ABSTRACT Phonon heat conduction has to be described by the Boltzmann transport equation (BTE) when sizes or sources are comparable to or smaller than the phonon mean free paths (MFPs). When domains much larger than MFPs are to be treated or when regions with large and small MFPs coexist, the computation time associated with full BTE treatment becomes large, calling for a multiscale strategy to describe the total domain and decreasing the computation time. Here, we describe an iterative method to couple the BTE, under the Equation of Phonon Radiative Transfer approximation solved by means of the deterministic Discrete Ordinate Method, to a Finite-Element Modeling commercial solver of the heat equation. Small-size elements are embedded in domains where the BTE is solved, and the BTE domains are connected to a domain where large-size elements are located and where the heat equation is applied. It is found that an overlapping zone between the two types of domains is required for convergence, and the accuracy is analyzed as a function of the size of the BTE domain. Conditions for fast convergence are discussed, leading to the computation time being divided by more than five on a study case in 2D Cartesian geometry. The simple method could be generalized to other types of solvers of the Boltzmann and heat equations.

中文翻译：

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多尺度声子热传导介观玻尔兹曼输运方程与宏观热扩散方程的耦合

摘要 当大小或来源与声子平均自由程 (MFP) 相当或小于声子平均自由程 (MFP) 时，必须用玻尔兹曼传输方程 (BTE) 来描述声子热传导。当要处理比 MFPs 大得多的域时，或者当大小 MFPs 的区域共存时，与完全 BTE 处理相关的计算时间变大，需要多尺度策略来描述总域并减少计算时间。在这里，我们描述了一种迭代方法，在通过确定性离散纵坐标法求解的声子辐射传递近似方程下，将 BTE 耦合到热方程的有限元建模商业求解器。小尺寸元素嵌入在解决 BTE 的域中，并且 BTE 域连接到大尺寸单元所在的域和应用热方程的域。发现需要两类域之间的重叠区域才能收敛，并将精度作为BTE域大小的函数进行分析。讨论了快速收敛的条件，导致在二维笛卡尔几何的研究案例中计算时间被除以超过 5。这种简单的方法可以推广到其他类型的玻尔兹曼方程和热方程求解器。导致计算时间在二维笛卡尔几何的研究案例中被除以超过 5。这种简单的方法可以推广到其他类型的玻尔兹曼方程和热方程求解器。导致计算时间在二维笛卡尔几何的研究案例中被除以超过 5。这种简单的方法可以推广到其他类型的玻尔兹曼方程和热方程求解器。