Uncertainty propagation, entropy, and relative entropy in the homogenization of some particulate composites

Abstract

The main idea of this work is an application of probabilistic entropy and also relative entropy in the numerical analysis of uncertainty propagation in the homogenization of some composite materials. The homogenization method is based on the determination of deformation energy for the representative volume elements computed with the use of some specific finite element method experiments. Uncertainty propagation concerns material and geometrical design parameters of particulate composites and is performed thanks to an application of polynomial responses; probabilistic moments of the effective tensor are computed via the iterative generalized stochastic perturbation technique and the semi‐analytical probabilistic method. Probabilistic entropy is determined according to Shannon theory, while relative entropies equations employed here follow mathematical models created by Kullback & Leibler, Jeffreys, Hellinger, and Bhattacharyya. Deterministic analyses have been performed with the use of the system ABAQUS, while all the remaining procedures have been programmed in the computer algebra system MAPLE.

Probabilistic Relative Entropy in Homogenization of Fibrous Metal Matrix Composites (MMCs)

“…Some specific Dirichlet boundary conditions imposed throughout all outer surfaces of this RVE represent uniaxial, biaxial, and transverse deformations, while periodicity conditions are fulfilled on the remaining ones; any von Neumann conditions apply. Using classical notation available in the literature, one writes this energy U (and its equation) as [ 27 ] where Ω denotes the RVE (cf. Figure 1 ), and are the homogenized and the original composite constitutive tensor, stands for the strain tensor (adjacent to the adopted geometrical equations), and represents unitary or zero strains depending on a specific component to be determined.…”

“…Some specific Dirichlet boundary conditions imposed throughout all outer surfaces of this RVE represent uniaxial, biaxial, and transverse deformations, while periodicity conditions are fulfilled on the remaining ones; any von Neumann conditions apply. Using classical notation available in the literature, one writes this energy U (and its equation) as [27]…”

“…They are implemented using numerical recovery of the polynomial bases [ 26 ] linking effective elasticity tensor components with the uncertainty source thanks to several FEMhomogenization tests delivered in the ABAQUS system. These techniques enable the determination of the basic probabilistic characteristics of the homogenized tensor such as expectations and standard deviations, while uncertainty propagation is studied thanks to different relative entropy models available from probability theory [ 27 ]. These include the Kullback–Leibler model [ 28 ], Bhattacharyya theory [ 29 ], Hellinger idea [ 30 ], as well as its Jeffreys symmetrization [ 31 , 32 ].…”

On probabilistic entropies application in uniaxial deformation of hyper‐elastic materials

“…A deformation of a hyper-elastic specimen is treated as Gaussian random variable having a priori given expectation varying within the given experimental bounds and some specific standard deviation relevant to the experimental error level. Probabilistic computational analysis is delivered here with the use of the stochastic perturbation technique [3] and this analysis includes numerical determination of strain-dependent fluctuations of the first two probabilistic characteristics, that is expectations and coefficients of variations of the resulting tensile stress and has been focused on Shannon entropy calculation [4]. Finally, a relative entropy proposed by Bhattacharyya [5] is introduced to quantify a distance in-between increasing tensile stress value and its admissible counterpart.…”

Relative entropy-based reliability assessment of cable structures

“…Such a relative entropy quantifies a distance of two different probability distributions [5] and may measure an interference of R and E(t) in this case. It can be represented under the same assumptions by the following relation [9]:…”