# Development of Post Processing for Wave Propagation Problem: Response Filtering Method
## 4 Numerical Examples
However, the larger filtering length does not always guarantee better accuracy improvement (see Table 4). It should be noted that the RFM also plays an important role in the frequency domain. To explain this, the frequency components of the numerical solution without the RFM and analytical solution are presented in Figure 2.0  
Figure 12 shows that there is an enormous difference between the frequency components of the analytical solution and the numerical solution without the RFM. In particular, the analytical solution contains all of the frequency components, whereas the numerical solutions contain only the low-frequency components. Generally, a finer sampling time is required to consider higher frequencies, thereby increasing the computational cost. However, even finer sampling does not always guarantee inclusion of the higher-frequency components that appear in the analytical solution, as shown in Figure 2.1 In this study, however, it was discovered that a higher frequency range can be covered by using the RFM without modification of the sampling time. Figure 2.2 Schematic diagram of the analytical solution. As shown in Figure 13a, when the RFM is not applied, only the low-frequency components are present (black line). However, with the current RFM, several high-frequency components are generated (red line). In addition, increasing the filtering length creates additional new frequency components \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Newmark’s Method** & & & & & & & & & \\ \hline Filtering ratio (\%) & w/o RFM & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 \\ Error indicator \(\xi(\times 10^{-3})\) & 2.5622 & 2.5463 & 2.4712 & 2.3883 & 2.3330 & 2.0453 & 2.0096 & 2.0617 \\ Number of over (under) shoots & 40 & 40 & 40 & 34 & 28 & 22 & 22 & 22 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Accuracy improvement of the RFM with varying filtering length in Example 1. Figure 11: Changes before and after filtering with different filtering lengths and comparison with the analytical solution for Example 1: (**a**) without filtering, (**b**) with \(FR=0.5\%\), (**c**) with \(FR=1\%\), and (**d**) with \(FR=2\%\). Figure 12: Decomposition of the analytical solution and the numerical solutions without RFM in the frequency domain for Example 1. (see Figure 13b), and if the filtering length is greater than a certain length (in this example, greater than 8), the filtered numerical solutions can expand to cover all the frequency components (see Figure 13c). Note that the frequency components of the filtered numerical solution are changed without changing the sampling time. In general, the frequency components of the numerical solution are affected by the sampling time in a discrete system. In other words, with a chosen sampling time, the higher-frequency component is also determined. Applying the RFM can increase the higher frequency, which cannot be obtained by changing the sampling time. **Example 2**. : _Complex geometric example_  
Figure 13: Changes to the numerical solution after filtering with different filtering ratio and comparison with the analytical solution for Example 1 in the frequency domain: (**a**) filtered solution with \(FR=0.5\%\), (**b**) with \(FR=1\%\), and (**c**) with \(FR=2\%\). As the second example, Figure 14 considers a structure with many branches. Each bar has the same material properties as the previous problems: \(E=30\times 10^{6}\) psi, \(\rho=7.4\times 10^{-4}\) lb-s\({}^{2}\)/in\({}^{4}\), \(\
u=0\). A force is applied at the right arm, and the other arms are clamped. The stress at the center of the analysis domain is computed with and without the RFM. Figure 14 shows the detailed shape of the complex beam structure with three types of rectangular beams. The example structure is discretized using \(0.5\) in \(\times\)\(0.5\) in Q4 elements. It is not clear whether exact solution to this problem can be calculated.