Structural Optimization of Thin Elastic Plates: The Three Dimensional Approach

Abstract

The natural way to find the most compliant design of an elastic plate, is to consider the three-dimensional elastic structures which minimize the work of the loading term, and pass to the limit when the thickness of the design region tends to zero. In this paper, we study the asymptotic of such compliance problem, imposing that the volume fraction remains fixed. No additional topological constraint is assumed on the admissible configurations. We determine the limit problem in different equivalent formulations, and we provide a system of necessary and sufficient optimality conditions. These results were announced in [18]. Furthermore, we investigate the vanishing volume fraction limit, which turns out to be consistent with the results in [16,17]. Finally, some explicit computation of optimal plates are given.

The optimal compliance problem for thin torsion rods: A 3D-1D analysis leading to Cheeger-type solutions

“…The analogous of problem (3) in the 3D-2D case (namely when the design region is of the formD × δI), has been studied in some of our recent papers, see [1][2][3][4]. In this respect, we stress that the 3D-1D case is not a purely technical variant.…”

“…Recalling that j(z) = (λ/2)| tr(z)| 2 + η|z| 2 , some explicit computations givej(z) = 2η α |z α | 2 + (Y /2)|z 3 | 2 , where Y = η 3λ+2η λ+η is the Young modulus, written in terms of the Lamé coefficients λ, η. (4), and (5), and we let δ tend to zero. Then, for every fixed k ∈ R, the sequence φ δ j,G (k) converges to the limit φ(k) defined by…”

“…Une étude similaire à celle traitée ici a été effectuée dans le cadre de la réduction 3D-2D (optimisation de la compliance de plaques) dans [1][2][3][4]. Les résultats annoncés dans cette note sont cependant très différents et offrent des géométries optimales plus riches.…”

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

“…In [6], [7] and [8] the asymptotic behaviour of a 3D optimal elastic compliance problem is studied, as the thickness (or the cross section in the case of beams) tends to zero and the volume fraction in the design region remains unchanged. It is assumed that the material has a convex and 2-homogeneous potential and the analysis is performed in the smalldisplacement setting.…”

Optimal Thin Torsion Rods and Cheeger Sets

“…where δ > 0 is a small parameter, I = [−1/2, 1/2] is a bounded interval, and D ⊂ R 2 is an open bounded connected domain. The case when Q δ = D × δI corresponds to perform a 3d-2d dimension reduction in problem (1.2) and to study the optimal design of less compliant thin plates (see [7,8,9,10]). The case when Q δ = δD × I, which is quite far from being merely a technical variant of the previous one, involves a 3d-1d dimension reduction process: the matter is now the optimization of thin elastic rods.…”