# Numerical Design and Optimisation of Self-Compacting High Early-Strength Cement-Based Mortars
## 3 Response Models
### D-Flow Model  
For the D-flow model, the authors decided not to include the axial points (CC1, CC2, CC3, CC4, CC5, CC6, CC7, CC8, CC9, and CC10) because the D-flow values contained a wide range of spread flow diameters, as presented in Section 2.2, from 168 to 398 mm. The attempt to fit a quadratic model led to a significant lack of fit when axial runs were included, due essentially to non-linearities that a quadratic model is not able to explain. Moreover, \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **D-Flow (mm)** & **T-Funnel (s)** & **F,24 h (MPa)** & **Rc,24 h (MPa)** \\ \hline \multicolumn{5}{c}{All 50 mixtures} \\ \hline Minimum & 168.00 & 6.91 & 9.43 & 48.92 \\ Maximum & 398.00 & 114.06 & 12.79 & 78.42 \\ Mean & 323.31 & 22.39 & 11.17 & 61.61 \\ Standard deviation & 48.97 & 21.63 & 0.77 & 6.24 \\ Coefficient of variation (\%) & 15.15 & 96.62 & 6.93 & 10.14 \\ \hline \multicolumn{5}{c}{Central points (Ci)} \\ \hline Minimum & 339.50 & 13.56 & 11.03 & 59.31 \\ Maximum & 348.00 & 14.98 & 12.29 & 62.17 \\ Mean & 343.75 & 14.29 & 11.67 & 60.52 \\ Standard deviation & 3.31 & 0.47 & 0.48 & 1.10 \\ Coefficient of variation (\%) & 0.96 & 3.31 & 4.16 & 1.81 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The main statistics for all CCD points and central points. axial points are associated with high Cook's distances and leverage points. Further research should be put into finding a non-linear model for D-flow response. Table 5 presents the ANOVA for D-flow data considering the aforementioned conditions. The reduced quadratic model is significant. Examining the magnitude of the effects, it can be perceived that Vs/Vm (factor X4) and Vw/Vp (factor X3) are dominant compared to Sp/p (factor X2), Vw/Vc (factor X1), and Vs/Vs (factor X5). The analysis of variance confirms the previous interpretation. In this case, Vw/Vc, Sp/p, Vw/Vp, and Vs/Vm are significant terms of the model, i.e., the main effects Vw/Vc, Sp/p, Vw/Vp, and Vs/Vm, and the second order effects, (Vw/Vc) \(\times\) (Vw/Vp), (Vw/Vp) \(\times\) (Vfs/Vs), and (Vw/Vc)2, are those that most significantly influence the response Y1: D-flow. The predicted R2 of 0.9683 is in reasonable agreement with the adjusted R2 of 0.9802 (i.e., the difference is less than 0.2). The coefficients of the estimated regression model for D-flow (in coded values) are presented in the second column of Table 6. Since the second-order effect (Vw/Vp) \(\times\) (Vfs/Vs) is considered a significant factor, Vfs/Vs was added to the model. Furthermore, (Sp/p) \(\times\) (Vfs/Vs) was forced to enter the model to avoid a significant lack of fit. Table 6 presents the model's coefficients' standard error and confidence intervals. Figure 2a presents the normal probability plot of the model's residuals. It is reasonable to consider that there are no relevant normality issues in the data. The plot of externally studentised residuals is presented in Figure 2b. One can notice an observation falls outside the \(-\)2 to +2 band, corresponding to the spreading diameter measurement of F5 with a \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Factor** & **Coefficient Estimate** & **Standard Error** & **95\% CI Low** & **95\% CI High** \\ \hline Intercept & 343.75 & 1.95 & 339.76 & 347.74 \\ Vw/Vc & 8.92 & 0.9747 & 6.93 & 10.92 \\ Sp/p & 4.58 & 0.9747 & 2.58 & 6.57 \\ Vw/Vp & **19.42** & 0.9747 & 17.43 & 21.42 \\ Vs/Vm & **-34.92** & 0.9747 & \(-\)36.92 & \(-\)32.93 \\ Vfs/Vs & \(-\)1.42 & 0.9747 & \(-\)3.42 & 0.5748 \\ Vw/Vc \(\times\) Sp/p & \(-\)2.89 & 0.9747 & \(-\)4.89 & \(-\)0.894 \\ Vw/Vc \(\times\) Vw/Vp & \(-\)5.36 & 0.9747 & \(-\)7.36 & \(-\)3.36 \\ Vw/Vc \(\times\) Vs/Vm & 3.11 & 0.9747 & 1.11 & 5.11 \\ Sp/p \(\times\) Vs/Vs & 1.89 & 0.9747 & \(-\)0.106 & 3.89 \\ Vw/Vp \(\times\) Vs/Vs & 2.98 & 0.9747 & 0.9877 & 4.98 \\ (Vw/Vc)2 & **-20.67** & 2.18 & \(-\)25.14 & \(-\)16.21 \\ \hline \hline \end{tabular}  
* The three most significant parameters are typed in **bold**, and the most significant term is also underlined.  
\end{table}