Level set method with topological derivatives in shape optimization

International Journal of Computer Mathematics

Volumn 85, Issue 10, 2008, Pages 1491-1514

  Fulmanski, Piotr ^a   Laurain, Antoine ^b   Scheid, Jean François ^b   Sokoowski, Jan ^b  

^a UNIVERSITY OF LODZ   (Poland)

^b UNIVERSITÉ HENRI POINCARÉ NANCY 1   (France)

Author keywords

Level set method; Optimal design; Shape optimization; Topological derivative

Indexed keywords

BOUNDARY VARIATIONS; ELLIPTIC BOUNDARY VALUE PROBLEM; ENERGY FUNCTIONALS; LEVEL SET METHOD; LINEAR BOUNDARY VALUE PROBLEM; OPTIMAL DESIGN; SHAPE OPTIMIZATION PROBLEM; TOPOLOGICAL DERIVATIVES;

ASYMPTOTIC ANALYSIS; BOUNDARY VALUE PROBLEMS; COMPUTER AIDED ENGINEERING; DROP BREAKUP; LEVEL MEASUREMENT; TOPOLOGY;

SHAPE OPTIMIZATION;

EID: 49549101797     PISSN: 00207160     EISSN: 10290265     Source Type: Journal    
DOI: 10.1080/00207160802033350     Document Type: Conference Paper

References (31)

1. G. Allaire, F. De Gournay, F. Jouve, and A.M. loader, Structural optimization using topological and shape sensitivity via a level set method, Control Cybernet 34 (2005), pp. 59-80.
2. D. Bucur, Contrle Par Rapport au Domaine Dans Les E.D.P., Thse de doctorat de l'cole des mines de Paris, 1995.
3. D. Bucur and N. Varchon, Boundary variation for a Neumann problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4(29) (2000), pp. 807-821.
4. _, A duality approach for the boundary variation of Neumann problems, SIAM J. Math. Anal. 34(2) (2002), pp. 460-477.
5. M.C. Delfour and J.-P. Zolesio, Shapes and Geometries, Advances in Design and Control, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001.
6. G. Fremiot, Structure de la semi-d'erienne eul'erienne dans le cas de domains fissur'es et quelques applications, Ph.D Thesis of University Henri Poincar'e-Nancy 1, 2000.
7. P. Fulmanski, A. Laurain, J.-F. Scheid, and J. Soko?owski, A level set method in shape and topology optimization for variational inequalities, Int. J. Math. Comput. Sci. 17 (2007), pp. 413-430.
8. A. Henrot and M. Pierre, Variation et Optimisation de Formes: une Analyse Gomtrique, No 48 de Mathmatiques et Applications, Springer, Paris, 2005.
9. L. Jackowska, J. Soko?owski, A. ? Zochowski, and A. Henrot, On numerical solution of shape inverse problems. Comput. Optim. Appl. 23(2) (2002), pp. 231-255.
10. A.L. Jackowska, J. Soko?owski, and A. Zochowski, Topological optimization and inverse problems, Comput. Assist. Mech. Eng. Sci. 10(2) (2003), pp. 163-176.
11. A. Laurain, Singularly perturbed domains in shape optimization, Ph.D. Thesis, 2006, Universit'e de Nancy.
12. M. Masmoudi, The topological asymptotic, in Computationnal Methods for Control Applications, H. Kawarada and J. Periaux eds., International Series GAKUTO, 2002.
13. V. Mazya, S.A. Nazarov, and B. Plamenevskij, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, Vol. 1 and 2, Birkhauser Verlag, Basel, 2000, p. 435.
14. S.A. Nazarov, Asymptotic conditions at a point, self adjoint extensions of operators, and the method of matched asymptotic expansions, Amer. Math. Soc. Trans. (2), 198 (1999), pp. 77-125.
15. S.A. Nazarov, A.S. Slutskij, and J. Soko?owski, Topological derivative of the energy functional due to formation of a thin ligament on a spatial body. Folia Mathematicae, Acta Universitatis Lodziensis 12 (2005), pp. 39-72.
16. S.A. Nazarov and J. Soko?owski, Self adjoint extensions of differential operators in application to shape optimization, C. R. Mech. 331(10), (2003), pp. 667-672.
17. _, Selfadjoint extensions for the elasticity system in shape optimization, Bull. Polish Acad. Sci. - Math. 52 (2004), pp. 237-248.
18. _, Asymptotic analysis of shape functionals, J. Math. Pures Appl. 82 (2003), pp. 125-196.
19. _, The topological derivative of the Dirichlet integral due to formation of a thin ligament, Siberian Math. J. 45(2) (2004), pp. 341-355.
20. S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, NewYork, 2004.
21. S. Osher and J. Sethian, Fronts propagating with curvature-dependant speed: algorithrms based on Hamilton-Jacobi formulation, J. Comput. Phys. 79 (1988), pp. 12-49.
22. D. Peng, B. Merriman, S. Osher, H. Zhao, and M. Kang, A PDE-based fast local level set method, J. Comput. Phys. 155 (1999), pp. 410-438.
23. J. Sethian, Level Set Methods, Cambridge University Press, Cambridge, UK 1996.
24. J. Soko?owski and A. ? Zochowski, On the topological derivative in shape optimization, SIAM J. Control Optim. 37(4) (1999), pp. 1251-1272.
25. _, Topological derivatives of shape functionals for elasticity systems, Mech. Struct. Machines 29 (2001), pp. 333-351.
26. _, Optimality conditions for simultaneous topology and shape optimization, SIAM J. Control Optim. 42(4) (2003), pp. 1198-1221.
27. _, Topological derivatives for contact problems, Numer. Math. 102(1) (2005), pp. 145-179.
28. _, Topological derivatives for obstacle problems, in Free and Moving Boundaries: Analysis, Simulation and Control, Lecture Notes in Pure and Applied Mathematics Vol. 252, R. Glowinski and J.-P. Zolesio, eds., Chapman & Hall/CRC, Boca Raton, FL, 2007, pp. 309-320.
29. J. Sokołowski and J.-P. Zolesio, Introduction to Shape Optimization, Vol. 16 of Springer Series in Computational Mathematics, Springer Verlag, Berlin, 1992.
30. V. Sverak, On optimal shape design, J. Math. Pures Appl. 72(6) (1993), pp. 537-551.
31. G.N.Watson, Theory of Bessel Functions, The University Press, Cambridge, 1944.