Keywords: shape and topology optimization, unilateral problems, piecewise constant level set method, Uzawa method.

This paper considers the numerical solution of a structural optimization problem for an elastic body in unilateral contact with a rigid foundation. The structural optimization problem for the elastic body in contact consists in finding such a topology of the domain occupied by the body and the shape of its boundary that the normal contact stress along the boundary of the body is minimized. The volume of the body is bounded.

In structural optimization the standard level set method is employed in numerical algorithms for tracking the evolution of the domain boundary on a fixed mesh and finding an optimal domain [1,2]. The position of the domain boundary is described as an isocontour of a scalar level set function of a higher dimensionality. The evolution of the domain boundary is governed by the Hamilton-Jacobi equation. Recently, a piecewise constant level set method has been proposed to solve the elliptic shape recovery problems [3]. For a domain divided into an arbitrary number of subdomains a discontinuous piecewise constant level set function takes distinct constant values on each subdomain. This approach is free of the Hamilton-Jacobi equation.

The paper is concerned with the numerical shape and topology optimization of the elastic contact problems using the piecewise constant level set approach. It extends the results of [4]. The original structural optimization problem is approximated by a two-phase shape optimization problem in a larger computational domain. Different phases in this domain are represented by a one piecewise constant level set function only. Using this approach the original problem is reformulated as an equivalent constrained optimization problem in terms of this level set function. The necessary optimality condition is formulated. This problem is solved numerically using the projected Lagrangian method. Numerical examples are provided and discussed. They indicate that the proposed numerical algorithm allows for significant improvements of the structure from one iteration to the next and is more efficient than the algorithms based on the standard level set approach.

1

T. Yamada, K. Izui, S. Nishiwaki, A. Takezawa, "A topology optimization method based on the level set method incorporating a fictitious interface energy", Comput. Methods Appl. Mech. Engrg., 199, 2876-2891, 2010. doi:10.1016/j.cma.2010.05.013

2

A. Myslinski, "Level Set Method for Optimization of Contact Problems", Engineering Analysis with Boundary Elements, 32, 986-994, 2008. doi:10.1016/j.enganabound.2007.12.008

3

S. Zhu, Q. Wu, C. Liu, "Shape and topology optimization for elliptic boundary value problems using a piecewise constant level set method", Applied Numerical Mathematics, 61, 752-767, 2011. doi:10.1016/j.apnum.2011.01.005

4

A. Myslinski, "Structural Optimization of Elastic Contact Problems using Piecewise Constant Level Set Method", CD-ROM Proceedings of 9th World Congress of Structural and Multidisciplinary Optimization, June 13-17, 2011, Shizuoka, Japan, ISSMO, 2011.

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