Shape optimization of a linearly elastic rolling structure under unilateral contact using Nitsche’s method and cut finite elements

Abstract

The main motivation of this work is to develop a numerical strategy for the shape optimization of a rolling elastic structure under contact with respect to a uniform rolling criterion. A first objective is to highlight the influence on the treatment of the contact terms. To do so, we present a numerical comparison between a penalty-based approach and the use of Nitsche's method which is known to have good consistency properties. A second task concerns the construction of an objective functional to force the uniform rolling criterion. Here, we present and compare two different strategies that will lead to quite similar results. All the numerical experiments proposed in this paper were performed using a fictitious domain approach coupled with a level set representation of the shape and the use of a cut finite element method to approximate the elastic equation.

Shape sensitivity analysis of an elastic contact problem: Convergence of the Nitsche based finite element approximation

“…Remark 14. Although we cannot demonstrate a convergence result from the discrete adjoint state to its continuous counterpart, at least for θ = 1, the use of ph Ω in (20) allows to properly define the gradient of the discrete energy J h which can be use to minimize J h using a gradient algorithm, as we proposed in [30].…”

“…, where u h Ω ∈ V h solution of ( 14), a first approach is to derive the adjoint state of the discrete formulation, for instance using a Lagrangian approach. This is presented in [30] and leads to the following formulation:…”

“…In the recent work [30], we are interested in the optimization of an elastic structure under contact conditions while trying to minimize criteria that couple compliance terms and additional terms allowing pressure uniformizations. We propose the use of Nitsche-based methods [31] to efficiently discretize the contact terms.…”

“…The optimization of the elastic structure is also performed using gradient descent strategy where the gradient is estimated via the adjoint state method applied directly on the discrete formulation of the problem. Although the proposed method allows us to obtain convincing structure optimization, no results of convergence analysis about the discretization of the adjoint state problem were given in [30]. The aim of this paper is therefore to analyze and propose a first result in this direction.…”

“…We recall then, in Section 3, some results about the conical directional differentiability of the solution to the contact problem and the link with the shape gradient mainly following [16,32]. In a second step, as in [30], we present in Section 4 the discretization of the adjoint state problem consisting in applying the adjoint state method on the discrete Nitsche version of the direct problem. Unfortunately, we note a lack of consistency of this approach.…”