Generative design of funicular shells via physics-guided diffusion

Funicular shells carry loads predominantly through membrane action, achieving high structural efficiency with little material. Their computational design is difficult: the space of feasible forms is large, and classical optimisation explores it through local, problem-specific searches. Diffusion models offer an alternative by sampling from a learned distribution of plausible geometries, but coupling them with structural constraints is challenging, because physical guidance must act on noisy intermediate states whereas continuum-mechanical operators are defined on smooth geometries. Evaluating a non-linear shell operator on a denoised point estimate therefore yields a response that differs from the conditional mechanical response of the current diffusion state. We propose a modular two-branch Architect–Engineer pipeline. The Architect is a velocity-parameterised denoising diffusion model that learns an unconditional geometric prior over shell elevation maps. The Engineer is a time-conditioned Physics-Based Parallel UNet surrogate trained directly on noisy diffusion states, so that it predicts the mechanical response associated with the states visited during reverse sampling. The two models are trained independently and coupled only at inference, through a differentiable Membrane-Factor guidance pathway modulated by a bell-shaped schedule that concentrates the correction in the mid reverse process. On a dataset of funicular shells, the guided sampler raises the mean predicted Membrane Factor from 0.67 to 0.96, with 89% of samples above the 0.90 high-funicularity threshold, while preserving more geometric diversity than constant guidance. A guidance-strength sweep reveals a smooth transition between diversity-preserving and quality-prioritising regimes without architectural or training changes.