Elastic Lattices Inspired by Ulam-Warburton Cellular Automaton

TL;DR

This paper introduces a novel class of aperiodic elastic lattices inspired by cellular automata, specifically the Ulam-Warburton automaton, revealing unique vibrational properties and localized modes that could enable new wave phenomena.

Contribution

It presents the design and analysis of UWCA-inspired elastic lattices, demonstrating their distinctive vibrational characteristics and potential for novel wave behavior.

Findings

Symmetric eigenfrequency spectra observed

Presence of strongly localized corner modes

Repeated natural frequencies with high multiplicity

Abstract

Periodic lattices have been widely explored for decades, owing to their peculiar vibrational behavior. On the other hand, certain types of aperiodic lattices have enabled new phenomena that may not be otherwise attainable in periodic ones. In this paper, a new class of aperiodic lattices inspired by cellular automaton is introduced. Cellular automata were originally developed as a machine replication algorithm and it has been intensively explored in computer science. These algorithms yield structures that are not necessarily periodic, yet follow well-defined rules that lead to interesting patterns. The concept is utilized here to build elastic lattices following such rules, and Ulam-Warburton Cellular Automaton (UWCA) is demonstrated as an example. Starting from a square monatomic lattice, an UWCA lattice is constructed and its vibrational behavior is analyzed, showing unique dynamical properties, including symmetric eigenfrequency spectra, repeated natural frequencies of large multiplicity, and the emergence of strongly localized corner modes. It is envisioned that computer-algorithm-inspired lattices may unlock new wave phenomena that could outperform existing lattice designs.