Alignment conditions for NURBS-based design of mixed tension-compression grid shells

In architecture, shapes of surfaces that can withstand gravity with no bending action are considered ideal for shell structures. Those shells have special geometries through which they can stream gravitational force toward the ground via stresses strictly tangent to the surface, making them highly efficient. The process of finding these special forms is called form-finding. Recently, [Miki and Mitchell 2022] presented a method to reliably produce mixed tension-compression continuum shells, a type of shells known to be especially difficult to form-find. The key to this method was to use the concept of the Airy stress function to derive a valid bending-free shell shape by iterating on both the shell shape and the Airy stress function; this turns a problem that is over-constrained in general into a problem with many solutions. In [Miki and Mitchell 2022], it was proposed that the method could also be used to design grid shells by tracing curves on a continuum shell such that the resulting grid has bars that are both bending-free and form flat panels, a property useful for construction of real grid shells made of glass and steel. However, this special type of grid is not guaranteed to exist in general on a mixed-tension compression shell, even when the shell is in bending-free equilibrium [Miki and Mitchell 2023]. Additional conditions must be imposed on the shell shape to guarantee the existence of simultaneously bending-free and conjugate grid directions. The current study resolves the existence issue by adding alignment conditions. We consider several practical curve alignment conditions: alignment with the lines of curvature of the shell, approximate alignment with a bidirectional set of user-prescribed guide curves, and exact alignment with a single direction of user-prescribed guide curves. We report that the variable projection method originally used to solve the form-finding problem in the work of [Miki and Mitchell 2022] can be successfully extended to solve the newly introduced alignment conditions, and conclude with results for several practical design examples. To our knowledge, this is the first method that can take a user-input grid and find a "nearby" grid that is both flat-panelled and in bending-free equilibrium for the general case of mixed tension-compression grid shells.