Realizations of Isostatic Material Frameworks

TL;DR

This paper investigates the set of all equivalent realizations of 2D isostatic frameworks, introduces algorithms for their enumeration, and explores applications in atomic clusters, glasses, and jamming.

Contribution

It presents new algorithms for enumerating all equivalent realizations of 2D isostatic frameworks and analyzes their properties and applications.

Findings

An even number of realizations preserve edge lengths and connectivity.

Algorithms reduce computational complexity for large systems.

Applications demonstrated in atomic clusters, glasses, and jamming.

Abstract

This paper studies the set of equivalent realizations of isostatic frameworks in two dimensions, and algorithms for finding all such realizations. We show that an isostatic framework has an even number of equivalent realizations that preserve edge lengths and connectivity. We enumerate the complete set of equivalent realizations for a toy framework with pinned boundary in two dimensions and study the impact of boundary length on the emergence of these realizations. To ameliorate the computational complexity of finding a solution to a large multivariate quadratic system corresponding to the constraints; alternative methods - based on constraint reduction and distance-based covering map or Cayley parameterization of the search space - are presented. The application of these methods is studied on atomic clusters, a model two-dimensional glasses, and jamming.