On Shape Optimization and Related Issues

  • Roland Glowinski
  • Jiwen He
Part of the Progress in Systems and Control Theory book series (PSCT, volume 24)


The main goal of this article is to review, briefly, some of the issues associated with shape optimization for systems modeled by partial differential equations. The practical calculation of the objective function gradient is one of these issues and it clearly includes the use of Automatic Differentiation (AD) techniques for derivative computations. Also, we shall take advantage of this article to describe some recent results concerning the controllability of the Kuramoto-Sivashinsky equation since these results seem to justify the well-known claim that under certain conditions “chaos may enhance controllability.”


Adjoint Equation Transonic Flow Automatic Differentiation Direct Search Method Shape Optimization Problem 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. Cea and E. Haug (eds.). Optimization of Distributed Parameter Structures, Sijhtoff and Noordhoff, Alphen aan der Rijn, the Netherlands, 1981.MATHGoogle Scholar
  2. [2]
    R. Glowinski and O. Pironneau. On the numerical computation of the minimum-drag profile in laminar flow, J. Fluid Mech., 72(2):385–389, 1975.MATHCrossRefGoogle Scholar
  3. [3]
    R. Glowinski and O. Pironneau. Toward the computation of minimum drag profiles in viscous laminar flow, Applied Math. Modeling, 1:58–66, 1976.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    O. Pironneau. Optimal Shape Design for Elliptic Systems,Springer-Verlag, Berlin, 1983.Google Scholar
  5. [5]
    F. Angrand, R. Glowinski, J. Periaux, P. Perrier and O. Pironneau. Optimum design for potential flow, in Proceedings of the Third International Conference on Finite Elements in Flow Problems, Banff, Alberta, Canada, Vol I, D.H. Norrie ed., 400–412, June 1980.Google Scholar
  6. [6]
    A. Jameson. Aerodynamic design, in Computational Science for the 21st Century, M.O. Bristeau, G. Etgen, W. Fitzgibbon, J.L. Lions, J. Périaux and M.F. Wheeler, eds., J. Wiley, Chichester, 619–633, 1997,.Google Scholar
  7. [7]
    S. Ta’asan, G. Kuruvila and M. Salas. Aerodynamic design and optimization in one shot, AIAA paper 92–0025, 1992.Google Scholar
  8. [8]
    A. Dervieux, J.A. Desideri and N. Marco. Aspects of optimal control in aerodynamics, in Computational Science for the 21st Century, M.O. Bristeau, G. Etgen, W. Fitzgibbon, J.L. Lions, J. Périaux and M.F. Wheeler, eds., J. Wiley, Chichester, 608–618, 1997.Google Scholar
  9. [9]
    B. Stoufflet. A few issues of future simulation tools in computational fluid dynamics for aeronautical applications, in Computational Science for the 21st Century, M.O. Bristeau, G. Etgen, W. Fitzgibbon, J.L. Lions, J. Périaux and M.F. Wheeler, eds., J. Wiley, Chichester, 392–400, 1997.Google Scholar
  10. [10]
    J. Haslinger and P. Neittanmäki. Finite Element Approximation for Optimal Shape Design: Theory and Applications, Wiley, New York, NY, 1988.MATHGoogle Scholar
  11. [11]
    J. Sokolowski and J.P. Zolesio. Introduction to Shape Optimization, Springer-Verlag, Berlin, 1992.MATHGoogle Scholar
  12. [12]
    D. Begis and R. Glowinski. Application de la méthode des éléments finis à la résolution d’un problème de domaine optimal, in Computing Methods in Applied Sciences and Engineering, R. Glowinski and J.L. Lions, eds., Lecture Notes in Computer Sciences, Vol. 11, Springer-Verlag, Berlin, 403–434, 1974.CrossRefGoogle Scholar
  13. [13]
    D. Begis and R. Glowinski. Application de la méthode des éléments finis à l’approximation d’un problème de domaine optimal. Méthode de résolution des problèmes approchés, Applied Math. and Optimization, 2:30–169, 1975.MathSciNetGoogle Scholar
  14. [14]
    M.O. Bristeau, R. Glowinski, J. Périaux, O. Pironneau and G. Poirier. On the numerical solution of nonlinear problems in fluid dynamics by least-squares and finite element methods (II). Application to transonic flow simulations, Computer Methods in Applied Mech. and Eng., 51:363–394, 1985.MATHCrossRefGoogle Scholar
  15. [15]
    G. Sivashinsky. Nonlinear analysis of hydrodynamic instability in laminar flames, Part I. Derivation of basic equations, Acta Astronaut, 4:1177–1206, 1977.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    G. Sivashinsky. On flame propagation under conditions of stoichiometry, SIAM J. Appl. Math., 39:67–82, 1980.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    G. Sivashinsky and D.M. Michelson. On irregular wavy flow of a liquid down a vertical plane, Progr. Theoret. Phys., 63:2112–2114, 1980.CrossRefGoogle Scholar
  18. [18]
    S. Gama, U. Frisch and H. Scholl. The two-dimensional Navier-Stokes equations with a large scale instability of the Kuramoto-Sivashinsky type: numerical exploration on the Connection Machine, Journal of Scientific Computing, 6(4):425–452, 1991.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    M. Gorman, M. El-Hamdi and K.A. Robbins. Experimental observations of ordered states of cellular flames, Combust. Sci. and Tech., 98:37–45, 1994.CrossRefGoogle Scholar
