On the existence of optimal shapes in architecture

We consider shape optimization problems for elasticity systems in architecture. A typical objective in this context is to identify a structure of maximal stability that is close to an initially proposed one. For structures without external forces on varying parts, classical methods allow proving the existence of optimal shapes within well-known classes of bounded uniformly Lipschitz domains. We discuss this for maximally stable roof structures. We then introduce a more general framework that includes external forces on varying parts (for instance, caused by loads of snow on roofs) and prove the existence of optimal shapes, now in a subclass of bounded uniformly Lipschitz domains, endowed with generalized surface measures on their boundaries. These optimal shapes realize the infimum of the corresponding energy of the system. Generalizing further to yet another, very new framework, now involving classes of bounded uniform domains with fractal measures on their boundaries, we finally prove the existence of optimal architectural shapes that actually realize the minimum of the energy. As a by-product we establish the well-posedness of the elasticity system on such domains. In an auxiliary result we show the convergence of energy functionals along a sequence of suitably converging domains. This result is helpful for an efficient approximation of an optimal shape by shapes that can be constructed in practice. © 2021 Elsevier Inc.

Hinz M.1 , Magoulès F.2, 3 , Rozanova-Pierrat A.2, 4 , Rynkovskaya M. 5 , Teplyaev A.6
Elsevier Inc.
  • 1 Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, Bielefeld, 33501, Germany
  • 2 MICS, CentraleSupélec, Université Paris Saclay, 3 rue Joliot Curie, Gif-sur-Yvette, 91190, France
  • 3 Faculty of Engineering and Information Technology, University of Pécs, Pécs, Hungary
  • 4 MICS, Univérsité Paris-Saclay, Campus de Gif-sur-Yvette, Plateau de Moulon, 3 rue Joliot Curie, Gif-sur-Yvette, 91190, France
  • 5 Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation
  • 6 Department of Mathematics, University of Connecticut, Storrs, CT 06269-1009, United States
Elasticity system; Fractals; Mosco convergence; Shape optimization; Stable roof structures under snow loads; Trace and extension
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