标题： 关于离散单轴多晶材料有效输运的谱理论 显示英文标题

标题： Spectral theory of effective transport for discrete uniaxial polycrystalline materials

摘要： 我们之前已经证明，包括电导率和热导率、扩散系数、复介电常数以及磁导率在内的单轴多晶材料的宏观输运系数具有涉及自伴随机算子谱测度的斯蒂尔吉斯积分表示。 这些积分表示来自于涉及这些自伴算子的物理场的预解式表示，例如与局部电导率矩阵$\boldsymbol{\sigma}$和电阻率矩阵$\boldsymbol{\rho}$相关的导电介质中的电场$\boldsymbol{E}$和电流密度$\boldsymbol{J}$。 在这篇文章中，我们提供了对该数学框架的一种离散矩阵分析，这与连续理论相平行。 我们展示了算子的离散化产生了由投影矩阵组成的实对称随机矩阵。 我们推导出离散的预解式表示法用于$\boldsymbol{E}$和$\boldsymbol{J}$，其中涉及的矩阵能够导致$\boldsymbol{E}$和$\boldsymbol{J}$的特征向量展开。 我们推导出有效电导率和电阻率矩阵$\boldsymbol{\sigma}^*$和$\boldsymbol{\rho}^*$的分量的离散斯蒂尔吉斯积分表示法，其中涉及的谱测度由实对称随机矩阵的实特征值及其标准正交特征向量显式给出。 我们提供了一种投影方法，利用投影矩阵的性质来表明谱测度可以通过更小的矩阵计算得出，从而得到一种更高效且更稳定的数值算法来计算整体输运系数和物理场。 我们通过数值计算模型二维和三维各向同性多晶介质（具有棋盘微几何结构）的谱测度和电流密度来展示这一算法。 显示英文摘要

摘要： We previously demonstrated that the bulk transport coefficients of uniaxial polycrystalline materials, including electrical and thermal conductivity, diffusivity, complex permittivity, and magnetic permeability, have Stieltjes integral representations involving spectral measures of self-adjoint random operators. The integral representations follow from resolvent representations of physical fields involving these self-adjoint operators, such as the electric field $\boldsymbol{E}$ and current density $\boldsymbol{J}$ associated with conductive media with local conductivity $\boldsymbol{\sigma}$ and resistivity $\boldsymbol{\rho}$ matrices. In this article, we provide a discrete matrix analysis of this mathematical framework which parallels the continuum theory. We show that discretizations of the operators yield real-symmetric random matrices which are composed of projection matrices. We derive discrete resolvent representations for $\boldsymbol{E}$ and $\boldsymbol{J}$ involving the matrices which lead to eigenvector expansions of $\boldsymbol{E}$ and $\boldsymbol{J}$. We derive discrete Stieltjes integral representations for the components of the effective conductivity and resistivity matrices, $\boldsymbol{\sigma}^*$ and $\boldsymbol{\rho}^*$, involving spectral measures for the real-symmetric random matrices, which are given explicitly in terms of their real eigenvalues and orthonormal eigenvectors. We provide a projection method that uses properties of the projection matrices to show that the spectral measure can be computed by much smaller matrices, which leads to a more efficient and stable numerical algorithm for the computation of bulk transport coefficients and physical fields. We demonstrate this algorithm by numerically computing the spectral measure and current density for model 2D and 3D isotropic polycrystalline media with checkerboard microgeometry.