Shape Optimization for Free Boundary Problems – Analysis and Numerics

  • Karsten EpplerEmail author
  • Helmut Harbrecht
Part of the International Series of Numerical Mathematics book series (ISNM, volume 160)


In this paper the solution of a Bernoulli type free boundary problem by means of shape optimization is considered. Four different formulations are compared from an analytical and numerical point of view. By analyzing the shape Hessian in case of matching data it is distinguished between well-posed and ill-posed formulations. A nonlinear Ritz-Galerkin method is applied for the discretization of the shape optimization problem. In case of well-posedness existence and convergence of the approximate shapes is proven. In combination with a fast boundary element method efficient first and second-order shape optimization algorithms are obtained.


Shape optimization free boundary problems sufficient optimality conditions boundary element method 


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  1. 1.
    A. Acker. On the geometric form of Bernoulli configurations. Math. Meth. Appl. Sci. 10 (1988) 1–14.CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    H.W. Alt and L.A. Caffarelli. Existence and regularity for a minimum problem with free boundary. J. reine angew. Math. 325 (1981) 105–144.zbMATHMathSciNetGoogle Scholar
  3. 3.
    N.V. Banichuk and B.L. Karihaloo. Minimum-weight design of multi-purpose cylindrical bars. International Journal of solids and Structures 12 (1976) 267–273.CrossRefzbMATHGoogle Scholar
  4. 4.
    M. Brühl. Explicit characterization of inclusions in electrical impedance tomography. SIAM J. Math. Anal. 32 (2001) 1327–1341.CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    M. Brühl and M. Hanke. Numerical implementation of two noniterative methods for locating inclusions by impedance tomography. Inverse Problems 16 (2000) 1029–1042.CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    R. Chapko and R. Kress. A hybrid method for inverse boundary value problems in potential theory. J. Inverse Ill-Posed Probl. 13 (2005) 27–40.CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    O. Colaud and A. Henrot. Numerical approximation of a free boundary problem arising in electromagnetic shaping. SIAM J. Numer. Anal. 31 (1994) 1109–1127.CrossRefMathSciNetGoogle Scholar
  8. 8.
    W. Dahmen, H. Harbrecht and R. Schneider. Compression techniques for boundary integral equations – optimal complexity estimates. SIAM J. Numer. Anal. 43 (2006) 2251–2271.CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    M. Delfour and J.-P. Zolesio. Shapes and Geometries. SIAM, Philadelphia, 2001.zbMATHGoogle Scholar
  10. 10.
    K. Eppler. Boundary integral representations of second derivatives in shape optimization. Discuss. Math. Differ. Incl. Control Optim. 20 (2000) 63–78.zbMATHMathSciNetGoogle Scholar
  11. 11.
    K. Eppler. Optimal shape design for elliptic equations via BIE-methods. Appl. Math. Comput. Sci. 10 (2000) 487–516.zbMATHMathSciNetGoogle Scholar
  12. 12.
    K. Eppler and H. Harbrecht. Numerical solution of elliptic shape optimization problems using wavelet-based BEM. Optim. Methods Softw. 18 (2003) 105–123.CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    K. Eppler and H. Harbrecht. A regularized Newton method in electrical impedance tomography using shape Hessian information. Control Cybern. 34 (2005) 203–225.zbMATHMathSciNetGoogle Scholar
  14. 14.
    K. Eppler and H. Harbrecht. Efficient treatment of stationary free boundary problems. Appl. Numer. Math. 56 (2006) 1326–1339.CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    K. Eppler and H. Harbrecht. Wavelet based boundary element methods in exterior electromagnetic shaping. Eng. Anal. Bound. Elem. 32 (2008) 645–657.CrossRefGoogle Scholar
  16. 16.
    K. Eppler and H. Harbrecht. Tracking Neumann data for stationary free boundary problems. SIAM J. Control Optim. 48 (2009) 2901–2916.CrossRefMathSciNetGoogle Scholar
  17. 17.
    K. Eppler and H. Harbrecht. Tracking the Dirichlet data in 𝐿 2 is an ill-posed problem. J. Optim. Theory Appl. 145 (2010) 17–35.MathSciNetGoogle Scholar
  18. 18.
