Shapley values and machine learning to characterize metamaterials for seismic applications

TL;DR

This work addresses the challenge of efficiently quantifying how metamaterial input parameters affect seismic band-gap characteristics. It combines a Shapley-value–based sensitivity framework with data-driven regression models to rank parameter importance and predict two QoIs: the first-frequency cut-off and the width of the first band-gap, for both Bragg-scattering (phononic) and local-resonance (sonic) metamaterials. The authors demonstrate that parameter dominance shifts with design ranges and show that random-forest models deliver high-accuracy QoI predictions with low computational cost, while Shapley analysis provides interpretable parameter rankings and design guidance without requiring large datasets. The findings offer a practical pathway for tailoring seismic metamaterials to shield infrastructure, leveraging minimal data and fast ML predictions to guide material selection and geometry in Bragg and local-resonant designs.

Abstract

Given the damages from earthquakes, seismic isolation of critical infrastructure is vital to mitigate losses due to seismic events. A promising approach for seismic isolation systems is metamaterials-based wave barriers. Metamaterials -- engineered composites -- manipulate the propagation and attenuation of seismic waves. Borrowing ideas from phononic and sonic crystals, the central goal of a metamaterials-based wave barrier is to create band gaps that cover the frequencies of seismic waves. The two quantities of interest (QoIs) that characterize band-gaps are the first-frequency cutoff and the band-gap's width. Researchers often use analytical (band-gap analysis), experimental (shake table tests), and statistical (global variance) approaches to tailor the QoIs. However, these approaches are expensive and compute-intensive. So, a pressing need exists for alternative easy-to-use methods to quantify the correlation between input (design) parameters and QoIs. To quantify such a correlation, in this paper, we will use Shapley values, a technique from the cooperative game theory. In addition, we will develop machine learning models that can predict the QoIs for a given set of input (material and geometrical) parameters.

Figures (14)

• Figure 1: This figure shows the layered arrangement of base materials in a phononic metamaterial. The left figure shows a unit cell, which, in the case of a phononic crystal, is repeated infinite number of times.
• Figure 2: This figure shows the arrangement of base materials in a sonic crystal. The left depicts a unit cell, which is repeated to realize sonic crystal in two dimensions.
• Figure 3: This figure presents the Shapley value analysis results for the mathematical model \ref{['Eqn:Shapley_continuous']}. The graph plots the $x_1$ dominance in percentage. Thus, in the regions with lower percentages [$\leq 50$], the parameter $x_2$ is dominant.
• Figure 4: This figure shows the arrangement of the base materials in the phononic metamaterial for the exploratory data analysis. The base material for layer 2 is rubber; only the geometrical properties of layer 2 are changed, while (dimensionless) unit cell width is fixed at 1. The properties of layer 1 are varied by altering the ratios. The changing parameters for the design are Young's modulus ratio, density ratio, and thickness ratio. These ratios are calculated as $x_{layer 1}:x_{layer 2}$ for a given property $x$.
• Figure 5: Shapley value analysis for a phononic metamaterial. This figure presents the parameter ranking for the first QoI---decreasing the first frequency cut-off point. The dominant and secondary parameters are shown in subfigures (a) and (b), respectively. For this QoI, elasticity modulus has no effect over the design. For the lower ranges of thickness ratio [$\leq 9.5$] and density ratio [$\leq 2$], thickness ratio governs the design. Above this range, the density ratio is the dominant parameter. However, this dominance zone of thickness ratio reduces as the density ratio increases.