Skip to main content

Moreau–Yosida regularization in shape optimization with geometric constraints

Abstract

In the context of shape optimization with geometric constraints we employ the method of mappings (perturbation of identity) to obtain an optimal control problem with a nonlinear state equation on a fixed reference domain. The Lagrange multiplier associated with the geometric shape constraint has a low regularity (similar to state constrained problems), which we circumvent by penalization and a continuation scheme. We employ a Moreau–Yosida-type regularization and assume a second-order condition to hold. The regularized problems can then be solved with a semismooth Newton method and we study the properties of the regularized solutions and the rate of convergence towards a solution of the original problem. A model for the value function in the spirit of Hintermüller and Kunisch (SIAM J Control Optim 45(4): 1198–1221, 2006) is introduced and used in an update strategy for the regularization parameter. The theoretical findings are supported by numerical tests.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3

References

  1. 1.

    Angrand, F.: Optimum design for potential flows. Int. J. Numer. Methods Fluids 3, 265–282 (1983)

    Article  MATH  Google Scholar 

  2. 2.

    Antonietti, P., Borzì, A., Verani, M.: Multigrid shape optimization governed by elliptic pdes. SIAM J. Control Optim. 51(2), 1417–1440 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Brandenburg, C.: Adjoint-based adaptive multilevel shape optimization based on goal-oriented error estimators for the instationary navier-stokes equations. Ph.D. thesis, TU Darmstadt (2011)

  4. 4.

    Brandenburg, C., Lindemann, F., Ulbrich, M., Ulbrich, S.: A continuous adjoint approach to shape optimization for navier-stokes flow. In: Kunisch, K., Sprekels, J., Leugering, G., Tröltzsch, F. (eds.) ptimal Control of Coupled Systems of Partial Differential Equations, International Series of Numerical Mathematics, vol. 158, pp. 35–56. Birkhäuser, Basel (2009)

    Chapter  Google Scholar 

  5. 5.

    Brandenburg, C., Lindemann, F., Ulbrich, M., Ulbrich, S.: Advanced numerical methods for pde constrained optimization with application to optimal design in navier stokes flow. In: Leugering, G., Engell, S., Griewank, A., Hinze, M., Rannacher, R., Schulz, V., Ulbrich, M., Ulbrich, S. (eds.) Constrained Optimization and Optimal Control for Partial Differential Equations, International Series of Numerical Mathematics, vol. 160, pp. 257–275. Birkhäuser, Basel (2011)

    Google Scholar 

  6. 6.

    Butt, R.: Optimal shape design for a nozzle problem. The ANZIAM J. 35, 71–86 (1993)

  7. 7.

    Casas, E., Tröltzsch, F.: A general theorem on error estimates with application to a quasilinear elliptic optimal control problem. Comput. Optim. Appl. 53(1), 173–206 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, London (1983)

    MATH  Google Scholar 

  9. 9.

    Delfour, M.C., Zolésio, J.P.: Shapes and Geometries, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2011)

  10. 10.

    Geiger, C., Kanzow, C.: Theorie und Numerik restringierter Optimierungsaufgaben. Springer-Lehrbuch Masterclass, Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  11. 11.

    Goto, Y., Fujii, N.: Second-order numerical method for domain optimization problems. J. Optim. Theory Appl. 67(3), 533–550 (1990)

  12. 12.

    Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman Publishing Inc., Marshfield, MA (1985)

  13. 13.

    Grisvard, P.: Singularités en elasticité. Arch. Ration. Mech. Anal. 107(2), 157–180 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Grisvard, P.: Singularities in Boundary Value Problems. Masson, Paris (1992)

    MATH  Google Scholar 

  15. 15.

    Guillaume, P., Masmoudi, M.: Computation of high order derivatives in optimal shape design. Numer. Math. 67(2), 231–250 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Haslinger, J., Mäkinen, R.A.E.: Introduction to Shape Optimization. Society for Industrial and Applied Mathematics, Philadelphia (2003)

  17. 17.

    Hintermüller, M., Kunisch, K.: Feasible and noninterior path following in constrained minimization with low multiplier regularity. SIAM J. Control Optim. 45(4), 1198–1221 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Hintermüller, M., Kunisch, K.: Path-following methods for a class of constrained minimization problems in function space. SIAM J. Optim. 17(1), 159–187 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Hintermüller, M., Ring, W.: A second order shape optimization approach for image segmentation. SIAM J. Appl. Math. 64(2), 442–467 (2004)

    Article  Google Scholar 

  20. 20.

    Hintermüller, M., Schiela, A., Wollner, W.: The length of the primal-dual path in moreau-yosida-based path-following for state constrained optimal control. Tech. rep., Universität Hamburg (2012)

  21. 21.

    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer, Berlin (2009)

    MATH  Google Scholar 

  22. 22.

