## 3 Results and Statistical Analysis
### Effect of Interaction of Parameters on Sa
#### 3.2.3 Scan Speed and Scan Spacing  
\(\bullet\)\(45^{\circ}\) down-facing surface  
It is quite clear from the first graph that at all scan spacings, the \(Sa\) value increases with an increase in scan speed. The second graph shows that at all the different scan speeds the \(Sa\) value will decrease up to a certain scan spacing value (80 \(\upmu\)m) after which the \(Sa\) value increases once again. This is consistent with the observations made between laser power and scan spacing as well.  
\(\bullet\)\(35^{\circ}\) down-facing surface  
In Figure 9, it can be seen that the effect of each scan spacing on the \(Sa\) shows an increase up to a certain scan speed, after which it begins to decrease once again. This point differs for each scan spacing, as can be seen in the graphs.  
Figure 8: Interaction effects of parameters with the measured \(Sa\) for \(45^{\circ}\) down-facing surfaces.  
The data processing and statistical modelling were completed using MATLAB (R2018b, MathWorks). A linear regression model with interaction effects was used to describe the relationship between parameters (laser power, scan speed and scan spacing) and _Sa_. The linear model was fit to the measured raw data, and this model was then used to generate the interaction plots as seen in Figures 8 and 9. These plots show the estimated effect on the response from changing each variable value, averaging out the effect of other parameters. This plot also shows the estimated effect when the other variable values are fixed at certain values as seen in Figures 8 and 9.  
This same model also makes it possible to generate the prediction slice plots as seen in Figures 10 and 11. The prediction slice plots in Figures 10 and 11 show the main effects for all parameter values. The green line in each panel shows the change in the response variable as a function of the parameter values when all other predictor values are kept constant. The dashed red curves in each panel depict the 95% confidence bounds for the predicted response variable.  
The plots shown in Figures 8 and 9 illustrate the effect of each predictor on the _Sa_ model, with the objective being to minimize the obtainable surface roughness for the down-facing surface by understanding the effect of the down-facing parameters. It is now possible to use this predictive model in order to obtain parameters when minimizing the _Sa_. Obtaining a smoother surface by simply controlling the process parameters extends the current capabilities of printing inclined and down-facing surfaces without requiring the printing of support structures, thereby contributing to a reduction in post-processing times. To obtain an as-printed _Sa_ of 12 \(\upmu\)m would illustrate an improvement over current capabilities when printing with default parameters. Figure 9 suggests a set of parameters in order to achieve a _Sa_ of 12.73 \(\upmu\)m, the corresponding normalized values for each process parameter were: Laser power = \(-\)0.35 (115 W), scan speed = \(-\)1 (465 mm/s) and scan spacing = \(-\)0.02 (approx. 75 \(\upmu\)m). For the 35\({}^{\circ}\) down-facing surface, a prediction of 20 \(\upmu\)m would illustrate an improvement over current capabilities and with use of the developed model it was possible to predict process parameters to achieve this, as seen in Figure 11. The normalized value of parameters suggested were: Laser power = 0 (150 W), scan speed = 0.18164 (968 mm/s) and scan spacing = 0.56836 (89 \(\upmu\)m). The predicted _Sa_ and measured _Sa_ values were determined to have an average error of 5% for 45\({}^{\circ}\) down-facing surfaces and 6.3% for 35\({}^{\circ}\) down-facing surfaces, which are  
Figure 9: Interaction effects of parameters with the measured _Sa_ for 35\({}^{\circ}\) down-facing surface.  
considered to be acceptable at this stage. However, further experiments and measurements are being conducted in order to further feed and develop the model.