A Unified Representation of Folded Surfaces via Fourier Series

Abstract

Folded surfaces find application in many technical fields, as they offer a variety of solutions to engineering design problems at different scales. An important realm of application is represented by structural design, where folded surfaces permit to achieve stiff yet lightweight shell structures. In despite of the great variety of fold patterns and their suitability as mechanical structures, the lack of a generalized geometrical representation is a limiting shortcoming for their efficient modelling and simulation. This paper bridges this gap introducing a continuous representation of folded surfaces, which is able to reduce the multiplicity of possible fold patterns to a unified mathematical description. This is achieved by using 2-dimensional Fourier series constructed on a Bravais Lattice. Within this framework, a typical design task like geometry manipulation turns out to be straightforward. At the same time, the continuous mathematical formulation applied to folded surfaces enables the direct derivation of differential geometry entities, which facilitates in turn the mechanical modelling of shell structures. In this study, we demonstrate the validity of the proposed approach by the use of two case studies. The first one achieves a consistent representation of an existing doubly-curved shell structure. In the second, we deliver an analytical solution for bending of a trapezoidal sheet.