Some obstacle problems for partially hinged plates and related optimization issues

TL;DR

This paper investigates optimization problems for partially hinged plates, like bridge decks, considering real and artificial obstacles to improve stability and prevent collisions, with theoretical existence results and qualitative analysis.

Contribution

It introduces novel optimization frameworks for obstacle placement and density distribution in partially hinged plates, addressing stability and collision avoidance.

Findings

Existence of optimal solutions for both obstacle and density problems

Qualitative properties of optimal distributions analyzed

Framework applicable to bridge stability enhancement

Abstract

We study optimization problems for partially hinged rectangular plates, modeling bridge roadways, in the presence of real and artificial obstacles. Real obstacles represent structural constraints to avoid, while artificial ones are introduced to enhance stability. For the former, aiming to prevent collisions, we set up a worst-case optimization problem in which we minimize the amplitude of oscillations with respect to the density distribution; for the latter, aiming to improve the torsional stability, we minimize, with respect to the obstacles, the maximum of a gap function quantifying the displacement between the long edges of the plate. For both problems, existence results are provided, along with a discussion about qualitative properties of optimal density distributions and obstacles.