# Enhanced Upward Translations for Systems with Clusters
## 2 General Considerations on Clustering as Applied to Upward Translations  
Our objective is to increase the accuracy of the starting S2M translations when solving a problem for systems with multiple concentrators of fields, such as crack tips, common vertices of grains, edges of thin inclusions, corner points at boundaries, etc. The areas of high concentration are associated with clusters, and the accuracy of their translated fields influences the accuracy of final results. As known (e.g., [31], [[https://en.wikipedia.org/wiki/Cluster_analysis](https://en.wikipedia.org/wiki/Cluster_analysis)]([https://en.wikipedia.org/wiki/Cluster_analysis](https://en.wikipedia.org/wiki/Cluster_analysis)), accessed on 15 July 2024), in general, the notion of a \"cluster\", cannot be precisely defined. Its exact definition is problem-oriented. Hence, we need to delineate the specific features of our problem, which direct to an exact definition of a cluster and to a corresponding algorithm. These features are as follows. A typical BEM employs discretization of boundaries and approximation of the densities by form-functions with nodes at assigned points. The nodal points serve to build a hierarchical tree of a FMM. The total far field, generated by elements of a leaf and corresponding to a particular kernel of the starting BIE, may be evaluated as the sum of products of the kernel at a nodal point by an intensity at this point. The intensity itself is defined as the product of the nodal value of the density (scalar or vector) by integral of appropriate form-function over the element containing this node. Thus, the far field of a leaf is presented by the sum of far fields of point sources located in the leaf. The intensities of the sources for quantities, defined by boundary conditions, are known. They are accounted for in the right-hand side of an algebraic system to be solved. The intensities of the remaining sources are unknown. For large systems, they are found iteratively using matrix-to-vector multiplication on each iteration. The matrix in the product consists of influence coefficients, calculated for unit intensities. The vector of intensities, used in the matrix-to-vector product, is presumed known at the beginning of a current iteration (for the first iteration, rough approximate initial values are found by a simple algorithm, defined, for instance, by a preconditioner). Thus to the beginning of a current iteration performed by an FMM, the intensity and location of each source in a leaf are known. We see that distinguishing a cluster in a leaf may involve merely (i) the intensity of each source in a leaf, and (ii) its location. Knowing locations makes also known the distances between the sources. Only this information on sources is actually needed and used in S2M translations, which predefine the accuracy of a whole chain of up- and downward translations. As has been established in [24], the accuracy of conventional S2M translations strongly depends on the locations and intensities of sources. We aim to increase the accuracy of translations by constructing and appropriately employing clusters of closely located sources. A detailed analysis of the accuracy of S2M translations is provided in [24]. It shows that for a leaf with the equivalent surface (ES) of radius \(R_{e}\) (Figure 1a), the smallest error of S2M translations is observed for sources located in the middle of a leaf. The error increases as the sources approach the ES. The greatest error arises for sources located near the ES of a leaf. These are sources in the ring (spherical layer) of radii \(0.75R_{e}\) and \(R_{e}\) (shaded area in Figure 1a). Specifically, as established in the paper [24], the error grows from two orders for weakly singular operators to five orders for singular and hypersingular operators. The greatest error arises for sources located near the ES of a leaf. Actually, only this group of sources is essential for assembling them into clusters with the aim of increasing the accuracy of translations.