A Bounded Random Matrix Approach for Stochastic Upscaling
Sonjoy Das and Roger Ghanem
Multiscale Modeling and Simulation (Special Issue on Multiscale Modeling of Materials), v , No , pp. -- , 2009, doi:
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 are only adapted to the Wishart or matrix-variate Gamma probability model supported over the entire space of the 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, together with an associated sampling scheme, are constructed to synthesize realizations of this random matrix. An additional constraint of the mean matrix is also considered, and an appropriate probability model and sampling scheme are developed and illustrated numerically for this case. The present work also develops a new algorithm to compute the parameters of the Wishart or matrix-variate Gamma probability model, that is more consistent with the MaxEnt proposition.
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