## 1 Introduction
### Analytical and Asymptotic Methods  
We can now proceed with a brief review of the third analytical method based on the theory of multipoint correlation functions [25; 26; 27; 28]. This method is based on the complete theoretical statistical analysis. However, its practicality is limited by computational restrictions of correlation functions to the third order. For example, the work [29] contains a comprehensive study of fiber orientation. At the same time, the distribution of fibers is studied employing the two- and three-point probability distribution functions, where the second-order functions approximate the three-point functions. Therefore, this method leads to low-order formulas similar to SCMs. Another analytical approach is based on the series expansion of unknown functions. The problem is reduced to the solution of infinite systems of linear algebraic equations. This method was proposed by [PERSON] in 1892 for regular composites and systematically developed by [PERSON] [30] and [PERSON] et al. [31; 32; 33]. The method of functional equations [34] can be considered as a continuous representation of discrete methods for regular circular and spherical inclusions and its extension to random composites. This method in discrete form coincides with the [PERSON] and [PERSON] methods for regular arrays of circular and spherical inclusions. It led to the definition of the analytical representative volume element (\(a\)RVE) summarized in [34; 35]. The functional equation method can be considered the generalized alternating method of [PERSON] developed in [21; 22] for arbitrary inclusion shapes. The \(a\)RVE theory includes the most known special results for dispersed composites. It explicitly demonstrates the precision of the derived formulas dependent on concentration \(f\) and contrast parameters. Our main goal is the development of a comprehensive strategy for computing the effective properties of materials according to the existing homogenization theory and the new concept of \(a\)RVE. The method described in Section 1.1 yields results for the numerically given values of component constants and their respective locations. The approach presented in Section 1.2 results in approximate formulas in a low order. The latter produce acceptable results in the case of dilute and low-contrast composites. The aforementioned approaches fail when applied to a large number of inclusions or to symbolic geometric and physical parameters relevant to optimal design problems. In contrast, the method proposed in this paper leverages structural sums, a fundamental element of the \(a\)RVE theory. This method [36] leads to a more refined classification of different classes of composites, which was unattainable using previous approaches. Importantly, it has been used in [37; 38] using only a single structural sum, \(e_{2}\). It allows to describe the effective properties of nanocomposites arising from the mesoscopic and the macroscopic levels. In this paper, we extend the study by incorporating multiple structural sums, significantly enhancing the effectiveness of the method. Rather than relying on a standard FEM package or simplified formulas, we propose a refined strategy to consider a diverse range of dispersed random composites. In Section 2, we outline the principles of homogenization, which imply that the effective constants of dispersed random composites can be derived from a periodicity problem. The remainder of this paper focuses on the constructive implementation of the \(a\)RVE theory for two-dimensional (2D) dispersed composites. The main steps of the strategy are detailed in Section 3. Section 4 is concerned with deterministic composites, while structural sums are discussed in Section 5. The generation of random composites and polynomial approximations for their effective constants is covered in Section 6. Finally, Section 7 explores the asymptotically equivalent transformations of the obtained polynomials to expressions that describe percolation regimes.