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Khludnev, Alexander; Sokołowski, Jan; Szulc, Katarzyna

Shape and topological sensitivity analysis in domains with cracks. (English) Zbl 1224.49014

Appl. Math., Praha 55, No. 6, 433-469 (2010).
Summary: The framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. The equilibrium problem for the elastic body with cracks is considered. Inequality type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. Modelling of such problems in two spatial dimensions is presented with all necessary details for further applications in shape optimization in structural mechanics. In the paper, general results on the shape and topology sensitivity analysis of this problem are provided. The results are of interest of their own. In particular, the existence of the shape and topological derivatives of the energy functional is obtained. The results presented in the paper can be used for numerical solution of shape optimization and inverse problems in structural mechanics.

Cited in 8 Documents

MSC:

49J40 Variational inequalities
74K20 Plates
35J25 Boundary value problems for second-order elliptic equations
49K10 Optimality conditions for free problems in two or more independent variables
49Q10 Optimization of shapes other than minimal surfaces

Keywords:

crack with non-penetration; shape sensitivity; derivative of energy functional; topological derivative
PDFBibTeX XMLCite
\textit{A. Khludnev} et al., Appl. Math., Praha 55, No. 6, 433--469 (2010; Zbl 1224.49014)
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References:

[1] Z. Belhachmi, J.M. Sac-Epée, J. Sokołowski: Mixed finite element methods for smooth domain formulation of crack problems. SIAM J. Numer. Anal. 43 (2005), 1295–1320. · Zbl 1319.74027 · doi:10.1137/S0036142903429729
[2] G.A. Francfort, J.-J. Marigo: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (1998), 1319–1342. · Zbl 0966.74060 · doi:10.1016/S0022-5096(98)00034-9
[3] G. Fremiot, J. Sokołowski: Shape sensitivity analysis of problems with singularities. Shape Optimization and Optimal Design. Proc. IFIP coference, Cambridge 1999. Lect. Notes Pure Appl. Math. 216 (J. Cagnol et al., eds.). Marcel Dekker, New York, 2001.
[4] P. Fulmański, A. Lauraine, J.-F. Scheid, J. Sokołowski: A level set method in shape and topology optimization for variational inequalities. Int. J. Appl. Math. Comput. Sci., 17 (2007), 413–430. · Zbl 1237.49058 · doi:10.2478/v10006-007-0034-z
[5] S. Garreau, Ph. Guillaume, M. Masmoudi: The topological asymptotic for PDE systems: The elasticity case. SIAM J. Control Optimization 39 (2001), 1756–1778. · Zbl 0990.49028 · doi:10.1137/S0363012900369538
[6] I. Hlávček, A.A. Novotný, J. Sokołowski, A. \.Zochowski: On topological derivatives for elastic solids with uncertain input data. J. Optim. Theory Appl. 141 (2009), 569–595. · Zbl 1215.74065 · doi:10.1007/s10957-008-9490-3
[7] K.-H. Hoffmann, A.M. Khludnev: Fictitious domain method for the Signorini problem in linear elasticity. Adv. Math. Sci. Appl. 14 (2004), 465–481. · Zbl 1134.74368
[8] A.M. Khludnev: Invariant integrals in the problem of a crack on the interface between two media. J. Appl. Mech. and Tech. Phys. 46 (2005), 717–729;, Prikl. Mekh. Tekh. Fiz. 46 (2005), 123–137. (In Russian.) · Zbl 1125.74365
[9] A.M. Khludnev, V.A. Kovtunenko: Analysis of Cracks in Solids. WIT Press, Southampton-Boston, 2000. · Zbl 0954.35076
[10] A.M. Khludnev, V.A. Kovtunenko, A. Tani: Evolution of a crack with kink and non-penetration. J. Math. Soc. Japan 60 (2008), 1219–1253. · Zbl 1153.49040 · doi:10.2969/jmsj/06041219
[11] A.M. Khludnev, K. Ohtsuka, J. Sokołowski: On derivative of energy functional for elastic bodies with a crack and unilateral conditions. Q. Appl. Math. 60 (2002), 99–109. · Zbl 1075.74040
