Origami folds have found a large range of applications in Engineering as, for instance, solar panels for satellites, or the folding of airbags for optimal deployment or metamaterials. A homogenization process turning origami folds into smooth surfaces, developed in [Nassar et al, 2017], is first discussed. Then, its application to two specific folds is presented alongside the PDEs characterizing the associated smooth surfaces. The talk will then focus on the PDEs describing Miura surfaces by studying existence and uniqueness of solutions and by proposing a numerical method to approximate them. Finally, some numerical examples are presented.

The unfolding of a ladybird's wings, the trapping mechanism used by a flytrap, the design of self-deployable space shades, and the constructions of curved origami are diverse examples where strategically placed material defects are leveraged to generate large and robust deformations. With these applications in mind, we derive plate models incorporating the possibly of curved folds as the limit of thin three-dimensional hyper-elastic materials with defects. This results in a fourth order geometric partial differential equation for the plate deformations further restricted to be isometries. The latter nonconvex constraint encodes the plates inability to undergo shear nor stretch and is critical to justify large deformations. We explore the rigidity of the folding process by taking advantage of the natural moving frames induced by piecewise isometries along the creases. We then deduce relations between the crease geodesic curvature, normal curvature, torsion, and folding angle. Regarding the numerical approximation, we propose a locally discontinuous Galerkin method. The second order derivatives present in the energy are replaced by weakly converging discrete reconstructions. Furthermore, the isometry constraint is linearized and incorporated within a gradient flow. We show that the sequence of resulting equilibrium deformations converges to a minimizer of the exact energy (and, in particular, to an isometry) as the discretization parameters tend to infinity. This theory does not require additional smoothness on the plate deformations besides having a finite energy. The capabilities and efficiency of the proposed algorithm is documented throughout the presentation by illustrating the behavior of the model on relevant examples.