## 5 Numerical Examples
### Solid Plate Numerical Example
The analysis results in the Z direction are compared in Figure 7.  
We used R-squared (R2) and the relative average absolute error (RAAE) [27] to illustrate accuracy. The R2 and RAAE are computed using:  
\[R^{2}=1-\sum\limits_{i=1}^{n}{(\mathbf{y}_{i}-\hat{\mathbf{y}}_{i})^{2}}/\sum\limits_{i =1}^{n}{(\mathbf{y}_{i}-\overline{\mathbf{y}}_{i})^{2}} \tag{81}\]  
\[\text{RAAE}=\frac{\sum\limits_{i=1}^{n}{|\mathbf{y}_{i}-\hat{\mathbf{y}}_{i}|}}{n\times \text{STD}} \tag{82}\]  
where \(n\) is the length of response data; \(\mathbf{y}_{i}\) and \(\hat{\mathbf{y}}_{i}\) are the acceleration response results of the FEM and IBS methods, respectively; \(\overline{\mathbf{y}}_{i}\) and STD are, respectively, the mean and standard deviation of the FEM result. The accuracy of NMI-IBS is presented in Table 1.  
The results of the two interpolation schemes are slightly different. However, as shown in Figure 6 and Table 1, displacement and velocity achieve strong agreement with the reference. The acceleration result is not as good as that of displacement but still shows relatively high accuracy. This could be affected by the correlation function (or weight function) and the distribution of the interface support nodes. Under the distribution of the interface support nodes in Figure 4 and \(\alpha=1\), the R2 of the acceleration in the Z direction with the varying radius of the compact support domain \(h_{c}\) is shown in Table 2.  
As shown in Table 2, the accuracy of the moving Kriging interpolation scheme changes slightly when \(h_{c}\) changes. Comparatively, the accuracy of the moving least squares interpolation scheme is much more sensitive with \(h_{c}\), even though they both perform quite well when \(h_{c}\) changes in an appropriate range. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **R2** & & **RAAE** \\ \hline  & **NMI-IBS-MK** & **NMI-IBS-MSL** & **NMI-IBS-MK** & **NMI-IBS-MSL** \\ \hline Displacement & 0.99987 & 0.99986 & 0.00984 & 0.00875 \\ Velocity & 0.99961 & 0.99972 & 0.01283 & 0.00932 \\ Acceleration & 0.97904 & 0.98493 & 0.10491 & 0.08743 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Accuracy of the NMI-IBS for response of the solid plate in Z direction. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\mathbf{h}_{c}\) & **60** & **85** & **100** & **200** & **300** & **400** & **500** & **600** \\ \hline NMI-IBS-MK & 0.96423 & 0.97903 & 0.97313 & 0.98650 & 0.98296 & 0.97836 & 0.98239 & — \\ \hline NMI-IBS-MSL & — & 0.98491 & 0.98317 & 0.97977 & 0.95931 & 0.92501 & 0.90041 & 0.88771 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: R2 of acceleration between the NMI-IBS and the reference in Z direction with various \(h_{c}\). Figure 7: Comparison between the NMI-IBS and the reference for the solid plate. However, both interpolation schemes meet numerical problems when \(h_{c}\) varies outside the desired range. For the moving least squares interpolation, it cannot form its interpolation function when the compact support domain is too small, such as \(h_{c}=60\), in this configuration of the interface. Meanwhile, the interpolation function of the moving Kriging interpolation no longer satisfies the normalization condition when a relatively large compact support domain is chosen, such as \(h_{c}=600\). Once the normalization condition is not satisfied (i.e., \(\sum\limits_{i=1}^{N1}G_{u(i,j)}^{(1)}\
eq 1\)), the force equilibrium cannot be achieved and the results become incorrect, as shown in Figure 8.  
The coefficient \(\alpha\) in Equation (64) is used to fit the model. Now, we set \(h_{c}=85\), the R2 of the acceleration in the Z direction, with the varying \(\alpha\), as shown in Table 3.  
The results in Table 3 show that the performance of the moving Kriging interpolation scheme is more sensitive with \(\alpha\). However, as shown in Tables 2 and 3, both interpolation schemes perform well enough when \(\alpha\) and \(h_{c}\) remain within an appropriate range.