## 2 Theoretical Background  
This study was performed according to the flow chart in Figure 6. First, a theoretical model for estimating the heat source was developed to specify the temperature of the welding center. The heat source was calculated by a relational expression of process variables (rotation speed, welding speed, and the tool's downward force), design variables (workpiece and tool dimensions), and unspecified variables that are difficult to calculate theoretically or experimentally (welding efficiency and friction coefficient).  
The optimal process was derived by performing FSW on the fillet joint specimen units according to each process condition to derive the process variables for calculating the numerical model. Then, heat transfer analysis was performed by using this numerical model after inputting the optimal process condition variables. In addition, temperature measurement experiments were conducted for the optimal process conditions, and the errors in the numerical model were reduced through test correlation by comparing the two results and adjusting the heat flux of the heat source. Heat transfer analysis was performed after deriving the temperature of the heat source at the welding center of the fillet joint, and the full-frame structure's behavior was predicted by conducting thermal elasto-plastic analysis based on the heat transfer analysis results.  
The frictional heat generated between the tool and the base material is the heat source of the welding center when performing FSW and this is a significant factor in determining the strength of the weld joint. Therefore, conventional methods used to predict the heat source of the weld usually attempted to calculate the total heat source by combining the frictional heat generated from each part of the tool [9; 10]. As shown in Figure 7a, frictional heat is generated from three areas where the tool meets the material: Shoulder, pin lateral, and pin bottom. In the case of butt welding, the contact surface where the tool meets the base metal is the entire bottom of the tool as shown in Figure 7b. However, in this study, the fillet joint of the base material and the shoulder of the tool make partial contact, so the tool's overall friction area is 66% (shoulder 55% and pin lateral 95%) compared to the butt joint. Figure 7 describes the friction area of the butt joint tool and the fillet joint tool, and detailed contact areas are summarized in Table 1.  
Figure 6: Flow chart of research procedure.  
This study applied the heat source estimation model proposed by [PERSON] in 2003, assuming that the heat source is a constant heat flux generated from the friction surface [11]. This model is defined as shown in Equation (1), and considers all of the heat sources generated from the tool's shoulder, pin lateral, and pin bottom. The [PERSON] model shows similar calculations to the actual results under full sliding conditions, so this study assumed that the shoulder inclined at a 45\({}^{\circ}\) angle was flat, and in terms of the reduced friction area and unspecified variables such as the welding efficiency and friction coefficient, the errors of the heat transfer simulation results were reduced by test correlation with the temperature measurement experiment results for the optimal process. Table 2 shows the tool's design variables for heat source calculation.  
\[\mathrm{Q_{t}=Q_{Shouder\_bottom}+Q_{Pin\_lateral}+Q_{Pin\_bottom}=\eta\{2\pi \omega\tau(R_{s}{}^{3}-R_{p}{}^{3})/3+2\pi\omega\tau R_{p}{}^{3}/3+2\pi\omega R_{ p}{}^{3}H_{p}\}} \tag{1}\]  
\[\tau=\mu P=\mu F_{N}/(\tau\,R_{s}{}^{2}) \tag{2}\]