Differentiable force density method for the design of lightweight structures

Lightweight roofs, bridges, and towers span long distances with slender cross-sections due to their mechanically efficient shapes. Funicular geometry, spatial configurations where external loads are carried primarily through internal axial forces, can endow these structures with such efficiency. However, converting funicular geometry into a feasible design that satisfies architectural, fabrication, and structural constraints remains challenging due to nonlinear mechanical behaviors and high-dimensional search spaces. This paper presents the differentiable force density method (DFDM), a form-finding method for the automated design of efficient and feasible shapes for lightweight structures modeled as pin-jointed bar systems. The DFDM leverages gradient-based optimization to compute funicular geometries subject to design constraints, while accounting for large-displacement effects and shape-dependent loads, such as self-weight and pressure. By extending automatic differentiation with analytical adjoints, the DFDM solves high-dimensional inverse problems to planarize gridshell panels, simulate pneumatic membranes, and retrofit masonry vaults up to four orders of magnitude faster than gradient-free algorithms and one order of magnitude faster than general-purpose automatic differentiation. Its full differentiability enables load-finding and support-finding. These rich parametrizations allow the optimization of loads and support positions jointly with force densities to discover better-performing solutions than with force density optimization alone. This work unifies structural simulations and differentiable programming, opening up pathways to expedite the development of meter-scale systems that lower material consumption in the built environment.