New Design Framework Enables Programmable Metamaterials with On-demand Topological States

Metamaterials are a class of engineered solids that derive their key properties not from their compositions but rather from their geometry, that is, their complex internal architecture. Built from repeating networks of simple building blocks such as rods, beams, or plates, these metamaterials display mechanical properties that transcend those of their individual components. Metamaterials allow control of  how waves of sound, vibration, or light travel through them, often manipulating these phenomena in ways that are not achievable working with ordinary materials. In recent years, scientists have discovered that topology—a branch of mathematics concerned with shapes—can be used to describe and classify these unusual wave behaviors and explain many exotic functionalities of metamaterials. Understanding and engineering the effects enabled by topology has opened an entirely new frontier for mechanical design.

Despite the promise of this emerging intersection between metamaterials engineering and topology, one major challenge remains to be addressed: how to fully automate systematic design of metamaterials that exhibit desired topological properties. A new study published in the Proceedings of the National Academy of Sciences (PNAS), researchers at the University of Minnesota, in collaboration with colleagues at the University of Illinois Urbana-Champaign and the University of Michigan, has achieved a substantial step forward toward this goal by establishing a computational design framework that enables the discovery of lattice configurations with targeted topological properties.

The study, led by Professor Stefano Gonella of the Department of Civil, Environmental, and Geo-Engineering at Minnesota, and Pegah Azizi, a doctoral researcher in his lab and the paper’s first author, introduces an inverse-design framework that integrates notions of topological quantum chemistry—a theoretical framework originally developed to characterize and classify quantum phenomena—with topology optimization, a powerful algorithmic design tool used across engineering to design structural materials with desired functionalities. 

“Our work offers a much-needed synthesis between classical metamaterial functionalities and topological states of matter,” said Gonella. “We effectively were able to distill and express mathematically some key principles governing wave topology and encode them directly into the optimization algorithms that are used to generate lattice configurations, using them as drivers of the optimization problem. In doing so, we enabled automated generation of configurations with specific topological properties. The ability to encode this new set of topology-protected functionalities in metamaterials design can contribute to bringing metamaterials technology one step closer to widespread adoption in engineering design.”

It is important to recognize that while topological materials have been widely investigated in the physics community, only a small fraction of the theoretically possible configurations have been realized experimentally. This research aims at bridging this gap between theory and applications by constructing topology-informed design and optimization algorithms, physically fabricating the configurations output by the optimizers, and experimentally confirming the emergence of the desired functionalities via laser vibrometry testing.

“The remarkable aspect of our approach is that, within a single design framework, we can generate a plethora of configurations just by tweaking some parameters in the optimization problem,” said Azizi. “While the different configurations obtained share the same topological character targeted by optimization, they display different and possibly complementary secondary characteristics, including a variety of mechanical and dynamical properties—such as broadband or low-frequency filtering and tailored wave speeds. I believe that such ease and flexibility of design is a significant contribution to the field, providing a pathway to engineering physically realizable structures that combine advanced wave-manipulation capabilities with practical considerations such as load-bearing performance and ease of fabrication.”  

Gonella explains, “From a philosophical perspective, the ultimate goal is even more ambitious.  Our end game is to exploit this design versatility to build entire libraries of configurations and organize the behavior of metamaterials according to rational categorization principles, thus achieving a general and systematic treatment akin to how, for instance, elements are organized in periodic tables in the chemistry world.”

“One of the most exciting parts of this work was seeing ideas from different fields—quantum physics, mechanics, and optimization—come together in a single framework,” said Azizi. “We began with a simple question: Can we make the design of topological metamaterials as straightforward as possible? The answer was yes, by distilling complex physics into clear, practical rules that could be encoded directly into optimization algorithms. What thrilled me even more was watching how small changes in the parameters produced entirely different designs—some resembling the well-known kagome lattice, others taking on unexpected, flower-like shapes. I see this work as just the beginning of a journey, opening the door to new ways of applying topological quantum chemistry to create mechanical metamaterials for uses we haven’t even imagined yet.”

The research was supported by the National Science Foundation (NSF), the Air Force Office of Scientific Research (AFOSR), and the Office of Naval Research (ONR), reflecting the growing national interest in topological and architected materials for applications in vibration control, energy harvesting, and robust structural design.

About the Paper: “Lattice Materials with Topological States Optimized on Demand,” was published in PNAS in August 2025. The research team includes Pegah Azizi (University of Minnesota), Rahul Dev Kundu and Weichen Li (University of Illinois Urbana-Champaign), Kai Sun (PI at University of Michigan), Xiaojia Shelly Zhang (PI at University of Illinois Urbana-Champaign), and Stefano Gonella (PI at University of Minnesota and project leader).