Design and computation of origami-inspired structures and metamaterials

About This Grant

Origami is the art of folding paper into intricate forms. Structures composed of origami patterns have been used for decades in the space industry as they are very compact when folded and can unfold into intricate shapes. More recently, Origami structure have been used to produce inexpensive mechanical metamaterials. Mechanical metamaterials are novel materials that present mechanical properties that are not common to usual materials. However, the design possibilities offered by origami structures remain presently mostly unexplored. This project will develop models and numerical methods to compute new origami patterns and study their deformation. The tools developed in this project will enable engineers to design new origami patterns with new properties and therefore create new metamaterials and foldable structures. Possible applications include designing structures that unfold into a target shape or designing micro-structures to obtain a desired macroscopic property. This project will contribute to the study of the direct and inverse problems of designing origami structures. In the direct problem, one chooses a given periodic folding pattern and derives Partial Differential Equations (PDEs) describing the kinematics and energy of the limit surface. One then wants to study and approximate the solutions of PDE constrained optimization problems where the PDEs are nonlinear and can change type (between elliptic and hyperbolic) and degenerate. This project will use careful regularizations and nonconforming finite element discretizations in order to approximate the solutions of these difficult problems. The inverse problem consists in determining a crease pattern that will allow to fold from a flat state into a given target surface. Determining if a given pattern is flat foldable is known to be NP-hard. This project proposes to represent possible fold lines by damage in an elastic sheet and then to adapt the method of Ambrosio and Tortorelli to approximate minimizers of the Mumford--Shah functional. This will produce folding patterns on an initially flat surface which will be able to fold into the target surface. As paper deforms isometrically, this project intends to explore the approximation of nonzero Gauss curvature target surfaces to determine if notable properties emerge. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.