# Enhanced Upward Translations for Systems with Clusters
## 2 General Considerations on Clustering as Applied to Upward Translations
Since they are within the layer of the thickness \(h=0.25R_{e}\), the distance \(r\) of each of them to the leaf center exceeds \(0.75R_{e}\). This implies that there is no sense in taking the cluster radius \(r_{cl}\) exceeding the thickness \(h\) of the spherical layer; thus \(r_{cl}<0.25R_{e}\). For the average location of the source centers on the middle surface of the spherical layer, the reasonable value of the cluster radius is \(r_{cl}<0.5h=0.125R_{e}\). The inequality \(r>0.75R_{e}\) may serve to distinguish the sources, for which it is reasonable to improve S2M translation. Thus merely these sources may be assembled into clusters. The remaining leaf sources may be clustered or may stay ungrouped. In the latter case, their far fields are translated conventionally by using S2M translations, described, for instance, in [18; 20]. For them, the errors of conventional translations are much less than those for sources in the spherical layer. In general, a BEM may employ a number of kernels. We consider any of them, and further discussion refers merely to those leaf sources, whose far fields are generated by a particular kernel \(\mathbf{K}(\mathbf{x},\mathbf{y})\). For brevity, these sources will be called leaf sources. Their number in a leaf is denoted \(N_{L}\). The _first step_ of creating clusters consists of establishing the _hierarchy_ of leaf sources. To this purpose, the intensity (scalar or vector) of a source is characterized by its norm. The latter is taken from Euclidian. Thus when the intensity is a scalar, the norm equals its absolute value; for a vector, it is the square root from the sum of squared components. The norm \(q=|\mathbf{q}|\) of intensity \(\mathbf{q}\) gives the source magnitude. Of essence is that a mag  
Figure 1: (**a**) A leaf surrounded by equivalent surface (ES) of radius \(R_{e}\), (**b**) a typical cluster of radius \(r_{cl}\) with sources near the ES. nitude is a non-negative scalar. This serves to establish the hierarchy of the leaf sources by their assembling in the order of decreasing magnitudes \(q_{1}\geq q_{2}\geq\ldots\geq q_{N_{L}}\geq 0\), \(q_{j}=\Big{|}\mathbf{q}_{j}\Big{|}j=1,\ldots,\ N_{L}\). The ordering is efficiently performed by using a standard procedure, say [PERSON]. This path of establishing hierarchy has been effectively applied for zero-order moments [25] to reduce the total number of M2L translations. It is equally applicable to the leaf sources subjected to S2M translations. The _second step_ of clustering consists of using the established hierarchy for successive _grouping_ of the sources into clusters. The grouping is performed as follows. The source with the highest magnitude becomes the object around which the first cluster is grouped. This is the leading source for the cluster. The grouping may be performed in a standard way, by using distance connectivity. Indeed, with known distances between sources, the metric distance matrix [31], is actually known. Thus we may distinguish as a cluster the leading source itself and all the sources, whose distances to it do not exceed an assigned cluster radius \(r_{cl}\) (Figure 1). The latter may be chosen as convenient. As mentioned, for a leaf with the equivalent surface (ES) of radius \(R_{e}\), it is appropriate to take the cluster radius \(r_{cl}\) not exceeding \(0.125\ R_{e}\). To the moment, the cluster radius \(r_{cl}\) is assumed set. A similar process is applied to the remaining sources in the hierarchical ordering of sources. This defines the second cluster, grouped around the second leading source and separated from the first cluster. The process is continued until all the sources of the leaf are distributed between clusters. Finally, the collection of leaf sources is represented by the collection of clusters. Thus \(N_{L}=\sum_{p=1}^{N_{cl}}M_{clp}\), where \(N_{cl}\) is the total number of clusters, \(M_{clp}\) is the number of sources in the \(p\)-th cluster. The _third step_ consists of appropriately _characterizing_ a cluster.