## 3 Results  
### Regression Analysis  
The regression equations for each cell type and for each response factor are listed in Table 2. These equations are mathematical representations that help predict the crucial responses of the surface area to volume ratio and the temperature difference in terms of the factors T and X. Substituting any values for T and X in these linear equations gives the predicted results of each response for each cell type. These equations help quantify the direction, magnitude, and statistical significance of the relationship between variables. Next, the main effect plots and Pareto charts are observed to evaluate the effect of the factors and are shown in Tables 3 and 4. These visualizations provide crucial insights into how lattice thickness (T) and cell size (X) influence SA/V\({}_{\text{S}}\) and \(\Delta\)T for different TPMS structures. The main effect plots show the effect of each factor on the response, direction of the effect, and relative importance of the factors. \begin{table}
\begin{tabular}{c c c c c} \hline  & & **Factors** & & **Responses** \\ \hline \multicolumn{2}{c}{**Cell Type (CT)**} & **Lattice Thickness (T), mm** & **Cell Size (X), mm3** & **SA/V\({}_{\text{S}}\)** & \(\Delta\)T = T\({}_{\text{T}}-10^{\circ}\) C** \\ \hline \multirow{5}{*}{5} & Schwartz & Thin: 1 mm & Small: \(10\times 10\times 10\) & 1.3 & 1.3 \\  & Schwartz & Thin: 1 mm & Medium: \(14\times 14\times 14\) & 1.3 & 1.5 \\  & Schwartz & Thin: 1 mm & Large: \(18\times 18\times 18\) & 1.4 & 1.9 \\  & Schwartz & Moderate: 1.5 mm & Small: \(10\times 10\times 10\) & 0.9 & 0.8 \\  & Schwartz & Moderate: 1.5 mm & Medium: \(14\times 14\times 14\) & 1.0 & 1 \\  & Schwartz & Moderate: 1.5 mm & Large: \(18\times 18\times 18\) & 1.0 & 1.3 \\  & Schwartz & Thick: 2 mm & Small: \(10\times 10\times 10\) & 0.6 & 0.3 \\  & Schwartz & Thick: 2 mm & Medium: \(14\times 14\times 14\) & 0.7 & 0.7 \\  & Schwartz & Thick: 2 mm & Large: \(18\times 18\times 18\) & 0.8 & 0.9 \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._ \begin{table}
\begin{tabular}{c c c c} \hline  & & **SA/Vs** & **\(\Delta\)T** \\ \hline
1 & Gyroid & \(2.820-0.564\) T \(-0.0264\) X + \(0.01008\) T \(\times\) X & \(3.333-0.733\) T \(-0.0315\) X + \(0.0143\) T \(\times\) X \\
2 & Diamond & \(3.034-0.674\) T \(-0.0314\) X + \(0.01395\) T \(\times\) X & \(2.640-0.571\) T + \(0.0077\) X + \(0.0036\) T \(\times\) X \\
3 & Lidinoid & \(2.812-0.628\) T \(-0.0258\) X + \(0.0124\) T \(\times\) X & \(1.213-0.210\) T + \(0.0256\) X \(-0.00357\) T \(\times\) X \\
4 & SplitP & \(3.307-0.786\) T \(-0.0192\) X + \(0.0142\) T \(\times\) X & \(3.157-0.876\) T + \(0.0060\) X + \(0.0143\) T \(\times\) X \\
5 & Schwartz & \(2.070-0.877\) T \(-0.0089\) X + \(0.0177\) T \(\times\) X & \(1.486-0.933\) T + \(0.0708\) X + \(0.0000\) T \(\times\) X \\ \hline \end{tabular}
\end{table} 
Table 2: Regression equations of all five cell types for each of the response factors. Main effect plots for SA/V\({}_{\text{S}}\) and \(\Delta\)T: All the cell types have very similar main effect plots for both responses except for Schwartz. From Gyroid to SplitP, the lattice thickness plot shows a steep slop that represents that this factor is significant. On the other hand, the cell size plot is mostly flat line which means it is not a significant factor. With the increase in factor level, the response decreases, and the slope of T goes downward showing a negative effect. In the case of cell size, the response increased in some cases but not significantly. Thus, the relative importance of these factors can be understood. Pareto charts for SA/V\({}_{\text{S}}\): Lattice thickness is the most significant factor for all the cell types for SA/V\({}_{\text{S}}\). The lattice thickness bar crosses the cumulative line (red vertical line) that represents the 80/20 rule of the Pareto principle for all the cell types. This means that changing the lattice thickness would have a dramatic influence on the resulting surface area to volume ratio but changing the cell size will not affect the result so much.