On the existence of optimal shapes in architecture

Michael Hinz (1) , Frédéric Magoulès (2, 3) , Anna Rozanova-Pierrat (2) , Marina Rynkovskaya (4) , Alexander Teplyaev (5)

1 Universität Bielefeld = Bielefeld University
2 MICS - Mathématiques et Informatique pour la Complexité et les Systèmes
3 University of Pécs
4 RUDN - Peoples Friendship University of Russia [RUDN University]
5 UCONN - University of Connecticut

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.

Mots clés: Elasticity system; shape optimization; trace and extension; fractals; Mosco convergence

Domaines: Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]; Physique mathématique [math-ph]; Optimisation et contrôle [math.OC]

Michael Hinz, Frédéric Magoulès, Anna Rozanova-Pierrat, Marina Rynkovskaya, Alexander Teplyaev. On the existence of optimal shapes in architecture. 2020. ⟨hal-02956458v1⟩