Statistical ensemble error bounds for homogenized microheterogeneous solids

Abstract

Typically, in order to characterize the homogenized effective macroscopic response of new materials possessing random heterogeneous microstructure, a relation between averages|Ω| Ω · dΩ, and where σ and are the stress and strain tensor fields within a statistically representative volume element (SRVE) of volume |Ω|. The quantity, I E * , is known as the effective property, and is the elasticity tensor used in usual macroscale analyses. In order to generate homogenized responses computationally, a series of detailed boundary value representations resolving the heterogeneous microstructure, posed over the SRVE's domain, must be solved. This requires an enormous numerical effort that can overwhelm most computational facilities. A natural way of generating an approximation to the SRVE's response is by first computing the response of smaller (subrepresentative) samples, each with a different random realization of the microstructural type under investigation, and then to ensemble average the results afterwards. Compared to a direct simulation of an SRVE, testing many small samples is a computationally inexpensive process since the number of floating point operations is greatly reduced, as well as the fact that the samples' responses can be computed trivially in parallel. However, there is an inherent error in this process. Clearly the population's ensemble average is not the SRVE's response. However, as shown in this work, the moments on the distribution of the population can be used to generate rigorous upper and lower error bounds on the quality of the ensemble-generated response. Two-sided bounds are given on the SRVE response in terms of the ensemble average, its standard deviation and its skewness.

On the optimality of the window method in computational homogenization

“…While the above bounds monitor only the mean response of the sample population, ordering relations for higher-order statistical measures may also be constructed (Zohdi, 2005). Moreover, these bounds can be argued to be arbitrarily refinable, i.e.…”

Variational bounds for thermal fields in media with heterogeneous microstructure

“…Note that the Λ bound depends only on double-struckK 0 , double-struckK true˜ , and θ 0 . Early variants of this class of bound for more structured elastic media can be found in Zohdi [15] and Zohdi et al [16].…”

Determination of the Effective Elastic Modulus for Nodular Cast Iron Using the Boundary Element Method

“…There is no definitive way in which to establish an RVE [16]. ; the most frequently used procedure currently involves either fixing the nominal RVE size and then increasing the number of graphite nodules (Figure 3) or increasing the size of the RVE so that the number of heterogeneities changes by a fixed percentage.…”

“…In this work, the same strategy from [13] and [16] was used to study the effective property in an NCI RVE. Figure 12 presents the convergence curve of E* for a 3D model.…”

“…Through this method, the statistical data of many RVEs can be obtained, and by averaging the values of each distribution of spherical inclusions, it is possible to find the global response of the material [16]. According to Zohdi [16], this technique is more trustworthy than performing individual direct simulations. Buroni and Marczak [13] developed a homogenization procedure using a 2D BEM as a numerical approach to determine the effective modulus of NCI.…”