Computational Optimization and Applications

, Volume 46, Issue 3, pp 535–569 | Cite as

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

  • M. HintermüllerEmail author
  • A. Laurain


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.


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


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  1. 1.
    Agmon, S.: Lectures on Elliptic Boundary Value Problems. Van Nostrand Mathematical Study, vol. 2. Van Nostrand, Toronto (1965) zbMATHGoogle 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) zbMATHMathSciNetGoogle 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) zbMATHGoogle Scholar
  4. 4.
    Eschenauer, H., Schumacher, A.: Bubble method for topology and shape optimization of structures. Struct. Optim. 8, 42–51 (1994) CrossRefGoogle 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) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Hackbusch, W.: Elliptic Differential Equations. Springer Series in Computational Mathematics, vol. 18. Springer, Berlin (1992) zbMATHGoogle Scholar
  7. 7.
    Hackbusch, W., Mittelmann, H.: On multigrid methods for variational inequalities. Numer. Math. 42, 65–76 (1983) zbMATHCrossRefGoogle 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) zbMATHCrossRefMathSciNetGoogle 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) zbMATHCrossRefMathSciNetGoogle 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) zbMATHCrossRefMathSciNetGoogle 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) zbMATHCrossRefGoogle Scholar
  12. 12.
    Hoppe, R.H.W.: Multigrid methods for variational inequalities. SIAM J. Numer. Anal. 24, 1046–1065 (1987) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hoppe, R.H.W.: Two-sided approximations for unilateral variational inequalities by multigrid methods. Optimization 18, 867–881 (1987) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hoppe, R.H.W., Kornhuber, R.: Adaptive multilevel methods for obstacle problems. SIAM J. Numer. Anal. 31, 301–323 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic, New York (1980) zbMATHGoogle Scholar
  16. 16.
    Kornhuber, R.: Monotone multigrid methods for elliptic variational inequalities I. Numer. Math. 69, 167–184 (1994) zbMATHMathSciNetGoogle Scholar
  17. 17.
    Kornhuber, R.: Monotone multigrid methods for elliptic variational inequalities II. Numer. Math. 72, 481–499 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kornhuber, R.: Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems. Teubner, Stuttgart (1997) zbMATHGoogle 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) zbMATHCrossRefMathSciNetGoogle 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) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods, 2nd edn. Cambridge University Press, Cambridge (1999) zbMATHGoogle 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) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Sokolowski, J., Zolesio, J.-P.: Introduction to Shape Optimization. Springer Series in Computational Mathematics, vol. 16. Springer, Berlin (1992) zbMATHGoogle 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) zbMATHMathSciNetGoogle Scholar
  28. 28.
    Tai, X.-C.: Rate of convergence for some constraint decomposition methods for nonlinear variational inequalities. Numer. Math. 93, 755–786 (2003) zbMATHCrossRefMathSciNetGoogle 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) CrossRefMathSciNetGoogle Scholar
  30. 30.
    Wloka, J.: Partielle Differentialgleichungen. Teubner, Stuttgart (1982) zbMATHGoogle 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) CrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.University of SussexBrightonUK
  2. 2.Department of Mathematics, and Scientific Computing, Institute of Mathematics and Scientific ComputingUniversity of GrazGrazAustria
  3. 3.Institute of Mathematics and Scientific ComputingUniversity of GrazGrazAustria

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