Journal Club for May 2024: Inverse Design of Mechanical Metamaterials with Nonlinear Properties

Inverse Design of Mechanical Metamaterials with Nonlinear Properties

Bolei Deng, Georgia Institue of Technology

1.   Introduction

Mechanical metamaterials have recently emerged as platforms for engineering systems whose mechanical behaviors are governed by geometry rather than composition [1-4]. While initial efforts have concentrated on metamaterials with negative properties in the linear regime [5-8], recent advancements have demonstrated that nonlinear responses can be achieved. These responses, often accompanied by large internal rotations, are triggered by integrating slender elements prone to elastic instabilities into the designs [9]. Such nonlinear behaviors not only exhibit rich physics but also enable advanced functionalities like shape morphing [10-11], energy absorption [12-15], and programmability [16]. Despite the ability to tune these functionalities by altering the geometry, identifying architectures that achieve a specific nonlinear response remains a complex challenge.

2.   Challenges in Inverse Design of Nonlinear Metamaterials

The inverse design of nonlinear mechanical metamaterials faces two major challenges:

     1) Nonlinear simulations are time-consuming, and the inverse design process requires a large number of these simulations.

     2) The mapping from geometry to property is highly rugged, filled with local minima, which makes it difficult for traditional gradient-based methods to find the global optimum.

3.   Current solutions

There are many approaches to address these challenges, and here I will briefly introduce two AI-based strategies that I am familiar with. The field is still in its early stages, which is why I believe it worth further discussion.

3.1   Neural network as surrogate models

Recently, we proposed a rather brutal force solution to these challenges in order to inversely design nonlinear properties of mechanical metamaterials [17]. We focus on the static response of rotating unit systems—an iconic category of mechanical metamaterials—under large deformations. By adjusting the vertices of polygons, we can tune the shape of these units, which in turn leads to varied nonlinear stress-strain responses. Our objective is to determine whether we can identify the meso-structure that produces a user specified stress-strain curve.

A simple initial idea to tackle the first challenge—namely, the lengthy simulation times—is to employ a neural network as a surrogate model. This neural network is quite effective, boasting an accuracy rate of less than 5% error and offering substantial speed improvements. Now that we have this convenient, fast tool for forward evaluation, let's take a moment to visualize the mapping from metamaterial geometry to stress-strain curve in the figure below. It’s easy to see that this mapping is quite jagged, filled with numerous local minima. Traditional gradient-based methods would easily get lost in this challenging landscape.

To navigate through this complex space, we use an evolution strategy, an optimization algorithm inspired by natural selection. This strategy is immune to local minima due to its heuristic nature, making it especially suitable for this task. The main drawback of the evolution strategy is its requirement for a massive number of forward evaluations. For instance, each round of our evolutionary iterations involves assessing the stress-strain curves of 10,000 new metamaterial designs, which would typically take our simulator about a month to process. Fortunately, we have a neural network acting as a surrogate model that can perform the same evaluations in just 0.2 seconds. Clearly, the neural network and evolution strategy are a powerful combination in the inverse design of nonlinear properties.

This algorithm that combines the evolution strategy with the neural network successfully designs metamaterials with specific stress-strain curves, as shown in the figure below. These target curves are markedly different from those in our initial dataset. They include a linear stiff response that is theoretically the stiffest the system can achieve, a soft response that is 20% softer than the softest in our dataset, a non-monotonic curve, and a stress-strain curve with a sudden drop. Interestingly, this straightforward algorithm manages to identify metamaterials that produce stress-strain curves very close to the targeted ones.

The work we've discussed represents the most basic approach to merging neural networks—or data-driven models—with genetic algorithms; that is, the network is trained just once and then serves as a surrogate model for the evolution strategy. There’s a lot of potential to spice up this method, which may enhance performance. For instance, at the optimization's conclusion, the data verified by the simulator could be used to further refine the neural network. This enhanced network can then be deployed in another evolution strategy cycle. This iterative process, discussed another recent publication [18], has been proven to offer robust performance. It can identify certain optimal designs with an order of magnitude fewer simulations.

