Programme

This presentation reviews recent advances in the form-finding of tension-compression mixed shells. Although purely compressive stress states are traditionally considered ideal for shell structures, I propose that allowing a mix of tension and compression can expand the range of feasible shell geometries. The key challenge lies in the fact that the equilibrium problem becomes a hyperbolic boundary value problem, which is notoriously difficult to solve. I point out that the introduction of Airy’s stress function reveals that the equilibrium equation is a bilinear partial differential equation (PDE). I then indicate that this PDE can be solved using the Variable Projection (VarPro) method—developed specifically for bilinear problems. I also demonstrate that the alignment of stress and curvature directions is governed by a bilinear PDE, which can be solved concurrently with the equilibrium equation using the VarPro method.

Compliant shell mechanisms are creased and corrugated thin-walled structures that can drastically change shape to move, deploy, or adapt to a changing environment. They have found use cases in the context of recent space programs and in other domains ranging from biomedical technology to architecture. Not unlike slender beams, thin shells prefer bending over stretching. Ideally, thin shells deform isometrically should isometric deformations exist. The problem of finding, or disproving the existence of, isometric deformations for various surfaces preoccupied many mathematicians and mechanicians. The most noteworthy results undoubtedly pertain to three broad categories of surfaces: developable surfaces, convex surfaces, and axisymmetric surfaces. In the modern context of computer graphics, discrete differential geometry and “Origami science,” more focus has been directed towards tri- and quad-based polyhedral surfaces. In this lecture, we report on recent results that characterize the isometric deformations of periodic surfaces, be them smooth or piecewise smooth with straight or curved creases.

In this talk I will discuss computational design strategies for different classes of deployable gridshells. I will show how geometric analysis, physics-based simulation, and numerical optimization can be combined to create effective design tools for gridshells with straight or curved flexible beams, as well as for segmented gridshells that are assembled using weaving techniques. In several design studies, large-scale demonstrators, and commissioned installations I will highlight the unique benefits of deployable gridshells, but also indicate challenges that motivate future research.