Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectral theory of effective transport for discrete uniaxial polycrystalline materials

N. Benjamin Murphy, Daniel Hallman, Elena Cherkaev, Kenneth M. Golden

TL;DR

This work develops a discrete matrix formulation of the spectral theory for effective transport in uniaxial polycrystalline materials, aligning with the continuum Stieltjes framework. By expressing local conductivity as a linear combination of real-symmetric projection matrices, the authors derive discrete resolvent representations and eigenvector expansions for the electric and current fields, leading to spectral measure formulations of bulk coefficients. A projection method reduces the dimensionality of the spectral problem, yielding substantial computational savings while preserving accuracy, and is validated through 2D and 3D checkerboard microgeometries. The approach offers a robust foundation for both forward calculations of transport properties and potential inverse bounds based on spectral data. Overall, the discrete theory closely parallels the continuum theory and provides practical, scalable tools for polycrystalline transport analysis.

Abstract

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 $\boldsymbolσ$ and resistivity $\boldsymbolρ$ 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, $\boldsymbolσ^*$ and $\boldsymbolρ^*$, 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.

Spectral theory of effective transport for discrete uniaxial polycrystalline materials

TL;DR

This work develops a discrete matrix formulation of the spectral theory for effective transport in uniaxial polycrystalline materials, aligning with the continuum Stieltjes framework. By expressing local conductivity as a linear combination of real-symmetric projection matrices, the authors derive discrete resolvent representations and eigenvector expansions for the electric and current fields, leading to spectral measure formulations of bulk coefficients. A projection method reduces the dimensionality of the spectral problem, yielding substantial computational savings while preserving accuracy, and is validated through 2D and 3D checkerboard microgeometries. The approach offers a robust foundation for both forward calculations of transport properties and potential inverse bounds based on spectral data. Overall, the discrete theory closely parallels the continuum theory and provides practical, scalable tools for polycrystalline transport analysis.

Abstract

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 and current density associated with conductive media with local conductivity and resistivity 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 and involving the matrices which lead to eigenvector expansions of and . We derive discrete Stieltjes integral representations for the components of the effective conductivity and resistivity matrices, and , 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.

Paper Structure

This paper contains 9 sections, 5 theorems, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Stieltjes integrals for bulk transport coefficients for continuous media
  3. Stieltjes integrals for bulk transport coefficients for discrete media
  4. Existence of transport fields for discrete media
  5. Resolvent representations for transport fields
  6. Stieltjes integrals for bulk transport coefficients
  7. Projection method
  8. Numerical Results
  9. Conclusions

Key Result

Theorem 3.1

Let $A$ be a real valued matrix of size $m\times n$ then where $\oplus$ indicates $\mathcal{R}(A)$ is orthogonal to $\mathcal{K}{(A^T)}$, i.e., $\mathcal{R}(A)\perp\mathcal{K}{(A^T)}$, for example.

Figures (1)

  • Figure 1: Spectral measures and current density fields for 2D (top row) and 3D (bottom row) uniaxial polycrystalline media. (a) Spectral measure weights $m_i$ versus associated eigenvalue $\lambda_i$ of the matrix $M$ with corresponding current density field $|\boldsymbol{J}|$ in log10 scale shown in (c). The red horizontal lines in (a) correspond to the smallest weight $m_i$ across the entire spectrum. The orientation of each crystallite (d) is taken to be uniformly distributed on the interval $[-\pi, \pi]$ from the $x$-axis (2D) and from the $x$, $y$, and $z$-axes (3D). The crystallite orientations in 3D are achieved through composed rotations along the $x$, $y$, and $z$ axes in a random succession. (b) Corresponding spectral functions $\mu(\lambda)$ (histogram representations) of the spectral measure $\mu$ in (a), averaged over 2000 distinct geometric configurations. Each configuration is distinguished solely by the isotropic orientation angles $\theta_j \sim U[-\pi,\pi]$. In 2D, the geometric isotropy of the orientation angles allows the spectral function to align with the self-duality theoretical result $\mu_{kk}(\lambda)=(1/\pi)\sqrt{(1-\lambda)/\lambda}$. The electrical conductivity is taken to be $\sigma_1 = 51.074+ i45.160$ in the $x$ direction and $\sigma_2 = 3.070+i0.0019$ in the $y$ and $z$ directions. $E_0$ is taken to be oriented along the $y$-axis.

Theorems & Definitions (7)

  • Theorem 3.1: Fundamental theorem of linear algebra
  • Theorem 3.2
  • proof
  • Corollary 3.1
  • proof
  • Theorem 4.1
  • Corollary 4.1