Asymptotic analysis and design of shell-based thermal lattice metamaterials

We present a rigorous asymptotic analysis framework for investigating the thermal conductivity of shell lattice metamaterials, extending prior work from mechanical stiffness to heat transfer. Central to our analysis is a new metric, the asymptotic directional conductivity (ADC), which captures the leading-order influence of the middle surface geometry on the effective thermal conductivity in the vanishing-thickness limit. A convergence theorem is established for evaluating ADC, along with a sharp upper bound and the necessary and sufficient condition for achieving this bound. These results provide the first theoretical justification for the optimal thermal conductivity of triply periodic minimal surfaces. Furthermore, we show that ADC yields a third-order approximation to the effective conductivity of shell lattices at low volume fractions. To support practical design applications, we develop a discrete algorithm for computing and optimizing ADC over arbitrary periodic surfaces. Numerical results confirm the theoretical predictions and demonstrate the robustness and effectiveness of the proposed optimization algorithm.