Optimal complexity of deployable compressive structures

The usual use of fractals involves self-similar geometrical objects to fill a space, where the self-similar iterations may continue ad infinitum. This is the first paper to propose the use of self-similar mechanical objects that fill an alloted space, while achieving an invariance property as the self-similar iterations continue (e.g. invariant strength). Moreover, for compressive loads, this paper shows how to achieve minimal mass and invariant strength from self-similar structures. The topology optimization procedure uses self-similar iteration until minimal mass is achieved, and this problem is completely solved, with global optimal solutions given in closed form. The optimal topology remains independent of the magnitude of the load. Mass is minimized subject to yield and/or buckling constraints. Formulas are also given to optimize the complexity of the structure, and the optimal complexity turns out to be finite. That is, a continuum is never the optimal structural for a compressive load under any constraints on the physical dimension (diameter). After each additional self-similar iteration, the number of bars and strings increase, but, for a certain choice of unit topology shown, the total mass of bars and strings decreases. For certain structures, the string mass monotonically increases with iteration, while the bar mass monotonically reduces, leading to minimal total mass in a finite number of iterations, and hence a finite optimal complexity for the structure. The number of iterations required to achieve minimal mass is given explicitly in closed form by a formula relating the chosen unit geometry and the material properties. It runs out that the optimal structures produced by our theory fall in the category of structures we call tensegrity. Hence our self-similar algorithms can generate tensegrity fractals.