Finding the overall thermo-mechanical properties (heat conductivity, elastic modulus, thermal expansion coefficients, etc.) of heterogeneous materials, such as solid propellants, is complicated because direct computational analysis of a large material body is very expensive when the necessary resolution is dictated by microstructure.
We will present a mathematical/numerical framework that allows us to compute overall properties directly from tomographically characterized material systems. We start by taking samples using micro-computer tomography and then statistically characterize them to reconstruct a statistically optimal unit cell that is much smaller than the entire domain, yet has the same statistical makeup as that of the actual material. Using this unit cell, we compute the first-, second- and third-order probability functions and use these statistical descriptors with a homogenization theory to obtain upper and lower bounds on material properties. To accomplish this, we propose third-order accurate extended Hashin-Shtrikman variational principles and their discretizations, constructed to reduce the spatial complexity of a polarization field that captures the inhomogeneous and/or potentially anisotropic aspects of physical processes of interest. We study two different reformulated models and perform error analysis to assess the quality of the polarization filed approximation.
Numerical evaluation of the polarization filed requires calculation of complex integrands that include multiplications of probability functions and the second derivative of Green's function. To compute these integrands accurately and efficiently, we introduce an adaptive sparse Smolyak integration method with hierarchical bases and modify it to use spherical coordinates, which are natural for our problem. To speed up the integration procedure, we exploit parallel computation.
Finally, we show several examples involving to particulate composites and discuss recent progress and future research directions.