Article Outline
Introduction
Definitions
Formulation
Three-Dimensional Anisotropic Elastic Body with a Small Cavity
Contact Problem for Plane Elasticity
Solution of the Elasticity System in the Ring
Cases
Plane Isotropic Elasticity System
Three-Dimensional Isotropic Elasticity Systems
Conclusions
References
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References
Allaire G, de Gournay F, Jouve F, Toader A-M (2005) Structural optimization using topological and shape sensitivity via a level set method. Control Cybern 34:59–80
Allaire G, Jouve F (2007) Minimum stress optimal design with the level set method, to appear in Int J Boundary Elem Appl
Amstutz S, Horchani I, Masmoudi M (2005) Crack detection by the topological gradient method. Control Cybern 34:119–138
Amstutz S, Andrä H (2006) A new algorithm for topology optimization using a level-set method. J Comput Phys 216(2):573–588
Auroux D, Masmoudi M Image processing by topological asymptotic expansion. J Math Imag Vis, to appear
Bonnet M, Guzina BB (2004) Sounding of finite solid bodies by way of topological derivative. Int J Numer Methods Eng 61:2344–2373
Burger M, Hackl B, Ring W (2004) Incorporating Topological Derivatives into Level Set Methods. J Comput Phys 194:344–362
Eschenauer HA, Kobelev VV, Schumacher A (1994) Bubble method for topology and shape optimization of structures. Struct Optim 8:42–51
Fulmanski P, Laurain A, Scheid J-F, Sokolowski J (2007) A levelset method in shape and topology optimization for variational inequalities. Int J Appl Math Comput Sci 17(3):413–430
Fulmanski P, Laurain A, Scheid J-F, Sokolowski J (2007) Level set method with topological derivatives in shape optimization Les Prépublications de l'Institut Elie Cartan
Garreau S, Guillaume P, Masmoudi M (2001) The topological asymptotic for PDE systems: the elasticity case. SIAM J Control Optim 39(6):1756–1778
Hahn HG (1985) Elastizitätstheorie. Teubner, Stuttgart
Hintermüller M (2005) Fast level-set based algorithms using shape and topological sensitivity information. Control Cybern 34(1):305–324
Hintermüller M (2006) A combined shape Newton and topology optimization technique in real-time image segmentation. In: Biegler L, Ghattas O, Heinkenschloss M, Keyes D, van Bloemen Waanders B (eds) Real-Time PDE-Constrained Optimization, SIAM
Il'in AM (1992) Matching of Asymptotic Expansions of Solutions of Boundary Value Problems. AMS, p 281
Jackowska-Strumiłło L, Sokołowski J, Żochowski A, Henrot A (2002) On numerical solution of shape inverse problems. Comput Optim Appl 23:231–255
Kamotski IV, Nazarov SA (1998) Spectral problems in singular perturbed domains and selfadjoint extensions of differential operators. Trudy St.-Petersburg Mat Obshch 6:151–212 (Engl. transl. in: Proceedings of the St. Petersburg Mathematical Society (2000), Am Math Soc Transl Ser 2 6:127–181, 199, Am Math Soc, Providence, RI)
Laurain A (2006) Domaines singulierements perturbes en optimisation de formes. Ph.D. dissertation, Institut Elie Cartan, Nancy, http://tel.archives-ouvertes.fr/tel-00139595
Leugering G, Sokolowski J Topological derivatives for ellipticv problems on graphs. In: Novotny A, Tarocco E, de Souza Neto E (eds) The book for Raul Feijoo, to be published by International Center for Numerical, Methods in Engineering (CIMNE), Barcelona, in press
Lewinski T, Sokolowski J (1998) Optimal shells formed on a sphere. The topological derivative method. RR-3495, INRIA-Lorraine
Lewinski T, Sokolowski J (2000) Topological derivative for nucleation of non-circular voids Contemporary Mathematics. Am Math Soc 268:341–361
Lewinski T, Sokolowski J (2003) Energy change due to appearing of cavities in elastic solids. Int J Solids Struct 40:1765–1803
Lewinski T, Sokolowski J, Żochowski A (1999) Justification of the bubble method for the compliance minimization problems of plates and spherical shells. CD-Rom, 3rd World Congress of Structural and Multidisciplinary Optimization (WCSMO-3) Buffalo/Niagara Falls, NY, May 17–21
Mazja WG, Nazarov SA, Plamenevskii BA (1991) Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten. 1, 2. Akademie-Verlag, Berlin. (English transl.: Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, vol 1, 2. Birkhäuser, Basel (2000))
Muskhelishvili NI (1953) Some Basic Problems on the Mathematical Theory of Elasticity. Noordhoff, Groningen
Nazarov SA, Plamenevsky BA (1994) Elliptic Problems in Domains with Piecewise Smooth Boundaries. De Gruyter Exposition in Mathematics 13. Walter de Gruyter, Berlin
Nazarov SA, Sokołowski J (2003) Asymptotic analysis of shape functionals. J Math Pures Appl 82(2):125–196
Nazarov SA, Sokołowski J (2004) Topological derivative of the Dirichlet integral due to formation of a thin ligaments. Siberian Math J 45:341–355
Nazarov SA, Sokołowski J (2006) Self adjoint extensons for the Neumann laplacian in application to shape optimization. Act Math Sin Engl Ser 22:879–906
Nazarov SA, Slutskij AS, Sokołowski J (2005) Topological derivative of the energy functional due to formation of a thin ligament on a spatial body. Folia Math 12:39–72
Novotny AA (2003) Anàlise de Sensibilidade Topológica. Ph.D. Thesis. LNCC/MCT, Petrópolis
Novotny AA, Feijo RA, Padra C, Taroco E (2005) Topological-Shape Sensitivity Analysis Applied to Topology Design of Kirchhoff's Plate Bending Problem. Control Cybern 34(1):339–361
Rao M, Sokołowski J (2000) Tangent sets in Banach spaces and applications to variational inequalities. Les prépublications de l'Institut Élie Cartan 42
Rocha de Faria J, Novotny AA, Feijo RA, Taroco E, Padra C (2007) Second Order Topological Sensitivity Analysis. Int J Solids Struct 44(14–15):4958–4977
Sokołowski J, Żochowski A (1999) On topological derivative in shape optimization. SIAM J Control Optim 37(4):1251–1272
Sokołowski J, Żochowski A (1999) Topological derivative for optimal control problems. Control Cybern 28(3):611–626
Sokołowski J, Żochowski A (1999) Topological derivatives for elliptic problems. Inverse Problems 15(1):123–134
Sokołowski J, Żochowski A (2001) Topological derivatives of shape functionals for elasticity systems. Mech Struct Mach 29(3):333–351
Sokołowski J, Żochowski A (2003) Optimality conditions for simultaneous topology and shape optimization. SIAM J Control Optim 42:1198–1221
Sokołowski J, Żochowski A (2005) Modelling of topological derivatives for contact problems. Num Math 102:145–179
Sokołowski J, Żochowski A (2007) Asymptotic analysis and topological derivatives for shape and topology optimization of elasticity problems in two spatial dimensions. Prépublication IECN, 16
Sokołowski J, Zolesio J-P (1992) Introduction to Shape Optimization. Shape Sensitivity Analysis. Springer, Berlin
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Sokolowski, J., Zochowski, A. (2008). Topological Derivative in Shape Optimization . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_682
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DOI: https://doi.org/10.1007/978-0-387-74759-0_682
Publisher Name: Springer, Boston, MA
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