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International Journal of Differential Equations
Volume 2015, Article ID 954836, 10 pages
http://dx.doi.org/10.1155/2015/954836
Research Article

On the Second-Order Shape Derivative of the Kohn-Vogelius Objective Functional Using the Velocity Method

Department of Mathematics and Computer Science, College of Science, University of the Philippines Baguio, Governor Pack Road, 2600 Baguio City, Philippines

Received 31 July 2015; Revised 11 November 2015; Accepted 11 November 2015

Academic Editor: Julio D. Rossi

Copyright © 2015 Jerico B. Bacani and Julius Fergy T. Rabago. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. K. K. Choi and N.-H. Kim, Structural Sensitivity Analysis and Optimization 1: Linear Systems, Mechanical Engineering Series, Springer, New York, NY, USA, 2005. View at Publisher · View at Google Scholar
  2. K. Choi and N. Kim, Structural Sensitivity Analysis and Optimization: Nonlinear Systems and Applications, Mechanical Engineering Series, Springer, 2005.
  3. B. Mohammadi and O. Pironneau, Applied Shape Optimization for Fluids, Clardendon Press, Oxford, UK, 2001. View at MathSciNet
  4. J. Haslinger and R. A. E. Mäkinen, “Introduction to shape optimization: theory, approximation, and computation,” in Advances in Design and Control, Society for Industrial and Applied Mathematics, 2003. View at Publisher · View at Google Scholar
  5. M. C. Delfour and J.-P. Zolésio, Shapes and Geometries, SIAM, Philadelphia, Pa, USA, 2001. View at MathSciNet
  6. J. Sokolowski and J.-P. Zolésio, Introduction to Shape Optimization: Shape Sensitivity Analysis, vol. 16 of Springer Series in Computational Mathematics, Springer, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  7. B. Abda, F. Bouchon, G. Peichl, M. Sayeh, and R. Touzani, “A new formulation for the Bernoulli problem,” in Proceedings of the 5th International Conference on Inverse Problems, Control and Shape Optimization, pp. 1–19, Cartagena, Spain, 2010.
  8. H. Kasumba and K. Kunisch, “On computation of the shape Hessian of the cost functional without shape sensitivity of the state variable,” SFB-Report 2012-012, 2012. View at Google Scholar
  9. J. B. Bacani and G. H. Peichl, “On the first-order shape derivative of the Kohn-Vogelius cost functional of the Bernoulli problem,” Abstract and Applied Analysis, vol. 2013, Article ID 384320, 19 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  10. F. Bouchon, S. Clain, and R. Touzani, “Numerical solution of the free boundary Bernoulli problem using a level set formulation,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 36-38, pp. 3934–3948, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. M. Flucher and M. Rumpf, “Bernoulli's free-boundary problem, qualitative theory and numerical approximation,” Journal für die Reine und Angewandte Mathematik, vol. 486, pp. 165–204, 2003. View at Google Scholar
  12. J. Haslinger, T. Kozubek, K. Kunisch, and G. Peichl, “Shape optimization and fictitious domain approach for solving free boundary problems of Bernoulli type,” Computational Optimization and Applications, vol. 26, no. 3, pp. 231–251, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. K. Ito, K. Kunisch, and G. H. Peichl, “Variational approach to shape derivative for a class of Bernoulli problem,” Journal of Mathematical Analysis and Applications, vol. 314, no. 1, pp. 126–149, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. K. Eppler and H. Harbrecht, “Shape optimization for free boundary problems—analysis and numerics,” in Constrained Optimization and Optimal Control for Partial Differential Equations, vol. 160 of International Series of Numerical Mathematics, pp. 277–288, Springer, Basel, Switzerland, 2012. View at Publisher · View at Google Scholar
  15. K. Eppler and H. Harbrecht, “Tracking Neumann data for stationary free boundary problems,” SIAM Journal on Control and Optimization, vol. 48, no. 5, pp. 2901–2916, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. F. Murat and J. Simon, “Etude de problemes d'optimal design,” in Optimization Techniques Modeling and Optimization in the Service of Man Part 2, vol. 41 of Lecture Notes in Computer Science, pp. 54–62, Springer, Berlin, Germany, 1976. View at Publisher · View at Google Scholar
  17. F. Murat and J. Simon, “Sur le contrôle par un domaine géométrique,” Report of L.A. 189 76015, Université Paris VI, 1976. View at Google Scholar
  18. N. Fujii, “Domain optimization problems with a boundary value problem as a constraint,” in Control of Distributed Parameter Systems 1986, pp. 5–9, Pergamon Press, Oxford, UK, 1986. View at Google Scholar
  19. J. Simon, “Second variations for domain optimization problems,” in Control of Distributed Parameter Systems, Birkhäuser, 1988. View at Google Scholar
  20. M. C. Delfour and J.-P. Zolésio, “Computation of the shape Hessian by a Lagrangian method,” in Control of Distributed Parameter Systems, pp. 215–220, IFAC, Perpignan, France, 1989. View at Google Scholar
  21. M. C. Delfour and J.-P. Zolésio, “Shape Hessian by the velocity method: a Lagrangian approach,” in Stabilization of Flexible Structures, vol. 147 of Lecture Notes in Control and Information Sciences, pp. 255–279, Springer, 1990. View at Google Scholar
  22. K. Eppler and H. Harbrecht, “On a Kohn-Vogelius like formulation of free boundary problems,” Computational Optimization and Applications, vol. 52, no. 1, pp. 69–85, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. J. B. Bacani and G. H. Peichl, “Solving the exterior Bernoulli problem using the shape derivative approach,” in Mathematics and Computing 2013, R. N. Mohapatra, D. Giri, P. K. Saxena, and P. D. Srivastava, Eds., vol. 91 of Springer Proceedings in Mathematics & Statistics, pp. 251–269, Springer, New Delhi, India, 2014. View at Google Scholar
  24. T. Tiihonen, “Shape optimization and trial methods for free boundary problems,” RAIRO: Modélisation Mathématique et Analyse Numérique, vol. 31, no. 7, pp. 805–825, 1997. View at Google Scholar · View at MathSciNet
  25. J. B. Bacani and G. H. Peichl, “The second-order shape derivative of kohn-vogelius-type cost functional using the boundary differentiation approach,” Mathematics, vol. 2, no. 4, pp. 196–217, 2014. View at Publisher · View at Google Scholar
  26. M. C. Delfour and J.-P. Zolésio, “Velocity method and Lagrangian formulation for the computation of the shape Hessian,” SIAM Journal on Control and Optimization, vol. 29, no. 6, pp. 1414–1442, 1991. View at Publisher · View at Google Scholar · View at Scopus
  27. J. Haslinger, K. Ito, T. Kozubek, K. Kunisch, and G. Peichl, “On the shape derivative for problems of Bernoulli type,” Interfaces and Free Boundaries, vol. 11, no. 2, pp. 317–330, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  28. J. Sokolowski and J.-P. Zolésio, Introduction to Shape Optimization, Springer, Berlin, Germany, 1991.
  29. J. B. Bacani, Methods of shape optimization in free boundary problems [Ph.D. thesis], Karl-Franzens-Universität Graz, Graz, Austria, 2013.