Elastic Lattices Inspired by Ulam-Warburton Cellular Automaton

TL;DR

A new class of aperiodic lattices inspired by cellular automaton is introduced and it is envisioned that computer-algorithm-inspired lattices may unlock new wave phenomena that could outperform existing lattice designs.

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.

Figures (4)

• Figure 1: Elastic lattices inspired by Ulam-Warbutron Cellular Automaton (UWCA): The first seven generations of the considered UWCA lattice are depicted. The initial (zeroth) generation starts as an infinite square monatomic elastic lattice with pinned masses $m$ (or in an "OFF" state), interconnected via springs of stiffness $k$. The first generation is enabled by unpinning the central mass, which is interpreted here as the "ON" state. Subsequent generations are determined by unpinning masses that share a spring connection with exactly one unpinned (or an "ON") mass, as dictated by the UWCA algorithm.
• Figure 2: Degrees of freedom numbering of the seventh generation UWCA lattice: This schematic illustrates the degrees of freedom order for the seventh generation of the UWCA lattice, which comprises a total of forty-nine degrees of freedom (i.e., $n = 49$).
• Figure 3: Eigenvalue analysis and growth of UWCA Lattices: (a) The natural frequency spectra of elastic lattices constructed based on the first eleven generations of UWCA. A schematic of the free (or "ON") masses for each generation is depicted for reference. (b) The eigenvalue spectra of the UWCA lattice for an increasing number of generations, exhibiting symmetry about the eigenvalue $\lambda = 4$. Among those eigenvalues, there are corner modes (colored in orange), which occur at odd numbers of generations, starting at seven and increasing in increments of four. (c) The degrees of freedom (denoted $n$) versus the number of generation (denoted $n_g$) is presented, demonstrating a considerable increase in the degrees of freedom as UWCA pattern grows with higher generations.
• Figure 4: Corner modes in UWCA lattices: The response of the first eight generations of UWCA lattices with corners of $\perp$ shape, thus exhibiting strongly-localized corner modes, are shown for a frequency near $\omega = \sqrt{5} \omega_0$ (or $\lambda = 5$). Corner modes are excited by applying a force at the junction connecting the $\perp$ shaped corner to the bulk of the lattice. These junctions are colored as red for ease of reference, and the normalized displacement response is indicated by the color bar. The accompanying bar chart in the middle of the figure shows the number of corners of $\perp$ shape versus the integer $r$. The number of corners differs with various values of $r$ (and generation number $n_g = 4r-1$), but they reset to four whenever $r = 2^z+1$ following a local maximum at $r = 2^z$, where $z$ is an integer larger than 1.