# Topology optimization of systems governed by variational inequalities

• Volume: 30, Issue: 2, page 237-252
• ISSN: 1509-9407

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## Abstract

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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.

## How to cite

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Andrzej Myśliński. "Topology optimization of systems governed by variational inequalities." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 30.2 (2010): 237-252. <http://eudml.org/doc/271166>.

@article{AndrzejMyśliński2010,
abstract = {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.},
author = {Andrzej Myśliński},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {variational inequality; topology optimization},
language = {eng},
number = {2},
pages = {237-252},
title = {Topology optimization of systems governed by variational inequalities},
url = {http://eudml.org/doc/271166},
volume = {30},
year = {2010},
}

TY - JOUR
AU - Andrzej Myśliński
TI - Topology optimization of systems governed by variational inequalities
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2010
VL - 30
IS - 2
SP - 237
EP - 252
AB - 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.
LA - eng
KW - variational inequality; topology optimization
UR - http://eudml.org/doc/271166
ER -

## References

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10. [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. Zbl1167.74487
11. [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. [12] J. Sokołowski and A. Żochowski, Topological derivatives for optimization of plane elasticity contact problems, Engineering Analysis with Boundary Elements 32 (2008), 900-908. Zbl1244.74108
13. [13] J. Sokołowski and J.P. Zolesio, Introduction to shape optimization. Shape sensitivity analysis, Berlin, Springer, 1992. Zbl0761.73003

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