Phononic materials with effectively scale-separated hierarchical features using interpretable machine learning

TL;DR

This work tackles the challenge of designing hierarchical phononic materials with tunable multi-band gaps by extending the interpretable unit-cell template method to hierarchical templates. A two-level coarse-to-fine design enables scale separation, allowing coarse-scale bandgap objectives to be preserved while refining fine-scale features to realize additional high-frequency bandgaps, all within a much larger, not previously explored design space. The method demonstrates robust scale-separation: coarse objectives remain intact as fine features are added, with quantified robustness via fine-scale transfer precision and substantial computational savings from sequential design. Numerical and experimental validations—featuring FE dispersion analysis and a 3D-validated, 8×8 steel lattice with LDV measurements—confirm that the hierarchical templates can realize prescribed bandgaps in real structures, offering a flexible route to multidimensional wave control in metamaterials.

Abstract

Manipulating the dispersive characteristics of vibrational waves is beneficial for many applications, e.g., high-precision instruments. architected hierarchical phononic materials have sparked promise tunability of elastodynamic waves and vibrations over multiple frequency ranges. In this article, hierarchical unit-cells are obtained, where features at each length scale result in a band gap within a targeted frequency range. Our novel approach, the ``hierarchical unit-cell template method,'' is an interpretable machine-learning approach that uncovers global unit-cell shape/topology patterns corresponding to predefined band-gap objectives. A scale-separation effect is observed where the coarse-scale band-gap objective is mostly unaffected by the fine-scale features despite the closeness of their length scales, thus enabling an efficient hierarchical algorithm. Moreover, the hierarchical patterns revealed are not predefined or self-similar hierarchies as common in current hierarchical phononic materials. Thus, our approach offers a flexible and efficient method for the exploration of new regions in the hierarchical design space, extracting minimal effective patterns for inverse design in applications targeting multiple frequency ranges.

Figures (11)

• Figure 1: (a) A schematic for a unit-cell dispersion relation indicating multiband design objectives for coarse-scale and fine-scale patterns. (b) A design-space map for hierarchical materials with different classes indicated (included insets from left to right are from refs. mousanezhad2015honeycombliu2018fractalliang2020design, respectively). The horizontal axis represents the extent of difference in hierarchical length scales (from close hierarchical scales within the same order of magnitude on the left to significantly different hierarchical scales varying by several orders of magnitude on the right). The vertical axis represents the extent of independence of design processes at each hierarchical scale from the remaining scales. Models with fine-scale grids are needed to analyze hierarchical materials with close scales, which means that design processes for the hierarchical scales are interdependent (bottom-left corner). Conversely, hierarchical materials with significantly different scales are usually analyzed with multi-scale models and the design process at each hierarchical length scale can be performed relatively independently from the remaining scales (top-right corner). Our developed approach, "hierarchical templates," fits in the top-left corner of the design-space map as it represents handling hierarchical materials with close scales but, as we show, the design process for different scales can still be carried out independently (consecutively from coarse to fine).
• Figure 2: Outline for the hierarchical unit-cell template method: On the left, a coarse-scale template with fixed pixels as well as free pixels is illustrated. Note that the free (green) pixels can be assigned the properties of a solid material or void and the target coarse-scale bandgap objective would still be achieved. As a reminder, the coarse unit-cell template was designed to satisfy a coarse-scale design objective, e.g., a specific bandgap exhibited by the periodic wave-propagating medium. Upon further refining the green free-pixel regions, fine-scale templates are obtained by our method to satisfy an additional design objective, e.g., another bandgap in a different frequency range. Thus, the hierarchical template, including both coarse and fine-scale grids, satisfies both design objectives.
• Figure 3: (a) Optimal set of coarse-scale templates obtained such that the design objective of having a single bandgap on a minimum of $60\%$ of the normalized-frequency interval of [0.28, 0.42]. (b) Influence of perturbing each fixed pixel on the templates precision, with darker pixels indicating higher importance.
• Figure 4: First example of a coarse-scale template obtained such that the design objective of having a single bandgap on a minimum of $60\%$ of the normalized-frequency interval of [0.28, 0.42] (gray region) is satisfied. Two unit-cells from this template, representing minimum and maximum solid fraction, along with their respective dispersion curves are shown. A second bandgap objective (highlighted by the red dashed region) is later defined for the hierarchical template derived from this coarse template (see \ref{['fig:Fig_03']}) and is shown to be unmet at the coarse scale.
• Figure 5: Second example of a coarse-scale template obtained such that the design objective of having a single bandgap on a minimum of $60\%$ of the normalized-frequency interval of [0.28, 0.42] (gray region) is satisfied. Two unit-cells generated from this template, representing minimum and maximum solid fraction, along with their respective dispersion curves are shown. A second bandgap objective (highlighted by the red region) is later defined for the hierarchical template derived from this coarse template (see \ref{['fig:Fig_04']}) and is shown to be unmet at the coarse scale. Note that the free pixels highlighted with bright red outlines cannot be flipped to solid as it would lead to zero-dimensional points of connection at the corners.