## 4 Modeling and Stiffness Analysis  
### Sandwich Structural Parameters Optimization  
The multi-objective optimization has a rather different perspective compared with single-objective optimization. There is only one global optimum in single-objective optimization, but in multi-objective optimization there is a set of solutions, named the Pareto-optimal set, which are considered to be equally important; all of them constitute global optimum solutions [32; 33; 34]. Non-dominated Sorting Genetic Algorithm II (NSGA-II) is regarded as an effective and efficient multi-objective optimization technique in various applications [33; 34]. To optimize the sandwich structures, multi-objective optimization of sandwich structural parameters based on an improved NSGA-II algorithm were conducted using MATLAB R2013a software according to Equations (3)-(5), where the lower and upper bounds of the design variables were: \(0.1\leq t_{s}\leq 1.0\), \(1\leq n\leq 10\) and \(20\leq c\leq 30\) respectively. The detailed optimization process of the sandwich structural parameters will be reported in another investigation. Table 4 listed two of the optimal results from the Pareto-optimal set of the sandwich structural parameters in combination with commercial specifications of skin layer and core layer materials for further framework structural optimization.  
Figure 3 presented the schematic diagrams of the two types of sandwich panels, where, [\(0^{\circ}-45^{\circ}\) \(45^{\circ}\) \(0^{\circ}\)] and [\(0^{\circ}\) \(0^{\circ}-45^{\circ}\) \(0^{\circ}\) \(0^{\circ}\) \(45^{\circ}\) \(0^{\circ}\) \(0^{\circ}\)] represent orientation angle of CF in per layer. Zero degree orientation angle is for axial load (\(z\)-direction load in the present study), and \(\pm 45^{\circ}\)orientation angles are for shear load (\(x\)-direction or \(y\)-direction load in the present study).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Name**} & \multirow{2}{*}{**Material Type**} & \multicolumn{3}{c}{**Parameters**} & \multicolumn{2}{c}{**Properties**} \\ \cline{3-6}  & & \(n\) & \(t_{s}\) (mm) & \(c\) (mm) & \(EI\) (Nm\({}^{2}\)) & \(m\) (kg/m\({}^{2}\)) \\ \hline Composite plate & CF + PVC & 4 & 0.57 & 24 & 0.27 & 8.104 \\ Reinforced composite plate & CF + PVC & 8 & 0.57 & 20 & 0.47 & 14.770 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Two of the optimal results of the sandwich structure of the portal.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Objective** & **Function** & **Description** \\ \hline Minimize & Mass & Defined by Equation (4) \\ Minimize & Material costs & Defined by Equation (5) \\ Maximize & Stiffness & Defined by Equation (3) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Objective functions defined within the size optimization.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Material Type**} & \multicolumn{3}{c}{**Material Properties**} & \multirow{2}{*}{\(C_{kg}\) (USD/kg)} \\ \cline{2-3}  & \(E\) (GPa) & \(G\) (GPa) & \(\rho\) (kg/m\({}^{3}\)) & \\ \hline CF & 200 & 4.4 & 1650 & 60 \\ GF & 16.7 & 2.237 & 1704 & 15 \\ Al & 70 & 26.1 & 2700 & 6 \\ PVC foam & 0.13 & 0.035 & 100 & 55 \\ PUR foam & 0.055 & 0.021 & 150 & 22 \\ \hline \hline \end{tabular} Note: Data were taken from the literature [30; 31], and costs were converted from SEK to USD.  
\end{table}
Table 3: Materials properties and the costs used within the optimization studies.