## Materials and Methods
### FE Analysis Methods
#### 2.3.2 FE Analysis of Cu Tube Expansion  
Expansion Ball Geometry  
The effect of the expansion ball shape was investigated using an expansion ball without grooves on the surface (Type 1) and one with grooves (Type 2). The groove size was based on the size of the tube's inner grooves. The Type 2 ball has 40 grooves for distributing the applied load by increasing the contact area between the ball and tube surface. The Type 3 ball was generated by reducing the grooves' top width (\(W\)) from 0.2 mm to 0.1 mm to analyze the influence of the contact area. For Type 4, an additional free gap (\(H\) = 0.1 mm) was made between the ball and tube-groove peak point on Type 3 to remove the contact point between the tube-groove and ball. Type 5 was produced by decreasing the number of expansion ball grooves of Type 3 from 40 to 20. Type 6 and Type 7 were designed with surface radii of \(R\) = 5.3 mm and \(R\) = 7.9 mm, respectively, and without expansion ball grooves to analyze the influence of the radius of curvature \(R\) on the tube expansion performance (Figure 4 and Table 2).  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Expansion Ball Type** & **Groove Number** & **W (mm)** & **H (mm)** & **R (mm)** \\ \hline Type 1 & 0 & 0 & 0 & 5.7 \\ Type 2 & 40 & 0.2 & 0 & 5.7 \\ Type 3 & 40 & 0.1 & 0 & 5.7 \\ Type 4 & 40 & 0.1 & 0.1 & 5.7 \\ Type 5 & 20 & 0.1 & 0 & 5.7 \\ Type 6 & 0 & 0 & 0 & 5.3 \\ Type 7 & 0 & 0 & 0 & 7.3 \\ \hline \end{tabular}
\end{table}
Table 2: Design variables for different expansion ball shapes.  
Figure 3: (**a**) Effects of friction coefficients (0.2, 0.3, 0.47) on expansion forces for the S1N group; and (**b**) comparisons of expansion forces between the expansion experiments and FE analyses for the S1N and G1N groups.  
Similar to the validation FE analysis of the Al tube expansion (Section 2.3.1), a hexahedron with eight nodes was used for the mesh in Cu tube models. Accordingly, 42,634 nodes and 30,220 elements were produced, and a 1/20 cyclical symmetry condition was applied based on the x-axis. The tube ends were constrained to the direction of expansion (x-axis in Figure 5) to replicate no tube shrinkage condition during actual tube expansion. The fin combined with the tube was constrained to the upper side (the y-direction in Figure 5). The same expansion ball speed of 90 mm/s as in the actual tube expansion process was applied, along with a friction coefficient of 0.36 between the expansion ball and Cu tube [12].  
After the tube expansion analysis, reference points P1 and P2 were selected at the peak and valley of the tube's inner groove, respectively. The tube's inner groove deformation (\(\Delta\)P2 \(-\)\(\Delta\)P1) was determined by the difference between P1 and P2 before and after measuring the displacement of the two points. In the FE analyses, the expansion force was computed using the average of the steady-state values, which occurred at approximately 0.03-0.11 s in the axial direction of expansion. The radial force was measured in the radial direction of the tube expansion (Figure 5).  
Figure 4: (**a**) Three-dimensional models of a smooth expansion ball (Type 1); (**b**) an expansion ball with expansion grooves on its surface for distributing concentrated load on the tube’s inner grooves during the tube expansion (Types 2–5); (**c**) schematic of expansion balls with the radius of curvature \(R\) as the design variable (Types 6 and 7); and (**d**) definitions of width (\(W\)) of the expansion ball groove and gap (\(H\)) between the valley of the expansion ball groove and the peak of the tube’s inner groove.