# Assimilation of Dynamic Combined Finite Discrete Element Methods Using the Ensemble Kalman Filter
## 6 Conclusions  
Accurate simulation of fracture propagation for a number of materials is an important problem in the engineering community in general. There are a number of state-of-the-art  
Figure 6: Estimated parameter values for \(\sigma_{t}^{max}\) and \(E_{t}\) provided by the EnKF data assimilation using SHPB time–stress curve observations, where the blue line is the average parameter value at each iteration, and the green dashed line is the final value at Iteration 3. The EnKF is applied three times, each time providing an estimated parameter value. The assimilation provides a consistent estimate for \(\sigma_{t}^{max}\), where the parameter value is seen to converge around 0.15. In the case of \(E_{t}\), the assimilation seems to be providing monotonically decreasing values of the parameter value, which might not be reliable. Figure 7: Time–stress curves for each assimilation iteration, starting with the initial ensemble of 25 members (top-left plot, blue lines), first assimilation iteration (top-right plots, green lines), second iteration (bottom-left plot, yellow lines), and third iteration (bottom-right plot, magenta lines), as well as the observations (black line in all plots). numerical models in the community for fracture simulations, and among them is the HOSS model. In the HOSS model, as well as in other models, a number of important processes are determined by the value of key model parameters, such as material softening or degradation behavior, and energy dissipation and strength in both normal and tangential directions. Oftentimes, the appropriate value of these parameters becomes elusive for a number of materials, which poses a difficulty for the accurate simulation of fracture behavior in these materials. In this work, we present a systematic and reliable framework to estimate two key model parameters for HOSS using data assimilation for a Split Hopkinson Pressure Bar experiment for granite material. This was achieved via the implementation of an ensemble Kalman filter assimilation method. The main advantage of the EnKF is its ease of implementation and reliability to provide an estimate of the parameter values for the SHPB observational data. Given that the parameters are set only at the beginning of the HOSS simulation, and have an impact throughout the simulation, the assimilation approach is based on the comparison of the stress versus time curve from HOSS against the one obtained from SHPB experiments. In essence, the EnKF assimilation utilizes this stress versus time curve as a state vector, computes the correlation between the stress versus time curve and the parameters, and exploits that correlation to adjust the values of the parameters so as to agree with the SHPB stress versus time curve. The EnKF assimilation, as explained, was applied three times (called, in this work, EnKF iterations) to obtain the parameter values. A previous sensitivity analysis test [17] indicated that the two main parameters that play a significant role in determining the stress versus time curve in the HOSS experiment are the strength and energy parameters in the tangential direction (\(\sigma_{t}\) and \(E_{t}\)). Therefore, only these two parameters are estimated using the EnKF assimilation method. To determine the accuracy and reliability of the assimilation method, we performed a twin experiment test. In this twin experiment, we generated synthetic data by sampling from a stress versus time curve from a HOSS simulation, where we fix the parameter values to a chosen reference value. Afterwards, we assimilate the synthetic observations to determine whether the assimilation method can recover the reference values used to generate the observations. Our twin experiment indicates that the assimilation can, in fact, recover the reference values (see Section 5.2). The stress versus time curve from a SHPB experiment is used to estimate the model parameters \(\sigma_{t}\) and \(E_{t}\) using EnKF data assimilation.