## 3 Results and Discussions
### Parameter Sensitivity Analysis for Surface Roughness Using Taguchi Orthogonal Array  
In P\(\mu\)SL process, a 3D part is built in a layer-by-layer fashion. Since each layer has its own characteristic side profile as a result of photopolymerization reaction, when they are stacked together repeatedly, a distinctive surface roughness arises. This is called the staircase effect. We performed a systematic analysis to study effect of P\(\mu\)SL process parameters on the surface roughness of a 3D printed structure. Since there are many parameters involved, we employed design of experiment (DOE) method proposed by [PERSON] to reduce the number of experiments to be performed. This method is known as Taguchi method of orthogonal arrays (OA) [37; 38; 39]. Figure 8 shows the steps involved in DOE for this study.  
Based on the result obtained in the previous section, we selected (PI), (PA), (O), layer thickness (LT), and curing time (CT) to control surface roughness. We set three levels for each parameter, or  
Figure 8: Steps for developing a robust DOE.  
Figure 7: Effect of printing parameters on (**a**) curing depth and (**b**) curing width.  
'factor'. Therefore, the total number of possible combinations of parameters obtainable is \(3^{5}=243\). From standard Taguchi OA tables, L27(313) meets the criteria of our analysis where there are five factors with three levels. This dramatically reduces the number of experiment necessary to only 27. Following this, we performed 27 simulations and numerically extracted surface roughness from each simulation. Detailed method to extract surface profile from photopolymerization simulation is described in Appendix D. The factors and their levels, along with surface roughness as response, are shown in Table 5. Surface roughness is measured by RMSE. After obtaining RMSE from each simulation, sensitivity analysis was performed to find the optimal levels for each factor. Signal-to-noise (SN) ratios measure how the response (RMSE value in this study) varies relative to a target value (the expectation set to find optimal factor levels). Since minimum surface roughness would give high quality AM parts,'smaller the better' criterion was chosen to evaluate the factor levels. The formula to calculate SN ratio for'smaller the better' criterion is given by,  
\[\text{SN ratio }=10*\log_{10}\left(\frac{\sum Y^{2}}{n}\right), \tag{14}\]  
where \(Y\) is the response of given factor level combination--i.e., RMSE for this case--and \(n\) is the number of responses in given factor level combination. Using equation (14), SN ratios for each experiment were obtained and listed in Table 5.  
Based on this result, main effects plots for SN ratios in Figure 9 were generated. Also, a response table (Table 6) was generated, from which the parameters that have the largest effect on the response can be identified. In Figure 9, the average SN ratio of the response is presented by the dotted line. Since the goal here is to find values for each parameter that maximize the SN ratio, the optimal values for each parameter can be determined to achieve the minimum surface roughness. The rank represents which factor affects surface roughness the most. Rank is based on the delta value, which is  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Simulation**} & \multicolumn{5}{c}{**Factors**} & \multirow{2}{*}{**RMSE** **(\(\mathbf{\mu}\))**} \\ \cline{2-2} \cline{4-7}  & & & & & **Layer Thickness** & **Curing Time** & **(\(\mathbf{\mu}\))** & **SN Ratio** \\ \hline
1 & 1 & 0.05 & 10 & 100 & 2 & 1 & 0 \\
2 & 1 & 0.05 & 10 & 100 & 3  
the difference between the highest and the lowest average SN ratio value for each factor. From the response table, the parameter that affects surface roughness the most is PA concentration, and oxygen concentration has the least impact on surface roughness.