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2010 | 30 | 2 | 237-252
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Topology optimization of systems governed by variational inequalities

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EN
Abstrakty
EN
This paper deals with the formulation of the necessary optimality condition for a topology optimization problem of an elastic body in unilateral contact with a rigid foundation. In the contact problem of Tresca, a given friction is governed by an elliptic variational inequality of the second order. The optimization problem consists in finding such topology of the domain occupied by the body that the normal contact stress along the contact boundary of the body is minimized. The topological derivative of the cost functional is calculated and a necessary optimality condition is formulated. The calculated topological derivative is also used in the numerical algorithm to find a descent direction by inserting voids in the domain occupied by the body. Numerical examples are provided.
Twórcy
  • Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland
Bibliografia
  • [1] G. Allaire, Shape optimization by the homogenization method, New York, Springer, 2002. doi: 10.1007/978-1-4684-9286-6
  • [2] A. Chambolle, A density result in two-dimensional linearized elasticity and applications, Arch. Ration. Mech. Anal. 167 (2003), 211-233.
  • [3] P. Fulmański, A. Laurain, J.F. Scheid and J. Sokołowski, A level set method in shape and topology optimization for variational inequalities, Int. J. Appl. Math. Comput. Sci. 17 (2007), 413-430. doi: 10.2478/v10006-007-0034-z
  • [4] S. Garreau, Ph. Guillaume and M. Masmoudi, The topological asymptotic for PDE systems: the elasticity case, SIAM J. Control Optim. 39 (2001), 1756-1778.
  • [5] I. Hlavaček, J. Haslinger, J. Nečas and J. Lovišek, Solving of Variational Inequalities in Mechanics, Mir, Moscow (in Russian), 1986.
  • [6] S. Hüber, G. Stadler and B. Wohlmuth, A primal-dual active set algorithm for three-dimensional contact problems with Coulomb friction, SIAM J. Sci. Comput. 30 (2008), 572-596.
  • [7] A. Myśliński, Level set method for optimization of contact problems, Engineering Analysis with Boundary Elements 32 (2008), 986-994.
  • [8] A. Myśliński, Optimization of contact problems using a topology derivative method, in: B.H.V. Topping, M. Papadrakakis, (Editors), Proceedings of the Ninth International Conference on Computational Structures Technology, Civil-Comp Press, Stirlingshire, UK, Paper 177 (doi: 10.4203/ccp.88.177), 2008.
  • [9] S.A. Nazarov and J. Sokołowski, Asymptotic Analysis of Shape Functionals, J. Math. Pures Appl. 82 (2) (2003), 125-196. doi: 10.1016/S0021-7824(03)00004-7
  • [10] A.A. Novotny, R.A. Feijóo, C. Padra and E. Tarocco, Topological derivative for linear elastic plate bending problems, Control and Cybernetics 34 (2005), 339-361.
  • [11] J. Sokołowski and A. Żochowski, On topological derivative in shape optimization, in: Optimal Shape Design and Modelling, T. Lewiński, O. Sigmund, J. Sokołowski, A. Żochowski, (Editors), Academic Printing House EXIT, Warsaw, 2004, 55-143.
  • [12] J. Sokołowski and A. Żochowski, Topological derivatives for optimization of plane elasticity contact problems, Engineering Analysis with Boundary Elements 32 (2008), 900-908.
  • [13] J. Sokołowski and J.P. Zolesio, Introduction to shape optimization. Shape sensitivity analysis, Berlin, Springer, 1992.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1122
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