2021/04/14 by Bołbotowski, Karol
#74P05 #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2104.07148
This work puts forward a form finding problem of designing a least-volume vault that is a surface structure spanning over a plane region, which via pure compression transfers a vertically tracking load to the supporting boundary. Through a duality scheme, developed recently for the design of pre-stressed membranes, the optimal vault problem is reduced to a pair of mutually dual convex problems (P), (P^*) formulated on the 2D reference region. The vault constructed upon solutions of those problems is proved to be both of minimum volume and minimum compliance; analytical examples of optimal vaults are given. Through a measure-theoretic approach, thus found optimal vaults are proved to solve the Prager problem of designing a 3D structure that by compression carries a transmissible load. The ground structure method applied to the convex problems furnishes a pair of discrete, conic quadratic programs (PX), (PX^*) leading to optimal design of grid-shells. By adopting the member-adding adaptive technique this pair is efficiently tackled numerically, which is demonstrated on a number of examples where highly precise grid-shell approximations of optimal vaults are found.