Multi-Region Matrix Interpolation for Dynamic Analysis of Aperiodic Structures under Large Model Parameter Perturbations

TL;DR

This paper tackles efficient frequency-response estimation for dynamic mechanical metamaterials with large parametric perturbations and aperiodic substructures by building a surrogate on matrix interpolation atop Craig-Bampton reduction. It reveals that the common modal projection can induce ill-conditioning via mode crossing, and introduces a sampling-based procedure to delineate the well-conditioned parameter region $\\bm{\\Theta}^o$, complemented by a region-specific interpolation framework. By combining SVM-driven region identification with PCA for dimensionality reduction and Kriging for interpolation, the method extends the usable parameter space and delivers accurate predictions within each region. The approach is validated on a unit-cell benchmark and a beam-like metamaterial with attenuation bands, showing superior performance over traditional Lagrange interpolation under large perturbations and highlighting practical benefits for design exploration, uncertainty quantification, and optimization in metamaterials.

Abstract

This work introduces a surrogate-based model for efficiently estimating the frequency response of dynamic mechanical metamaterials, particularly when dealing with large parametric perturbations and aperiodic substructures. The research builds upon a previous matrix interpolation method applied on top of a Craig-Bampton modal reduction, allowing the variations of geometrical features without the need to remesh and recompute Finite Element matrices. This existing procedure has significant limitations since it requires a common modal projection, which inherently restricts the allowable perturbation size of the model parameters, thereby limiting the model parameter space where matrices can be effectively interpolated. The present work offers three contributions: (1) It provides structural dynamic insight into the restrictions imposed by the common modal projection, demonstrating that ill-conditioning can be controlled, (2) it proposes an efficient, sampling-based procedure to identify the non-regular boundaries of the usable region in the model parameter space, and (3) it enhances the surrogate model to accommodate larger model parameter perturbations by proposing a multi-region interpolation strategy. The efficacy of this proposed framework is verified through two illustrative examples. The first example, involving a unit cell with a square plate and circular core, validates the approach for a single well-conditioned projection region. The second example, using a beam-like structure with vibration attenuation bands, demonstrates the true advantage of the multi-region approach, where predictions from traditional Lagrange interpolation deviated significantly with increasing perturbations, while the proposed method maintained high accuracy across different perturbation levels.

Figures (25)

• Figure 1: a) Example metamaterial structure and b) illustration of the substructure model parameter vector $\bm{\theta}^i$ and its components $x^i$ and $y^i$.
• Figure 2: Schematic representation of the surrogate model presented in mencik2024improved for predicting substructural Craig-Bampton matrices.
• Figure 3: Substructure used to study the effect of mode switching on the ill-conditioning of $\hat{\bm{\Phi}^{p,o}}$.
• Figure 4: Rank of $(\mathbf{R}^o)^T\bm{\Phi}^1$ for different numbers of retained modes while randomly switching all vibration modes of $\bm{\Phi}^1$ above the $21^{\text{st}}$ and below the $80^{\text{th}}$.
• Figure 5: (a) Rank of $(\mathbf{R}^o)^T\bm{\Phi}^p$ for each support point $k_2^p$. (b) values of the $45^{\text{th}}$ and $46^{\text{th}}$ natural frequencies each support point. Values adopted for $\bm{\theta}^o$ corresponds to: $m=5$ g, $k_1=1$ kN/mm, and $k_2=0.9$ kN/mm.