# Development of Post Processing for Wave Propagation Problem: Response Filtering Method
## 2 Finite Element Analysis for the Transient Stress Response  
### Transient Finite Element Simulation  
The transient response of a mechanical structure can be analyzed using the finite element procedure as follows [3,5,6]:  
\[\ddot{\mathbf{U}}_{i}=\mathbf{M}^{-1}\Big{[}\mathbf{F}_{i}-\mathbf{C}\dot{ \mathbf{U}}_{i}-\mathbf{K}\mathbf{U}_{i}\Big{]} \tag{1}\]  
where the mass, damping, and stiffness matrices are denoted by \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\), respectively, and the force vector at the \(i^{\text{th}}\) time step is denoted by \(\mathbf{F}_{i}\), and the displacement, velocity, and acceleration vectors at the \(i^{\text{th}}\) time step are denoted by \(\mathbf{U}_{i}\), \(\dot{\mathbf{U}}_{i}\), and \(\ddot{\mathbf{U}}_{i}\), respectively. Many numerical schemes can be applied to solve the above [PERSON]'s law. This study implements both explicit and implicit integration method to investigate performance of the present RFM. Central-difference method and Newmark's method are adopted as representation of explicit and implicit integration methods. Central-difference method approximates velocity and acceleration by  
\[\ddot{\mathbf{U}}_{i}=\frac{1}{2\Delta t}(\mathbf{U}_{i+1}-\mathbf{U}_{i-1}) \tag{2}\]  
\[\ddot{\mathbf{U}}_{i}=\frac{1}{\Delta t^{2}}(\mathbf{U}_{i+1}-2\mathbf{U}_{i} +\mathbf{U}_{i-1}) \tag{3}\]  
and \(\mathbf{U}_{i+1}\) and \(\mathbf{U}_{i-1}\) can be obtained by Taylor series about time \(i\Delta t\):  
\[\mathbf{U}_{i+1}=\mathbf{U}_{i}+\frac{\Delta t}{1!}\dot{\mathbf{U}}_{i}+\frac {\Delta t^{2}}{2!}\ddot{\mathbf{U}}_{i}+\frac{\Delta t^{3}}{3!}\ddot{\mathbf{ U}}_{i}+\cdots \tag{4}\]  
\[\mathbf{U}_{i-1}=\mathbf{U}_{i}-\frac{\Delta t}{1!}\dot{\mathbf{U}}_{i}+\frac {\Delta t^{2}}{2!}\ddot{\mathbf{U}}_{i}-\frac{\Delta t^{3}}{3!}\ddot{\mathbf{ U}}_{i}+\cdots \tag{5}\]  
Combining Equations (2)-(5) with the second-order accuracy provides the following equation:  
\[\mathbf{U}_{i+1}=\bigg{[}\frac{1}{\Delta t^{2}}\mathbf{M}+\frac{1}{2\Delta t }\mathbf{C}\bigg{]}^{-1}\bigg{(}\mathbf{F}_{i}-\mathbf{K}\mathbf{U}_{i}+\frac {1}{\Delta t^{2}}\mathbf{M}(2\mathbf{U}_{i}-\mathbf{U}_{i-1})+\frac{1}{2 \Delta t}\mathbf{C}\mathbf{U}_{i-1}\bigg{)} \tag{6}\]  
In [PERSON]'s method [3,5,6], displacement vector of \(i+1^{th}\) time step can be obtained as follows:  
\[\dot{\mathbf{U}}_{i+1}=\dot{\mathbf{U}}_{i}+(\Delta t)\Big{[}(1-\gamma)\ddot{ \mathbf{U}}_{i}+\gamma\ddot{\mathbf{U}}_{i+1}\Big{]} \tag{7}\]  
\[\mathbf{U}_{i+1}=\mathbf{U}_{i}+(\Delta t)\dot{\mathbf{U}}_{i}+(\Delta t)^{2} \bigg{[}\bigg{(}\frac{1}{2}-\beta\bigg{)}\ddot{\mathbf{U}}_{i}+\beta\ddot{ \mathbf{U}}_{i+1}\bigg{]} \tag{8}\]  
where \(\Delta t\) is the time step, and the two parameters of [PERSON]'s scheme are \(\beta\) and \(\gamma\)[6]. With the above approximations of [PERSON]'s method, the following discretized equations in the time domain can be obtained:  
\[\mathbf{K}^{{}^{\prime}}\mathbf{U}_{i+1}=\mathbf{F}^{{}^{\prime}}_{i+1}, \tag{9}\]  
\[\mathbf{K}^{{}^{\prime}}=\frac{\mathbf{M}}{\Delta t^{2}\beta}+\frac{\gamma}{ \Delta t\beta}\mathbf{C}+\mathbf{K}, \tag{10}\]\[ \begin{array}{ll}\mathbf{F}^{{}^{\prime}}_{i+1}&=\mathbf{F}_{i+1}+\left[\frac{ \mathbf{M}+\Delta t\gamma\mathbf{C}}{\Delta t^{2}\beta}\right]\mathbf{U}_{i}+ \left[\frac{\mathbf{M}+\Delta t\gamma\mathbf{C}}{\Delta t\beta}-\mathbf{C} \right]\tilde{\mathbf{U}}_{i}\\ &+\left[\left(\frac{1}{2\beta}-1\right)(\mathbf{M}+\Delta t\gamma\mathbf{C})- \Delta t(1-\gamma)\mathbf{C}\right]\tilde{\mathbf{U}}_{i}.\end{array}  \tag{11}\]  
In this study, the coefficients of [PERSON]'s method, \(\beta\) and \(\gamma\), are set to \(\frac{1}{4}\) and \(\frac{1}{2}\), respectively for unconditional stability. The dynamic stress at the \(i\)th time step can be calculated from the displacements at the \(i\)th time step as follows:  
\[\boldsymbol{\sigma}_{i}=\left[ \begin{array}{c}\sigma_{x}\\ \sigma_{y}\\ \tau_{xy}\end{array} \right]_{i}=\mathbf{DBU}_{i}. \tag{12}\]  
\[\mathbf{D}=\frac{E}{1-\
u^{2}}\left[ \begin{array}{ccc}1&\
u&0\\ \
u&1&0\\ 0&0&\frac{1-\
u}{2}\end{array}