Advertisement

Solving the Exterior Bernoulli Problem Using the Shape Derivative Approach

  • Jerico B. BacaniEmail author
  • Gunther Peichl
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 91)

Abstract

In this paper, we are interested in solving the exterior Bernoulli free boundary problem by minimizing a particular cost functional \(J\) over a class of admissible domains subject to two well-posed PDE constraints: a Dirichlet boundary value problem and a Neumann boundary value problem. The main result for this paper is the thorough computation of the first-order shape derivative of \(J\) using the shape derivatives of the state variables. At first, the material derivatives of the states are rigorously justified. Then the equation and the boundary conditions satisfied by the corresponding shape derivatives are derived directly from the definition of the shape derivative and the variational equation for the material derivative. It becomes apparent that the analysis of the shape derivatives of the states requires more regular domains. Finally, it is noted that the shape gradient agrees with the structure predicted by the Hadamard structure theorem.

Keywords

Shape optimization Free boundary problem Overdetermined boundary value problem Material derivative Shape derivative 

AMS Subject Classifications:

35R35 35N25 49K20 49Q12 

References

  1. 1.
    Abda, B., Bouchon, F., Peichl, G., Sayeh, M., Touzani, R.: A Dirichlet-Neumann cost functional approach for the Bernoulli problem. J. Eng. Math. 81, 157–176 (2013)CrossRefGoogle Scholar
  2. 2.
    Afraites, L., Dambrine, M., Kateb, D.: On second-order shape optimization methods for electrical impedance tomography. Preprint, HAL-00140211, version 1, pp. 1–28 (2007)Google Scholar
  3. 3.
    Bacani, J.B.: Methods of shape optimization in free boundary problems. Ph.D. Thesis, Karl-Franzens-Universitaet Graz (2013)Google Scholar
  4. 4.
    Bacani, J.B., Peichl, G.: On the first-order shape derivative of the Kohn-Vogelius cost functional of the Bernoulli problem. Abstr. Appl. Anal. 2013, 19 (2013). Article ID 384320. doi: 10.1155/2013/384320
  5. 5.
    Caffarelli, L.A., Salsa, S.: A Geometric Approach to Free Boundary Problems. American Mathematical Society, Providence (2005)Google Scholar
  6. 6.
    Crank, J.: Free and Moving Boundary Problems. Oxford University Press Inc., New York (1984)zbMATHGoogle Scholar
  7. 7.
    Delfour, M.C., Zolesio, J.P.: Shapes and Geometries. SIAM, Philadelphia (2001)zbMATHGoogle Scholar
  8. 8.
    Delfour, M.C., Zolesio, J.P.: Anatomy of the shape Hessian. Annali di Matematica pura ed applicata 159, 315–339 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Flucher, M., Rumpf, M.: Bernoulli’s free-boundary problem, qualitative theory and numerical approximation. J. Reine Angew. Math. 486, 165–204 (1997)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Friedman, A.: Free boundary problems in science and technology. Not. AMS 47, 854–861 (2000)zbMATHGoogle Scholar
  11. 11.
    Fujii, N.: Second variation and its application in domain optimization problem, control of distributed parameter systems. In: Proceedings of the 4th IFAC Symposium, vol. 24, pp. 346–360. Pergamon Press (1986)Google Scholar
  12. 12.
    Haslinger, J., Ito, K., Kozubek, T., Kunisch, K., Peichl, G.: On the shape derivative for problems of Bernoulli type. Interfaces Free Boundaries 1, 317–330 (2009)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Haslinger, J., Mäkinen, R.A.E.: Introduction to Shape Optimization (Theory, Approximation, and Computation). SIAM Advances and Control, Philadelphia (2003)Google Scholar
  14. 14.
    Henrot, A., Pierre, M.: Variation et Optimisation de Formes. Springer, Berlin (2005)CrossRefzbMATHGoogle Scholar
  15. 15.
    Ito, K., Kunisch, K., Peichl, G.: Variational approach to shape derivatives for a class of Bernoulli problems. J. Math. Anal. Appl. 314, 126–149 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Kohn, R., Vogelius, M.: Determining conductivity by boundary measurements. Commun. Pure Appl. Math. 37, 289–298 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Lamboley, J., Pierre, M.: Structure of shape derivatives around irregular do- mains and applications. eprint arXiv:math/0609526, pp. 1–14 (2006)
  18. 18.
    Masanao, T., Fujii, N.: Second-order necessary conditions for domain optimization problems in elastic structures, Part 1: surface traction given as a field. J. Optim. Theor. Appl. 72, 355–382 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Simon, J.: Second variations for domain optimization problems. Int. Ser. Numer. Math. 91, 361–378 (1989)Google Scholar
  20. 20.
    Sokolowski, J., Zolesio, J.: Introduction to Shape Optimization. Springer, Berlin (1991)Google Scholar
  21. 21.
    Tiihonen, T.: Shape optimization and trial methods for free boundary problems. RAIRO Modelisation mathematique et analyse numerique 31, 805–825 (1997)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Toivanen, J.I., Haslinger, J., Mäkinen, R.A.E.: Shape optimization of systems governed by Bernoulli free boundary problems. Comput. Methods Appl. Mech. Eng. 197, 3803–3815 (2008)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer India 2014

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceCollege of Science, University of the Philippines BaguioBaguioPhilippines
  2. 2.Institute for Mathematics and Scientific Computing, University of GrazGrazAustria

Personalised recommendations