# Assimilation of Dynamic Combined Finite Discrete Element Methods Using the Ensemble Kalman Filter
## 3 SHPB Experiment and HOSS Model Setup  
The SHPB experiment is used to indirectly determine the tensile strength of materials at high rates of strain. In this type of experimental setup, a cylindrical sample is positioned between two slender metal bars (see Figure 1). The load is generated by the impact of a projectile on the free end of the incident bar. This impact creates a stress wave that travels along the incident bar, and after a certain amount of time it reaches the sample, imposing a compressive load on it. A portion of the wave is transmitted to the sample, while the rest is reflected back into the incident bar [33; 34]. Similar wave reflection behavior is observed between the sample and the transmission bar. In order to measure the amplitudes of the wave trains, strain gauges are placed in both bars. This allows for the determination of the initial load in the form of a velocity versus time series, and also of the contact force between the bars and the sample, \(f(t)\), which in turn allows for the calculation of the stress at the middle of the sample as follows [30]:  
\[\sigma_{n}=\frac{2f(t)}{\pi dh}, \tag{2}\]  
where \(h\) and \(d\) are the sample's thickness and the diameter, respectively.  
Observational data from the SHPB experiment described above is obtained for granite. The elastic properties and the density of the material are listed in Table 1. More details  
data assimilation methods that can be applied. The methods range from deterministic to Monte-Carlo-based methods which take advantage of the model dynamics. There are several assimilation methods, and the most widely used are the variational methods (3D-Var and 4D-Var) [36], Kalman filtering [37], as well as the Monte Carlo techniques [38; 39; 40; 41].  
In this work, the ensemble Kalman filter has been selected to assimilate data into the HOSS model. The main motivation for selecting an EnKF method to calibrate the model parameters in HOSS was because the EnKF is one of the most computationally inexpensive and easy to implement [42]. There are other methods that can potentially be used for the model calibration, such as control theory. Unfortunately, due to the complexity of the model and its simulation requirements, these methods might be hard to implement. The EnKF has been shown to be computationally feasible and efficient for parameter estimation problems, since it is a compromise between Monte Carlo methods and Bayesian theory for Gaussian distributions [42]. This compromise enables the EnKF to use statistical information efficiently and provide an estimate of the probability distribution function of the model parameter. Practically, the computational efficiency of the EnKF translates to the number of model simulations needed to achieve convergence. A typical number of model simulation that the EnKF needs is around 30, whereas other methods might need significantly more simulations, without even providing a probability distribution of the model parameters. This advantage makes the EnKF an attractive alternative to estimate the parameters for a complex model like HOSS. A description of the EnKF method is presented in the following subsection.