Differentiable Flexible Mechanical Metamaterials

🚀 The goal of this framework is to push the boundary of designable dynamic behaviors achievable by nonlinear mechanical metamaterials.

🤖 Through the power of differentiable simulations, the framework removes the typical limitations of metamaterial design e.g. periodicity, intuition-based design, manual tuning, single-task design, etc., and enables the automated discovery of non-periodic, multi-functional, and reprogrammable metamaterials in the nonlinear dynamic regime.

This repository provides a fully differentiable dynamic design framework for 2D flexible mechanical metamaterials i.e. a network of rigid units connected by flexible ligaments. The main physical ingredients of the model are:

• 🥌 Rigid-body kinematics of the units.
• 🎈 Elastic ligaments modeled by customizable energy functions.
• 💥 Energy-based contact interactions between rigid units.

With these ingredients, flexible mechanical metamaterials define a rich space of nonlinear dynamic behaviors that can be navigated by the framework.

🔭 From a high-level perspective, the framework facilitates the construction of the mapping between design parameters and the desired behavior of the metamaterial system.

By leverging JAX, this complex mapping is implemented in a differentiable fashion, thus allowing gradients to flow through the entire dynamic simulation. In particular, differentiability is provided with respect to:

• Geometric paramaters: arbitrary parametrizations can be defined in the geometry module.
• Ligament paramaters: energy functions can be defined in the energy module.
• Damping parameters: linear viscous damping as defined in the loading module.
• Driving parameters: arbitrary driving functions can be applied to any degree of freedom.
• Loading parameters: arbitrary loading functions can be applied to any degree of freedom.
• Initial conditions: initial positions and velocities of the system.
• and any other paramater present in the ControlParams data structure.

The main entry point of the simulator is the setup_dynamic_solver(...) in the dynamics module. This function takes all the fixed mappings (geometry, energy, loading, etc.) and returns a differentiable function that simulates the metamaterial dynamics. Arbitrary forward problems can be defined by chaining this simulator function with any desired objective function.

This repository contains all the code developed for the paper:

G. Bordiga, E. Medina, S. Jafarzadeh, C. Boesch, R. P. Adams, V. Tournat, K. Bertoldi. Automated discovery of reprogrammable nonlinear dynamic metamaterials. Nature Materials. (2024).

The framework has been used to design a variety of mechanical metamaterials with different functionalities.

💾 All data generated or used for the paper can be downloaded from . To access and visualize the data:

• Extract data.zip in the root directory of the repository.
• Load and visualize optimization results for all the solved problems using the provided notebooks.
• Load and visualize experimental data using the notebooks in the exp folder.

Assuming you have access to the repo and ssh keys are set up in your GitHub account, you can install the package with

pip install git ssh://[email protected]/bertoldi-collab/DifFlexMM.git

Clone the repository, cd into the DifFlexMM folder, and install with

pip install -e .

If you use this code in your research or anywhere, please cite the paper:

@article{bordiga_2024,
    title   = {Automated Discovery of Reprogrammable Nonlinear Dynamic Metamaterials},
    author  = {Bordiga, Giovanni and Medina, Eder and Jafarzadeh, Sina and B{\"o}sch, Cyrill and Adams, Ryan P. and Tournat, Vincent and Bertoldi, Katia},
    year    = {2024},
    journal = {Nature Materials},
    volume  = {23},
    number  = {11},
    pages   = {1486--1494},
    issn    = {1476-4660},
    doi     = {10.1038/s41563-024-02008-6},
}