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Exact Solutions in Optimal Design Problems for Stationary Diffusion Equation

Acta Applicandae Mathematicae volume 161, pages 71–88 (2019)Cite this article

Abstract

We consider two-phase multiple state optimal design problems for stationary diffusion equation. Both phases are taken to be isotropic, and the goal is to find the optimal distribution of materials within domain, with prescribed amounts, that minimizes a weighted sum of energies. In the case of one state equation, it is known that the proper relaxation of the problem via the homogenization theory is equivalent to a simpler relaxed problem, stated only in terms of the local proportion of given materials.

We prove an analogous result for multiple state problems if the number of states is less than the space dimension. In spherically symmetric case, the result holds for arbitrary number of states, and the optimality conditions of a simpler relaxation problem, which are necessary and sufficient, enable us to explicitly calculate the unique solution of proper relaxation for some examples. In contrary to maximization problems, these solutions are not classical.

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References

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

    Book  MATH  Google Scholar 

  2. Antonić, N., Vrdoljak, M.: Sequential laminates in multiple state optimal design problems. Math. Probl. Eng. 2006, 68695 (2006). https://doi.org/10.1155/MPE/2006/68695

    MathSciNet  MATH  Google Scholar 

  3. Antonić, N., Vrdoljak, M.: Gradient methods for multiple state optimal design problems. Ann. Univ. Ferrara, Sez. 7: Sci. Mat. 53, 177–187 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Burazin, K.: On unique solutions of multiple-state optimal design problems on an annulus. J. Optim. Theory Appl. 177, 329–344 (2018). https://doi.org/10.1007/s10957-018-1284-7

    Article  MathSciNet  MATH  Google Scholar 

  5. Burazin, K., Crnjac, I., Vrdoljak, M.: Variant of optimality criteria method for multiple state optimal design problems. Commun. Math. Sci., accepted for publication, 18 pp.

  6. Casado-Diaz, J.: Smoothness properties for the optimal mixture of two isotropic materials: the compliance and eigenvalue problems. SIAM J. Control Optim. 53, 2319–2349 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Casado-Diaz, J.: Some smoothness results for the optimal design of a two-composite material which minimizes the energy. Calc. Var. Partial Differ. Equ. 53, 649–673 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. SIAM, Philadelphia (1999)

    Book  MATH  Google Scholar 

  9. Goodman, J., Kohn, R.V., Reyna, L.: Numerical study of a relaxed variational problem from optimal design. Comput. Methods Appl. Mech. Eng. 57, 107–127 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. Elsevier, Amsterdam (1979)

    MATH  Google Scholar 

  11. Lurie, K.A., Cherkaev, A.V.: Exact estimates of conductivity of composites formed by two isotropically conducting madia, taken in prescribed proportion. Proc. R. Soc. Edinb. 99A, 71–87 (1984)

    Article  MATH  Google Scholar 

  12. Murat, F., Tartar, L.: Calcul des Variations et Homogénéisation. In: Les Méthodes de l’Homogenisation Théorie et Applications en Physique. Collect. Dir. Études Rech. Élec. France, vol. 57, pp. 319–369. Eyrolles, Paris (1985)

    Google Scholar 

  13. Tartar, L.: Estimations fines des coefficients homogénéisés. In: Krée, P. (ed.) Ennio DeGiorgi colloquium. Res. Notes Math., vol. 125, pp. 168–187. Pitman, London (1985)

    Google Scholar 

  14. Tartar, L.: The appearance of oscillations in optimization problems. In: Non-Classical Continuum Mechanics, Durham, 1986. London Math. Soc. Lecture Note Ser., vol. 122, pp. 129–150. Cambridge University Press, Cambridge (1987)

    Chapter  Google Scholar 

  15. Tartar, L.: Remarks on homogenization method in optimal design problems. In: Homogenization and Applications to Material Sciences, Nice, 1995. GAKUTO Internat. Ser. Math. Sci. Appl., vol. 9, pp. 393–412. Gakkokotosho, Tokyo (1995)

    Google Scholar 

  16. Tartar, L.: An introduction to the homogenization method in optimal design. In: Cellina, A., Ornelas, A. (eds.) Optimal Shape Design, Troia, 1998. Lecture Notes in Math., vol. 1740, pp. 47–156. Springer, Berlin (2000)

    Chapter  Google Scholar 

  17. Vrdoljak, M.: On Hashin–Shtrikman bounds for mixtures of two isotropic materials. Nonlinear Anal., Real World Appl. 11, 4597–4606 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Vrdoljak, M.: Classical optimal design in two-phase conductivity problems. SIAM J. Control Optim. 54(4), 2020–2035 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Osijek, Osijek, Croatia

    Krešimir Burazin

  2. Department of Mathematics, Faculty of Science, University of Zagreb, Zagreb, Croatia

    Marko Vrdoljak

Authors
  1. Krešimir Burazin
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  2. Marko Vrdoljak
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Correspondence to Marko Vrdoljak.

Additional information

This work has been supported in part by Croatian Science Foundation under the project 9780 WeConMApp.

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Burazin, K., Vrdoljak, M. Exact Solutions in Optimal Design Problems for Stationary Diffusion Equation. Acta Appl Math 161, 71–88 (2019). https://doi.org/10.1007/s10440-018-0204-z

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  • DOI: https://doi.org/10.1007/s10440-018-0204-z

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