[12] A.M. Khludnev, J. Sokołowski: Modelling and Control in Solid Mechanics. International Series of Numerical Mathematics. Birkhauser, Basel, 1997.
[13] A.M. Khludnev, J. Sokołowski: Griffith formulae for elasticity systems with unilateral conditions in domains with cracks. Eur. J. Mech. A, Solids 19 (2000), 105–119. · Zbl 0966.74061 · doi:10.1016/S0997-7538(00)00138-8
[14] A.M. Khludnev, J. Sokołowski: On differentiation of energy functionals in the crack theory with possible contact between crack faces. J. Appl. Math. Mech. 64 (2000), 464–475.
[15] A.M. Khludnev, J. Sokołowski: Smooth domain method for crack problem. Q. Appl. Math. 62 (2004), 401–422. · Zbl 1067.74056
[16] D. Knees, C. Zanini, A. Mielke: Crack growth in polyconvex materials. Physica D 239 (2010), 1470–1484. · Zbl 1201.49013 · doi:10.1016/j.physd.2009.02.008
[17] V.A. Kovtunenko: Invariant integrals in nonlinear problem for a crack with possible contact between crack faces. J. Appl. Math. Mech. 67 (2003), 109–123. · Zbl 1067.74562 · doi:10.1016/S0021-8928(03)00021-2
[18] V.A. Kovtunenko: Numerical simulation of the non-linear crack problem with non-penetration. Math. Methods Appl. Sci. 27 (2004), 163–179. · Zbl 1099.74056 · doi:10.1002/mma.449
[19] A. Laurain: Structure of shape derivatives in non-smooth domains and applications. Adv. Math. Sci. Appl. 15 (2005), 199–226. · Zbl 1085.49018
[20] N.P. Lazarev: Differentiation of energy functional in the equilibrium problem for a body with a crack and Signorini boundary conditions. J. Appl. Industr. Math. 5 (2002), 139–147. · Zbl 1075.74558
[21] T. Lewiński, J. Sokołowski: Energy change due to the appearance of cavities in elastic solids. Int. J. Solids Struct. 40 (2003), 1765–1803. · Zbl 1035.74009 · doi:10.1016/S0020-7683(02)00641-8
[22] W.G. Maz’ja, S.A. Nazarov, B.A. Plamenevskii: Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, Vol. I, II. Birkhauser, Basel, 2000. · Zbl 1127.35301
[23] N. I. Muskhelishvili: Some Basic Problems of the Mathematical Theory of Elasticity. P. Noordhoff, Groningen, 1952.
[24] S.A. Nazarov, J. Sokołowski: Asymptotic analysis of shape functionals. J. Math. Pures Appl. Ser. 82 (2003), 125–196. · Zbl 1031.35020 · doi:10.1016/S0021-7824(03)00004-7
[25] E.M. Rudoy: Differentiation of energy functionals in two-dimensional elasticity theory for solids with curvilinear cracks. J. Appl. Mech. Techn. Phys. 45 (2004), 843–852. · Zbl 1087.74008 · doi:10.1023/B:JAMT.0000046033.10086.86
[26] E.M. Rudoy: Differentiation of energy functions in the three-dimensional theory of elasticity for bodies with surfaces cracks. J. Appl. Ind. Math. 1 (2007), 95–104. · doi:10.1134/S1990478907010103
[27] E.M. Rudoy: Differentiation of energy functionals in the problem of a curvilinear crack with possible contact between the shores. Mech. Solids 42 (2007), 935–946. · doi:10.3103/S0025654407060118
[28] J. Sokołowski, J-P. Zolesio: Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer Series in Computational Mathematics, Vol. 16. Springer, Berlin, 1992. · Zbl 0761.73003
[29] J. Sokołowski, A. \.Zochowski: On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999), 1251–1272. · Zbl 0940.49026 · doi:10.1137/S0363012997323230
[30] J. Sokołowski, A. \.Zochowski: Optimality conditions for simultaneous topology and shape optimization. SIAM J. Control Optim. 42 (2003), 1198–1221. · Zbl 1045.49028 · doi:10.1137/S0363012901384430
[31] J. Sokołowski, A. \.Zochowski: Modelling of topological derivatives for contact problems. Numer. Math. 102 (2005), 145–179. · Zbl 1077.74039 · doi:10.1007/s00211-005-0635-0
[32] J. Sokołowski, A. \.Zochowski: Asymptotic analysis and topological derivatives for shape and topology optimization of elasticity problems in two spatial dimensions. Prépublication IECN 16, 2007.
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