**1. Introduction**

A particle system is a complex system composed of many discrete materials widely existing in nature and applied to industrial production [1]. In particular, it is of great engineering significance and academic value to discuss and analyze the relationship between motion characteristics and the internal stress of particles.

The discrete element method (DEM) is widely used to study the macroscopic and microscopic physical properties of granular materials. A particle is regarded as a discrete element if its motion satisfies Newton's second law in the DEM. The DEM can quickly obtain the overall motion information of particles (position, velocity, etc.). Kačianauskas et al. [2] proposed a parallel three-dimensional DEM simulation of polydisperse materials described by normal size distribution. Fang et al. [3] developed a CUDA-GPU parallel algorithm based on super-quadric elements, which is applicable to and reliable for the large-scale engineering applications of non-spherical granular systems. Sun et al. [4] simulate the interactions of the bionic subsoilers and an ordinary subsoiler (O-S) with the soil based on the DEM. However, particles are regarded as discrete elements, and the whole particles are considered as a whole because of ignoring the internal changes of particles. One cannot obtain the internal information of particles (internal stress distribution, deformation, etc.).

When calculating the physical information inside the particles, the finite element method (FEM) is widely used [5–7]. It is a numerical solution method for elastic mechanics problems that developed rapidly with the advancement of computer power. It is applied in continuum mechanics to obtain the deformation, stress, natural frequency, and vibration mode of the structure [8]. The analyzed objects have been extended from elastic materials to plastic, viscoelastic, viscoplastic, and composite materials and from the continuum to a discontinuity [9]. The FEM is widely used in the internal stress simulation of objects. Fang et al. [3] studied the particle–wall collision process using by the FEM, which could solve local stress and strain rate. Kabir et al. [10] used the explicit FEM to simulate the flow phenomenon of particles. The results of shear behavior, particle kinetic energy, and particle stresses within the shear cell with time were given. Wagner et al. [11] proposed a new particle flow simulation method based on the extended FEM (x-FEM), which simulates moving particles without re-meshing. Krok et al. [12] conducted a systematic finite element analysis of the thermo-mechanical behavior of pharmaceutical powders during the molding process using the finite element solver ABAQUS.

The DEM and FEM have their advantages, so the combination of DEM and FEM is widely recognized and studied [13–15]. Guo and Zhao [16] proposed a multiscale framework to simulate the mechanical behavior of granular media based on DEM and FEM. A DEM assembly with the memory of its loading history is embedded in the Gauss integral points of the finite element mesh. The DEM assembly receives the global deformation at its Gauss point from the FEM as input boundary conditions in this new multiscale framework. Onate and Rojek [17] conducted a dynamic analysis of geological mechanics problems based on the combination of DEM and FEM. The combined models can employ spherical rigid and finite elements to discretize different parts of the system. Zárate and Oñate [18] proposed a new numerical method to predict the occurrence and evolution of fractures in continuous media, which combines the FEM with DEM. Munjiza and John [19] further studied the sensitivity to the mesh size of the combined single and smeared crack model in the context of the combined finite–discrete element method. Azevedo and Lemos [20] applied the hybrid method to analyze large structures. This method uses DEM to discrete the fracture zone and a discretization based on the FEM for the surrounding areas. Argilaga et al. [21] proposed a multiscale model based on an FEM DEM approach. The method uses discrete elements in a standard finite element framework, and it has proven to be an effective way to treat real-scale engineering problems. However, it should be emphasized that the coupling of the discrete element method and finite element method still needs further research. For example, the solution of coupled motion, internal stress, and other contents need to be further discussed and analyzed.

The discrete element-embedded finite element model (DEFEM) is proposed, and it can be used to calculate particle motion and heat transfer [22–24]. This method is applied to the calculation of temperature gradient and deformation in particles. In this paper, the DEFEM is extended based on the concept of the embedded discrete element *DEFEM Method and Its Application in Pebble Flows DOI: http://dx.doi.org/10.5772/intechopen.109347*

(EDE). The computation of deformation displacement is based on the concepts of displacement decomposition (translational and rotational motions and deformation displacement). The DEFEM mainly adopts the Lagrange finite element method to obtain the coupling solution about the stress and motion of particles. The finite element software verifies the relevant algorithms. The motion characteristics and deformation of particles are discussed, and the stress distribution and force chain structure in particle accumulation are obtained.
