Real-time design of architectural structures with differentiable mechanics and neural networks

TL;DR

The paper addresses real-time, physics-consistent inverse design for long-span architectural structures modeled as pin-jointed bar systems. It introduces a physics-in-loop surrogate that couples a neural encoder with a differentiable force-density method (FDM) simulator to map target geometries to bar stiffnesses $\mathbf{q}$ and recover $\mathbf{X}$ under equilibrium, guaranteeing mechanical integrity. The approach emphasizes an encoder that outputs positive stiffness with a force-sign bias, and a differentiable FDM-based decoder that enforces $\mathbf{K}(\mathbf{q})\mathbf{X}-\mathbf{P}(\mathbf{X})=\mathbf{0}$, enabling end-to-end training to minimize the geometry mismatch while maintaining physics. Evaluations on masonry shells and cable-net towers show substantial speedups over direct optimization with comparable accuracy, and superior generalization to unseen geometries; integration into CAD software and fabrication of a prototype demonstrate practical impact. Overall, the framework enables safe, interactive design exploration of complex architectural geometries with real-time feedback and physics guarantees.

Abstract

Designing mechanically efficient geometry for architectural structures like shells, towers, and bridges, is an expensive iterative process. Existing techniques for solving such inverse problems rely on traditional optimization methods, which are slow and computationally expensive, limiting iteration speed and design exploration. Neural networks would seem to offer a solution via data-driven amortized optimization, but they often require extensive fine-tuning and cannot ensure that important design criteria, such as mechanical integrity, are met. In this work, we combine neural networks with a differentiable mechanics simulator to develop a model that accelerates the solution of shape approximation problems for architectural structures represented as bar systems. This model explicitly guarantees compliance with mechanical constraints while generating designs that closely match target geometries. We validate our approach in two tasks, the design of masonry shells and cable-net towers. Our model achieves better accuracy and generalization than fully neural alternatives, and comparable accuracy to direct optimization but in real time, enabling fast and reliable design exploration. We further demonstrate its advantages by integrating it into 3D modeling software and fabricating a physical prototype. Our work opens up new opportunities for accelerated mechanical design enhanced by neural networks for the built environment.

Figures (17)

• Figure 1: Architecture of our model to amortize the generation of mechanically efficient geometry. A neural network first maps a target shape $\hat{\mathbf{X}}$ sampled from a family of shapes $\hat{\mathcal{X}}$ to a stiffness space $\mathbf{q}$. The stiffnesses, in tandem with the boundary conditions $\mathbf{b}$, are then decoded by a mechanical simulator into a physics-constrained shape $\mathbf{X}$ that approximates the target. The predicted shape can be used as the base geometry to design efficient structures like gridshells or masonry shells.
• Figure 2: A bar system. In the callout, the stiffness component $[\mathbf{K}(\mathbf{q})]_{ij}$ is indicated at a bar connecting nodes with positions $\mathbf{x}_{i}$ and $\mathbf{x}_{j}$. A load $\mathbf{p}_i$ is applied at $\mathbf{x}_{i}$. In this system, the nodes on the perimeter are fixed.
• Figure 3: Shape matching. To generate a mechanically efficient shape $\mathbf{X}(\mathbf{q})$ that approximates an arbitrary target $\hat{\mathbf{X}}$, traditional methods like direct optimization find bar stiffnesses $\mathbf{q}$ that minimize the shape loss $\mathcal{L}_{\text{shape}}$, but only after several iterations $t$. Our model amortizes this computationally taxing process during inference while guaranteeing physics, enabling real-time and sound design.
• Figure 4: Loss curves of the shell design task. (a) Our model learns a meaningful representation that minimizes the shape loss $\mathcal{L}_{\text{shape}}$ while fully satisfying the mechanics of compression-only shells, as $\mathcal{L}_{\text{physics}}$ is zero within numerical precision throughout training. (b) Shape loss of our model and the PINN baseline on test data interpolated between doubly-symmetric ($\delta=0$) and asymmetric ($\delta=1$) geometries. Our model's accuracy decays at a lower rate than the PINN's accuracy.
• Figure 5: Shape matching for shell design. While the NN and PINN models approximate the targets, they cannot suppress the residual forces (pink arrows). The stiffnesses $\mathbf{q}$ and the shapes predicted by our model are similar to direct optimization's, indicating our model learns a good task representation.