Geodesic
A shape-topological control of variational inequalities
Eurasian mathematical journal, Tome 7 (2016) no. 3, pp. 41-52.

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A shape-topological control of singularly perturbed variational inequalities is considered in the abstract framework for state-constrained optimization problems. Aiming at asymptotic analysis, singular perturbation theory is applied to the geometry-dependent objective function and results in a shape-topological derivative. This concept is illustrated analytically in a one-dimensional example problem which is controlled by an inhomogeneity posed in a domain with moving boundary. 
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V. A. Kovtunenko; G. Leugering. A shape-topological control of variational inequalities. Eurasian mathematical journal, Tome 7 (2016) no. 3, pp. 41-52. http://geodesic.mathdoc.fr/item/EMJ_2016_7_3_a5/

[1] G. Allaire, F. Jouve, A.-M. Toader, “Structural optimization using sensitivity analysis and a level-set method”, J. Comput. Phys., 194 (2004), 363–393 | DOI | MR | Zbl

[2] H. Ammari, P. Garapon, F. Jouve, H. Kang, M. Lim, S. Yu, “A new optimal control approach for the reconstruction of extended inclusions”, SIAM J. Control Optim., 51 (2013), 1372–1394 | DOI | MR | Zbl

[3] S. Amstutz, T. Takahashi, B. Vexler, “Topological sensitivity analysis for time-dependent problems”, ESAIM: COCV 14, 2008, 427–455 | DOI | MR | Zbl

[4] G. Barbatis, V. I. Burenkov, P. D. Lamberti, “Stability estimates for resolvents, eigenvalues, and eigenfunctions of elliptic operators on variable domains”, Around the Research of Vladimir Maz'ya II, ed. A. Laptev, Springer, New York, 2010, 23–60 | DOI | MR | Zbl

[5] P. Fulmanski, A. Laurain, J.-F. Scheid, J. Sokolowski, “A level set method in shape and topology optimization for variational inequalities”, Int. J. Appl. Math. Comput. Sci., 17 (2007), 413–430 | DOI | MR | Zbl

[6] M. Hintermüller, V. A. Kovtunenko, “From shape variation to topology changes in constrained minimization: a velocity method based concept”, Optim. Methods Softw., 26 (2011), 513–532 | DOI | MR | Zbl

[7] A. M. Il'in, Matching of asymptotic expansions of solutions of boundary value problems, AMS, 1992 | MR | Zbl

[8] H. Itou, A. M. Khludnev, E. M. Rudoy, A. Tani, “Asymptotic behaviour at a tip of a rigid line inclusion in linearized elasticity”, Z. Angew. Math. Mech., 92 (2012), 716–730 | DOI | MR | Zbl

[9] A. M. Khludnev, V. A. Kovtunenko, Analysis of cracks in solids, WIT-Press, Southampton, Boston, 2000

[10] A. M. Khludnev, V. A. Kovtunenko, A. Tani, “Evolution of a crack with kink and non-penetration”, J. Math. Soc. Japan, 60:4 (2008), 1219–1253 | DOI | MR | Zbl

[11] A. M. Khludnev, V. A. Kovtunenko, A. Tani, “On the topological derivative due to kink of a crack with non-penetration. Anti-plane model”, J. Math. Pures Appl., 94 (2010), 571–596 | DOI | MR | Zbl

[12] A. Khludnev, G. Leugering, “On elastic bodies with thin rigid inclusions and cracks”, Math. Methods Appl. Sci., 33 (2010), 1955–1967 | MR | Zbl

[13] A. Khludnev, G. Leugering, M. Specovius-Neugebauer, “Optimal control of inclusion and crack shapes in elastic bodies”, J. Optim. Theory Appl., 155 (2012), 54–78 | DOI | MR | Zbl

[14] A. M. Khludnev, J. Sokolowski, “The Griffith's formula and the Rice–Cherepanov integral for crack problems with unilateral conditions in nonsmooth domains”, European J. Appl. Math., 10 (1999), 379–394 | DOI | MR | Zbl

[15] V. A. Kovtunenko, K. Kunisch, “Problem of crack perturbation based on level sets and velocities”, Z. angew. Math. Mech., 87 (2007), 809–830 | DOI | MR | Zbl

[16] V. A. Kovtunenko, K. Kunisch, “High precision identification of an object: optimality conditions based concept of imaging”, SIAM J. Control Optim., 52 (2014), 773–796 | DOI | MR | Zbl

[17] V. A. Kovtunenko, G. Leugering, “A shape-topological control problem for nonlinear crack-defect interaction: the anti-plane variational model”, SIAM J. Control Optim., 54 (2016), 1329–1351 | DOI | MR | Zbl

[18] G. Leugering, J. Sokolowski, A. Zochowski, “Shape-topological differentiability of energy functionals for unilateral problems in domains with cracks and applications”, Optimization with PDE constraints, ESF Networking Program ‘OPTPDE’, ed. R. Hoppe, 2014, 203–221 | MR

[19] G. Leugering, J. Sokolowski, A. Zochowski, “Control of crack propagation by shape-topological optimization”, Discrete and Continuous Dynamical Systems-Series A (DCDS-A), 35 (2015), 2625–2657 | DOI | MR | Zbl

[20] V. G. Maz'ya, S. A. Nazarov, B. A. Plamenevski, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, Birkhäuser, Basel, 2000

[21] G. Mishuris, G. Kuhn, “Comparative study of an interface crack for different wedge-interface models”, Arch. Appl. Mech., 71 (2001), 764–780 | DOI | Zbl

[22] Zh. O. Oralbekova, K. T. Iskakov, A. L. Karchevsky, “Existence of the residual functional derivative with respect to a coordinate of gap point of medium”, Appl. Comput. Math., 12 (2013), 222–233 | MR | Zbl

[23] N. Ovcharova, J. Gwinner, “From solvability and approximation of variational inequalities to solution of nondifferentiable optimization problems in contact mechanics”, Optimization, 64 (2015), 1683–1702 | DOI | MR | Zbl

[24] J. Sokolowski, J.-P. Zolesio, Introduction to shape optimization. Shape sensitivity analysis, Springer-Verlag, Berlin, 1992 | MR | Zbl