Optimal design of a plate of variable thickness: a variational approach in dimension one

Abstract

Abstract. For a typical design problem of a plate of variable thickness, we analyze the onedimensional situation through a variational reformulation to discover that, in contrast with the higher dimensional case, there are optimal solutions. Another typical interpretation of this simplification is that of the optimal shape of a bending beam. The mechanism employed for the existence issue is the direct method for the new formulation. Optimality conditions are then pursued.Mathematical subject classification: 49J15, 49J45, 49K15, 74K20, 74P05.

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

“…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. Another point in many of these works is that the only cost functional considered is the compliance written before, along with some other variants of order zero (no derivatives of u).…”