Computing irregular hypar-based quad-mesh patterns for segmented timber shells

Hyperbolic paraboloids, “hypars,” are special types of ruled surfaces. Their geometric properties provide them with loadbearing and stabilizing capacities, as well as distinct esthetic qualities. These attributes become evident in numerous applications in buildings, in many of which concrete or timber is used for the construction of the hypars. Hypars could also be relevant in the context of circular construction and design for disassembly, and the upcycling of construction waste. Due to the geometric simplicity of straight lines, which generate ruled surfaces, hypar-based structures can be designed and built with relatively simple means. They can consist of self-similar or even identical elements, which could facilitate their reuse.

Compared to other types of ruled surfaces, such as conoids, hypars have the advantage of being doubly ruled, meaning that structural grids of straight elements can be formed. This paper investigates another interesting property, which is the possibility of creating flat-quad meshes by diagonally connecting the intersection points of the generatrices. This property has been previously described by other scholars, some of which explored its applicability for glass-clad steel grid shells. In this research, we focus on its potential for segmented timber shells that can serve as stand-alone structures, or as modular and reusable building parts, such as façade or roof components. The reusability of such modular units could be achieved by using reversible joints between them.

More specifically, our research investigates the design space of construction systems based on such components via computational design and optimization algorithms, such as the memory limited Broyden–Fletcher–Goldfarb–Shanno (LBFGS) algorithm with automatic computation of the gradient, within the Julia programming environment. By applying principles and methods of differential geometry, we study hypars with irregular tilings, enabling the integration of panels with diverse proportions, shapes and sizes, as they can occur in wood production waste. By reducing construction waste, the work aims at reducing the negative environmental impact of the building construction sector. Moreover, irregular tilings could enable a more customized design of acoustic qualities and offer visual variety in segmented hypar based timber structures.

The here presented studies show that the proposed optimization method provides a good fit of many tiles to rhombi, particularly when the steepness is not too large. We also show that optimizing towards rectangles provides better results. Overall, the results support the initial assumption that irregular rulings could be a means of adapting to both homogeneous and diverse material stocks.