3D–2D dimensional reduction for a nonlinear optimal design problem with perimeter penalization

Abstract

A 3D-2D dimension reduction for a nonlinear optimal design problem with a perimeter penalization is performed in the realm of Γ-convergence, providing an integral representation for the limit functional. RésuméRéduction dimensionnelle 3D−2D d'un problème non linéaire d'optimisation de forme avec pénalisation sur le périmètre On effectue dans ce travail une réduction dimensionnelle 3D-2D d'un problème non linéaire d'optimisation de forme avec une pénalisation du périmètre. Une représentation intégrale de la fonctionnelle limite est obtenue.

Relaxation for Optimal Design Problems with Non-standard Growth

“…In the following we adopt the standard scaling (see [12] and the references quoted therein) which maps x ≡ (x 1 , x 2 , x 3 ) ∈ Ω(ε) → ( 1 ε x 1 , 1 ε x 2 , x 3 ) ∈ Ω := B(0, 1) × (0, l), in order to state the problem in a fixed domain (see (5.3) below). We also denote by ∇ α u and D α χ, respectively, the partial derivatives of u and χ with respect to x α ≡ (x 1 , x 2 ).…”

“…In the model under consideration, the sequence χ ε ∈ BV (Ω; {0, 1}) represents the design regions, whereas u ε ∈ W 1,q (Ω; R 3 ) is the sequence of deformations, which are possibly clamped at the extremities of the string. Standard arguments in dimension reduction (see [2] and [12]) ensure that energy bounded sequences (see the term in square brackets of (5.3)), converge (up to a subsequence), in the relevant topology, to fields (χ, u) such that D α χ and ∇ α u are null, thus they can be identified, with an abuse of notation, with fields (χ, u) ∈ BV ((0, l); {0, 1}) × W 1,p ((0, l); R 3 ). In what follows we use this notation.…”

“…The characterisation of this measure was provided by obtaining an integral representation for F OD (χ, u; A), where A is an open subset of Ω. The case of non-convex W i with superlinear growth was addressed in the context of thin films in [12].…”

Relaxation for an optimal design problem with linear growth and perimeter penalization

“…finding the solution (u, E) and describing the regularity properties of the optimal set E. In this paper we consider the minimization of a similar functional, where the energy density | • | 2 has been replaced by the more general W i , i = 1, 2, without any convexity assumptions and with linear growth, and since the lower-order terms g 1 (x, u) and g 2 (x, u) do not play any role in the asymptotics, we omit them in our subsequent analysis. The case of W i , i = 1, 2, not convex with superlinear growth has been studied in the context of thin films in [16].…”

Orlicz equi-integrability for scaled gradients

“…of the type j Á j p ) for the energy density, by convex functions [satisfying suitable properties, as (5) and (6)]. We refer to the recent works [18,19] aimed to describe thin structures and their bending phenomena, and to the forthcoming paper [16], where optimal design questions are addressed in the same spirit of [5,6]. We believe that our result can have further applications like those to fluid mechanics and multiscale problems (we refer to [21], where homogenization of integral functionals was treated, in a very similar setting to ours).…”