Advertisement

Optimal shape design for navier-strokes flow

  • Juan A. Bello
  • Enrique Fernandez-Cara
  • Universidad de Sevilla
  • Jacques Simon
  • C.N.R.S. et Laboratorie de Mathématiques Appliquées
III Optimal Control III.2 Distributed Parameter Systems
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 180)

Abstract

The computation of optimal profiles, i.e. those minimizing the drag, has been investigated by several suthors. frequently, the drag has been approximated by the viscous energy which is dissipated in the fluid. For instance, O. Pironneau computes in [9,10] the “derivative” of this quantity adapting Hadamard’s normal variations techniques. F. Murat and J. Simon use in [6] formal calculus to deduce an expression for the derivative. More recently, J. Simon has considered the problem from a rigorous viewpoint in the case of a Stokes flow (cf. [13]). See als [4], for some theoretical and numerical considerations. This paper deals with optimal profiles in Navier-Stokes regime. Let us introduce an initital body Δ and assume that an admissible variation of Δ is represented by a vector field u. We prove that the mapping uJ(Δ+u), where J(Δ+u) is the energy sssociated to the body Δ+u, is Fréchet-differentiable. We also apply some results from [11] to the computation of the derivative.

Keywords

Total Derivative Stoke Flow Pair Function Formal Calculus Local Derivative 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Agmon, S.-Douglis, A.-Niremberg, L.: Estimates Near the Boundary for Solutions of Elliptic Partial Differential Equations Satisfying General Boundary Conditions, I and II. Comm. Pure and Applied Math., Vol. XII, 623–727 (1959); Vol. XVII, 35–92 (1964).CrossRefGoogle Scholar
  2. [2]
    Bello, J.A.: Thesis. University of Sevilla, to appear.Google Scholar
  3. [3]
    Bello, J.A.-Simon, J.: To appear.Google Scholar
  4. [4]
    Fernández Cara, E.: Optimal design in fluid Mechanics. In “Control of Partial Differential Equations”, Proceedings of IFIP Conference in Santiago de Compostela, 1987, A. Bermúdez Ed., p. 120–131, Lecture Notes in Control and Information Sciences No. 114, Springer-Verlag, 1989.Google Scholar
  5. [5]
    Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, London 1969.zbMATHGoogle Scholar
  6. [6]
    Murat F.-Simon J.: Quelques résultats sur le contrôle par un domaine géometrique. Rapport du L.A. 189 No. 74003. Université Paris VI (1974).Google Scholar
  7. [7]
    Murat F.-Simon J.: Sur le contrôle par un domaine géometrique. Rapport du L.A. 189 No. 76015. Université Paris VI (1976).Google Scholar
  8. [8]
    Nečas, J.: Les Méthodes Directes en Théorie des Equations Elliptiques. Masson, Paris 1967.Google Scholar
  9. [9]
    Pironneau, O.: On optimum design in fluid Mechanics. J. Fluid. Mech. (1974), Vol. 64, part. I, pp. 97–110.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Pironneau, O.: Optimal Shape Design for Elliptic Systems. Springer-Verlag, New-York, 1984.zbMATHGoogle Scholar
  11. [11]
    Simon, J.: Differentiation with respect to the Domain in Boundary Value Problems. Numer. Funct. Anal. and Optimiz., 2 (7 and 8), 649–687 (1980).zbMATHCrossRefGoogle Scholar
  12. [12]
    Simon, J.: Diferenciación de Problemas de Contorno respecto del Dominio. Lectures in the University of Sevilla, 1989.Google Scholar
  13. [13]
    Simon, J.: Domain variation for drag in Stokes flow. Proceedings of IFIP Conference in Shanghaï, 1990, Li Xunjing Ed., Lecture Notes in Control and Information Sciences, to appear.Google Scholar

Copyright information

© International Federation for Information Processing 1992

Authors and Affiliations

  • Juan A. Bello
    • 1
  • Enrique Fernandez-Cara
    • 1
  • Universidad de Sevilla
  • Jacques Simon
    • 2
  • C.N.R.S. et Laboratorie de Mathématiques Appliquées
  1. 1.Departmento de Análisis MatemáticoUniversidad de SevillaSevillaSpain
  2. 2.Université Blaise Pascal (Clermot-Ferrand 2)Aubiere CedexFrance

Personalised recommendations