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2012 | 22 | 2 | 269-280
Tytuł artykułu

Topology optimization of quasistatic contact problems

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
This paper deals with the formulation of a necessary optimality condition for a topology optimization problem for an elastic contact problem with Tresca friction. In the paper a quasistatic contact model is considered, rather than a stationary one used in the literature. The functional approximating the normal contact stress is chosen as the shape functional. The aim of the topology optimization problem considered is to find the optimal material distribution inside a design domain occupied by the body in unilateral contact with the rigid foundation to obtain the optimally shaped domain for which the normal contact stress along the contact boundary is minimized. The volume of the body is assumed to be bounded. Using the material derivative and asymptotic expansion methods as well as the results concerning the differentiability of solutions to quasistatic variational inequalities, the topological derivative of the shape functional is calculated and a necessary optimality condition is formulated.
Rocznik
Tom
22
Numer
2
Strony
269-280
Opis fizyczny
Daty
wydano
2012
otrzymano
2011-03-21
poprawiono
2011-09-08
Twórcy
  • Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447 Warsaw, Poland
Bibliografia
  • Allaire, G. (2002). Shape Optimization by the Homogenization Method, Springer, New York, NY.
  • Allaire G., Jouve, F. and Toader, A., (2004). Structural optimization using sensitivity analysis and a level let method, Journal of Computational Physics 194(1): 363-393.
  • Ammari, H., Kang, H. and Lee, H. (2009). Layer Potential Techniques in Spectral Analysis, Mathematical Surveys and Monographs, Vol. 153, AMS, Providence, RI.
  • Amstuz, S., Takahashi T., Vexler, B. (2008). Topological sensitivity analysis for time-dependent problems, ESAIM: Control, Optimisation, and Calculus of Variations 14(3): 427-455.
  • Ayyad, Y. and Sofonea, M. (2007). Analysis of two dynamic frictionless contact problems for elastic-visco-plastic materials, Electronic Journal of Differential Equations (55): 1-17.
  • Bendsoe, M.P and Sigmund, O. (2003). Topology Optimization: Theory, Methods, and Applications, Springer, Berlin.
  • Chambolle, A. (2003). A density result in two-dimensional linearized elasticity and applications, Archive for Rational Mechanics and Analysis 167(3): 211-233.
  • Chudzikiewicz, A. and Myśliński, A. (2009). Thermoelastic wheel-rail contact problem with elastic graded materials, 8th International Conference on Contact Mechanics and Wear of Rail/Wheel Systems, Firenze, Italy, pp. 795-801.
  • Duvaut, G. and Lions, J.L. (1972). Les inequations en mecanique et en physique, Dunod, Paris.
  • Denkowski, Z. and Migórski, S. (1998). Optimal shape design problems for a class of systems described by hemivariational inequalities, Journal of Global Optimization 12(1): 37-59.
  • Eck, C., Jarušek, J. and Krbeč, M. (2005). Unilateral Contact Problems: Variational Methods and Existence Theorems, Pure and Applied Mathematics, Vol. 270, CRC Press, New York, NY.
  • Eschenauer, H.A., Kobolev V.V. and Schumacher, A. (1994). Bubble method for topology and shape optimization of structures, Structural Optimization 8(1): 42-51.
  • Fulmański, P., Laurain, A., Scheid, J.F. and Sokołowski, J. (2007). A level set method in shape and topology optimization for variational inequalities, International Journal of Applied Mathematics and Computer Science 17(3): 413-430, DOI: 10.2478/v10006-007-0034-z.
  • Garreau, S., Guillaume, Ph. and Masmoudi, M. (2001). The topological asymptotic for PDE systems: The elasticity case, SIAM Journal on Control Optimization 39(6): 1756-1778.
  • Han, W. and Sofonea, M. (2002). Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, AMS/IP Studies in Advanced Mathematics, Vol. 30, AMS/IP, Providence, RI.
  • Haslinger, J. and Mäkinen, R. (2003). Introduction to Shape Optimization. Theory, Approximation, and Computation, SIAM Publications, Philadelphia, PA.
  • Hüber, S., Stadler, G. and Wohlmuth, B. (2008). A primal-dual active set algorithm for three-dimensional contact problems with Coulomb friction, SIAM Journal on Scientific Computation 30(2): 572-596.
  • Jarušek, J., Krbec, M., Rao, M. and Sokołowski, J. (2003). Conical differentiability for evolution variational inequalities, Journal of Differential Equations 193(1): 131-146.
  • Kowalewski, A., Lasiecka, I. and Sokołowski, J. (2010). Sensitivity analysis of hyperbolic optimal control problems, Computational Optimization and Applications, DOI: 10.1007/s10589-010-9375-x.
  • Myśliński, A. (2006). Shape Optimization of Nonlinear Distributed Parameter Systems, Academic Publishing House EXIT, Warsaw.
  • Myśliński, A. (2008). Level set method for optimization of contact problems, Engineering Analysis with Boundary Elements 32(11): 986-994.
  • Myśliński, A. (2010). Topology optimization of systems governed by variational inequalities, Discussiones Mathematicae: Differential Inclusions, Control and Optimization 30(2): 237-252.
  • Nazarov, S.A. and Sokołowski, J. (2003). Asymptotic Analysis of Shape Functionals, Journal de Mathématiques Pures et Appliquées 82(2): 125-196.
  • Novotny, A.A., Feijóo, R.A., Padra, C. and Tarocco, E. (2005). Topological derivative for linear elastic plate bending problems, Control and Cybernetics 34(1): 339-361.
  • Rocca, R. and Cocu, M. (2001). Existence and approximation of a solution to quasistatic Signorini problem with local friction, International Journal of Engineering Science 39(11): 1233-1255.
  • Sokołowski, J. and Zolesio, J.P. (1992). Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer, Berlin.
  • Sokołowski, J. and Żochowski, A. (1999). On the topological derivative in shape optimization, SIAM Journal on Control and Optimization 37(4): 1251-1272.
  • Sokołowski, J. and Żochowski, A. (2004). On topological derivative in shape optimization, in T. Lewiński, O. Sigmund, J. Sokołowski and A. Żochowski (Eds.), Optimal Shape Design and Modelling, Academic Publishing House EXIT, Warsaw, pp. 55-143.
  • Sokołowski, J. and Żochowski, A. (2005). Modelling of topological derivatives for contact problems, Numerische Mathematik 102(1): 145-179.
  • Sokołowski, J. and Żochowski, A. (2008). Topological derivatives for optimization of plane elasticity contact problems, Engineering Analysis with Boundary Elements 32(11): 900-908.
  • Strömberg, N. and Klabring, A. (2010). Topology optimization of structures in unilateral contact, Structural Multidisciplinary Optimization 41(1): 57-64.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-amcv22i2p269bwm
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