Material-minimizing forms and structures

Three-dimensional structures in building construction and architecture are realized with conflicting goals in mind: engineering considerations and financial constraints easily are at odds with creative aims. It would therefore be very beneficial if optimization and side conditions involving statics and geometry could play a role already in early stages of design, and could be incorporated in design tools in an unobtrusive and interactive way. This paper, which is concerned with a prominent class of structures, is a substantial step towards this goal. We combine the classical work of Maxwell, Michell, and Airy with differential-geometric considerations and obtain a geometric understanding of "optimality" of surface-like lightweight structures. It turns out that total absolute curvature plays an important role. We enable the modeling of structures of minimal weight which in addition have properties relevant for building construction and design, like planar panels, dominance of axial forces over bending, and geometric alignment constraints.