On a shape control problem for the stationary Navier-Stokes equations

Max D. Gunzburger; Hongchul Kim; Sandro Manservisi

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2000)

  • Volume: 34, Issue: 6, page 1233-1258
  • ISSN: 0764-583X

How to cite

top

Gunzburger, Max D., Kim, Hongchul, and Manservisi, Sandro. "On a shape control problem for the stationary Navier-Stokes equations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 34.6 (2000): 1233-1258. <http://eudml.org/doc/194035>.

@article{Gunzburger2000,
author = {Gunzburger, Max D., Kim, Hongchul, Manservisi, Sandro},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {stationary Navier-Stokes equations; viscous drag minimization; optimal shape control problem; two-dimensional channel; adjoint method; Lagrangian multiplier method; optimality system; shape gradient},
language = {eng},
number = {6},
pages = {1233-1258},
publisher = {Dunod},
title = {On a shape control problem for the stationary Navier-Stokes equations},
url = {http://eudml.org/doc/194035},
volume = {34},
year = {2000},
}

TY - JOUR
AU - Gunzburger, Max D.
AU - Kim, Hongchul
AU - Manservisi, Sandro
TI - On a shape control problem for the stationary Navier-Stokes equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2000
PB - Dunod
VL - 34
IS - 6
SP - 1233
EP - 1258
LA - eng
KW - stationary Navier-Stokes equations; viscous drag minimization; optimal shape control problem; two-dimensional channel; adjoint method; Lagrangian multiplier method; optimality system; shape gradient
UR - http://eudml.org/doc/194035
ER -

