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Project KingOfMonsters is a project where we develop a series of NURBS-based form-finding and structural analysis tools that run on Rhinoceros and Grasshopper.

It is known that the form-finding problem is very likely NOT to give any solutions when there are tension-compression mixed zones. This project is motivated to aid this situation, and we have developed a stress-function - shell coupled solver that finds a correct pair of stress function and a shell that produces a well-posed form-finding problem even if there are tension-compression mixed zones.

Our paper about the tension-compression mixed form-finding method has been accepted for presentation at Siggraph Asia 2022 (journal track).

Interactive Exploration of Tension-compression Mixed Shells

MASAAKI MIKI, The University of Tokyo, Japan

TOBY MITCHELL, Skidmore, Owings and Merrill, USA

**Figure 1** Solution pairs of stress function and shell for problems presented in Section \ref{sec:computationalresults} obtained with the proposed method. For each problem, two solution pairs are obtained by placing the user-defined point handles at different heights. Pink: Airy's stress functions with negative Gaussian curvature zones; green: tension-compression mixed shells. The proposed method refines both the stress function and the shell simultaneously until a solution pair is finally found. Metallic blue spheres represent user-defined point handles through which the shapes of the shells can be manipulated with user operation. Illuminated edges labeled $\flat, \dagger, \sharp$, and $\S$ represent four types of boundary conditions especially configured for the stress function and the shell.

**Abstract**

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 pure-compression 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.

paper (low-res) [movie (YouTube)]

The high-res version should be available at ACM Digital Library soon.

The tool, named Godzilla and runs on Rhinoceros and Grasshopper, can be downloaded at [Godzilla(Food4Rhino)]