Tuning collective actuation of active solids by optimizing activity localization

TL;DR

This work tackles how to achieve mode-selective actuation in active elastic solids by localizing activity to specific lattice regions. It leverages agent-based simulations on ordered and disordered lattices, revealing that energy distribution among normal modes deviates from equipartition under elasto-active feedback and can be steered by spatially confining activity. An optimization algorithm based on Metropolis Monte Carlo relocates active nodes to maximize or minimize the energy in a target mode, showing superior performance in disordered lattices and competitive results in ordered ones. The authors establish a design principle via activation susceptibility, linking activation paths to mode polarization geometries, and discuss extensions to lower-coordination networks and joint disorder-activation optimization with potential practical impact for tunable metamaterials.

Abstract

Active solids, more specifically elastic lattices embedded with polar active units, exhibit collective actuation when the elasto-active feedback, generically present in such systems, exceeds some critical value. The dynamics then condensates on a small fraction of the vibrational modes, the selection of which obeys non trivial rules rooted in the nonlinear part of the dynamics. So far the complexity of the selection mechanism has limited the design of specific actuation. Here we investigate numerically how, localizing the activity on a fraction of modes, one can select non-trivial collective actuation. We perform numerical simulations of an agent based model on triangular and disordered lattices and vary the concentration and the localization of the active agents on the lattices nodes. Both contribute to the distribution of the elastic energy across the modes. We then introduce an algorithm, which, for a given fraction of active nodes, evolves the localization of the activity in such a way that the energy distribution on a few targeted modes is maximized -- or minimized. We illustrate on a specific targeted actuation, how the algorithm performs as compared to manually chosen localization of the activity. While, in the case of the ordered lattice, a well educated guess performs better than the algorithm, the latter outperform the manual trials in the case of the disordered lattice. Finally, the analysis of the results in the case of the ordered lattice leads us to introduce a design principle based on a measure of the susceptibility of the modes to be activated along certain activation paths.

Figures (13)

• Figure 1: Ordered and disordered lattices with randomly distributed driven nodes:(a) triangular lattice with $N=127$ ($R = 7$) and 25 randomly distributed driven nodes, resulting in $\phi_{\rm dn}~$$= 0.2$. (b) a disordered lattice with $N=126$ and $\phi_{\rm dn}~$$= 0.2$. In both cases, the external layer of nodes is pinned, as indicated by dark grey points in the figure. White points represent empty nodes and red points thermally or activity driven nodes.
• Figure 2: Distribution of energy in thermally and actively driven lattices:(a) Average squared amplitude of the modes $\overline{P_k^2} \propto \omega_k ^{\alpha}$vs$\omega_k$ for a thermally driven, an actively driven ordered, and an actively driven disordered lattice as reported in the legend. Each point corresponds to one mode. The dashed line corresponds to the fit $\overline{P_k^2} \propto \omega_k ^{-\alpha}$. While $\alpha=2$ in the thermal case, it is much larger in the active case. $N=127, \phi_{dn}=100\%$. (b) Dependence of $\alpha$ on the fraction of driven nodes $\phi_{dn}$ for $N=127$. (c) Dependence of $\alpha$ on the system size $N$ for $\phi_{dn}=100\%$. The data are averaged over 50 initial conditions and 12 different disordered lattices where used. ($\Pi = 1.3$).
• Figure 3: Selective actuation of modes by symmetry classes, using different spatial distribution of active nodes: The top row indicate the localization of the active nodes. Black points represent pinned edges, red points active nodes and open circles are empty nodes. In (a) active particles are located in all the nodes of the lattice, (b) in the 4th ring, (c) in a semi-circle of the 4th ring and (d) in an horizontal line. The bottom row indicates the energy repartition among the modes grouped by symmetry classes, as indicated by the colors of the legend. The extreme right column (e) indicates this distribution when all nodes are thermally driven, as a reference. For each spatial distribution (column (a), (b), (c) (d)), the distribution of the energy is represented for increasing values of $\Pi$. When $\Pi$ is too small, collective actuation does not take place and the system remains frozen, as indicated by the blue overlay. Data shown here are averaged over 100 runs with random initial polarization; lattice size $N=127$.
• Figure 4: Monte Carlo move: schematic example of two configurations, $|\sigma\rangle$ and $|\sigma^{'}\rangle$, which differ by one Monte Carlo step, where an active node is made inactive in favor of an other node, keeping the fraction $\phi_{dn}$ constant.
• Figure 5: Performance of the optimization algorithm: Violin plot showing the distributions of the final projections onto mode 3, denoted $C_3^{|\sigma\rangle} = \overline{P_3^2}$, obtained after 1200 Monte Carlo steps (last column on the right). These distributions are compared across 6 preset configurations with spatial distributions of the active nodes, indicated in red on the top row. The white points represent the mean of the distribution, while the black bars indicate the dispersion around the mean. The density of active nodes, $\phi_{dn} = 24/127$ is identical across all cases. The results are shown for both ordered hexagonal lattices (top) and disordered lattices (bottom). In the ordered networks, 30 distinct initial conditions (ICs) were used for each configuration, except for the optimization case, for which were used 350 distinct ICs. In the disordered networks, 30 ICs were simulated for 30 different networks in all configurations, except for the optimized cases, which involved only 10 networks. $R=7$, $N=127$ and $\Pi = 1.3$.