References

top
  1. [1] F. Abergel and R. Temam, On some control problems in fluid mechanics. Theor. Comp. Fluid Dyn. 1 (1990) 303-326. Zbl0708.76106
  2. [2] R. Adams, Sobolev Spaces, Academic Press, New York (1975). Zbl0314.46030MR450957
  3. [3] V. Alekseev, V. Tikhomirov and S. Fomin, Optimal Control. Consultants Bureau, New York (1987). Zbl0689.49001MR924574
  4. [4] G. Armugan and O. Pironneau, On the problem of riblets as a drag reduction device. Optimal Control Appl. Methods 10 (1989) 93-112. Zbl0667.49002MR997236
  5. [5] I. Babuska, The finite element method with Lagrangian multipliers. Numer. Math. 16 (1973) 179-192. Zbl0258.65108MR359352
  6. [6] D. Bedivan, Existence of a solution for complete least squares optimal shape problems. Numer. Funct. Anal. Optim. 18 (1997) 495-505. Zbl0895.35042MR1467658
  7. [7] D. Bedivan and G. Fix, An extension theorem for the space Hdiv. Appl. Math. Lett. (to appear). Zbl0915.46024
  8. [8] D. Begis and R. Glowinski, Application de la méthode des éléments finis à l'approximation d'un problème de domaine optimal. Méthodes de résolution des problèmes approchés. Appl. Math. Optim. 2 (1975) 130-169. Zbl0323.90063MR443372
  9. [9] D. Chenais, On the existence of a solution in a domain identification problem. J. Math. Anal. Appl 52 (1975) 189-219. Zbl0317.49005MR385666
  10. [10] P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). Zbl0383.65058MR520174
  11. [11] P. Ciarlet, Introduction to Numerical Linear Algebra and Optimization. Cambridge University, Cambridge (1989). Zbl0672.65001MR1015713
  12. [12] E. Dean, Q. Dinh, R. Glowinski, J. He, T. Pan and J. Periaux, Least squares domain embedding methods for Neumann problems: applications to fluid dynamics, in Domain Decomposition Methods for Partial Differential Equations, D. Keyes et al. Eds., SIAM, Philadelphia (1992). Zbl0768.76045MR1189595
  13. [13] N. Di Cesare, O. Pironneau and E. Polak, Consistent approximations for an optimal design problem. Report 98005, Labotatoire d'Analyse Numérique, Paris (1998). 
  14. [14] N. Fujii, Lower semi-continuity in domain optimization problems. J. Optim. Theory Appl. 57 (1988) 407-422. Zbl0629.49006MR972916
  15. [15] V. Girault and R. Raviart, The Finite Element Method for Navier-Stokes Equations: Theory and Algorithms. Springer, New York (1986). Zbl0585.65077MR851383
  16. [16] R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984). Zbl0536.65054MR737005
  17. [17] R. Glowinski and O. Pironneau, Toward the computation of minimum drag profile in viscous laminar flow. Appl. Math. Model. 1 (1976) 58-66. Zbl0361.76035MR455851
  18. [18] M. Gunzburger, L. Hou and T. Svobodny, Analysis and finite element approximations of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls. RAIRO Modél. Math. Anal. Numér. 25 (1991) 711-748. Zbl0737.76045MR1135991
  19. [19] M. Gunzburger, L. Hou and T. Svobodny, Optimal control and optimization of viscous, incompressible flow, in Incompressible Computational Fluid Dynamics, M. Gunzburger and R. Nicolaides Eds., Cambridge University, New York (1993) 109-150. Zbl1189.76447
  20. [20] M. Gunzburger and H. Kim, Existence of a shape control problem for the stationary Navier-Stokes equations. SIAM J. Control Optim. 36 (1998) 895-909. Zbl0917.49004MR1613877
  21. [21] M. Gunzburger and S. Manservisi, Analysis and approximation of the velocity tracking problem for Navier-Stokes equations with distributed control. SIAM J. Numer. Anal. 37 (2000) 1481-1512. Zbl0963.35150MR1759904
  22. [22] M. Gunzburger and S. Manservisi, The velocity tracking problem for Navier-Stokes flows with bounded distributed control. SIAM J. Control Optim. 37 (1999) 1913-1945. Zbl0938.35118MR1720145
  23. [23] M. Gunzburger and S. Manservisi, A variational inequality formulation of an inverse elasticity problem. Comput. Methods Appl. Mech. Engrg. 189 (2000) 803-823. Zbl0969.76025MR1755697
  24. [24] M. Gunzburger and S. Manservisi, Some numerical computations of optimal shapes for Navier-Stokes flows (in préparation). Zbl0963.35150
  25. [25] J. Haslinger, K.H. Hoffmann and M. Kocvara, Control fictitious domain method for solving optimal shape design problems. RAIRO Modél Math. Anal. Numér. 27 (1993) 157-182. Zbl0772.65043MR1211614
  26. [26] J. Haslinger and P. Neittaanmaki, Finite Element Approximation for Optimal Shape, Material and Topology Design, 2nd edn. Wiley, Chichester (1996). Zbl0845.73001MR1419500
  27. [27] K. Kunisch and G. Pensil, Shape optimization for mixed boundary value problems based on an embedding domain method (to appear). Zbl0914.49027
  28. [28] O. Pironneau, Optimal Shape Design in Fluid Mechanics. Thesis, University of Paris, France (1976). 
  29. [29] O. Pironneau, On optimal design in fluid mechanics. J. Fluid. Mech. 64 (1974) 97-110. Zbl0281.76020MR347229
  30. [30] O. Pironneau, Optimal Shape Design for Elliptic Systems. Springer, Berlin (1984). Zbl0534.49001MR725856
  31. [31] R. Showalter, Hilbert Space Methods for Partial Differential Equations. Electron. J. Differential Equations (1994) http://ejde.math.swt.edu/mono-toc.html Zbl0991.35001MR1302484
  32. [32] J. Simon, Domain variation for Stokes fiow, in Lecture Notes in Control and Inform. Sci. 159, X. Li and J. Yang Eds., Springer, Berlin (1990) 28-42. Zbl0801.76075MR1129956
  33. [33] J. Simon, Domain variation for drag Stokes flows, in Lecture notes in Control and Inform. Sci.114, A. Bermudez Ed., Springer, Berlin (1987) 277-283. Zbl0801.76075
  34. [34] T. Slawig, Domain Optimization for the Stationary Stokes and Navier-Stokes Equations by Embedding Domain Technique. Thesis, TU Berlin, Berlin (1998). Zbl0935.49020
  35. [35] J. Sokolowski and J. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer, Berlin (1992). Zbl0761.73003MR1215733
  36. [36] S. Stojanovic, Non-smooth analysis and shape optimization in flow problems. IMA Preprint Series 1046, IMA, Minneapolis (1992). 
  37. [37] R. Temam, Navier-Stokes equation. North-Holland, Amsterdam (1979). Zbl0426.35003
  38. [38] R. Temam, Navier-Stokes equations and Nonlinear Functional Analysis. SIAM, Philadelphia (1993). Zbl0833.35110MR1318914
  39. [39] V. Tikhomirov, Fundamental Principles of the Theory of Extremal Problems. Wiley, Chichester (1986). Zbl0595.49001MR866483

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.