A Bounded Random Matrix Approach for Stochastic Upscaling

A maximum entropy (MaxEnt) based probabilistic approach is developed to model mechanical systems characterized by symmetric positive-definite matrices bounded from below and above. These matrices are typically encountered in the constitutive modeling of heterogeneous materials, where the bounds are deduced by employing the principles of minimum complementary energy and minimum potential energy. Current random matrix approaches in the context of computational stochastic mechanics are adapted only to the Wishart or matrix-variate Gamma probability model supported over the entire space of symmetric positive-definite matrices and therefore unable to exploit additional information available through the lower and upper bounds when appropriate. Specifically, for a given material, the constitutive matrix is construed as a random matrix. A probability measure that reflects the constraints consistent with the energy-based bounds is constructed, and an associated sampling scheme is described to synthesize realizatio...