JAX FDM

Auto-differentiable and hardware accelerated force density method.

Crafted with care in the AI Lab at Princeton University ❤️🇺🇸

Lightweight structures span long distances with slender cross-sections due to their mechanically efficient shapes. However, simulating these structures and turning them into feasible designs that satisfy additional technical constraints remains challenging. This is because of the geometrically nonlinear mechanical behaviors of these structure and the high-dimensional search spaces that describe them.

JAX FDM solves inverse design problems for lightweight structures modeled as pin-jointed bar systems using the force density method (FDM) and gradient-based optimization. It streamlines the integration of mechanical simulations into deep learning models for machine learning research.

The full documentation lives at arpastrana.github.io/jax_fdm.

Features

• Legendary form-finding solver. JAX FDM computes static equilibrium states for pin-jointed bar systems with the force density method (FDM), the time-tested solver for geometrically nonlinear systems backed up by over 50 years of peer-reviewed research 📚.

• Derivatives, JIT compilation, and parallelization. JAX FDM is written in JAX, a library for high-performance numerical computing and machine learning research, and it thus inherits many of JAX's perks: calculate derivatives, parallelize, and just-in-time (JIT) compile entire structural simulations written in Python code, and run them on a CPU, a GPU, or a TPU 🤯.
• Autotune those force densities, loads, and supports. A lightweight structure should fulfill additional technical requirements to become a feasible system for real-world construction. This requires finding the parameters that lead to a specific constrained equilibrium state satisfying these conditions. Formulate such an inverse problem with JAX FDM, and let one of its gradient-based optimizers solve it by automatically tweaking the system's force densities, applied loads, and support positions 🕺🏻.
• A rich bank of goals, constraints, and loss functions. No two structures are alike. JAX FDM allows you to model a custom design task with its (growing!) collection of goals, constraints, and loss functions via a simple, object-oriented API. The available goals and constraints in the framework are granular and applicable to an entire structure; to a subset of its nodes (i.e., vertices), edges, and combinations thereof 💡.

• Structural simulations as another layer in a neural network. As an auto-differentiable library, JAX FDM can be seamlessly added as a layer in a differentiable function approximator like a neural network that can be then trained end-to-end. Let the neural network learn the underlying physics of static equilibrium directly from the simulation, instead of resorting to laborious techniques like data augmentation 🤖.

JAX FDM is a research project under development. Expect sharp edges and possibly some API breaking changes as we continue to support a broader set of features.

Installation

Install JAX FDM with a one-liner via pip:

pip install jax-fdm

This pulls in COMPAS 2.x and the other core dependencies automatically. JAX FDM supports Python 3.11 to 3.12, and builds on JAX, SciPy, Equinox, and the COMPAS framework. For the optional extras (3D and notebook viewers, a 2D plotter, the IPOPT optimizer, development tools) and platform notes for Windows, see the installation guide.

Quick example

Suppose you are interested in generating a form in static equilibrium for a 5-meter span arch subjected to vertical point loads of 0.3 kN. The arch has to be a compression-only structure. You model the arch as a jax_fdm network built from scratch: a straight line of nodes evenly spaced along the span, joined edge to edge. Then, you apply a force density of -1 to all of its edges, and compute the required shape with the force density method.

from jax_fdm.datastructures import FDNetwork
from jax_fdm.equilibrium import fdm


length = 5.0
num_segments = 10
segment_length = length / num_segments

xs = [-length / 2.0 + i * segment_length for i in range(num_segments + 1)]
nodes = [[x, 0.0, 0.0] for x in xs]
edges = [(i, i + 1) for i in range(num_segments)]

network = FDNetwork.from_nodes_and_edges(nodes, edges)
network.edges_forcedensities(q=-1.0)
network.nodes_supports(keys=[node for node in network.nodes() if network.is_leaf(node)])
network.nodes_loads([0.0, 0.0, -0.3])

f_network = fdm(network)

Continue this example (adding constraints, optimizing the form, and visualizing the result) in the docs, which also collects runnable Colab notebooks and more advanced example scripts.

Citation

If you found this library to be useful in academic or industry work, please consider (1) starring the project on Github, and (2) citing it:

@article{pastrana_dfdm_2026,
         title = {Differentiable force density method for the design of lightweight structures},
         author = {Pastrana, Rafael and Oktay, Deniz and Bletzinger, Kai-Uwe and Adams, Ryan P. and Adriaenssens, Sigrid},
         date = {2026},
         journaltitle = {Computer Methods in Applied Mechanics and Engineering},
         volume = {458},
         pages = {118783},
         issn = {00457825},
         doi = {10.1016/j.cma.2026.118783}}

@inproceedings{pastrana_jaxfdm_2023,
               title = {{{JAX FDM}}: {{A}} differentiable solver for inverse form-Finding},
               booktitle = {Differentiable {{Almost Everything Workshop}} of the 40th {{International Conference}} on {{Machine Learning}}},
               author = {Pastrana, Rafael and Oktay, Deniz and Adams, Ryan P. and Adriaenssens, Sigrid},
               year = {2023},
               address = {Hawaii, USA},
               url = {https://openreview.net/forum?id=Uu9OPgh24d}}

Acknowledgements

This work has been supported by the U.S. National Science Foundation under grant OAC-2118201 and the Institute for Data Driven Dynamical Design.

See also

COMPAS CEM: Inverse design of 3D trusses with the combinatorial equilibrium modeling (CEM) framework.

JAX CEM: The combinatorial equilibrium modeling (CEM) framework in JAX.

JAX: Composable transformations of Python+NumPy programs.

License

MIT