A Bounded Random Matrix Approach for Stochastic Upscaling

Abstract

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 based nonparametric approach is 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 energybased bounds, together with an associated sampling scheme, are constructed to synthesize realizations of this random matrix. An additional constraint of the ensemble mean matrix is also considered, and an appropriate probability model and sampling scheme are developed and illustrated numerically for this case.

A probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures

“…The induced probability measure is specified by the MaxEnt procedure and constraints synthesized from experimental data. The paper generalizes results previously obtained on bounded random elasticity matrices (see [2]) so as to facilitate the development of an associated random field model. This paper is organized as follows.…”

“…(29) [31] for a detailed discussion). This set of constraints has already been considered in [2]. Let…”

“…A straightforward derivation then yields: [28]. In this context, an algorithmic scheme for computing the multipliers [Λ [L] ], λ and λ u is provided in [2].…”

“…Invoking the energy-based boundedness constraints on the random elasticity tensor established by Huet (see [10]), a random matrix approach has been derived later on in [2] (see Section 2.3.1). Noticing that the two above approaches induce anisotropic statistical fluctuations which may not be encountered in practice (and in particular, in geophysical applications), Ta and his coworkers proposed a refinement of the probabilistic model derived by Soize by introducing a new parameter controlling the anisotropy index apart from the level of fluctuations [36].…”

A stochastic computational multiscale approach; Application to MEMS resonators

“…Random positive-definite matrices can be generated by the means of the Cholesky decomposition [8]. In order to ensure the existence of the expectation of the norm of the generated tensor inverses [29,30], we introduce a lower bound, as proposed in [31], when generating the meso-scale material tensors. Bounds were also introduced in the random field generator in the other works [32,33].…”