# Assimilation of Dynamic Combined Finite Discrete Element Methods Using the Ensemble Kalman Filter
## 1 Introduction  
The accurate simulation of fracture initiation, propagation, and arrest in materials is an important problem for a number of applications, ranging from the study of micro-scale behavior of materials to the analysis of large-scale fracture initiation and propagation. Many numerical solutions have been proposed to simulate fractures; one of the most widely accepted is the Combined Finite-Discrete Element Method (FDEM) [1, 2]. FDEM, which merges solutions developed under both finite elements and discrete element methods, is mainly aimed at modelling failing, fracturing, and/or fragmenting solids. This is achieved by the usage of Finite Element formulations to resolve stresses in the material combined with Discrete Element formulations to resolve fracture and contact problems [2]. A review of the current fracture models based on FDEM is presented by [3]. Furthermore, FDEM has been widely used in real-world problems, such as dam and rock slope stability [4, 5], hydraulic fracture and fracture propagation on shale [6, 7, 8], fracture coalescence [9, 10], rock mass strength [11], dynamic fault rupture [12, 13, 14], and granular gauge response [15, 16]. As with many models, several material-specific parameters (i.e., material properties) are needed for a FDEM simulation, such as density, Young's modulus, Poisson's ratio, and strength parameters for failure modes I (normal) and II (tangential). Additionally, the specific fracture energies for these two modes are also needed. Closely related to the specific fracture energies is the use (within FDEM) of a displacement-softening curve torepresent the strength of degradation as micro-fractures form in the material. In previous work, the sensitivity of the FDEM model to perturbations in these parameters was analyzed [17, 18], where the authors found a relatively small number of parameters influencing the behaviour of the stress versus time curve obtained from FDEM. To accurately simulate fracture behavior, the characterization of displacement-softening parameters need to be correctly specified. However, experimental information that could be used to determine these parameters is very often not easy to find, and modelers are forced to establish them in an ad hoc manner or based on their own experience. Data assimilation is a method that fuses observational data into models to improve the nowcast and/or forecast of the model [19, 20]. There are several types of data assimilation methods, from statistical based on Bayesian theory, to optimization-based methods. Data assimilation methods are used in several geophysical sciences, including climate, atmospheric weather prediction, space weather, and ocean sciences, to name a few. Additionally, assimilation methods are increasingly being implemented in geomaterials and fracture propagation to improve the simulation of geophysical models (weather, climate, subsurface flow, etc.) [19, 20, 21]. The calibration of model parameters is of relevance for a number of applications. For example, it is relevant to the simulation of complex masonry construction problems [22, 23] where non-linear analysis of the response of the structures are needed. In these cases, the problem itself is complex enough, and the uncertainty on the parameter determination can be quite high. This is also the case for the simulation of underground processes, such as hydraulic fracturing and earthquake ruptures, where knowledge about material properties is limited to areas where core sampling can be extracted and analyzed. This provides a starting point for analysts to populate their models, but in many cases this is not sufficient, and further calibration is needed. Sensitivity analysis [17] and assimilation methods [18] provide a robust and systematic way of analyzing the system under consideration as a whole, while identifying the key parameters and establishing a confidence interval for them.