Approximation of Young measures by functions and application to a problem of optimal design for plates with variable thickness

Abstract

Given a parametrised measure and a family of continuous functions (φn), we construct a sequence of functions (uk) such that, as k→∞, the functions φn(uk) converge to the corresponding moments of the measure,in the weak * topology. Using the sequence (uk) corresponding to a dense family of continuous functions, a proof of the fundamental theorem for Young measures is given.We apply these techniques to an optimal design problem for plates with variable thickness. The relaxation of the compliance functional involves three continuous functions of the thickness. We characterise a set of admissible generalised thicknesses, on which the relaxed functional attains its minimum.

Structural Optimization of Thin Elastic Plates: The Three Dimensional Approach

“…to add in our starting problem (1.6) the constraint that Ω takes the form Ω = {h − (x ′ ) < x 3 < h + (x ′ )}. This variant of the problem has some similarities with the one considered in [8,9,10]. However, the comparison with our approach discloses deep differences.…”

“…The search of optimal designs in this context has been the object of several studies. Without any attempt of being complete, we refer to [4,5,8,9,10,23,24,26,32]. In these works it is assumed that any section of the plate is a segment and that the thickness variations are smooth enough for the classical dimension reduction analysis to be applied.…”

A Review of an Optimal Design Problem for a Plate of Variable Thickness

“…Several later works emphasized this perspective and proved various types of results always trying to minimize in various ways the number of design variables needed to describe minimizing profiles. In many of these contributions, Young measures associated with minimizing profiles were used in one way or another (see [1], [3], [4], [11], [15]). Second, in some other situations, existence of optimal profiles has been shown despite the fact of the just-mentioned difficulties (see [14], [16], [17]), coming to a situation where it is not completely clear when, depending on the ingredients, one can trust existence results or else anticipate highly oscillating optimal profiles.…”

Characterization of Two-Scale Gradient Young Measures and Application to Homogenization

“…A key feature of these measures is their capacity to capture the oscillations of minimizing sequences of non convex variational problems, and many applications arise e.g. in models of elastic crystals (see Chipot & Kinderlehrer [16] and Fonseca [19]), phase transition (see Ball & James [8]), optimal design (see Bonnetier & Conca [11], Maestre & Pedregal [24] and Pedregal [32]). The special properties of Young measures generated by sequences of gradients of Sobolev functions have been studied by Kinderlehrer & Pedregal [21,22] and are relevant in the applications to nonlinear elasticity.…”