# Development of Post Processing for Wave Propagation Problem: Response Filtering Method
## 3 Response Filtering Method for Improved Transient Stress
### Response Filtering Method
The filtering length, filtering ratio and number of the filtered regions are \(FL\), \(FR\), and \(NR\), respectively. In the present study, the weight factors are set as equal to one. For example, the filtered stress in (17) with an \(FL\) value of two becomes the following:  
\[\widetilde{\sigma}=\left[\underbrace{\frac{\sigma_{1}+\sigma_{2}}{2},\frac{ \sigma_{1}+\sigma_{2}}{2}}_{\widetilde{\sigma}_{1}},\underbrace{\frac{\sigma_ {3}+\sigma_{4}}{2},\frac{\sigma_{3}+\sigma_{4}}{2}}_{\widetilde{\sigma}_{2}},\cdots,\underbrace{\frac{\sigma_{N-1}+\sigma_{N}}{2},\frac{\sigma_{N-1}+ \sigma_{N}}{2}}_{\widetilde{\sigma}_{FL}}\right] \tag{20}\]This simple filtering approach, which is similar to the stress average in static FE analysis, is carried out in the time domain rather than in the spatial domain. To validate the performance of the proposed method, the benchmark problem in Section 2 can be revisited. The transient stresses of the benchmark problem in Figure 2 are filtered using the proposed RFM by varying the ratio of filter data, \(FR\), in Figure 4. Moreover, the following norm-based absolute error for the stress (response) is employed to evaluate the performance of the present method for transient analysis:  
\[\xi=\int\!\left|\overline{\sigma}(t)-\sigma(t)\right|dt\ \text{or}\ \int\left| \overline{\sigma}(t)-\overline{\sigma}(t)\right|\!dt, \tag{21}\]  
where the exact (analytical), numerical, and filtered numerical solutions at time \(t\) are denoted by \(\overline{\sigma}(t)\) and \(\overline{\sigma}(t)\), respectively. Figure 4 presents the tendency of the accuracy improvement of the RFM with varying filtering ratio, \(FR\). Table 1 shows the performance of RFM with respect to filtering ratio (\(FR\)), the error valuer in Equation (21) is reduced by increasing \(FR\) by around 2% and also the number of oscillations is reduced as shown in Table 2. The table shows that the RFM could show similar performance regardless of integration methods. Approximately 2% of \(FR\) may show significant improvement in accuracy under different numerical conditions. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{10}{c}{**(a)**} \\ \hline \multicolumn{10}{c}{Central-difference method} \\ \hline Filtering ratio (\%) & w/o RFM & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 \\ Error indicator \(\xi(\times 10^{-3})\) & 1.8472 & 1.8283 & 1.7097 & 1.5967 & 1.3808 & 1.2124 & 1.2241 & 1.1895 \\ Number of over (under) shoots & 67 & 65 & 58 & 45 & 28 & 21 & 19 & 19 \\ \hline [PERSON]’s method & & & & & & & & \\ Filtering ratio (\%) & w/o RFM & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 \\ Error indicator \(\xi(\times 10^{-3})\) & 1.7223 & 1.7053 & 1.5941 & 1.4387 & 1.3123 & 1.1205 & 1.1937 & 1.1340 \\ Number of over (under) shoots & 65 & 65 & 65 & 49 & 27 & 20 & 20 & 19 \\ \hline \multicolumn{10}{c}{**(b)**} \\ \hline Central-difference method & & & & & & & & & \\ Filtering ratio (\%) & w/o RFM & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 \\ Error indicator \(\xi(\times 10^{-3})\) & 1.7883 & 1.7539 & 1.6543 & 1.4968 & 1.3046 & 1.3640 & 1.2136 & 1.0409 \\ Number of over (under) shoots & 65 & 64 & 59 & 41 & 26 & 19 & 19 & 17 \\ \hline Newmark’s method & & & & & & & & \\ Filtering ratio (\%) & w/o RFM & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 \\ Error indicator \(\xi(\times 10^{-3})\) & 1.7876 & 1.7515 & 1.6510 & 1.4962 & 1.3064 & 1.3618 & 1.2165 & 1.0495 \\ Number of over (under) shoots & 65 & 64 & 59 & 41 & 26 & 19 & 19 & 17 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: (**a**) Accuracy improvement of the RFM with different numerical integration method (at 20\({}^{\text{th}}\) element, \(\Delta t=\Delta t_{1}=4.0\times 10^{-7}\)s). (**b**) Accuracy improvement of the response filtering method with different numerical integration method (at 20\({}^{\text{th}}\) element, \(\Delta t=\Delta t_{1}\times 10^{-1}\)s).