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Advanced Numerical Methods for PDE Constrained Optimization with Application to Optimal Design in Navier Stokes Flow

  • Christian BrandenburgEmail author
  • Florian Lindemann
  • Michael Ulbrich
  • Stefan Ulbrich
Chapter
Part of the International Series of Numerical Mathematics book series (ISNM, volume 160)

Abstract

We present an approach to shape optimization which is based on transformation to a reference domain with continuous adjoint computations. This method is applied to the instationary Navier-Stokes equations for which we discuss the appropriate setting and discuss Fréchet differentiability of the velocity field with respect to domain transformations. Goal-oriented error estimation is used for an adaptive refinement strategy. Finally, we give some numerical results.

Keywords

Shape optimization Navier-Stokes equations PDE-constrained optimization goal-oriented error estimation 

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References

  1. 1.
    I. Babuška and A. Miller, The post-processing approach in the finite element method. Part 1: Calculation of displacements, stresses and other higher derivatives of the displacements, Int. J. Numer. Methods Eng., 20, pp. 1085–1109, 1984.CrossRefzbMATHGoogle Scholar
  2. 2.
    R. Becker and H. Kapp and R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: Basic concepts, SIAM J. Control Optim., 39(1), pp. 113–132, 2000.CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numerica 10, pp. 1–102, 2001.CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    J.A. Bello, E. Fernandez-Cara, J. Lemoine, and J. Simon, The differentiability of the drag with respect to the variations of a Lipschitz domain in Navier-Stokes flow, SIAM J. Control Optim., 35(2), pp. 626–640, 1997.CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    M. Berggren, Numerical solution of a flow-control problem: vorticity reduction by dynamic boundary action, SIAM J. Sci. Comput. 19, no. 3, pp. 829–860, 1998.CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    C. Brandenburg, F. Lindemann, M. Ulbrich, and S. Ulbrich, A continuous adjoint approach to shape optimization for Navier Stokes flow, in Optimal control of coupled systems of partial differential equations, K. Kunisch, G. Leugering, and J. Sprekels, Eds., Int. Ser. Numer. Math. 158, Birkhäuser, Basel, pp. 35–56, 2009.CrossRefGoogle Scholar
  7. 7.
    C. Brandenburg and S. Ulbrich, Goal-oriented error estimation for shape optimization with the instationary Navier-Stokes equations, Preprint in preparation, Fachbereich Mathematik, TU Darmstadt, 2010.Google Scholar
  8. 8.
    K.K. Choi and N.-H. Kim, Structural sensitivity analysis and optimization 2: Nonlinear systems and applications, Mechanical Engineering Series, Springer, 2005.Google Scholar
  9. 9.
    M.C. Delfour and J.-P. Zolésio, Shapes and geometries: Analysis, differential calculus, and optimization, SIAM series on Advances in Design and Control, 2001.Google Scholar
  10. 10.
    J. Doormaal and G.D. Raithby, Enhancements of the SIMPLE method for predicting incompressible fluid flows, Num. Heat Transfer, 7, pp. 147–163, 1984.zbMATHGoogle Scholar
  11. 11.
    K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Introduction to adaptive methods for differential equations, Acta Numerica, pp. 105–158, 1995.Google Scholar
  12. 12.
    K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Computational differential equations, Cambridge University Press, 1996.Google Scholar
  13. 13.
    M. Geissert, H. Heck, and M. Hieber, On the equation div 𝑢 = 𝑔 and Bogovski’s operator in Sobolev spaces of negative order, in Partial differential equations and functional analysis, E. Koelink, J. van Neerven, B. de Pagter, and G. Sweers, eds., Oper. Theory Adv. Appl., 168, Birkhäuser, Basel, pp. 113–121, 2006.Google Scholar
  14. 14.
    P. Guillaume and M. Masmoudi, Computation of high-order derivatives in optimal shape design, Numer. Math., 67, pp. 231–250, 1994.CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    J. Haslinger and R.A.E. Mäkinen, Introduction to shape optimization: Theory, approximation, and computation, SIAM series on Advances in Design and Control, 2003.Google Scholar
  16. 16.
    M.A. Heroux, J.M. Willenbring, and R. Heaphy, Trilinos developers guide, Sandia National Laboratories, SAND2003-1898, 2003.Google Scholar
  17. 17.
    M. Hinze, Optimal and instantaneous control of the instationary Navier-Stokes equations, Habilitation, TU Dresden, 2002.Google Scholar
  18. 18.
    J. Hoffman and C. Johnson, Adaptive Finite Element Methods for Incompressible Fluid Flow, Error estimation and solution adaptive discretization in CFD: Lecture Notes in Computational Science and Engineering, Springer Verlag, 2002.Google Scholar
  19. 19.
    J. Hoffman and C. Johnson, A new approach to computational turbulence modeling, Comput. Meth. Appl. Mech. Engrg., 195, pp. 2865–2880, 2006.CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    F. Lindemann, M. Ulbrich, and S. Ulbrich, Fréchet differentiability of time-dependent incompressible Navier-Stokes flow with respect to domain variations, Preprint in preparation, Fakultät für Mathematik, TU München, 2010.Google Scholar
  21. 21.
    A. Logg and G.N. Wells, DOLFIN: Automated finite element computing, ACMTransactions on Mathematical Software, 37(2), 2010.Google Scholar
  22. 22.
    K. Long, Sundance: A rapid prototyping tool for parallel PDE-constrained optimization, in Large-scale pde-constrained optimization, L.T. Biegler, M. Heinkenschloss, O. Ghattas, and B. van Bloemen Wanders, eds., Lecture Notes in Computational Science and Engineering, 30, Springer, pp. 331–341, 2003.Google Scholar
  23. 23.
    D. Meidner and B. Vexler, Adaptive space-time finite element methods for parabolic optimization problems, SIAM J. Control Optim., 46, pp. 116–142, 2007.CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    B. Mohammadi and O. Pironneau, Applied shape optimization for fluids, Oxford University Press, 2001.Google Scholar
  25. 25.
    M.E. Mortensen, Geometric modeling, Wiley, 1985.Google Scholar
  26. 26.
    F. Murat and S. Simon, Etudes de problems d’optimal design, Lectures Notes in Computer Science, 41, pp. 54–62, 1976.Google Scholar
  27. 27.
    M. Schäfer and S. Turek, Benchmarkc omputations of laminar flow around a cylinder, Preprints SFB 359, No. 96-03, Universit¨at Heidelberg, 1996.Google Scholar
  28. 28.
    J. Sokolowski and J.-P. Zolésio, Introduction to shape optimization, Series in Computational Mathematics, Springer, 1992.Google Scholar
  29. 29.
    T. Slawig, PDE-constrained control using Femlab – Control of the Navier-Stokes equations, Numerical Algorithms, 42, pp. 107–126, 2006.CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    R. Temam, Navier-Stokes equations: Theory and numerical analysis 3rd Edition, Elsevier Science Publishers, 1984.Google Scholar
  31. 31.
    B. Vexler and W. Wollner, Adaptive finite elements for elliptic optimization problems with control constraints, SIAM J. Control Optim., 47, pp. 509–534, 2008.CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    A. Wächter and L.T. Biegler, On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming, Math. Program., 106, pp. 25–57, 2006.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • Christian Brandenburg
    • 1
    Email author
  • Florian Lindemann
    • 2
  • Michael Ulbrich
    • 2
  • Stefan Ulbrich
    • 1
  1. 1.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Lehrstuhl für Mathematische Optimierung Zentrum Mathematik, M1TU MünchenGarching bei MünchenGermany

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