# Enhanced Upward Translations for Systems with Clusters
## 5 Simplified Enhanced Translations  
Each of the modified translations, introduced to increase the accuracy of calculated far fields, requires evaluation the matrix of the system (9) and its inversion. Depending on the location of the cluster center, the matrices are different for various clusters. The calculation of the matrices and their inversion for each of the clusters is an additional computational burden. Despite it is performed only once, of value is to avoid it, for the price of partial loss of accuracy. The notion of the zero error of the modified translation for a cluster consisting of a single (leading) source suggests the wanted simplification. Indeed, as mentioned, for such a cluster, the modified equivalent intensities \(\mathbf{q}_{mesp}\) (\(s=1,\ldots,N_{e}\)) are zero. Consequently, by (7) the modified far field is zero. Then (6) implies that the actual far field \(\mathbf{f}_{p}(\mathbf{x})\) of such a cluster \(p\) is promptly found as  
\[\mathbf{f}_{p}(\mathbf{x})=\mathbf{K}\Big{(}\mathbf{x},\mathbf{y}_{clp}\Big{)}\mathbf{q}_{clp} \tag{12}\]  
Thus, for this cluster, there is no need for evaluation and inversion of the matrix of the system (9). Equation (12) expresses the evident fact that the far field is generated by the particular leaf source, which has served as the leading source for the \(p\)-th cluster. This trivial result implies that in a general case, when the modified (residual) far field (6) of a cluster \(p\) is notably less than the field of the source representing it, the total far field (5) of all its sources may be quite accurately calculated by using (12). It remains to establish exactly when the simplification (12) is applicable. The following rough estimation and numerical data on the maximal relative errors serve to obtain an answer for typical kernels. In many applied problems, like those of fracture mechanics, a cluster is comprised of a large flaw, surrounded by many small cracks. The far field, generated by the single vector representing the cluster, is actually the same as the sum of far fields of all its entries. Even in very unfavorable cases, the upper bound of relative maximal error is of order \(E=r_{cl}/R\), where \(R\) is the distance from the cluster center. In particular, the estimation refers to a cluster composed of merely two sources of equal intensities located at the maximal distance \(2r_{cl}\) between them. In 3D, the minimal distance between the equivalent surface of a leaf and the check surface of its parent is \(2.44R_{e}\), where \(R_{e}\) is the radius of the child equivalent surface. Then for \(r_{cl}<0.125\)\(R_{e}\), the upper bound of the relative error is \(E<0.125/2.44\approx 0.051\). For sources of equal intensities, which are more/less evenly distributed around the leading source, the actual maximal error is notably less. Table 8 evidently demonstrates this for a number of typical kernels. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\varepsilon\) & **0.1** & & **0.2** & \\ \hline \(r_{cl}\)/\(R_{e}\) & **0.03** & **0.07** & **0.03** & **0.07** & **0.12** \\ \hline \multicolumn{5}{c}{**2D Laplace hypersingular BIE and hypersingular kernel**} \\ \hline
**max** & 0.00054 & 0.00104 & 0.00052 & 0.00100 & 0.00161 \\ \hline
**average** & 0.00032 & 0.00065 & 0.00032 & 0.00064 & 0.00109 \\ \hline \multicolumn{5}{c}{**3D Laplace hypersingular BIE and hypersingular kernel**} \\ \hline
**max** & 0.00016 & 0.00033 & 0.00016 & 0.00032 & 0.00066 \\ \hline
**average** & 0.00013 & 0.00026 & 0.00013 & 0.00026 & 0.00054 \\ \hline \multicolumn{5}{c}{**3D Navier hypersingular BIE and hypersingular kernel**} \\ \hline
**max** & 0.00201 & 0.00582 & 0.00176 & 0.00505 & 0.01934 \\ \hline
**average** & 0.00053 & 0.00117 & 0.00053 & 0.00116 & 0.00299 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Maximal and average errors of the far field when representing the cluster by its representative source.