Abstract:
The homogenization for classifying composites and determining their effective properties is an important optimal
design problem of material sciences studied by mathematical modeling. The application of artificial intelligence (AI) and
machine learning (ML) in the theory of composite materials is discussed. One of the main problems is the choice of
characteristic ML features to describe multi-scale dispersed random composites and to predict their macroscopic
properties. The complexity drastically increases when confronted with tasks such as estimating the effective properties of
random composites, exploring optimal design scenarios with variable properties of components, or determining the optimal
location and shape of inclusions since the myriad use of numerical computations proves challenging due to constraints in
time and memory. In such instances, analytical, exact, or approximate formulas with the optimized parameters in symbolic
form are preferred because powerful calculus methods can be applied to select the optimal parameters. The present paper
is devoted to adequately choosing the parameters called structural sums, and corresponding analytical formulas. Such
a formula is often asymptotic, and its correctly determined asymptotic precision shows its application area. We consider
the question of the RVE size equivalent to the number of inclusions 𝑵𝑵 per periodicity cell. It can be investigated
numerically by solving a periodicity problem with 𝑵𝑵 increasing up to stable effective constants not depending on 𝑵𝑵 .
Though one can find works in literature following these lines, they concern special distributions of inclusions with the
numerical results performed for small 𝑵𝑵 and for a small number of statistically investigated samples. A comprehensive
study of 2D two-phase composites with equal circular inclusions is developed. It is demonstrated that using the
concentration of inclusions and a contrast parameter is insufficient to properly study dispersed composites. The method of
structural sums in combination with ML to improve model accuracy is applied. Based on the study, a new approach is
suggested for selecting optimal parameters to analyze and classify two-dimensional dispersed composite structures. The
included content fits 2020 Mathematics Subject Classification: 74Q15, 74-10.
