## 2 Development of the GPGPU-Accelerated 3D FDEM Code for the RC Structure
### Embedded Reinforcement Bar Model in 3D FDEM Concrete Model  
A reinforcement bar (rebar) model is integrated into TET4s and CE6s and implemented in 3D FDEM code to simulate rebar reinforcement in the RC structure. A one-dimensional rebar model is adopted, and it is placed within the TET4s. Figure 3 depicts the RC structural model overlapped with the rebar model. Each rebar object is defined by its start and end points, and the entire rebar is further discretized into two-node rebar elements (REs) and two-node joint rebar elements (RJEs) based on the intersection points between the TET4s and rebar model interfaces.  
The RE is embedded within the TET4 because the intersected RE's two points with the TET4 are fixed. Thus, the deformation of TET4 influences the deformation of RE, while the axial strain of RE, \(\epsilon_{axial}^{RE}\), is calculated using Equation (7) while the RE's axial force, \(F_{axial}^{RE}\), is derived by applying the linear-elastic constitutive model.  
\[\epsilon_{axial}^{RE}=\frac{l_{c}^{RE}-l_{i}^{RE}}{l_{i}^{RE}} \tag{7}\]  
\[F_{axial}^{RE}=A^{\text{Rebar}}\sigma_{axial}^{RE}=A^{\text{Rebar}}E^{\text{ Rebar}}\epsilon_{axial}^{RE} \tag{8}\]  
where \(A^{\text{Rebar}}\) and \(E^{\text{Rebar}}\) denote the cross-sectional area and Young's modulus of the each rebar. \(F_{axial}^{RE}\) is assembled as the internal force of the corresponding TET4. It is worth noting that each RE is embedded in its host TET4 during the FDEM simulation.  
Figure 1: Two types of finite elements ( TET4 and CE6) in 3D FDEM.  
Figure 2: Tensile (**left**) and shear (**right**) softening curves in the ICZM.  
When cracks initiate between REs, the initially zero-length two-node RJE begins to influence the mechanical behavior of concrete. The induced forces by RJEs consist of two components, i.e., axial force (\(F_{n}^{RJE}\)) and shear force (\(F_{s}^{RJE}\)). The \(F_{n}^{RJE}\) and \(F_{s}^{RJE}\) are derived by the axial stress (\(\sigma_{n}^{RJE}\)) and shear stress (\(\tau_{s}^{RJE}\)), which are determined as a function of the relative opening (\(o\)) and slip(\(s\)) between two nodes of RJE (Figure 4), respectively, as presented in Equations (8) and (9).  
\[F_{n}^{RJE}=A^{\text{Rebar}}\sigma_{n}^{RJE}=A^{\text{Rebar}}K_{n}^{\text{ Rebar}}o \tag{9}\]  
\[F_{s}^{RJE}=A^{\text{Rebar}}\tau_{s}^{RJE}=A^{\text{Rebar}}K_{s}^{\text{ Rebar}}s \tag{10}\]  
where \(K_{n}^{Rebar}\) and \(K_{s}^{Rebar}\) denote normal and tangential stiffness of each rebar, respectively. Equivalent constitutive models in [20; 21] were used for computing the \(\sigma_{n}^{RJE}\) and \(\tau_{s}^{RJE}\). \(\sigma_{n}^{RJE}\) follows the elastic-perfect plastic model, while \(\tau_{s}^{RJE}\) follows the elastic model. As can be seen in Figure 5, the \(\sigma_{n}^{RJE}\) and \(\tau_{s}^{RJE}\) in the RJE initially behave like elastic material, while the \(\sigma_{n}^{RJE}\) remains constant when it reaches the yield stress of the rebar. The yield stress is determined by a user as the mechanical properties of the rebar. Finally, the \(F_{n}^{RJE}\) and \(F_{s}^{RJE}\) are assembled into the cohesive force in the CE6.  
Figure 4: The opening and sliding definitions for the RJE (point B-C in the figure).  
Figure 3: The RC structure as a numerical representation (separation between TET4s is exaggerated for visualization).