The Inverse Micromechanics Problem given Dielectric Constants for Isotropic Composites with Spherical Inclusions

Abstract

In this article, convex optimization is introduced as a promising tool to study Eshelby based inverse micromechanics problems. The focus is on inverse micromechanics using the Eshelby-Mori-Tanaka model given the dielectric constants of the composite material and of all of its components. The model is exactly the same for the conductivity properties as well. This choice of model is made since the model is fairly simple and has a closed form analytical solution for the case of spheroidal inclusions as well. The forward or direct micromechanics problem deals with the determination of effective properties of a composite material given the properties of its components and microstructural information. The focus is on isotropic composites and the distribution of inclusions is assumed to be such that this holds. The inverse micromechanics problem considered in this paper deals with the determination of microstructural information given the properties of the composite material and all of its components. Since in this paper, isotropy of the composite and only spherical inclusions are considered, the goal is to determine just volume fractions of the components of the composite material. The inverse problem is formulated as a Linear Programming problem and is solved. Before this, the inverse problem and certain important variants of it are examined through the lens of convex optimization. Lastly, promising results regarding the relationship between dispersive materials, noise in measurements, and quality of obtained volumetric splits are showcased. The scope of the use of convex optimization in inverse micromechanics is discussed.

Figures (6)

• Figure 1: Performance of Problem \ref{['eqn: Final formulation - Multifreq Epigraph']} in terms of recovery of $\bm{\phi^\mathrm{t}}$ for MS-1 and MS-3 for $\delta_R=\delta_I=0.1$ and for $m=1$. For MS-1, measurement frequency is $2.0$ GHz and for MS-3, measurement frequency is $0.5$ MHz.
• Figure 2: Performance of Problem \ref{['eqn: Final formulation - Multifreq Epigraph']} (in terms of recovery of $\bm{\phi^\mathrm{t}}$) for MS-2 for $\delta_R=\delta_I=0.1$ and for $m=1$ (measurement at only a frequency of 2.0 GHz) and $m=5$ (measurements at 5 equally spaced frequencies in the band 0.4-3.0 GHz, including 0.4 GHz and 3.0 GHz).
• Figure 3: Performance of Problem \ref{['eqn: Final formulation - Multifreq Epigraph']} in terms of $\underset{30,000}{\mathrm{max}}\left\lVert\bm{\phi^\star}-\bm{\phi^\mathrm{t}}\right\rVert_\infty$ and $\underset{30,000}{\mathrm{max}}\left(t^\star\right)$ against number of frequencies at which measurements are taken ($m$) for $\delta_R=\delta_I=0.1$.
• Figure 4: Performance of Problem \ref{['eqn: Final formulation - Multifreq Epigraph']} in terms of $\underset{30,000}{\mathrm{min}}\left(\sigma_\mathrm{min}\left(\bm{G^{\hat{\varepsilon}_t}}\right)\right)$ against number of frequencies at which measurements are taken ($m$) for $\delta_R=\delta_I=0.1$.
• Figure 5: Forward micromechanics problem for an isotropic 2 component composite material.