A Minimaxmax Problem for Improving the Torsional Stability of Rectangular Plates

TL;DR

This paper introduces a novel minimaxmax shape optimization framework to enhance the torsional stability of rectangular plates by optimizing reinforcements and external forces, supported by theoretical analysis and numerical experiments.

Contribution

It develops a new worst-case optimization approach for torsional stability, considering reinforcement strategies and external force modifications, with existence results and numerical validation.

Findings

Existence of optimal reinforcements and external forces established.

Numerical experiments demonstrate the effectiveness of the proposed optimization.

Open problems and conjectures are proposed for future research.

Abstract

We use a gap function in order to compare the torsional performances of different reinforced plates under the action of external forces. Then, we address a shape optimization problem, whose target is to minimize the torsional displacements of the plate: this leads us to set up a minimaxmax problem, which includes a new kind of worst-case optimization. Two kinds of reinforcements are considered: one aims at strengthening the plate, the other aims at weakening the action of the external forces. For both of them, we study the existence of optima within suitable classes of external forces and reinforcements. Our results are complemented with numerical experiments and with a number of open problems and conjectures.