  20. [20]
    I. G. Kevrekidis, B. Nicolaenko and J. C. Scovel. Back in the saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation, SIAM J. Appl. Math., 50(3):760–790, 1990.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    R. Temam. Infinite-Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, NY, 1988.MATHGoogle Scholar
  22. [22]
    J. M. Hyman and B. Nicolaenko. The Kuramoto-Sivashinsky equation: a bridge between PDEs and dynamical systems, Physica-D, 18:13–126, 1986.MathSciNetGoogle Scholar
  23. [23]
    A. Nicolas-Carrizosa. A factorization approach to the Kuramoto-Sivashinsky equation, in Advances in Numerical Partial Differential Equations and in Optimization, S. Gomez, J.P. Hennart, R. Tapia, eds., SIAM, Philadelphia, 262–272, 1991.Google Scholar
  24. [24]
    E.J. Dean, R. Glowinski and D. Trevas. An approximate factorization/least squares solution method for a mixed finite element approximation of the Cahn-Hilliard equation, Jap. J. Ind. Appl. Math., 13(3):495–517, 1996.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    J. Nocedal. Theory of algorithms for unconstrained optimization, Acta Numerica 1992, Cambridge University Press, 199–242, 1992.Google Scholar
  26. [26]
    J.L. Lions. Personal communication, 1997.Google Scholar
  27. [27]
    G. Lebeau and L. Robbiano. Controle exact de l’équation de la chaleur, Comm. in P.D.E.,20:335–356, 1995.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    E. Ott, T. Sauer and J.A. Yorke (eds.). Coping with Chaos, J. Wiley, New York, N.Y., 1994.MATHGoogle Scholar
  29. [29]
    E. Zuazua. Approximate controllability of the semilinear heat equation: boundary control, in Computational Science for the 21st Century, M.O. Bristeau, G. Etgen, W. Fitzgibbon, J.L. Lions, J. Périaux and M.F. Wheeler, eds., J. Wiley, Chichester, 738–747, 1997.Google Scholar
  30. [30]
    J.W. He and R. Glowinski. Controllability and stabilization issues for the Kuramoto-Sivashinsky equation: A numerical approach, (in preparation).Google Scholar
  31. [31]
    P. Hovland, C. Bischof, D. Spigelman and M. Casella. Efficient derivative codes through automatic differentiation and interface contraction: an application in Biostatistics, SIAM J. Sci. Comp., 18(4):1056–1066, 1997.MATHCrossRefGoogle Scholar
  32. [32]
    N. Rostaing, S. Dalmas and A. Galligo. Automatic differentiation in Odyssee, Tellus, 45a:558–568, 1993.Google Scholar
  33. [33]
    B. Mohammadi. Optimal Shape Design, Reverse Mode of Automatic Differentiation and Turbulence, AIAA paper 97–0099, AIAA 35th Aerospace Science Meeting and Exhibit, Reno, Nevada, January 6–9, 1997.Google Scholar
  34. [34]
    R. Brent. Algorithms for Minimization without Derivatives, Prentice-Hall, Englewood Cliffs, N.J., 1973.MATHGoogle Scholar
  35. [35]
    J. Neider and R. Mead. A simplex method for function minimization, Comput. J., 7:308–313, 1965.CrossRefGoogle Scholar
  36. [36]
    J. Dennis and V. Torczon. Direct search methods on parallel machines, SIAM J. Optim., 1:448–474, 1991.MathSciNetMATHCrossRefGoogle Scholar
  37. [37]
    V. Torczon. On the convergence of the multidimensional direct search, SIAM J. Optim.,1:123–145, 1991.MathSciNetMATHCrossRefGoogle Scholar
  38. [38]
    S. Kirkpatrick, C.D. Gelatt and M.D. Vecchi. Science, 220:671, 1983.MathSciNetMATHCrossRefGoogle Scholar
  39. [39]
    D.E. Goldberg. Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley, Reading, MA, 1989.Google Scholar
  40. [40]
    H. Kawarada and H. Suito. Fuzzy Optimization Methods, in Computational Science for the 21st Century, M.O. Bristeau, G. Etgen, W. Fitzgibbon, J.L. Lions, J. Périaux and M.F. Wheeler, eds., J. Wiley, Chichester, 642–651, 1997.Google Scholar
  41. [41]
    R. Glowinski, T.W. Pan, A.J. Kearsley and J. Periaux. Numerical simulation and optimal shape for viscous flow by a fictitious domain method, Int. J. Num. Meth. Fluids, 20:695–711, 1995.MathSciNetMATHCrossRefGoogle Scholar
  42. [42]
    M. Berggren, R. Glowinski and J.L. Lions. A computational approach to controllability issues for flow-related models. (I): Pointwise control of the viscous Burgers equations, Int. J. of C.F.D., 7:237–252, 1996.MATHGoogle Scholar
  43. [43]
    A. Griewank. Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation, Optimization and Software, 1:35–54, 1992.CrossRefGoogle Scholar
  44. [44]
    K. Miettinen. Review of Nonlinear MCMD Methods, Report of the Laboratory of Scientific Computing, Department of Mathematics, University of Jyväskylä, Finland, 1997.Google Scholar
  45. [45]
    J. Dennis and R. Schnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, England Cliffs, NJ, 1983.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Roland Glowinski
    • 1
  • Jiwen He
    • 1
  1. 1.Department of MathematicsUniversity of HoustonHoustonUSA

Personalised recommendations