Truss metamaterials have been studied extensively for their remarkable properties, such as negative Poisson’s ratio, negative thermal expansion, high strength-to-weight ratio, and tunable wave propagation. However, designing truss metamaterials is not an easy task, as it involves exploring a vast and discrete design space and finding a suitable parameterization for various truss structures. 

Generalizable Parameterization: Unraveling the Potential of Truss Metamaterials

To address these challenges, we drew our inspiration from graphs that have been widely used in molecular chemistry by representing trusses as sets of nodes interconnected by solid beams, forming the nodes and edges of the graph, respectively. This representation allows for dynamic insertion and removal of nodes and connections, resulting in a vast dataset comprising diverse truss lattices, ranging from well-known topologies to unconventional designs.

The effective mechanical properties, computed through finite element homogenization, reveal a wide spectrum of Young’s moduli and shear moduli. By integrating graph-based representation with stochastic perturbation in structural and geometrical features, we have achieved unparalleled design flexibility while ensuring periodic tilability, substantially expanding the mechanical property design space.

Unified and Continuous Latent Space: A Gateway to Rapid Innovation

In the intricate realm of truss metamaterial design, the challenge lies in navigating a discrete and discontinuous design space. Truss lattices, each unique with varying node numbers and degrees of freedom, pose a formidable complexity.  

The foundation of our approach rested on a Variational Autoencoder (VAE) comprising two neural networks: an encoder and a decoder. The encoder learns to map from high-dimensional, discrete graph representations into compact, low-dimensional vectors. These latent vectors, representing truss structures, enabled an efficient exploration of the design space, transcending the rigid boundaries of traditional truss lattice formulations.

Furthermore, the latent space's association with specific mechanical properties through a neural network surrogate model revolutionized property predictions. By linking the latent vectors to the homogenized effective stiffness, similar structures clustered together in the continuous, smooth latent space, allowing for seamless transitions between distant designs.

One of the key highlights of the constructed continuous and smooth latent space is that it empowers designers to swiftly generate novel truss designs through simple yet powerful operations within this latent space.

Sampling in the latent space allows for the random generation of truss configurations. Each point represents a unique truss structure, offering a diverse array of design possibilities. Traversing the latent space involves moving systematically from one point to another, enabling engineers to explore gradual transitions between designs. Interpolating between structures allows for the seamless blending of different designs, leading to innovative designs that inherit the features of their parent designs.

Predicting the Unpredictable: Towards Extreme Properties Beyond Boundaries

The continuous, low-dimensional latent space serves as a robust representation, capturing the intricate relationships between structural features. Unlike simplistic memorization of training data, this constructed latent representation understands the underlying physics, enabling precise tailoring of truss behaviors by gradient-based optimization. This not only facilitates achieving desired properties but also enables extrapolation beyond the training data, broadening the scope of potential solutions.

We have showcased the model’s ability to generate manufacturable and counter-intuitive truss designs that exhibit exceptionally stiff, auxetic, and pentamode-like behaviors, surpassing the boundaries of the training domain. Crucially, the framework's applicability extends beyond linear mechanical responses. By delving into the realm of nonlinear mechanical metamaterials, the model demonstrates its efficiency in the inverse design of truss metamaterials with target nonlinear stress-strain responses. Remarkably, the model explores diverse truss designs even beyond the observed behaviors in the training dataset, underlining its potential for innovation and discovery in uncharted territories of metamaterial design.

In summary,  we have constructed a unified and continuous latent space that transcends the limitations of conventional design methodologies. The integration of gradient-based optimization offers a comprehensive solution to the complex challenge of truss metamaterial design. The generative modeling framework, which provides interpretability, versatility, and extrapolation capabilities, not only pushes the boundaries of what is achievable but also paves the way to uncover a broad range of novel materials and structures, opening new horizons for the field of metamaterial design.

More information about us can be found at https://www.mm.ethz.ch and https://www.mech-mat.com.