## 2 Methods
### Determination of Strains and Temperatures  
The evaluations of strains and temperatures for the tracks subjected to repeated heating and reheating during the laser forming were conducted by finite-element (FE) simulations by using the ABAQUSTM commercial software. To reduce the computation cost, only half of the plate was modeled and several adjacent tracks were analyzed in each simulation. In the model, the same fixture used in the real experiment was added around the center of the symmetric plane shown in Figure 3a. With a mesh refinement test, the whole specimen was meshed with approximately 54,500 eight-nodded linear hexahedral elements of type C3D8 RT, distributed uniformly in the whole region, as shown in Figure 3b. In the thickness direction (z-direction), the mesh was uniform and the number of elements were 6, in order to accurately capture the flux distribution of the moving laser beam. In the x and y-directions, the element sizes were \(148\times 10^{-6}\) m. If 10 nodes instead of 6 were used in the thickness direction, the temperature field and history were almost not affected, with the maximum temperature difference being less than 10 K. Decreasing the element length in the x and y-direction, the maximum temperature difference was less than 20 K. No change in the subsequent analysis was caused by these temperature differences. The heat flux followed a Gaussian distribution [6],  
\[\mathrm{Q}=\frac{2 AP}{\pi R^{2}}\exp(\frac{-2\Big{(}x^{2}+y^{2}\Big{)}}{R^{2} }), \tag{1}\]  
where \(P\) is the laser power (W), \(A\) is the absorption coefficient, \(R\) is the laser beam radius (m), and \(x\) and \(y\) are the projected distances (m) from the center point of the laser beam. For the heat conduction within the workpiece, the thermal conductivity \(k\) (W/mK) was considered to be temperature-dependent. The convection and radiation boundary conditions were set as  
\[q_{\mathrm{total}}=q_{\mathrm{con}}+q_{\mathrm{rad}}=h(T-T_{0})+\epsilon \epsilon(T-T_{0})^{4}, \tag{2}\]  
where \(h\) is the heat transfer coefficient (20 W/m\({}^{2}\) K [14]), \(T_{0}\) is the surrounding temperature (300 K), \(T\) is the surface temperature during the laser heating processing, \(\varepsilon\) is the emissivity (0.8 and 0.32 for the heated surfaces with graphite coating and uncoated domains, respectively), and \(\epsilon\) is the Stefan-Boltzmann constant (5.6703 \(\times\) 10\({}^{-8}\) W/m\({}^{2}\) K\({}^{4}\)). The temperature dependences of the other parameters were obtained from the material supplier and literature [15]. Figure 3: (**a**) Model with the clamp; and (**b**) the computational mesh used for 3D with plane-symmetry finite element analysis. The phase change was conducted according to the latent heat of fusion, which was provided separately by the enthalpy difference between the solidus and liquidus phases. The validation of the numerical model is explained in [12]. In the laser forming, the temperature could increase to the melting point of the workpiece material. Consequently, the influence of the strain rate on the material flow stress was increased at an increased temperature. Therefore, the strain rate associated with the strain had a large influence on the bend angle in the laser bending. The strains on the heating line (\(e_{hunting}\)) and spacing between scan tracks (\(e_{spacing}\)) contributed to the total strain \(e_{total}\)[16; 17; 18],  
\[e_{total}=e_{hunting}+e_{spacing}, \tag{3}\]  
The in-plane (\(e^{in-plane}\)) and out-of-plane (\(e^{out-of-plane}\)) strains were the individual components of the heating strain (\(e^{hating}\)) and the spacing strain (\(e^{spacing}\)) [18]. \[e_{hating}=e^{in-plane}+e^{out-of-plane} \tag{4}\]  
\[e_{spacing}=e^{in-plane}+e^{out-of-plane} \tag{5}\]  
The simulated model is shown in Figure 4a. The heating and spacing strains were calculated through numerical modeling at the centers of the regions shown in Figure 4b.