# Prediction of Transient Temperature Distributions for Laser Welding of Dissimilar Metals
## 3 Numerical Model  
The FE model was built using the heat transfer option in the solid section of COMSOL Multiphysics. The simulation is based on the basic time dependent conduction equation for heat transfer in solids in three dimensions. The heat source moves along the y-axis at a velocity \(v(y)\) presented in the moving coordinate system, which ties to the moving heat source. Conduction takes a major role in transferring the heat which is applied by the laser beam to the particles of the working materials. Suitable boundary conditions are used to solve the transient heat conduction equation. Equation (5) makes it understandable that heat conduction within the materials depends on other parameters such as density, heat capacity, and thermal conductivity:  
\[\rho c_{p}\frac{\partial T}{\partial t}+\rho c_{p}\left(\frac{dy}{dt}\right) \
abla T+\
abla\cdot\overset{\rightarrow}{q}=Q \tag{5}\]  
\[\overset{\rightarrow}{q}=-k\
abla T \tag{6}\]  
where \(\rho\) is the density, \(c_{p}\) is the specific heat at constant pressure, \(t\) is time, \(T\) is the temperature, \(k\) is the thermal conductivity, \(Q\) is volume heat rate and \(\overset{\rightarrow}{q}\) is the heat flux vector.  
Convection of heat between the surface and with the surroundings was considered as a part of heat lost during the laser welding process. Negligence of convection effect during simulation can be reflected as error when it is compared with any experimental data [51]. The heat loss from the top surfaces by natural convection was taken into account. The governing heat flux equation for convection is:  
\[q_{c}=h(T-T_{ext}) \tag{7}\]  
where \(h\) is convective heat transfer coefficient and \(T_{ext}\) is the surrounding temperature.  
Heat loss in another form, i.e., radiation, was also considered during the simulation. Part of the total heat applied on the working surface is lost through radiation process. The accuracy of the model can be improved and the energy balance becomes more realistic by introducing these losses:  
\[-\overset{\rightarrow}{n}\cdot q_{r}=\varepsilon\sigma\left(T_{ext}^{4}-T^{4}\right) \tag{8}\]  
where \(\overset{\rightarrow}{n}\) is normal outward vector, \(\varepsilon\) is the emissivity, and \(\sigma\) is [PERSON]'s constant.  
Mesh Convergence Analysis  
Meshing has a vital role in obtaining an accurate simulation model. COMSOL has a provision to make the meshing coarse or fine. Generally, the mesh at the vicinity and near the weld line should be finer than the rest of the two parts. Free tetrahedral meshing was chosen in the model builder as it is only fully an automatic 3D mesh generator and can be applied in any 3D geometry for various simulations with good accuracy. Figure 4 shows the free tetrahedron mesh formation of predefined extremely finer size with high optimization level. Near the weld line, the mesh was refined again for good simulation output. In the mesh used, there are 129,718 domain elements, 12,236 boundary elements, and 576 edge elements. Weld-bead element has an average quality of 0.8405. With a growth rate of 1.3, the mesh model has a maximum element size of 0.8 mm and a minimum of 0.008 mm.