On the existence of optimal shapes in architecture

TL;DR

This paper proves the existence of optimal shapes for elastic structures in architecture, including complex fractal boundaries, to maximize stability, with applications to snow load-resistant roofs.

Contribution

It establishes the existence of optimal shapes within classes of Lipschitz and fractal domains for elasticity problems, extending previous results to more general boundary classes.

Findings

Existence of optimal shapes in Lipschitz domains.

Existence of optimal shapes in fractal boundary domains.

Application to designing snow-resistant roof structures.

Abstract

We consider shape optimization problems for elasticity systems in architecture. A typical question in this context is to identify a structure of maximal stability close to an initially proposed one. We show the existence of such an optimally shaped structure within classes of bounded Lipschitz domains and within wider classes of bounded uniform domains with boundaries that may be fractal. In the first case the optimal shape realizes the infimum of a given energy functional over the class, in the second case it realizes the minimum. As a concrete application we discuss the existence of maximally stable roof structures under snow loads.