Reconstructing the thermal phonon transmission coefficient at solid interfaces in the phonon transport equation

TL;DR

This work advances the quantitative reconstruction of the frequency-dependent phonon-interface reflection coefficient $\\eta(\\omega)$ by casting thermal boundary resistance as an inverse problem for the linearized phonon transport (BGK) equation. The authors formulate a PDE-constrained least-squares problem using surface-temperature measurements and develop a stochastic gradient descent algorithm that leverages a Fréchet-derivative derived via an adjoint system. They establish a maximum principle and Lipschitz continuity of the Fréchet derivative to justify SGD convergence, and demonstrate numerical success for both parametric and nonparametric representations of $\\eta$, including robustness to noise. The results provide a practical framework for inferring thermal boundary resistance from nonintrusive surface measurements, with potential impact on material design and thermal management at interfaces.

Abstract

The ab initio model for heat propagation is the phonon transport equation, a Boltzmann-like kinetic equation. When two materials are put side by side, the heat that propagates from one material to the other experiences thermal boundary resistance. Mathematically, it is represented by the reflection coefficient of the phonon transport equation on the interface of the two materials. This coefficient takes different values at different phonon frequencies, between different materials. In experiments scientists measure the surface temperature of one material to infer the reflection coefficient as a function of phonon frequency. In this article, we formulate this inverse problem in an optimization framework and apply the stochastic gradient descent (SGD) method for finding the optimal solution. We furthermore prove the maximum principle and show the Lipschitz continuity of the Fréchet derivative. These properties allow us to justify the application of SGD in this setup.

Figures (12)

• Figure 1: Experiment setup: details can be found in hua2017experimental. In experiments, two solid materials are placed side by side, and heat is injected on the surface of aluminium. Temperature is also measured at the same location as a function of time.
• Figure 1: Plot of the Maxwellian $M$ as a function of $\omega$ (normalized). We compare it to the typically used Bose-Einstein distribution $1/(e^\omega-1)$ (normalized)and its approximation $e^{-\omega}$ (normalized) for quantum systems.
• Figure 1: Solution to the forward PDE ($\int gd\omega$ as a function of $(x,\mu)$ at different time frames) with input \ref{['eqn:boundary_num_forward']}. The four plots from top left to bottom right are the solution at time being $0.01$, $0.5$, $1$ and $3$ respectively.
• Figure 2: Solution to the forward PDE ($\int gd\mu$ as a function of $(x,\omega)$ at $t=0.01, 0.5, 1.5, 3$ respectively). The boundary input is centered at $\omega=1.5$.
• Figure 3: Evolution of the solution to the adjoint PDE. We plot $\int h d\omega$ as a function of $\mu$ and $x$ backwards in time. The time frame for the four plots are $t=5$, $4$, $2$ and $0.01$ respectively.

Theorems & Definitions (16)

• Remark 2.1
• Lemma 3.1
• Proof 1
• Theorem 3.2
• Proof 2
• Remark 3.3
• Theorem 4.1
• Proof 3
• Theorem 4.2
• Proposition 4.3