On the Shape Differentiability of Objectives: A Lagrangian Approach and the Brinkman Problem

Abstract

This paper establishes the shape derivative of geometry-dependent objective functions for use in constrained variational problems. Using a Lagrangian approach, our differentiablity result is based on the theorem of Delfour–Zolésio on directional derivatives with respect to a parameter of shape perturbation. As the key issue of the paper, we analyze the bijection under the kinematic transport of geometries that is needed for function spaces and feasible sets involved in variational problems. Our abstract theoretical result is applied to the Brinkman flow problem under incompressibility and mixed Dirichlet–Neumann boundary conditions, and provides an analytic formula of the shape derivative based on the velocity method.

Inverse problems for nonlinear quasi-hemivariational inequalities with application to mixed boundary value problems

“…In a similar way, we can prove that {u n } is bounded in V, and by passing to a subsequence if necessary, we have u n u in V, as n → ∞, for some u ∈ Γ(a) (see ( 14)). By the same reasoning as in (15), we conclude inf b∈A J κ (b) = J κ (a), that is, a ∈ A is a solution to problem 3.1. Hence the solution set of problem 3.1 is weakly compact.…”

“…(3) It would be important to derive optimality conditions for problem 4.1. This is an interesting open problem for the future research which is related to the recent shape differentiability result in the state-constrainted optimization, see [15]. Further, it would be also desirable to develop numerical techniques for the state-constrained optimization problem.…”

The energy release rate for non-penetrating crack in poroelastic body by fluid-driven fracture

“…Traits 1-4 satisfy all assumptions in Delfour and Zole´sio [28] (Chapter 10, Theorem 5.1), thus provide the following theorem (see the detailed proof in Gonza´lez et al [29]).…”

“…and get the variational equation ( 28) for u = u t , the stress t t := Ae(u t ) + t 0 À ap t I, and the contact force n T t t t n t + p re = l t according to equation (29). Conversely, from equation (28), it follows by convexity the minimum in equation ( 33)…”

“…For this task, we consider the problem in the incremental form ( 5) and (10), and endow it with a saddle-point formulation. Based on the Lagrange multiplier approach, we apply the formalism of directional differentiability for Lagrangians (see Delfour and Zole´sio [28]) and use rigorous asymptotic methods (see Gonza´lez et al [29]) to derive a shape derivative for the underlying Lagrangian function implying a free energy. The resulting formula describes the energy release rate under irreversible crack perturbations, which is useful for the Griffith criterion of quasi-static crack evolution (see Charlotte et al [30]).…”