# A Study on Aeroengine Conceptual Design Considering Multi-Mission Performance Reliability
## 5 Results and Discussion
### Validation of ANN  
The effectiveness of proposed HAPSOLM and the accuracy and efficiency of pre-trained ANN are validated in this section. In this study, the total number of the training samples is 280, which exceeds 30 times the number of key design variables and considered to be sufficient to adequately search the design variables space. These training samples are employed to respectively create four ANN surrogate models for reliability prediction. Started with the same initial values of connection weights and thresholds, ANN pre-training is carried out to compare the APSO with the standard PSO. Figure 8 displays the comparison results in terms of convergence characteristics of the standard PSO and APSO for approaching the ANN pre-training. It demonstrated that APSO can converge to a local optimum more quickly in the early phase. Besides, APSO is more capable to jump out of the local optima and find the other potential optimal region in the evolutionary processes, owing to the time-varying adaptive inertia weight. In the first stage of HAPSOLM, a better-conditional neural network is initialized that contains relatively high-precision connection weights and thresholds. In the second stage of HAPSOLM, we gained the final connection weights and thresholds of ANN by LM-based training, including:  
(1) ANN\({}_{1}\) for the reliability prediction of Fn under TkO condition:  
\[IW=\left[ \begin{array}{cccc}-3.229&-0.972&6.773&-4.346&-2.403\\ -2.273&-1.310&7.990&-8.926&9.541\\ 1.113&1.054&0.994&-4.923&-2.065\end{array} \right],b_{1}=\left[ \begin{array}[] {cccc}4.849\\ -2.438\\ -4.667\end{array} \right]LW=\left[ \begin{array}{cccc}0.085\\ -0.999\\ -0.089\end{array} \right]^{T},b_{2}=\left[0.007\right] \tag{33}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Operating Condition** & **Performance Parameter** & **Gasturb** & **MTF Model** & **Error (\%)** \\ \hline TkO & Fn (kN) & 69.91 & 68.21 & 2.43 \\ SbC & Fn (kN) & 19.01 & 18.48 & 2.79 \\  & SFC (kg/n*kN) & 89.77 & 92.02 & 2.44 \\  & Fn (kN) & 29.54 & 29.51 & 0.102 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Simulation comparison results of Gasturb® and turbofan model. MTF = mixed flow turbofan engine. ANN\({}_{2}\) for the reliability prediction of Fn under SpP condition:  
\[IW=\left[ \begin{array}{cccc}-0.257&-0.980&1.539&1.482&0.464\\ 6.206&0.044&-4.058&3.008&-10.78\\ 0.639&1.306&2.606&0.650&3.553\end{array} \right],b_{1}=\left[ \begin{array}{ c}3.962\\ 2.493\\ -4.693\end{array} \right],LW=\left[ \begin{array}{c}1.096\\ -0.992\\ -0.003\end{array} \right]^{T},b_{2}=[-1.079] \tag{34}\]  
ANN\({}_{3}\) for the reliability prediction of Fn under SbC condition:  
\[IW=\left[ \begin{array}{cccc}2.334&1.497&-8.591&9.910&-9.617\\ -7.756&0.221&11.08&-6.747&7.102\\ 36.63&-41.53&18.26&-17.09&12.48\end{array} \right],b_{1}=\left[ \begin{array}[ ]{c}2.679\\ 10.42\\ -29.34\end{array} \right],LW=\left[ \begin{array}{c}0.975\\ 0.105\\ -0.001\end{array} \right]^{T},b_{2}=[0.084] \tag{35}\]  
ANN\({}_{4}\) for the reliability prediction of SFC under SbC condition:  
\[IW=\left[ \begin{array}{cccc}0.472&2.906&-0.475&3.724&0.036\\ -2.941&-0.153&4.589&0.012&-0.289\\ -1.857&-1.439&3.815&-6.196&0.674\end{array} \right],b_{1}=\left[ \begin{array} []{c}-0.390\\ 2.601\\ -1.737\end{array} \right],LW=\left[ \begin{array}{c}0.589\\ 0.411\\ -0.418\end{array} \right]^{T},b_{2}=[-0.403] \tag{36}\]  
Another 20 data sets were utilized to validate the effectiveness of the HAPSOLM and the MC simulation results of testing samples are regarded as the expected outputs. Figure 8: Pre-training convergence performance of the standard Particle Swarm Optimization (PSO) and Adaptive Particle Swarm Optimization (APSO). samples according to reliability values from small to large; Figure 9 graphically compares the predictions of ANN and pre-trained ANN. It is seen that the prediction results of ANN and pre-trained ANN are basically consistent with MC results.