Skip to main content

A shape and topology optimization technique for solving a class of linear complementarity problems in function space

Abstract

A shape and topology optimization driven solution technique for a class of linear complementarity problems (LCPs) in function space is considered. The main motivating application is given by obstacle problems. Based on the LCP together with its corresponding interface conditions on the boundary between the coincidence or active set and the inactive set, the original problem is reformulated as a shape optimization problem. The topological sensitivity of the new objective functional is used to estimate the “topology” of the active set. Then, for local correction purposes near the interface, a level set based shape sensitivity technique is employed. A numerical algorithm is devised, and a report on numerical test runs ends the paper.

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

References

  1. 1.

    Agmon, S.: Lectures on Elliptic Boundary Value Problems. Van Nostrand Mathematical Study, vol. 2. Van Nostrand, Toronto (1965)

    MATH  Google Scholar 

  2. 2.

    Dambrine, M., Vial, G.: On the influence of a boundary perforation on the Dirichlet energy. Control Cybern. 34(1), 117–136 (2005)

    MATH  MathSciNet  Google Scholar 

  3. 3.

    Delfour, M., Zolesio, J.-P.: Shapes and Geometries. Analysis, Differential Calculus and Optimization. SIAM Advances in Design and Control. SIAM, Philadelphia (2001)

    MATH  Google Scholar 

  4. 4.

    Eschenauer, H., Schumacher, A.: Bubble method for topology and shape optimization of structures. Struct. Optim. 8, 42–51 (1994)

    Article  Google Scholar 

  5. 5.

    Garreau, S., Guillaume, P., Masmoudi, M.: The topological asymptotic for pde systems: the elasticity case. SIAM J. Control Optim. 39, 1756–1778 (2001)

    MATH  Article  MathSciNet  Google Scholar 

  6. 6.

    Hackbusch, W.: Elliptic Differential Equations. Springer Series in Computational Mathematics, vol. 18. Springer, Berlin (1992)

    MATH  Google Scholar 

  7. 7.

    Hackbusch, W., Mittelmann, H.: On multigrid methods for variational inequalities. Numer. Math. 42, 65–76 (1983)

    MATH  Article  Google Scholar 

  8. 8.

    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)

    MATH  Article  MathSciNet  Google Scholar 

  9. 9.

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

    MATH  Article  MathSciNet  Google Scholar 

  10. 10.

    Hintermüller, M., Ring, W.: A level set approach for the solution of a state-constrained optimal control problem. Numer. Math. 98, 135–166 (2004)

    MATH  Article  MathSciNet  Google Scholar 

  11. 11.

    Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13, 865–888 (2003)

    MATH  Article  Google Scholar 

  12. 12.

    Hoppe, R.H.W.: Multigrid methods for variational inequalities. SIAM J. Numer. Anal. 24, 1046–1065 (1987)

    MATH  Article  MathSciNet  Google Scholar 

  13. 13.

    Hoppe, R.H.W.: Two-sided approximations for unilateral variational inequalities by multigrid methods. Optimization 18, 867–881 (1987)

    MATH  Article  MathSciNet  Google Scholar 

  14. 14.

    Hoppe, R.H.W., Kornhuber, R.: Adaptive multilevel methods for obstacle problems. SIAM J. Numer. Anal. 31, 301–323 (1994)

    MATH  Article  MathSciNet  Google Scholar 

  15. 15.

    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic, New York (1980)

    MATH  Google Scholar 

  16. 16.

    Kornhuber, R.: Monotone multigrid methods for elliptic variational inequalities I. Numer. Math. 69, 167–184 (1994)

    MATH  MathSciNet  Google Scholar 

  17. 17.

    Kornhuber, R.: Monotone multigrid methods for elliptic variational inequalities II. Numer. Math. 72, 481–499 (1996)

    MATH  Article  MathSciNet  Google Scholar 

  18. 18.

    Kornhuber, R.: Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems. Teubner, Stuttgart (1997)

    MATH  Google Scholar 

  19. 19.

    Maz’ya, V., Nazarov, S.A., Plamenevskij, B.: Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, vols. 1, 2. Birkhäuser, Basel (2000)

    Google Scholar 

  20. 20.

    Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, New York (2004)

    Google Scholar 

  21. 21.

    Osher, S., Sethian, J.: Fronts propagating with curvature-dependant speed: algorithms based on Hamilton-Jacobi formulation. J. Comput. Phys. 79, 12–49 (1988)

    MATH  Article  MathSciNet  Google Scholar 

  22. 22.

    Peng, D., Merriman, B., Osher, S., Zhao, H., Kang, M.: A pde-based fast local level set method. J. Comput. Phys. 155, 410–438 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  23. 23.

    Sethian, J.A.: Level Set Methods and Fast Marching Methods, 2nd edn. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  24. 24.

    Smith, G.D.: Numerical Solution of Partial Differential Equations: Finite Difference Methods. Clarendon, Oxford (1993)

    Google Scholar 

  25. 25.

    Sokolowski, J., Zochowski, A.: On the topological derivative in shape optimization. SIAM J. Control Optim. 37(4), 1251–1272 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  26. 26.

    Sokolowski, J., Zolesio, J.-P.: Introduction to Shape Optimization. Springer Series in Computational Mathematics, vol. 16. Springer, Berlin (1992)

    MATH  Google Scholar 

  27. 27.

    Tai, X.-C.: Convergence rate analysis of domain decomposition methods for obstacle problems. East-West J. Numer. Math. 9, 233–252 (2001)

    MATH  MathSciNet  Google Scholar 

  28. 28.

    Tai, X.-C.: Rate of convergence for some constraint decomposition methods for nonlinear variational inequalities. Numer. Math. 93, 755–786 (2003)

    MATH  Article  MathSciNet  Google Scholar 

  29. 29.

    Tai, X.-C., Tseng, P.: Convergence rate analysis of an asynchronous space decomposition method for convex minimization. Math. Comput. 71, 1105–1135 (2001)

    Article  MathSciNet  Google Scholar 

  30. 30.

    Wloka, J.: Partielle Differentialgleichungen. Teubner, Stuttgart (1982)

    MATH  Google Scholar 

  31. 31.

    Zhao, H.K., Chan, T., Merriman, B., Osher, S.: A variational level set approach to multi-phase motion. J. Comput. Phys. 122, 179–195 (1996)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to M. Hintermüller.

Additional information

Both authors acknowledge support by the Austrian Ministry of Science and Education and the Austrian Science Fund FWF under START-grant Y305 “Interfaces and Free Boundaries”.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hintermüller, M., Laurain, A. A shape and topology optimization technique for solving a class of linear complementarity problems in function space. Comput Optim Appl 46, 535–569 (2010). https://doi.org/10.1007/s10589-008-9201-x

Download citation

Keywords

  • Function space
  • Level set method
  • Linear complementarity problem
  • Obstacle problem
  • Shape and topology optimization