Parametric generation of optimal structures through discrete exponential functions: unveiling connections between structural optimality and discrete isothermicity

Abstract

This study discusses an equilibrium state of structures endowed with integrability and relates the structural optimality for Michell’s classic problem and the isothermicity in discrete differential geometry. This discussion leads to a new approach for the parametric generation of quasi-optimal layouts of bar members. The layout of bar members is determined by taking the diagonals of a quadrilateral mesh constructed from a discrete exponential function. The configuration of the planar layout can be changed by adjusting the parameters of a discrete exponential function. In addition, the inverse stereographic projection allows for obtaining spherical shapes from the planar layouts, and the Möbius transformations enable the generation of eccentric near-optimal shapes. It is also demonstrated that the structural layouts generated in this study are the exact optimal or near-optimal solution to Michell’s optimization problem.