# Development of Post Processing for Wave Propagation Problem: Response Filtering Method
## 1 Introduction  
This study develops a new response filtering approach for recovering dynamic mechanical stress under impact loads. From a structural engineering point of view, a mechanical stress wave, due to structural impacts such as an impact force, explosion, or collision, that propagates through a structural medium is important. When this wave reaches weak parts of the structure, it can cause unexpected catastrophic failure or fracturing of structural components. Thus, it is important to compute and consider these stress waves in mechanical or civil engineering [1; 2; 3; 4]. With the use of state-of-the-art computational theory and computer hardware, numerical solutions can be regarded as acceptably close to the analytical solution. However, from a theoretical perspective, some discrepancies and errors remain, and verification of the convergence to the true solution is still important. Some efforts have been undertaken to reduce the differences and the numerical errors between the numerical solutions and analytical solutions [3; 5; 6]. To this end, this study develops a new post-processing approach for transient stress waves named the response filtering method (RFM) with an average operator. Numerical methods, e.g., the finite element (FE) method, finite volume method, or finite difference method, have been employed for the transient analysis of complex structures. The numerical solutions obtained using proper computational theory usually converge to the analytical solutions with a refined time step and refined mesh [3; 5; 6]. However, it has been observed that it is \(\widetilde{f}=\frac{f_{t}}{FL}\cdot m\pm f\) difficult for a numerical solution to describe an analytical solution that has significant discontinuity,even when proper meshes and time steps are adopted. In finite element (FE) numerical solutions, overshooting, undershooting, and oscillations occur around the discontinuity point known as the Gibbs phenomenon [1; 7; 8; 9]. Various studies have been conducted aiming to reduce these types of discrepancies between the analytical and numerical solutions. Some relevant research can be found to improve the numerical accuracy by introducing the numerical damping [10]. For instance, the discontinuous Galerkin method and variational integrator have been proposed to accurately describe discontinuities in the numerical solutions [11; 12; 13; 14; 15]. Other approaches have been proposed to reduce the dispersive and dissipative errors in the numerical solution of partial difference equations [3; 4; 16; 17]. [PERSON] and [PERSON] make a significant contribution to the finite element method for wave propagation problem. They present higher-order spatially discretized finite elements with an extra computational cost for the wave propagation problem [3; 16] and also present new implicit and explicit integration methods with Noh [4; 17]. Linear combinations of consistent and lumped mass matrices [18; 19; 20] and modified spatial integration rules for mass and stiffness matrices [21; 22] have been considered to ensure high-fidelity solutions. Comprehensive and considerable benchmark problems for wave propagation can be found in the work of [PERSON] et al. [23]. However, in numerical solutions, errors due to Gibb's phenomenon cannot be eliminated completely, and some improvements are required to the oscillation of the numerical solution [9]. A few post-processing methods have been developed to reduce these kinds of numerical errors [21; 23; 24]. The objective of this study is not to develop a new finite element or integration method but to develop a simple post-processing approach, i.e., the RFM, to reduce discrepancies between the numerical and analytical solutions. In this study, only one-dimensional problems are considered to calculate exact solutions (analytical solution) as references. The transient stress values in the discrete time domain are simply divided into several subset domains, and each subset domain is averaged.