Periodic body-and-bar frameworks

TL;DR

This paper characterizes the rigidity of periodic body-and-bar frameworks, providing a mathematical criterion and efficient algorithms for recognition, advancing understanding of crystalline material structures.

Contribution

It introduces a Maxwell-Laman type characterization for generic minimally rigid periodic body-and-bar frameworks, enabling polynomial-time recognition algorithms.

Findings

Established a Maxwell-Laman characterization for these frameworks.

Developed polynomial-time algorithms for recognition based on matroid partition and pebble games.

Enhanced understanding of the rigidity properties of crystalline material models.

Abstract

Abstractions of crystalline materials known as periodic body-and-bar frameworks are made of rigid bodies connected by fixed-length bars and subject to the action of a group of translations. In this paper, we give a Maxwell-Laman characterization for generic minimally rigid periodic body-and-bar frameworks. As a consequence we obtain efficient polynomial time algorithms for their recognition based on matroid partition and pebble games.