Shape derivatives for an augmented Lagrangian formulation of elastic contact problems

Abstract

This work deals with shape optimization of an elastic body in sliding contact (Signorini) with a rigid foundation. The mechanical problem is written under its augmented Lagrangian formulation, then solved using a classical iterative approach. For practical reasons we are interested in applying the optimization process with respect to an intermediate solution produced by the iterative method. Due to the projection operator involved at each iteration, the iterate solution is not classically shape differentiable. However, using an approach based on directional derivatives, we are able to prove that it is conically differentiable with respect to the shape, and express sufficient conditions for shape differentiability. Finally, from the analysis of the sequence of conical shape derivatives of the iterative process, conditions are established for the convergence to the conical derivative of the original contact problem.

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

“…We recall here some results coming mainly from [16,32]. The differentiation with respect to the domain aims at modifying the reference state of the domain Ω using the boundary method first described by J. Hadamard in [39] and then developed for instance in [40,41,42,4,43].…”

“…In view of Zolésio-Hadamard structure theorem, we make the usual choice to limit the geometric deformation fields Θ ∈ C 1 (R d ) along the direction of the normal n (see [16] for instance). The vector n is extended to C 1 (R d ) as ∂Ω is assumed to have C 1 regularity.…”

“…A way to get optimality conditions is to consider a sequence of penalized problems (see for instance [9,10,11] and for numerical applications see [12,13,14]). B. Chaudet and J. Deteix prove the conical differentiability of the solution to the contact problem using the penalization method in [15] and the augmented Lagrangian method in [16].…”

“…First of all, in Section 2, we recall the elastic formulation with the contact problem. 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.…”

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

“…A regularized approach is used in [6] and [5] and more recently [38] for different friction laws. See also the recent work [12] for the penalized approach and [13] for the augmented Lagrangian one.…”