# Development of Post Processing for Wave Propagation Problem: Response Filtering Method
## 2 Finite Element Analysis for the Transient Stress Response
### Wave Propagation Analysis and Gibb's Phenomenon  
Facilitated by the developments in hardware and software, finite element simulations of transient systems have been applied in many application areas [3]. Despite their successful applications, one of the remaining issues with transient finite element simulations is the Gibb's phenomenon, with its associated overshooting and undershooting phenomena [5]. This issue becomes a serious problem in structural analyses. To illustrate this problem, let us consider the stress analysis in Figure 2 as an example [5]. The right side of a 20-in-long rectangular bar (\(L_{\text{T}}\)) with a cross-sectional area (\(A\)) of 1 in\({}^{2}\) is fixed, and a step load of 100 lb is applied in the \(x\)-direction on the left side of the bar at \(t=0\). The Young's modulus (\(E\)), density (\(\rho\)), and Poisson's ratio (\(\
u\)) of the bar are \(30\times 10^{6}\) psi, \(7.4\times 10^{-4}\) lb-s\({}^{2}\)/in\({}^{4}\), and zero (for a one-dimensional problem), respectively. In this study, the analysis domain is discretized using a \(40\times 1\) mesh of 4-node rectangular elements (Q4). The mesh size of this example is chosen by following the parameters and setting of the reference [5]. In addition, it is worth to note that the issues of the stability and the accuracy of the numerical integration schemes should be considered, and the time of this example is chosen to consider the stabilities of the numerical integration schemes. The time step (sampling time), \(\Delta t\), is set to \(4.0\times 10^{-7}\) s smaller than the critical time ( \(\Delta t_{cr}=L_{c}/\
u\ \left(=2.483\times 10^{-6}\right)\), \(\
u:\sqrt{E/\rho}\), \(L_{c}\): characteristic length) for the stability of the central-difference method. Figure 2: Illustration of the mechanical stress wave problem: (**a**) the problem definition in [5], and (**b**) the force condition. The transient finite element simulation determines the transient solution with application of a sudden load to the cross section. The stress of the cross section in the steady-state solution becomes 100 psi. Figure 3 shows the mechanical stress value calculated at \(x=9.75\) in (20 th element). The solid and the dotted lines represent the analytical and numerical solutions, respectively. With mechanical properties we can calculate speed of wave propagation (\(\mathrm{v}=\sqrt{E/\rho}\)= \(2.012\times 10^{5}\) in \(\mathrm{/s}\)) and time stress curve. Before the arrival time (\(t_{1}=\frac{9.75}{2.012\times 10^{5}}=4.7214\times 10^{-5}\)s), the stress values at \(x=9.75\) in are zero because the stress wave has not yet arrived. After the first arrival time and before the next arrival time (\(t_{2}\)= \(\frac{9.75+2\times(20-9.75)}{2.012\times 10^{5}}=1.518\times 10^{-4}\)s), the analytical stress value becomes 100 psi. After \(1.5148\times 10^{-4}\)s, the analytical stress value increases to 200 psi owing to the arrival of the reflected wave at the right side. The steady-state stress is 100 psi. The stress values obtained with the explicit and implicit method are plotted with exact solution in Figure 3. The overall tendency is similar to that of the analytical solution, but some spurious oscillations naturally occur. It is observed that larger oscillations occur at the areas of sharp transition (\(t_{1}=4.7214\times 10^{-5}\)s and \(t_{2}=1.5158\times 10^{-4}\)s). Comparing the analytical and numerical solutions, it seems that the numerical solution obtained using the transient finite element simulation contains additional frequencies. However, later we will demonstrate that the numerical solution actually cannot carry valuable information in the frequency domain compared with the analytical solution. From a mathematical perspective, this phenomenon is referred to as Gibb's phenomenon and has been widely studied (see [4; 5] and the references therein). In finite element theory, considering this phenomenon, the stability issue has been researched [4; 5; 9; 14; 16; 20].