Research

My research primarily explores the application of Calculus of Variations and spectral theory in discrete lattice systems. My current projects on lattice systems can be categorized into the following areas:

• Effective elastic behavior of mechanism-based lattice systems
• Singular bar frameworks and tensegrity frameworks

• Effective elastic behavior of mechanism-based lattice systems

This project addresses problems from an emerging area of mechanics known as mechanism-based lattice systems. These systems often consist of periodically arranged building blocks, resembling elastic composites. However, compared to the traditional elastic composites, these mechanism-based lattice systems are more degenerate, since they can deform with zero elastic energy. Such deformations are called mechanisms.

One fascinating consequence of the presence of mechanisms in these lattice systems is their degenerate elastic behavior. In fact, these systems possess soft modes, which are large deformations that require only a small amount of elastic energy. Interestingly, the soft modes often result in global deformations that are entirely different from those of the mechanisms. A significant portion of my research is dedicated to developing an effective theoretical framework to study specialized mechanism-based lattice systems, where soft modes correspond to deformations that result in zero effective elastic energy.

A vivid and illustrative example for explaining our work on mechanism-based lattice systems is the Kagome lattice, which is a 2D tiling consisting of triangles and hexagons. Deriving an effective theory is not simple for the Kagome lattice since it has an infinite number of mechanisms. The following two figures show two different mechanisms of the Kagome lattice.

This is a joint work with my PhD advisor Robert V. Kohn at Courant Institute. The numerical work is a collaboration with Katia Bertoldi and Bolei Deng.

• Singular bar frameworks and tensegrity frameworks

Another research interest of mine is related to the singularity of bar frameworks. Singularity in bar frameworks means the failure of some infinitesimal flexes to integrate to fully nonlinear flexes. Singular bar frameworks can be categorized into two types: singular and flexible bar frameworks and singular and rigid bar frameworks. One research directions of mine involves constructing these singular structures.

- Designing singular and flexible bar frameworks

This is a joint work with Miranda Holmes-Cerfon, Christian D Santangelo and my student Mihnea Leonte at Columbia University. The code of our constrained saddle search approach to construct singular and flexible bar framework can be found here.

- Designing singular and rigid bar frameworks -- bar frameworks with higher-order rigidity

This is a joint work with Miranda Holmes-Cerfon and Christian D Santangelo.