Level set method for optimization of contact problems

Abstract

This paper deals with the numerical solution of topology and shape optimization problems of an elastic body in unilateral contact with a rigid foundation. The contact problem with the prescribed friction is described by an elliptic variational inequality of the second order governing a displacement field. The structural optimization problem consists in finding such shape of the boundary of the domain occupied by the body that the normal contact stress along the contact boundary of the body is minimized. The shape of this boundary and its evolution is described using the level set approach. Level set methods are numerically efficient and robust procedures for the tracking of interfaces. They allow domain boundary shape changes in the course of iteration. The evolution of the domain boundary and the corresponding level set function is governed by the Hamilton-Jacobi equation. The speed vector field driving the propagation of the level set function is given by the Eulerian derivative of an appropriately defined functional with respect to the free boundary. In this paper the necessary optimality condition is formulated. The level set method, based on the classical shape gradient, is coupled with the bubble or topological derivative method, which is precisely designed for introducing new holes in the optimization process. The holes are supposed to be filled by weak phase mimicking voids. Since both methods capture a shape on a fixed Eulerian mesh and rely on a notion of gradient computed through an adjoint analysis, the coupling of these two method yields an efficient algorithm. Moreover the finite element method is used as the discretization method. Numerical examples are provided and discussed.

Topology optimization of quasistatic contact problems

“…This constant is usually assumed to be the volume of the initial domain Ω and (34) is satisfied (Myśliński, 2008) …”

“…This paper is concerned with the application of a topological derivative approach to formulate a necessary optimality condition for a structural optimization problem for elliptic contact problems with Tresca friction. Unlike in the previous works of Myśliński (2008;2010), where the stationary contact model is used, here the quasistatic contact problem is considered. Quasistatic processes arise when the external forces applied to a system vary slowly in time.…”

“…In the literature, most papers (see Chambolle, 2003;270 A. Myśliński Denkowski and Migórski, 1998;Eschenauer et al, 1994;Fulmański et al, 2007;Garreau et al, 2001;Myśliński, 2008;2010;Nazarov and Sokołowski, 2003;Novotny et al, 2005;Sokołowski and Zolesio, 1999;Sokołowski anḋ Zochowski 2004;2008;Strömberg and Klabring, 2010) are devoted to asymptotic and topology sensitivity analysis for elliptic boundary value problems. A few papers only (e.g., Amstuz et al, 2008;Kowalewski et al, 2010) address this issue for the shape functionals depending on a solution to time-dependent boundary value problems.…”

“…The mathematical or engineering literature concerning this topic is rather extensive (see the references in the monographs of Eck et al (2005) as well as Han and Sofonea (2002)). Asymptotic and topology sensitivity analysis of solutions to unilateral stationary boundary value problems in elasticity is performed by Fulmański et al (2007), Myśliński (2010) and Sokołowski et al (2005;2008). Simultaneous shape and topology optimization of elastic structures where both the boundary perturbation and the nucleation of holes inside the domain occur is considered, among others, by Myśliński (2008) as well as Sokołowski andŻochowski (2004).…”

“…Asymptotic and topology sensitivity analysis of solutions to unilateral stationary boundary value problems in elasticity is performed by Fulmański et al (2007), Myśliński (2010) and Sokołowski et al (2005;2008). Simultaneous shape and topology optimization of elastic structures where both the boundary perturbation and the nucleation of holes inside the domain occur is considered, among others, by Myśliński (2008) as well as Sokołowski andŻochowski (2004). The main difficulty in topology sensitivity analysis of contact problems is associated with the nonlinearity of the non-penetration condition over the contact zone, which makes this boundary value problem non-smooth.…”