Abstract

We present a method for predicting thermal conductivity by deterministically solving the Boltzmann transport equation for gray phonons by utilizing arbitrary higher-order continuous finite elements on meshes which may also be unstructured and utilize curved surfaces. The self-adjoint angular flux (SAAF) formulation of the gray, steady-state, single relaxation time, phonon radiative transport (PRT) equation was spatially discretized using the continuous finite element method and angularly discretized using the discrete ordinates method. The solution discretization methodology was verified using a method of manufactured solution (MMS) spatial convergence test case and compared favorably to previous work. The angular phonon radiances, heat flux, and temperatures computed in this work compare favorably to previous literature in silicon thin films. Using local values of the temperature gradient and heat flux, the thermal conductivity as a function of position in a one-dimensional perfect crystal was evaluated using a Fourier’s Law representation and compared to kinetic theory. Our results show that in the interior of the simulation domain, our transport-based prediction of thermal conductivity converged on the kinetic theory estimation. We also find that near isothermal boundaries, the transport solution deviated from kinetic theory, implying non-equilibrium behavior in the thin-film limit and agreed with previous studies.

摘要

我们提出了一种通过利用网格上的任意高阶连续有限元确定性地求解灰色声子的玻尔兹曼传输方程来预测热导率的方法，网格也可能是非结构化的并利用曲面。使用连续有限元方法对灰度、稳态、单弛豫时间、声子辐射传输 (PRT) 方程的自伴随角通量 (SAAF) 公式进行空间离散，并使用离散纵坐标方法进行角度离散。使用制造解决方案 (MMS) 空间收敛测试案例的方法验证了解决方案离散化方法，并与以前的工作进行了比较。在这项工作中计算的角声子辐射、热通量和温度与之前的硅薄膜文献相比具有优势。使用温度梯度和热通量的局部值，使用傅立叶定律表示评估作为一维完美晶体中位置函数的热导率，并与动力学理论进行比较。我们的结果表明，在模拟域的内部，我们基于传输的热导率预测收敛于动力学理论估计。我们还发现在等温边界附近，输运解偏离了动力学理论，这意味着薄膜极限中的非平衡行为并与先前的研究一致。我们的结果表明，在模拟域的内部，我们基于传输的热导率预测收敛于动力学理论估计。我们还发现在等温边界附近，输运解偏离了动力学理论，这意味着薄膜极限中的非平衡行为并与先前的研究一致。我们的结果表明，在模拟域的内部，我们基于传输的热导率预测收敛于动力学理论估计。我们还发现在等温边界附近，输运解偏离了动力学理论，这意味着薄膜极限中的非平衡行为并与先前的研究一致。