3.2   Generative AI: diffusion model

Diffusion models have recently garnered significant interest for their impressive capability to craft photo-realistic images from textual descriptions, with DALL-E 2 being a notable example. They have even progressed to producing short video sequences with striking quality. When compared to other generative models like variational autoencoders and generative adversarial networks, diffusion models are lauded for delivering superior quality in the samples they generate and providing a more reliable training experience. These models function by progressively refining noise-added samples from an initial distribution, often a standard Gaussian.

Bastek et al. have recently developed a much more elegant approach to the inverse design of nonlinear properties in metamaterials by utilizing the cutting-edge diffusion model [19]. Their aim is similar: to inversely design unit cells of metamaterials that yield predetermined stress-strain curves. While generative models have previously been employed in the inverse design of meso-structures, they have predominantly focused on linear properties such as stiffness [20]. However, challenges arise with nonlinear properties in mechanics, as they often include discontinuities like contact and instabilities such as buckling. Under these conditions, it becomes nearly unfeasible to generate designs directly from the target due to the complex, indirect mapping required—from geometry to response and vice versa—without a comprehensive understanding of the complete deformation history and associated internal stress distributions that are crucial for the desired effective response.

Instead of direct learning the map from properties to designs, the authors generate the entire deformation pathway and the whole stress field and displacement field at each step. This ingenious approach has proven to be surprisingly effective, as it enables the direct generation of designs whose stress-strain curves closely align with the intended targets. Below, one of the most impressive inverse design results is showcased.

The triumph of their methodology can be analogized to the progression from still images to moving videos. Just as producing a video requires temporal consistency, solving nonlinear inverse design challenges necessitates mechanical consistency throughout the deformation steps. In their model, the desired nonlinear stress-strain response guides the denoising process across a series of mechanically deformed microstructural configurations, ultimately yielding the macroscopic stress-strain behavior that closely matches the predefined criteria.

4.   Conclusion and Outlooks

As we have explored, the integration of AI, particularly through the use of neural networks and diffusion models, is opening new opportunities in the design of mechanical metamaterials, especially those with nonlinear properties. The vast design space of meso-structures and the complex nature of nonlinear deformations challenge traditional analytical approaches, while simultaneously providing a fertile ground for machine learning models. The benefits are twofold: not only can metamaterials gain from advancements in AI, but AI can also enhance its capabilities by tackling these physical problems. By engaging with real-world data, AI can develop a more nuanced understanding of physical phenomena, addressing a significant limitation currently faced by generative AI models. Continued interdisciplinary collaboration and technological development will be crucial in overcoming existing challenges and unlocking the full potential of AI-driven metamaterial design.

5.   References

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[16] Chen, T., Pauly, M., and Reis, P.M., 2021. Reconfigurable kirigami polyhedra via tunable instabilities. Nature, 589, p.386.

[17] Deng, B., Zareei, A., Ding, X., Weaver, J.C., Rycroft, C.H. and Bertoldi, K., 2022. Inverse design of mechanical metamaterials with target nonlinear response via a neural accelerated evolution strategy. Advanced Materials, 34(41), p.2206238.

[18] Li, B., Deng, B., Shou, W., Oh, T.H., Hu, Y., Luo, Y., Shi, L. and Matusik, W., 2023. Computational discovery of microstructured composites with optimal strength-toughness trade-offs. arXiv preprint arXiv, 2302.

[19] Bastek, J.H. and Kochmann, D.M., 2023. Inverse design of nonlinear mechanical metamaterials via video denoising diffusion models. Nature Machine Intelligence, 5(12), pp.1466-1475.

[20] Mao, Y., He, Q. and Zhao, X., 2020. Designing complex architectured materials with generative adversarial networks. Science advances, 6(17), p.eaaz4169.