**4. Conclusions**

Analytical models and numerical model are used to predict the temperature in laser-based metal additive manufacturing configurations of either direct metal deposition or selective laser melting. In the past few decades, many researchers have been trying to understand the relationships between the process parameters and temperature using FEM. The numerical methods have low computational efficiency, and it cannot capture all the physical aspects of the metal AM processes. The lack of a physics-based analytical model that captures all the physical phenomena of the AM processes is sensible. The physics-based analytical modeling provides accurate results. The high computational efficiency and easy implementation are the other advantages of the analytical model for the additive manufacturing modeling.

In this work, an analytical model is proposed to predict the distribution of the temperature profile by considering the interaction of the layers during the laser metal additive manufacturing process. The material properties are assumed to be temperature dependent, and also the melting/solidification phase change is considered in this work. The temperature profile, the peak temperature, and the evolution of surface temperature are obtained from the proposed model. The analytical model of the temperature is based on the moving heat source assumption, as described in Section 2. The general differential equation of heat conduction is used to obtain the closed-form temperature solution using the separation of variables in a semi-infinite medium. The material is assumed to be homogeneous and isotropic. The predicted temperature from the analytical model are compared with the experimental values and FEM results. For further validation, a comparison of peak temperature considering the layer addition and without considering the layer addition is conducted and compared with experimental values.

The results of the temperature distribution considering the layering aspects of metal additive manufacturing showed better agreement with experimental values in comparison with the predicted temperature not considering the layer addition. The observations suggest that for a fixed laser power, the laser speed increases as the temperature decreases, since the material has less time to absorb the energy. Also, for a given scanning speed, the laser power increases as the maximum temperature increases.

A numerical model is also used to predict the temperature in the metal additive manufacturing process. The material properties are assumed to be temperature dependent. In the numerical model, the heat loss due to convection and radiation is considered. The temperature is well obtained using numerical models.

A comparison is conducted in order to capture the effect of considering the temperature sensitivity of material properties. A significant difference is observed between them. The main reason is that the thermal conductivity of Ti-6Al-4V is low, so the heat transfer rate decreases and causes the surface temperature to increase. However, by considering the temperature dependent material properties, the thermal conductivity increases by increasing the temperature. As a result, the heat transfer rate increases and causes the obtained surface temperature to be less compared to the case in which the thermal conductivity is constant.

The proposed model can also predict the melt pool size with the error margin being less than 7.6%. Hence, it eliminates the costly experiments and time-consuming FEM for predicting the melt pool size. This 2D model also shows that there is no need for doing 3D simulations in order to predict the melt pool size and geometry.

The proposed analytical model shows a good agreement with the experimental values. The proposed analytical model reduces the computational time to a fraction when compared to finite element analysis. The analytical model has also eliminated the costly experiments in order to understand the physical concepts of laser metal additive manufacturing. The influence of scanning speed and laser power on the temperature profile, surface temperature, and also peak temperature are investigated and the relations between them are established.

**Author Contributions:** E.M. conceived and developed the proposed analytical model, extracted and analyzed the data, and wrote the paper. J.N., P.B., O.F., and K.-N.C. provided general guidance. S.Y.L. provided general guidance and proofread the manuscript writing.

**Funding:** The funding is confidential.

**Conflicts of Interest:** The authors declare no conflict of interest.
