## 5 Conclusions  
Combining the powder feeding equation with the VOF volume function model to calculate the deposition layer morphology results in the establishment of a mathematical model of the three-dimensional coaxial powder feeding laser-directed energy deposition process and the simulation calculation of the single-channel and lap-forming processes. The deposition layer's height and breadth grew by 51.12% and 11.18%, respectively, when the powder feeding rate was raised from 9.6 g/min to 12.0 g/min. The deposition layer height and breadth grew by 30.67% and 25.31%, respectively, when the laser power was increased from 400 W to 700 W, or by 75%. The estimated height and breadth of the deposition layer are in excellent accord with experimental findings. Due to the thermal buildup of the first deposition channel, which absorbed more latent heat of melting, the melt pool expanded by 4.19 percent during lap processing in one-way parallel scanning as compared to single pass processing. During lap processing, the maximum surface temperature of 2038 K was somewhat lower than that of 2068 K during single-channel processing. During the scanning operation, the first deposition channel is likewise affected by the high-temperature condition of the lap deposition channel. There is a different degree of re-temperature at different locations, with the influence range from 1000 K to 1600 K and the re-temperature range from 100 K to 300 K. For the IN718 alloy, this affects the solid-state phase change after the solidification of the first deposition channel. On this basis, the deposition layer and temperature field are simulated for single-layer multi-channel overlap laser deposition. It is shown that the model may be used to forecast the deposition and temperature field for complicated processing procedures. The preceding findings help in the comprehension of the mechanism behind the deposition manufacturing process and the development of laser deposition manufacturing with high dimensional precision and intrinsic quality.  
Formal analysis, A.H.; Funding acquisition, Y.Y.; Investigation, T.W.; Project administration, Y.Y.; Supervision, Y.H.; Visualization, Y.W.; Writing--original draft, A.H.; Writing--review & editing, W.L. All authors have read and agreed to the published version of the manuscript.  
National Natural Science Foundation of China-Joint Fund Key Projects in Guangdong Province (U2001218); Project of Funds for International Cooperation and Exchange of the National Natural Science Foundation of China NSFC-RFBR (52111530035).  
Not applicable.  
The data that support the findings of this study are available from the corresponding author upon reasonable request.  
We thanks National Natural Science Foundation of China-Joint Fund Key Projects in Guangdong Province and Project of Funds for International Cooperation and Exchange of the National Natural Science Foundation of China NSFC-RFBR for the economic support.  
The authors declare that there are no conflict with this publication.  
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.