# Development of Post Processing for Wave Propagation Problem: Response Filtering Method
## 3 Response Filtering Method for Improved Transient Stress
### Response Filtering Method  
As its name implies, the RFM filters the transient stress values in the time domain. A standard FE simulation is conducted to compute the transient stress values. As illustrated in the previous section, the Gibb's phenomenon hinders our ability to obtain accurate solutions using this method. By investigating the responses and their characteristics, this study proposes the concept of averaging the erroneous transient stress values using the RFM. Through averaging, or filtering, it is expected that the oscillations can be cancelled out, and thus the accuracy of the computed transient stress values can  
Figure 3: Comparison of the mechanical stresses at \(x=9.75\) (20 th element) for the benchmark problem in Figure 2.  
be improved. This concept has been applied for the stress analysis of static structure systems and is implemented in various commercial software [25; 26]. As it is a relatively simple concept, it is easy to implement in the post-processing stage of any FE package. The present study is confined to averaging (or filtering) in the time domain in order to illustrate the benefit of the proposed RFM. Figure 1 illustrates a schematic diagram of the proposed RFM. First, the stress and displacement responses are computed in the time domain using a conventional numerical approach, i.e., [PERSON]'s method. The resulting transient solutions are approximations of the analytical solution, and oscillations inevitably occur due to the Gibb's phenomenon. The displacement or stress response at each time step in the numerical approach can be described by (12), and without the loss of generality, the stress values are employed to illustrate the RFM. \[\sigma=[\sigma_{1},\sigma_{2},\cdots\sigma_{N-1},\sigma_{N}]\ \ (N:\ \text{number of time steps}), \tag{15}\]  
where the (von Mises) stress of target element is denoted by \(\sigma_{i}\), and the subscript \(i\) denotes the time step and set of stresses with respect to time step is \(\sigma\). The transient stress values can be divided into \(N\!R\) regions and renumbered for convenience as follows:  
\[ \begin{array}{ll}\sigma&=[\underbrace{\sigma_{1},\sigma_{2},\cdots,\sigma_{ FL}}_{\text{the 1 st region}},\underbrace{\sigma_{FL+1},\cdots,\sigma_{2 FL}}_{\text{the 2 nd region}},\cdots,\underbrace{\sigma_{N- FL+1},\cdots,\sigma_{N}}_{\text{the }N\text{-th region}}\\ &=[\underbrace{\sigma_{1}^{1},\sigma_{2}^{1},\cdots,\sigma_{FL}^{1}}_{\text{ the 1 st region}},\underbrace{\sigma_{1}^{2},\cdots,\sigma_{FL}^{2}}_{\text{the 2 nd region}}, \cdots,\underbrace{\sigma_{1}^{N\!R},\cdots,\sigma_{FL}^{N\!R}}_{\text{the }N\text{-th region}}]. \end{array}  \tag{16}\]  
The \"region\" defined in this research refers the time step region in which the solutions are averaged. To avoid confusion in notation, the \(i\)th stress at the \(j\)th region is denoted by \(\sigma_{i}^{j}\). The filter length, which is equal to the number of stress values in each region, is denoted by \(FL\). The filtered stresses or responses are obtained by averaging with a weight factor as follows:  
\[\widetilde{\sigma}=[\underbrace{\widetilde{\sigma}^{1}}_{\text{the 1 st}},\cdots, \underbrace{\widetilde{\sigma}^{k}}_{\text{the }k\text{-th}},\cdots, \underbrace{\widetilde{\sigma}^{N\!R}}_{\text{the }N\text{-th}}], \tag{17}\]  
\[ \begin{array}{ll}\widetilde{\sigma}^{k}=[\sigma_{1}^{k},\cdots,\widetilde{ \sigma}_{j}^{k},\cdots,\widetilde{\sigma}_{FL}^{k}],\ (\widetilde{\sigma}_{1}^{k}=\widetilde{\sigma}_{j}^{k}= \widetilde{\sigma}_{FL}^{k}),\end{array}  \tag{18}\]  
\[\widetilde{\sigma}_{j}^{k}=\frac{\sum\limits_{i=1}^{FL}w_{i}^{k}\sigma_{i}^{k}} {\sum\limits_{i=1}^{FL}w_{i}^{k}},\ k=1,2,\cdots,N\!R,\ j=1,2,\cdots,FL\!\left(N =FL\times NR,\ FR=\frac{FL}{N}\times 100\right), \tag{19}\]  
where the set of original and filtered stresses (responses) at all time steps are denoted by \(\sigma\) and \(\widetilde{\sigma}\), respectively. The \(i\)th filtered stress and the weighting factor in the \(k\)th domain are denoted by \(\sigma_{j}^{k}\) and \(w_{j}^{k}\), respectively.