## 2 Materials and Methods
### Simulation Modeling of Temperature Field during Laser Direct Joining  
To simulate the temperature field during laser direct joining, the geometrical models shown in Figure 4 were established for the interface with and without polymer layer filling, respectively, and the material dimensions were the same as those in Section 2.1. To balance the computational accuracy and efficiency, two sizes of grids were chosen for meshing. Since the temperature gradient in the heated region \(\Gamma_{1}\) irradiated by the laser was high, a fine hexahedron with a size of 0.625 mm was used for meshing parallel to the thickness direction. The other region, \(\Gamma_{2}\), which had a low temperature gradient, was meshed with a coarse hexahedral of size 2.5 mm, and wedges were used for the transition between the two regions. In the material thickness direction, aluminum alloy was meshed using 0.5 mm, and the polymer layer was meshed using 0.25 mm. Considering the anisotropy and ply structure of the CFRTP composite, it was meshed using ply thickness of 0.125 mm. After establishing the meshed geometric models, it is necessary to set the governing equations and boundary conditions of the laser direct joining, including the material properties, the laser heat source, and the cooling conditions. Specific details can be found  
Figure 4: Meshed geometrical models for laser direct joining of CFRTP composite to aluminum alloy: (**a**) interface without polymer filling and (**b**) with polymer filling. Figure 3: Schematic of laser direct joining of CFRTP composite with aluminum alloy: (**a**) interface without polymer filling and (**b**) with polymer filling. in our previous study [25] and are briefly described here. The temperature field of the laser direct joining obeys the heat transfer equation in (1):  
\[\rho C_{p}\frac{\partial T}{\partial t}=k_{x}\frac{\partial^{2}T}{\partial x^{2} }+k_{y}\frac{\partial^{2}T}{\partial y^{2}}+k_{z}\frac{\partial^{2}T}{\partial z ^{2}} \tag{1}\]  
where \(\rho\) is the density; \(C_{p}\) is the specific heat capacity; \(T\) is the temperature; \(t\) is the time; and \(k_{x}\), \(k_{y}\), and \(k_{z}\) are the thermal conductivity in the \(x\), \(y\), and \(z\) directions, respectively. To solve the above heat transfer equation, the initial temperature \(T_{0}\) was set to be 20 \({}^{\circ}\)C, and the material properties were set based on Table 1. Additionally, based on the laser heat input and convective and radiative heat transfer on the outer surface of the material, the boundary conditions for the heated region \(\Gamma_{1}\) and other regions \(\Gamma_{2}\) were set as follows:  
\[k\frac{\partial T}{\partial n}=h(T_{s}-T_{0})+\varepsilon\sigma(T_{s}^{4}-T_{ 0}^{4})+q\text{ on }\Gamma_{1} \tag{2}\]  
\[k\frac{\partial T}{\partial n}=h(T_{s}-T_{0})+\varepsilon\sigma(T_{s}^{4}-T_{ 0}^{4})\text{ on }\Gamma_{2} \tag{3}\]  
where \(n\) is the normal vector of the outer surface; \(q\) is the laser heat flow density; \(h\) is the convective heat transfer coefficient, which was set to 5 \(W/(\text{m}^{2}\cdot\text{K})\); \(T_{s}\) is the surface temperature; \(\varepsilon\) is the emissivity; and \(\sigma\) is the Stefan Boltzmann constant, which was 5.67 \(\times\) 10\({}^{-8}\) W/(m\({}^{2}\cdot\text{K}^{4})\)). The emissivity coefficients of CFRTP composite and aluminum alloy were roughly set because of the little effect of radiative heat transfer on the temperature during the laser direct joining [34; 35]. Since the laser energy density was about 10\({}^{3}\cdot\)10\({}^{4}\) W/cm\({}^{2}\), and the aluminum alloy did not melt or had a small melting thickness, the Gaussian surface heat source was used to equalize the laser energy input.