    Panos M. Pardalos

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag

About this entry

Cite this entry

Sokolowski, J., Zochowski, A. (2008). Topological Derivative in Shape Optimization . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_682

Download citation

Publish with us

Policies and ethics

Search

Navigation