Skip to main content
SpringerLink
Log in

Optimal Design in Two-Dimensional Conductivity for a General Cost Depending on the Field

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We explore an optimal design problem in two-dimensional conductivity for a rather general cost depending on the underlying field. Through a typical variational reformulation that has been explored recently, we provide a simplified relaxed version which is amenable to numerical simulation, and prove that it is a true relaxation under a main structural hypothesis. Several important cases are covered including a linear cost in the gradient and a convex, isotropic functional (in particular, the pth power of the field for any \(p\geqq1\)). For the isotropic, non-quadratic case, our computations do not require an explicit form of the constrained quasiconvexification of the equivalent vector variational problem. That structural assumption ties together the underlying state equation and the integral cost.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Allaire G. (2002) Shape Optimization by the Homogenization Method. Springer-Verlag, Berlin

    MATH  Google Scholar 

  2. Aranda, E., Pedregal, P. Constrained envelope for a general class of design problems. Discrete Contin. Dyn. Syst., Supplement July 2003, 30–41 (2002)

  3. Ball, J.M. Some open problems in elasticity. Geometry, Mechanics and Dynamics (eds. P. Newton, P. Holmes, A. Weinstein), Springer, New York, pp 3–59, 2002

  4. Battacharya K., Dolzmann G. (2001) Relaxation of some multi-well problems. Proc. Roy. Soc. Edinburgn Sect. A 131, 279–320

    Google Scholar 

  5. Bellido J.C., Pedregal P. (2002) Explicit quasiconvexification of some cost functionals depending on derivatives of the state in optimal design. Discrete Contin. Dyn. Syst. 8, 967–982

    Article  MathSciNet  MATH  Google Scholar 

  6. Bousselsal M., Le Dret R. (2002) Remarks on the quasiconvex envelope of some functions depending on quadratic forms. Boll. Unione. Mat. Ital. Sez. B Artic. Ric. Mat. 5, 469–486

    MathSciNet  MATH  Google Scholar 

  7. Bousselsal M., Le Dret R. (2002) Relaxation of functionals involving homogeneous functions and invariance of envelopes. Chinese Ann. Math. Ser. B 23, 37–52

    Article  MathSciNet  MATH  Google Scholar 

  8. Cherkaev A. (2002) Variational Methods for Structural Optimization. Springer-Verlag, New York

    Google Scholar 

  9. Donoso, A., Pedregal, P. Optimal design of 2-D conducting graded materials by minimizing quadratic functionals in the field. Structural Opt., in press

  10. Grabovsky Y. (2001) Optimal design problems for two-phase conducting composites with weakly discontinuous objective functionals. Advan. in Appl. Math. 27, 683–704

    Article  MathSciNet  MATH  Google Scholar 

  11. Lipton R. (2004) Stress constrained G closure and relaxation of structural design problems. Quart. Appl. Math. 62, 295–321

    MathSciNet  MATH  Google Scholar 

  12. Lipton R. (2004) Bounds on the distribution of extreme values for the stress in composite materials. J. Mech. Phys. Solids 52, 1053–1069

    Article  MathSciNet  MATH  ADS  Google Scholar 

  13. Lipton R. (2003) Assessment of the local stress state through macroscopic variables. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 361, 921–946

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Lipton R. (2002) Relaxation through homogenization for optimal design problems with gradient constraints. J. Optim. Theory Appl. 114, 27–53

    Article  MathSciNet  MATH  Google Scholar 

  15. Lipton R. (2002) Design of functionally graded composite structures in the presence of stress constraints. Internat. J. Solids Structures 39, 2575–2586

    Article  MathSciNet  MATH  Google Scholar 

  16. Lipton, R., Velo, A. Optimal design of gradient fields with applications to electrostatics. Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, Vol XIV (Paris, 1997/1998), 509–532, Studies in Mathematics and its Applications, 31, North Holland, Amsterdam, 2002

  17. Murat, F. Contre-exemples pur divers probèmes ou le contrôe intervient dans les coefficients. Ann. Mat. Pura Appl. (4) 112, 49–68 (1977)

    Google Scholar 

  18. Pedregal P. (2000) Optimal design and constrained quasiconvexity. SIAM J. Math. Anal. 32, 854–869

    Article  MathSciNet  MATH  Google Scholar 

  19. Pedregal P. (2000) Variational Methods in Nonlinear Elasticity. SIAM, Philadelphia

    MATH  Google Scholar 

  20. Pedregal P. (2004) Constrained quasiconvexification of the square of the gradient of the state in optimal design. Quart. Appl. Math. 62, 459–470

    MathSciNet  MATH  Google Scholar 

  21. Tartar, L. Remarks on optimal design problems. Calculus of Variations, Homogenization and Continuum Mechanics (G. Buttazzo, G. Bouchitte and P. Suquet, eds). World Scientific, Singapore, 279–296, 1994

  22. Tartar, L. An Introduction to the Homogenization Method in Optimal design. Springer Lecture Notes in Mathematics, Vol. 1740, 47–15610, 2000

Download references

Author information

Authors and Affiliations

  1. ETSI Industriales, Universidad de Castilla-La Mancha, 13071, Ciudad Real, Spain

    Pablo Pedregal

Authors
  1. Pablo Pedregal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pablo Pedregal.

Additional information

Communicated by M. Ortiz

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pedregal, P. Optimal Design in Two-Dimensional Conductivity for a General Cost Depending on the Field. Arch Rational Mech Anal 182, 367–385 (2006). https://doi.org/10.1007/s00205-006-0007-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-006-0007-7

Keywords

Search

Navigation