## 3 Modeling Procedures of Extended Finite Element Method (XFEM)-Based Multiscale Analysis
### Multiscale Modeling Methodology  
In this study, the multiscale modeling methodology for RC columns is proposed. The flowchart of modeling methodology for multiscale simulation is presented in Figure 4. In general, the fiber beam element model is efficient in analyzing the global structural performance of RC columns. However, the detailed damage evolution process cannot be obtained. In order to describe the damage initiation and development at the foot of RC columns, solid elements are usually preferred in numerical analysis. Since the failure of RC columns usually occurs at the bottom of tested specimens, two-scale models are employed to investigate the global behavior and local failure pattern of tested specimens. For improving the computational efficiency, fiber beam elements are employed to model the upper part of RC columns. It can be seen that the two-scale model is the combination of the fiber element model and solid element model [16]. Multiscale models are established by modeling the lower part of two-scale models with mesoscale concrete. As shown in Figure 5, the multiscale concrete models are composed of fiber beam elements, solid elements, and mesoscale concrete with different aggregate distribution patterns and geometric shapes. For comparison, the CDP model is adopted for the material definition of mortar and the ITZ, as presented in Figure 5a. Moreover, the aggregates are assumed as elastic components. In Figure 5b, the solid element segment and mesoscale concrete segment are constructed using XFEM. Since the ITZ layers have been explicitly modeled, the number of elements increases significantly. Without loss of  
Figure 4: Flowchart of modeling methodology for multiscale simulation. generality, five sets of random aggregate samples are generated to investigate the effect of mesostructure variation of concrete on local and global behaviors of RC columns. Moreover, the separation of fiber element part and solid element part in two-scale models should be determined reasonably to balance computation efficiency and accuracy. For fiber beam elements, the constitutive models for steel bars and concrete are compiled using user material subroutines (Umat and Vumat) in ABAQUS, as detailed in Section 3.3. The height of the RC column modeled with solid elements \(l_{solid}\) can be determined according to the plastic segment length \(l_{p}\) calculated using Equation (4) [25]. \[l_{p}=0.08L+0.022d_{b}f_{y} \tag{4}\]  
where \(d_{b}\) and \(f_{y}\) stand for the diameter and yield strength of longitudinal reinforcement, respectively. Furthermore, the multiscale models composed of fiber beam element, solid element and mesoscale concrete are constructed to consider the mesostructure variation of concrete. According to the practical crack distribution and failure pattern of tested RC columns, the height of the mesoscale concrete segment is set to 0.2 \(L\). Figure 5: Multiscale modeling approaches. In this section, the behaviors of RC columns under monotonic and cyclic loadings, which are simulated using the fiber beam element model, solid element model and two-scale model composed of fiber beam elements and solid elements, are compared to verify the feasibility of the developed two-scale modeling approach. As illustrated in Figure 5a, the CDP-based two-scale and multiscale model are also established to further validate the accuracy of developed material subroutines for the fiber beam element and XFEM-based multiscale modeling approach. The parameters of the CDP model are shown in Table 2. The damage coefficients of the CDP model are calculated according to Equations (5) and (6) proposed by [PERSON] and [PERSON] [26]:  
\[d_{c} =1-\frac{\sigma_{c}/E_{c}}{\varepsilon_{c}^{pl}(1/b_{c}-1)+\sigma_{c}/E_{c}} \tag{5}\] \[d_{t} =1-\frac{\sigma_{t}/E_{c}}{\varepsilon_{t}^{pl}(1/b_{t}-1)+\sigma_{ t}/E_{c}} \tag{6}\]  
where \(E_{c}\) is the initial modulus, \(\sigma_{c}\) and \(\sigma_{t}\) stand for the compressive and tensile strengths of concrete.