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

Published online by Cambridge University Press:  15 April 2002

Max D. Gunzburger ,
Hongchul Kim  and
Sandro Manservisi
Max D. Gunzburger
Affiliation:
Department of Mathematics, Iowa State University, Ames IA, 50011-2064, USA. (gunzburg@math.iastate.edu) Supported in part by the Air Force Office of Scientific Research under grant number F49620-95-1-0407.
Hongchul Kim
Affiliation:
Department of Mathematics, Kangnŭng National University, Kangnŭng 210-702, Korea. (hongchul@knusun.kangnung.ac.kr)
Sandro Manservisi
Affiliation:
Department of Mathematics, Kaiserslautern University, Kaiserslautern, 67663, Germany. Current address: LIN, DIENCA, University of Bologna, Via dei colli 16, 40136 Bologna, Italy. (sandro.manservisi@mail.ing.unibo.it) Supported by the European community under grant XCT-97-0117.
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Abstract

An optimal shape control problem for the stationary Navier-Stokessystem is considered. An incompressible, viscous flow in atwo-dimensional channel is studied to determine the shape of part ofthe boundary that minimizes the viscous drag. Theadjoint method and the Lagrangian multiplier method are used to derivethe optimality system for the shapegradient of the design functional.

Keywords

Shape controloptimal designNavier-Stokes equationsdrag minimization.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 6 , November 2000 , pp. 1233 - 1258
Copyright
© EDP Sciences, SMAI, 2000

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References

Abergel, F. and Temam, R., On some control problems in fluid mechanics. Theor. Comp. Fluid Dyn. 1 (1990) 303-326. CrossRef
R. Adams, Sobolev Spaces. Academic Press, New York (1975).
V. Alekseev, V. Tikhomirov and S. Fomin, Optimal Control. Consultants Bureau, New York (1987).
Armugan, G. and Pironneau, O., On the problem of riblets as a drag reduction device. Optimal Control Appl. Methods 10 (1989) 93-112. CrossRef
Babuska, I., The finite element method with Lagrangian multipliers. Numer. Math. 16 (1973) 179-192. CrossRef
Bedivan, D., Existence of a solution for complete least squares optimal shape problems. Numer. Funct. Anal. Optim. 18 (1997) 495-505. CrossRef
D. Bedivan and G. Fix, An extension theorem for the space H div . Appl. Math. Lett. (to appear).
Begis, D. and Glowinski, R., 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. CrossRef
Chenais, D., On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52 (1975) 189-219. CrossRef
P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978).
P. Ciarlet, Introduction to Numerical Linear Algebra and Optimization. Cambridge University, Cambridge (1989).
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).
N. Di Cesare, O. Pironneau and E. Polak, Consistent approximations for an optimal design problem. Report 98005, Labotatoire d'Analyse Numérique, Paris (1998).
Fujii, N., Lower semi-continuity in domain optimization problems. J. Optim. Theory Appl. 57 (1988) 407-422. CrossRef
V. Girault and P. Raviart, The Finite Element Method for Navier-Stokes Equations: Theory and Algorithms. Springer, New York (1986).
R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984).
Glowinski, R. and Pironneau, O., Toward the computation of minimum drag profile in viscous laminar flow. Appl. Math. Model. 1 (1976) 58-66. CrossRef
Gunzburger, M., Hou, L. and Svobodny, T., 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. CrossRef
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.
Gunzburger, M. and Kim, H., Existence of a shape control problem for the stationary Navier-Stokes equations. SIAM J. Control Optim. 36 (1998) 895-909. CrossRef
Gunzburger, M. and Manservisi, S., Analysis and approximation of the velocity tracking problem for Navier-Stokes equations with distributed control. SIAM J. Numer. Anal. 37 (2000) 1481-1512. CrossRef
Gunzburger, M. and Manservisi, S., The velocity tracking problem for Navier-Stokes flows with bounded distributed control. SIAM J. Control Optim. 37 (1999) 1913-1945. CrossRef
Gunzburger, M. and Manservisi, S., A variational inequality formulation of an inverse elasticity problem. Comput. Methods Appl. Mech. Engrg. 189 (2000) 803-823. CrossRef
M. Gunzburger and S. Manservisi, Some numerical computations of optimal shapes for Navier-Stokes flows (in preparation).
Haslinger, J., Hoffmann, K.H. and Kocvara, M., Control fictitious domain method for solving optimal shape design problems. RAIRO Modél. Math. Anal. Numér. 27 (1993) 157-182. CrossRef
J. Haslinger and P. Neittaanmaki, Finite Element Approximation for Optimal Shape, Material and Topology Design, 2nd edn. Wiley, Chichester (1996).
K. Kunisch and G. Pensil, Shape optimization for mixed boundary value problems based on an embedding domain method (to appear).
O. Pironneau, Optimal Shape Design in Fluid Mechanics. Thesis, University of Paris, France (1976).
Pironneau, O., On optimal design in fluid mechanics. J. Fluid. Mech. 64 (1974) 97-110. CrossRef
O. Pironneau, Optimal Shape Design for Elliptic Systems. Springer, Berlin (1984).
R. Showalter, Hilbert Space Methods for Partial Differential Equations. Electron. J. Differential Equations (1994) http://ejde.math.swt.edu/mono-toc.html
J. Simon, Domain variation for Stokes flow, in Lecture Notes in Control and Inform. Sci. 159, X. Li and J. Yang Eds., Springer, Berlin (1990) 28-42.
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.
T. Slawig, Domain Optimization for the Stationary Stokes and Navier-Stokes Equations by Embedding Domain Technique. Thesis, TU Berlin, Berlin (1998).
J. Sokolowski and J. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer, Berlin (1992).
S. Stojanovic, Non-smooth analysis and shape optimization in flow problems. IMA Preprint Series 1046, IMA, Minneapolis (1992).
R. Temam, Navier-Stokes equation. North-Holland, Amsterdam (1979).
R. Temam, Navier-Stokes equations and Nonlinear Functional Analysis. SIAM, Philadelphia (1993).
V. Tikhomirov, Fundamental Principles of the Theory of Extremal Problems. Wiley, Chichester (1986).