## 3 Optimization Framework
### Formulation of the Optimization Problem  
The definition of the MDO problem includes the determination of design variables, constraints, and objectives. The main modeling parameters for the impeller and radial diffuser include the distribution of the blade angle of the mean camber line at the blade hub and blade tip and the distribution of thickness. Specifically, the distribution of blade angle can mainly determine the aerodynamic performance of the impeller and can dominate the characteristics of impeller strength and vibration simultaneously. Therefore, only the distribution of the blade angle of the impeller and diffuser is optimized in the present paper, and the distribution of thickness remains the same. During the optimization process, the control points of Bezier curves (shown in Figure 3) are selected as design variables, the values of which are listed in Table 1.  
Constraints are typically the requirements of aerodynamics, strength, vibration, etc. For aerodynamics, the blade shape should satisfy the criteria of aerodynamic design accurately, and the major constraints of aerodynamics of the current MDO are the mass flow, the pressure ratio, and the surge margin. From a strength perspective, the blade should not be broken under the working loads, and the maximum radial stress should therefore be less than the allowable stress in the present research. Low cycle fatigue is the most important factor affecting the life of the impeller disc. In the process of engine development, low cycle fatigue life is guaranteed to meet the user-specified design life requirements mainly through design and test. The TBO (time between overhaul) life of this engine is 1000 h. According to the requirement of TBO and flight load spectrum, the life of the impeller needs to reach 9000 cycles. Once the maximum von Mises stress is obtained, the life of the blade can be calculated according to the same classical methods with the material properties. Therefore, the maximum von Mises stress is employed as a constraint. When considering the vibration, the blade \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Design** & **Lower** & **Upper** & **Design** & **Lower** & **Upper** & **Design** & **Lower** & **Upper** \\
**Variable** & **Bound** & **Bound** & **Variable** & **Bound** & **Bound** & **Variable** & **Bound** & **Bound** \\ \hline \(G_{0}\) & \(-55\) & \(-38\) & \(J_{4}\) & \(-15\) & \(15\) & \(X_{9}\) & \(0.4\) & \(0.8\) \\ \(G_{1}\) & \(-47\) & \(-17\) & \(J_{5}\) & \(-10\) & \(10\) & \(X_{10}\) & \(0.6\) & \(1.0\) \\ \(G_{2}\) & \(-41\) & \(-11\) & \(X_{1}\) & \(0.1\) & \(0.3\) & \(R_{0}\) & \(-74\) & \(-68\) \\ \(G_{3}\) & \(-13\) & \(17\) & \(X_{2}\) & \(0.2\) & \(0.5\) & \(R_{1}\) & \(-67\) & \(-37\) \\ \(G_{4}\) & \(-9\) & \(21\) & \(X_{3}\) & \(0.3\) & \(0.6\) & \(R_{2}\) & \(-76\) & \(-46\) \\ \(G_{5}\) & \(-10\) & \(10\) & \(X_{4}\) & \(0.4\) & \(0.8\) & \(R_{3}\) & \(-75\) & \(-45\) \\ \(J_{0}\) & \(-68\) & \(-62\) & \(X_{5}\) & \(0.6\) & \(1.0\) & \(R_{4}\) & \(-65\) & \(-58\) \\ \(J_{1}\) & \(-78\) & \(-48\) & \(X_{6}\) & \(0.1\) & \(0.3\) & \(Y_{1}\) & \(0\) & \(0.5\) \\ \(J_{2}\) & \(-66\) & \(-36\) & \(X_{7}\) & \(0.2\) & \(0.5\) & \(Y_{2}\) & \(0.25\) & \(0.75\) \\ \(J_{3}\) & \(-45\) & \(-15\) & \(X_{8}\) & \(0.3\) & \(0.6\) & \(Y_{3}\) & \(0.5\) & \(1.0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design variables. Figure 8: Adaptive sampling point adding strategy. resonance should be avoided. After frequency analysis, the natural frequency of former orders of the blade can be gained and current investigation only focuses on the vibration frequency (dynamic frequency) of the four former orders of the blade. The index to evaluate regarding whether the blade can be kept away from resonance is the resonance margin. For the present MDO, the resonance margin should not be less than 10%. In the current research, the objective was to gain the optimized isentropic efficiency and minimize the number of blades. The optimization problem is presented as Equation (7).