Have a personal or library account? Click to login
A Level Set Method in Shape and Topology Optimization for Variational Inequalities Cover

A Level Set Method in Shape and Topology Optimization for Variational Inequalities

Open Access
|Oct 2007

References

  1. Allaire G., De Gournay F., Jouve F. and Toader A.M. (2005): Structural optimization using topological and shape sensitivity via a level set method.Control and Cybernetics, Vol. 34, No. 1, pp. 59-80.
  2. Amstutz S. and Andrä H. (2006): A new algorithm for topology optimization using a level-set method.Journal of Computer Physics, Vol. 216, No. 2, pp. 573-588.10.1016/j.jcp.2005.12.015
  3. Delfour M.C. and Zolesio J.-P. (2001): Shapes and Geometries. Philadelphia, PA: SIAM.
  4. Henrot A. and Pierre M. (2005): Variation et optimisation de formes: Une analyse géométrique. Berlin: Springer.10.1007/3-540-37689-5
  5. Jackowska L., Sokołowski J., Żochowski A. and Henrot A. (2002): On numerical solution of shape inverse problems.Computational Optimization and Applications, Vol. 23, No. 2, pp. 231-255.10.1023/A:1020528902875
  6. Jackowska A.L., Sokołowski J. and Żochowski A. (2003): Topological optimization and inverse problems.Computer Assisted Mechanics and Engineering Sciences, Vol. 10, No. 2, pp. 163-176.
  7. Jarusek J., Krbec M., Rao M. and Sokołowski J. (2003): Conical differentiability for evolution variational inequalities.Journal of Differential Equations, Vol. 193, No. 1, pp. 131-146.10.1016/S0022-0396(03)00136-0
  8. Laurain A. (2006): Singularly perturbed domains in shape optimization.—Ph.D. thesis, Université de Nancy.
  9. Masmoudi M. (2002): The topological asymptotic, In: Computational Methods for Control Applications (R. Glowinski, H. Kawarada and J. Periaux, Eds.). GAKUTO Int. Ser. Math. Sci. Appl., Vol. 16, pp. 53-72
  10. Maz'ya V., Nazarov S.A. and Plamenevskij B. (2000): Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Vols. 1 and 2, Basel: Birkhäuser, p. 435.
  11. Nazarov S.A. (1999): Asymptotic conditions at a point, self adjoint extensions of operators, and the method of matched asymptotic expansions.American Mathematical Society Translations, Vol. 198, No. 2, pp. 77-125.10.1090/trans2/193/05
  12. Nazarov S.A. and Sokołowski J. (2003a): Self adjoint extensions of differential operators in application to shape optimization.Comptes Rendus Mécanique, Vol. 331, No. 10, pp. 667-672.10.1016/j.crme.2003.07.003
  13. Nazarov S.A. and Sokołowski J. (2003b): Asymptotic analysis of shape functionals.Journal de Mathématiques pures et appliquées, Vol. 82, No. 2, pp. 125-196.10.1016/S0021-7824(03)00004-7
  14. Nazarov S.A. and Sokołowski J. (2004a): Self adjoint extensions for elasticity system in application to shape optimization.Bulletin of the Polish Academy of Sciences, Mathematics, Vol. 52, No. 3, pp. 237-248.10.4064/ba52-3-4
  15. Nazarov S.A. and Sokołowski J. (2004b): The topological derivative of the Dirichlet integral due to formation of a thin ligament.Siberian Mathematical Journal, Vol. 45, No. 2, pp. 341-355.10.1023/B:SIMJ.0000021290.61807.4b
  16. Nazarov S.A., Slutskij A.S. and Sokołowski J. (2005): Topological derivative of the energy functional due to formation of a thin ligament on a spatial body.Folia Mathematicae, Acta Universitatis Lodziensis, Vol. 12, pp. 39-72.
  17. Osher S. and Fedkiw R. (2004): Level Set Methods and Dynamic Implicit Surfaces. New York: Springer.10.1007/b98879
  18. Osher S. and Sethian J. (1988): Fronts propagating with curvature-dependant speed: Algorithms based on Hamilton-Jacobi formulation.Journal of Computational Physics, Vol. 79, No. 1, pp. 12-49.10.1016/0021-9991(88)90002-2
  19. Peng D., Merriman B., Osher S., Zhao S. and Kang M. (1999): A PDE-based fast local level set method.Journal of Computational Physics, Vol. 155, No. 2, pp. 410-438.10.1006/jcph.1999.6345
  20. Rao M. and Sokołowski J. (2000): Tangent sets in Banach spaces and applications to variational inequalities. Les prépublications de l'Institut Élie Cartan, No. 42.
  21. Sethian J. (1996): Level Set Methods. Cambridge: Cambridge University Press.
  22. Sokołowski J. and Zolesio J.-P. (1992): Introduction to shape optimization. Series in Computationnal Mathematics, Berlin: Springer Verlag, Vol. 16.
  23. Sokołowski J. and Żochowski A. (1999): On the topological derivative in shape optimization.SIAM Journal on Control and Optimization, Vol. 37, No. 4, pp. 1251-1272.10.1137/S0363012997323230
  24. Sokołowski J. and Żochowski A. (2001): Topological derivatives of shape functionals for elasticity systems.Mechanics of Structures and Machines, Vol. 29, No. 3, pp. 333-351.10.1007/978-3-0348-8148-7_19
  25. Sokołowski J. and Żochowski A. (2003): Optimality conditions for simultaneous topology and shape optimization.SIAM Journal on Control and Optimization, Vol. 42, No. 4, pp. 1198-1221.10.1137/S0363012901384430
  26. Sokołowski J. and Żochowski A. (2005a): Topological derivatives for contact problems.Numerische Mathematik, Vol. 102, No. 1, pp. 145-179.10.1007/s00211-005-0635-0
  27. Sokołowski J. and Żochowski A. (2005b): Topological derivatives for obstacle problems. Les prépublications de l'Institut Élie Cartan No. 12.
  28. Watson G.N. (1944): Theory of Bessel Functions. Cambridge: The University Press.
  29. Zhao H.K., Chan T., Merriman B. and Osher S. (1996): A variational level set approach to multi-phase motion.Journal of Computational Physics, Vol. 127, No. 1, pp. 179-195.10.1006/jcph.1996.0167
DOI: https://doi.org/10.2478/v10006-007-0034-z | Journal eISSN: 2083-8492 | Journal ISSN: 1641-876X
Language: English
Page range: 413 - 430
Published on: Oct 11, 2007
In partnership with: Paradigm Publishing Services
Publication frequency: 4 issues per year

© 2007 Piotr Fulmański, Antoine Laurain, Jean-Francois Scheid, Jan Sokołowski, published by University of Zielona Góra
This work is licensed under the Creative Commons License.

Volume 17 (2007): Issue 3 (September 2007)