Speaker
Description
It is well known that the compliance minimization of trusses is an ill-posed problem. Even for the simplest loads one is forced to keep adding joints and bars of decreasing cross sections to assure that the objective function decreases. In the limit, we find ourselves in the realm of Michell structures, which combine continuum medium with bars, usually curved. In the bending regime, the analogue problem is the one of finding an optimal grillage. So far, in the literature, the design of grillages has been addressed by adopting the methods established for Michell trusses, both from the theoretical and numerical standpoint. In this talk I will present an alternative perspective on the grillage optimization problem. Through a novel Rubinstein-Kantorovich-type duality, an optimal transport reformulation can be found. New theoretical insights follow. We prove that, when only point loads are applied, a classical optimal grillage (made up from finite number of beams) does exist. For more general loading, including uniform pressure, we obtain structure theorems for the optimal trajectories of bending moments: they concentrate on an infinite graph; in particular, there is no need for curved beams. The new perspective also unlocks the optimal transport numerical methods. They are not only much more efficient, but they also allow to exactly identify the position of the extra joints, without the need for discretizing the design space.
This talk will discuss the results of the joint paper with Guy Bouchitté, see \linebreak https://arxiv.org/abs/2412.00516 .
