# Enhanced Upward Translations for Systems with Clusters
## 4 Numerical Results for Typical Kernels of BIE
### 3D Navier BIE  
The calculations above show that, similar to the conclusions of [24], the errors of translation are greater for kernels of hypersingular BIE (11) than those of singular BIE (10). For 3D BIE, they are larger than 2D. Furthermore, the errors are greatest when translations are performed by means of a kernel depending on the direction of the normal at a source point. Thus, for the 3D Navier BIE, we consider the BIE (11) with emphasis on the hypersingular kernel. Translation by means of singular kernel \(Q_{nx,ik}(x,y)\). The kernel is \(Q_{nx,ik}(x,y)=\frac{1}{8\pi(1-v)}\frac{1}{P^{3}}n_{xj}\Big{[}(1-2v)(\delta_{ ij}r_{k}-r_{i}\delta_{jk}-r_{j}\delta_{ik})-\frac{3}{r^{2}}r_{i}r_{j}r_{k}\Big{]}\), where \(v\) is the Poisson ratio. The maximal and average _relative_ errors are provided in Table 6. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Conventional**} & \multicolumn{4}{c}{**Modified**} \\ \hline \(\varepsilon\) & **0.02** & **0.1** & **0.2** & **0.02** & **0.1** & **0.2** & \\ \hline \(r_{cl}/R_{e}\) & **0.01** & **0.07** & **0.07** & **0.12** & **0.01** & **0.07** & **0.07** & **0.12** \\ \hline \(r_{cl}/R_{e}\) & **0.01** & **0.07** & **0.07** & **0.12** & **0.01** & **0.07** & **0.07** & **0.12** \\ \hline \(\max\) & \(2.39\times 10^{-5}\) & \(1.11\times 10^{-5}\) & \(7.55\times 10^{-6}\) & \(7.52\times 10^{-6}\) & \(1.31\times 10^{-8}\) & \(5.88\times 10^{-8}\) & \(2.22\times 10^{-8}\) & \(7.98\times 10^{-8}\) \\ \hline \(\mathbf{average}\) & \(4.75\times 10^{-6}\) & \(2.24\times 10^{-6}\) & \(2.01\times 10^{-6}\) & \(2.01\times 10^{-6}\) & \(3.15\times 10^{-9}\) & \(1.80\times 10^{-8}\) & \(5.63\times 10^{-9}\) & \(1.79\times 10^{-8}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Maximal and average errors for translations by means of _singular_ kernel (3D _hypersingular_ Laplace BIE). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Conventional**} & \multicolumn{4}{c}{**Modified**} \\ \hline \(\varepsilon\) & **0.02** & **0.1** & **0.2** & **0.02** & **0.1** & **0.2** & \\ \hline \(r_{cl}/R_{e}\) & **0.01** & **0.07** & **0.07** & **0.12** & **0.01** & **0.07** & **0.07** & **0.12** \\ \hline \(\max\) & \(0.00012\) & \(0.00010\) & \(9.68\times 10^{-5}\) & \(9.68\times 10^{-5}\) & \(4.35\times 10^{-8}\) & \(2.42\times 10^{-7}\) & \(8.08\times 10^{-8}\) & \(2.29\times 10^{-7}\) \\ \hline \(\mathbf{average}\) & \(4.94\times 10^{-5}\) & \(4.80\times 10^{-5}\) & \(4.84\times 10^{-5}\) & \(4.85\times 10^{-5}\) & \(2.17\times 10^{-8}\) & \(1.04\times 10^{-7}\) & \(3.65\times 10^{-8}\) & \(1.05\times 10^{-7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Maximal and average errors for translations by means of _hypersingular_ kernel (3D _hypersingular_ Laplace BIE). Similar to the results for the Laplace equation, in all the cases, the accuracy of the modified translations is better than that of the conventional approach. When the relative radius of the cluster does not exceed 0.07, the gain in the accuracy is an order, at least. Translation by means of hypersingular kernel \(H_{nx,ny}(x,y)\). The kernel is \(H_{nx,ny\pm ik}(x,y)=\frac{2\mu}{8\pi(1-\
u)}\frac{1}{\tau^{3}}n_{xy}\{(1-4\
u) \delta_{ij}\eta_{yk}-(1-2\
u)(n_{yi}\delta_{jk}+n_{y}\delta_{ik})-\frac{3}{\tau^ {2}}[r_{s}n_{ys}((1-2\
u)\delta_{ij}r_{k}+\
u(r_{i}\delta_{jk}+rj\delta_{ik}))\] \[+\
u(n_{yi}r_{i}+n_{yi}r_{j})r_{k}+(1-2\
u)r_{i}r_{j}n_{ik}]+ \frac{15}{\tau}r_{s}n_{ys}r_{i}r_{j}r_{k}\}\). The kernel depends on the direction of the normal at a source point, where \(\mu\) is the shear modulus. This case is of special significance for fracture mechanics and, as appears from the results of [24], the accuracy of translations for it is worst. Even when using 386 nodes at ES, the maximal relative error of conventional S2M translations may reach 70% when a point source is located at a node of ES; only when using 602 nodes it decreases to 2%. Thus it is quite desirable to improve the accuracy.