    K. Eppler and H. Harbrecht. On a Kohn-Vogelius like formulation of free boundary problems. Comput. Optim. Appl. (to appear).Google Scholar
  19. 19.
    K. Eppler, H. Harbrecht, and M.S. Mommer. A new fictitious domain method in shape optimization. Comput. Optim. Appl., 40 (2008) 281–298.CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    K. Eppler, H. Harbrecht, and R. Schneider. On Convergence in Elliptic Shape Optimization. SIAM J. Control Optim. 45 (2007) 61–83.CrossRefMathSciNetGoogle Scholar
  21. 21.
    L. Greengard and V. Rokhlin. A fast algorithm for particle simulation. J. Comput. Phys. 73 (1987), 325–348.CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    W. Hackbusch and Z.P. Nowak. On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math. 54 (1989), 463–491.CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    A.V. Fiacco and G.P. McCormick. Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York, 1968.zbMATHGoogle Scholar
  24. 24.
    R. Fletcher. Practical Methods for Optimization, volume12. Wiley, New York, 1980.Google Scholar
  25. 25.
    M. Flucher and M. Rumpf. Bernoulli’s free-boundary problem, qualitative theory and numerical approximation. J. reine angew. Math. 486 (1997) 165–204.zbMATHMathSciNetGoogle Scholar
  26. 26.
    C. Grossmann and J. Terno. Numerikde r Optimierung. B.G. Teubner, Stuttgart, 1993.Google Scholar
  27. 27.
    H. Harbrecht. A Newton method for Bernoulli’s free boundary problem in three dimensions. Computing 82 (2008) 11–30.CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    J. Haslinger, T. Kozubek, K. Kunisch, and G. Peichl. Shape optimization and fictitious domain approach for solving free boundary problems of Bernoulli type. Comput. Optim. Appl. 26 (2003) 231–251.CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    J. Haslinger, K. Ito, T. Kozubek, K. Kunisch, and G. Peichl. On the shape derivative for problems of Bernoulli type. Interfaces and Free Boundaries 11 (2009) 317–330.CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    F. Hettlich and W. Rundell The determination of a discontinuity in a conductivity from a single boundary measurement. Inverse Problems 14 (1998) 67–82.CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    K. Ito, K. Kunisch, and G. Peichl. Variational approach to shape derivatives for a class of Bernoulli problems. J. Math. Anal. Appl. 314 (2006) 126–149.zbMATHMathSciNetGoogle Scholar
  32. 32.
    R. Kohn and M. Vogelius. Determining conductivity by boundary measurements. Comm. Pure Appl. Math. 37 (1984) 289–298.CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    F. Murat and J. Simon. Étude de problèmes d’optimal design. in Optimization Techniques, Modeling and Optimization in the Service of Man, edited by J. Céa, Lect. Notes Comput. Sci. 41, Springer-Verlag, Berlin, 54–62 (1976).Google Scholar
  34. 34.
    A. Novruzi and J.R. Roche. Newton’s method in shape optimisation: a threedimensional case. BIT 40 (2000) 102–120.CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    M. Pierre and J.-R. Roche Computation of free surfaces in the electromagnetic shaping of liquid metals by optimization algorithms. Eur. J. Mech. B/Fluids 10 (1991) 489–500.zbMATHMathSciNetGoogle Scholar
  36. 36.
    J.-R. Roche and J. Sokolowski Numerical methods for shape identification problems. Control Cybern. 25 (1996) 867–894.zbMATHMathSciNetGoogle Scholar
  37. 37.
    J. Simon. Differentiation with respect to the domain in boundary value problems. Numer. Funct. Anal. Optimization 2 (1980) 649–687.CrossRefzbMATHGoogle Scholar
  38. 38.
    J. Sokolowski and J.-P. Zolesio. Introduction to Shape Optimization. Springer, Berlin, 1992.CrossRefzbMATHGoogle Scholar
  39. 39.
    T. Tiihonen. Shape optimization and trial methods for free-boundary problems. RAIRO Model. Math. Anal. Numér. 31 (1997) 805–825.zbMATHMathSciNetGoogle Scholar

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© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Institut für Numerische MathematikTechnische Universität DresdenDresdenGermany
  2. 2.Mathematisches InstitutUniversität BaselBaselSchweiz

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