The challenge of achieving precise control over heat conduction has persisted for years. Recent advancements in thermal metamaterials have offered a promising avenue for addressing this challenge. Yet, predicting the behaviors of non-Fourier heat conduction within these metamaterials remains elusive. In this study, we leveraged the extended plane wave expansion method (EPWEM) to decipher the complex band structures inherent to two-dimensional (2D) thermal metamaterials. To ascertain the reliability of our approach, wave transmission in the time-domain and band structures derived from the finite element method (FEM) were juxtaposed against EPWEM results. Our findings revealed the presence of both directional and complete band-gaps in the complex band structures. Then a thermal bridge is predicted via the proposed complex band structure strategy and demonstrated numerically. By shedding light on non-Fourier heat conduction within thermal metamaterials, our analysis paves the way for groundbreaking applications in domains like laser hardening, laser cladding, metal additive manufacturing, and so on.

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