Keywords: contact problem, structural optimization, level set method.

The paper deals with the numerical solution of a topology optimization problem of an elastic body in unilateral contact with a rigid foundation. The contact problem with the prescribed friction is described by an static variational inequality of the second order governing a displacement field [1,2]. The optimization problem consists in finding such topology of the domain occupied by the body such that the normal contact stress is minimized.

Topology optimization deals with the optimal material distribution within the body resulting in its optimal shape [3,4,2]. The topological derivative is employed to account variations of the solutions due to a contact problem or cost functionals dependent on these solutions with respect to emerging small holes in the interior of the domain occupied by the body. The notion of topological derivative and results concerning its application in optimization of elastic structures are reported in a series of papers [3,5,6,7,4,2]. Among others, paper [2] deals with the calculation of topological derivatives of solutions to Signorini and elastic contact problems. The asymptotic expansion method combined with transformation of energy functional are employed to calculate these derivatives.

This paper deals with topology optimization of an elastic contact problems with the prescribed friction. The topology optimization problem for elastic contact problem is formulated. Topological derivative formulae of the shape functional is provided using material derivative and asymptotic expansion methods. This derivative is employed to formulate the necessary optimality condition for the topology optimization problem. A descent type numerical algorithm for the solution of this optimization problem is proposed. The circular small holes are inserted at grid points where this topological derivative has negative values. The finite element method is used as the discretization method. Numerical examples are provided and discussed.

1

A. Myslinski, "Level Set Method for Optimization of Contact Problems", Research Report No RB/78/2007, Systems Research Institute, Warsaw, Poland, 2007; Engineering Analysis with Boundary Elements - in press. doi:10.1016/j.enganabound.2007.12.008

2

J. Sokolowski, A. Zochowski, "Modelling of topological derivatives for contact problems", Numerische Mathematik, 102, 145 - 179, 2005. doi:10.1007/s00211-005-0635-0

3

S. Garreau, Ph. Guillaume, M. Masmoudi, "The topological asymptotic for PDE systems: the elasticity case", SIAM Journal on Control Optimization, 39, 1756 - 1778, 2001. doi:10.1137/S0363012900369538

4

J. Sokolowski, A. Zochowski, "On topological derivative in shape optimization", In: Optimal Shape Design and Modelling, T. Lewinski, O. Sigmund, J. Sokolowski, A. Zochowski eds. Academic Printing House EXIT. Warsaw, Poland, 55 - 143, 2004.

5

A.A Novotny, R.A. Feijoo, C. Padra, E. Tarocco, "Topological Derivative for Linear Elastic Plate Bending Problems", Control and Cybernetics, 34, 339 - 361, 2005.

6

J.A. Norato, M.P. Bendsoe, P. Haber, D.A. Tortorelli, "A topological derivative method for topology optimization", Structural Multidisciplinary Optimization, 33, 375 - 386, 2007. doi:10.1007/s00158-007-0094-6

7

J. Sokolowski, A. Zochowski, "Optimality Conditions for Simultaneous Topology and Shape Optimization", SIAM Journal on Control, 42, 1198 - 1221, 2003. doi:10.1137/S0363012901384430

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