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Original Articles

Level set method with topological derivatives in shape optimization

, , &
Pages 1491-1514 | Received 22 Aug 2006, Accepted 19 Nov 2007, Published online: 01 Sep 2008

References

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

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