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.
Highlights
The paper is concerned with 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 shape of the boundary of the domain occupied by the body that the normal contact stress along the boundary of the body is minimized
This paper extends results of [11] to contact problems with the prescribed friction
Summary
The paper is concerned with the numerical solution of a structural optimization problem for an elastic body in unilateral contact with a rigid foundation. The material derivative method is employed in monograph [2] to calculate the sensitivity of solutions to contact problems as well as the derivatives of domain depending functionals with respect to variations of the boundary of the domain occupied by the body. Simultaneous shape and topology optimization of Signorini and elastic frictionless contact problems are analyzed in papers [10,11] In these papers the level set method is incorporated in numerical algorithms. Shape as well as topological derivatives formulae of the cost functional are provided using the material derivative [2] and the asymptotic expansion [4] methods, respectively These derivatives are employed to formulate necessary optimality condition for simultaneous shape and topology optimization. This paper extends results of [11] to contact problems with the prescribed friction
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