# Dimensional Errors Due to Overhanging Features in Laser Powder Bed Fusion Parts Made of Ti-6 Al-4V
## 3 Results and Discussion  
### Data Processing and Analysis  
Data processing and statistical analysis were conducted using MATLAB (R2019a) (MathWorks, Natick, USA). A regression equation was developed to describe the relationship between the different process parameters and the dimensional error percentage. A number of regression formulas were  
Figure 3: Printed part that underwent image processing, where green lines indicate the detected edges that were used for measurement. created and tested, with the quadratic regression equation being the optimal one. A quadratic regression method was used to fit the second-order polynomial in Equation (1) to the measured data:  
\[y=b_{0}+\sum b_{i}x_{i}+\sum b_{ii}x_{ii}^{2}+\sum b_{ij}x_{i}x_{j}, \tag{1}\]  
Where \(y\) is the output variable which, in our case, is the dimensional error %; \(b_{0}\),\(b_{i}\), \(b_{ii}\), and \(b_{jj}\) are the regression coefficients or predictors; and xi is the value of the ith factor. The developed equation resulting from the quadratic regression formulation with a robust fit presented 21 terms for the five process parameters that were studied in this work and the values obtained for each coefficient are presented in Equation (2), where x1 is the laser power, x2 is the scan speed, x3 is the scan spacing, x4 is the scan pattern, x5 is the layer thickness, and \(y\) is the dimensional error %. \[ \begin{array}{ll}y=6.3668+&6.7157x_{1}-5.6775x_{2}-0.69749x_{3}+0.12742x_{4} -2.6591x_{5}\\ &-4.4385x_{1}x_{2}-0.73969x_{1}x_{3}-0.2506x_{1}x_{4}+0.055401x_{1}x_{5}\\ &-0.55883x_{2}x_{3}-0.62242x_{2}x_{4}+1.1935x_{2}x_{5}-0.0036447x_{3}x_{4}\\ &+0.07324x_{3}x_{5}-1.3799x_{4}x_{5}+1.0649x_{1}^{2}+2.2655x_{2}^{x}+3.2372x_{3 }^{2}\\ &+0.92257x_{4}^{2}\end{array}  \tag{2}\]  
This equation was then used to generate the interaction plots, as seen in Figures 4 and 5. The interaction effects depicted below show the estimated effects on the response from changing each variable value while averaging the effects of the other process parameters. Based on the interaction effects, some of the insights gained are as follows:  
* As seen in Figure 4a, increasing the scan speed tends to decrease the dimensional error at all laser powers; however, to different degrees. This can be explained as the meltpool and thus the associated heat-affected zone caused by high speeds is expected to be smaller than that caused by lower speeds. A smaller meltpool and heat-affected zone would reduce the seepage of molten material into the powder bed, which would mean that the dorsal formation and the associated dimensional inaccuracy would be reduced;
* As seen in Figure 4c, increasing laser powers at any scan spacing value increases the dimensional error. This can be explained as increasing the laser power causes more energy to be absorbed by the powder, causing greater overheating of loose powder and larger dorsal formations due to the larger melt pools, and thus a larger associated heat-affected zone;
* As seen in Figure 4f, increasing the scan speed at any scan spacing value decreases the dimensional error up to a certain point of around 80 um, after which it begins to increase once again;
* Figure 4d, e show that with an increasing laser power, the dimensional error increases and decreases with an increase in scan speed. The reasons for this have already been discussed in the previous points. However, it can also be noted that the scan spacing of 75 um always displayed a better dimensional accuracy than 50 um and 100 um;
* The variation in the effect of the different scan strategies is small, as the graphs in Figure 4a-c show that all three lines of 'Stripes', 'Rectangular Cell', and 'Hexagonal Cell' are quite close together and always follow the same trend. This can be explained due to the small size of the down-facing area.