A steel plate with an 8×8 grid of cells dampens elastic waves in two frequency corridors at once. This is not a lucky accident but the outcome of a calculation using a new scheme: first a coarse template for the low frequency, then fine-tuning within free pixels for the high frequency. A phononic material is an artificial structure that controls the propagation of elastic waves, much like a traffic light controls the flow of cars. Usually such a lattice has one characteristic length and one band gap, a frequency range where waves are attenuated. Here, there are two levels of nesting, and they are almost touching: the minimum size of the small element is only twice as small as the large one.

A translator gets distracted by every rustle in the hall, and a one-hour meeting stretches into two. The authors sought the opposite: how to make a phononic material that holds its main band gap even when small details appear inside. The unit-cell template method is a rule for arranging materials in a repeating block. First, on a coarse 10×10 grid, they find “parent” templates that yield a low-frequency band gap. Then, the free pixels of these templates are refined to a 20×20 grid, and within them they search for “child” templates for a second, higher band gap. The optimization uses integer programming with constraints on connectivity and minimum element size.

The key result is the scale-separation effect. Introducing small elements does not disrupt the low-frequency band gap, even though the scales are close. Numerically, the probability of preserving the large-scale goal when adding small details is 90% when the band gap width is 75% of the target range, and drops to 63% when the width is 35%. This means: the wider the gap, the more reliable the protection against the “whisper” of small structures. For a pair of target ranges (0.28–0.42 and 0.57–0.71 normalized frequency), a set of templates is obtained with 100% accuracy: all generated cells satisfy both goals. Template support covers from 50% to 100% of the admissible design space.

Before this work, it was unknown whether hierarchical phononic materials with close scales could be designed independently for each level without resorting to exhaustive search on a fine grid. Now it is shown: a coarse template optimized for a low-frequency band gap retains its goal when small-scale elements are added, provided the gap width is sufficiently large. This is achieved with an interpretable machine learning model that finds templates with fixed and free pixels, and by verifying robustness across all possible small-scale configurations. This approach reduces computational complexity from exponential to linear in the number of hierarchy levels. In practice, the method will be useful for designing vibration isolators and acoustic filters with multiple operating frequencies, for instance in precision instrumentation where vibrations from both a motor and human footsteps need to be dampened.