Interactive exploration of tension-compression mixed shells

Achieving a pure-compression stress state is considered central to the form-finding of shell structures. However, the pure-compression assumption restricts the geometry of the structure's plan in that any free boundary edges cannot bulge outward. Allowing both tension and compression is essential so that overhanging leaves can stretch out toward the sky. When performing tension-compression mixed form-finding, a problem with boundary condition (BC) compatibility arises. Since the form-finding equation is hyperbolic, boundary information propagates along the asymptotic lines of the stress function. If conflicting BC data is prescribed at either end of an asymptotic line, the problem becomes ill-posed. This requires a user of a form-finding method to know the solution in advance. By contrast, pure-tension or purecompression problems are elliptic and always give solutions under any BCs sufficient to restrain rigid motion. To solve the form-finding problem for tension-compression mixed shells, we focus on the Airy's stress function, which describes the stress field in a shell. Rather than taking the stress function as given, we instead treat both the stress function and the shell as unknowns. This doubles the solution variables, turning the problem to one that has an infinity of different solutions. By enforcing equilibrium in the shell interior and prescribing the correct matching pairs of BCs to both the stress function and the shell, a stress function and shell can be simultaneously found such that equilibrium is satisfied everywhere in the shell interior and thus automatically has compatible BCs by construction. The problem of a potentially over-constrained form-finding is thus avoided by expanding the solution space and creating an under-determined problem. By varying inputs and repeatedly searching for stress function-shell pairs that fall within the solution space, a user is allowed to interactively explore the possible forms of tension-compression mixed shells under the given plan of the shell.