    Hoppe, R.H.W., Linsenmann, C., Antil, H.: Adaptive path following primal dual interior point methods for shape optimization of linear and nonlinear stokes flow problems. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) Large-Scale Scientific Computing. Lecture Notes in Computer Science, vol. 4818, pp. 259–266. Springer, Berlin (2008)

    Chapter  Google Scholar 

  23. 23.

    Ito, K., Kunisch, K.: Semi-smooth newton methods for state-constrained optimal control problems. Syst. Control Lett. 50(3), 221–228 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Kiniger, B., Vexler, B.: A priori error estimates for finite element discretization of a shape optimization problem. Math. Modell. Numer. Anal. 47, 1733–1763 (2013)

  25. 25.

    Kluge, G., Stephani, H.: Theoretische Mechanik. Spektrum Akademischer Verlag, Heidelberg, Berlin, Oxford (1995)

  26. 26.

    Laumen, M.: Newton’s method for a class of optimal shape design problems. SIAM J. Optim. 10(2), 503–533 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Lindemann, F.: Theoretical and numerical aspects of shape optimization with navier-stokes flows. Ph.D. thesis, Technische Universität München, München (2012)

  28. 28.

    The MathWorks Inc, Natick, Massachusetts, United States: MATLAB R2012a

  29. 29.

    Meyer, C., Rösch, A., Tröltzsch, F.: Optimal control of pdes with regularized pointwise state constraints. Comput. Optim. Appl. 33(2–3), 209–228 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Meyer, C., Yousept, I.: Regularization of state-constrained elliptic optimal control problems with nonlocal radiation interface conditions. Comp. Optim. Appl. 44(2), 183–212 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Mohammadi, B., Pironneau, O.: Applied Shape Optimization for Fluids. Oxford University Press, Oxford (2001)

    MATH  Google Scholar 

  32. 32.

    Murat, F., Simon, J.: Etudes de problems d’optimal design. Lect. Notes Comput. Sci. 41, 54–62 (1976)

    Article  Google Scholar 

  33. 33.

    Murat, F., Simon, J.: Sur le contrôl par un domaine géométrique. Tech. rep., Universite P. et M. Curie (Paris IV) (1976)

  34. 34.

    Neitzel, I., Tröltzsch, F.: On convergence of regularization methods for nonlinear parabolic optimal control problems with control and state constraints. Control and Cybernetics 37(4), 1013–1043 (2008)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Nemec, M., Zingg, D.W., Pulliam, T.H.: Multipoint and multiobjective aerodynamic shape optimization. AIAA J. 42(6), 1057–1065 (2004)

    Article  Google Scholar 

  36. 36.

    Novruzi, A., Roche, J.R.: Second order derivatives, newton method, application to shape optimization. Rapport de Recherche 2555, Institut National de Recherche en Informatique et en Automatique (1995)

  37. 37.

    Novruzi, A., Roche, J.R.: Newton’s method in shape optimisation: A three-dimensional case. BIT Numer. Math. 40(1), 102–120 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  38. 38.

    Pironneau, O.: Optimal shape design for elliptic systems. In: Drenick, R., Kozin, F. (eds.) System Modeling and Optimization. Lecture Notes in Control and Information Sciences, vol. 38, pp. 42–66. Springer, Berlin, Heidelberg (1982)

  39. 39.

    Prüfert, U., Tröltzsch, F., Weiser, M.: The convergence of an interior point method for an elliptic control problem with mixed control-state constraints. Comput. Optim. Appl. 39(2), 183–218 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  40. 40.

    Rockafellar, R.T., Wets, R.J.B.: Variational analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317. Springer, Berlin (1998)

    Google Scholar 

  41. 41.

    Schiela, A.: Barrier methods for optimal control problems with state constraints. SIAM J. Optim. 20(2), 1002–1031 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  42. 42.

    Schiela, A., Günther, A.: An interior point algorithm with inexact step computation in function space for state constrained optimal control. Numer. Math. 119(2), 373–407 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  43. 43.

    Sokolowski, J., Zolésio, J.P.: Introduction to Shape Optimization, Series in Computational Mathematic. Springer, Berlin (1992)

    Book  Google Scholar 

  44. 44.

    Tröltzsch, F.: Regular lagrange multipliers for control problems with mixed pointwise control-state constraints. SIAM J. Optim. 15(2), 616–634 (2005)

    Article  MATH  Google Scholar 

  45. 45.

    Ulbrich, M.: Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. MOS-SIAM series on optimization (2011)

Download references

Acknowledgments

We gratefully acknowledge support from the International Research Training Group IGDK1754, funded by the German Science Foundation (DFG).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Moritz Keuthen.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Keuthen, M., Ulbrich, M. Moreau–Yosida regularization in shape optimization with geometric constraints. Comput Optim Appl 62, 181–216 (2015). https://doi.org/10.1007/s10589-014-9661-0

Download citation

Keywords

  • Shape optimization
  • Moreau–Yosida regularization
  • Method of mappings
  • Semismooth newton
  • Geometric constraints