## 3 Optimization Framework  
### Optimization Process  
To achieve a better aerodynamic performance and more economical processing costs, we propose a POD-based integrated optimization method. Figure 2 shows the process of the method, and the detailed procedure is as follows:  
1. Select the design variables and determine their lower and upper bounds. 2. Choose an appropriate design of experiment (DOE) to generate initial sample points. There are several types of DOEs, such as fractional factorial, full factorial, orthogonal array, Latin hypercube, and parameter study. In this study, the Latin hypercube was chosen to carry out 600 experiments, as it allows many points and more combinations can be studied for each factor, and the engineer has total freedom in selecting the number of designs to run as long as it is greater than the number of factors. 3. Utilize the parametric modeling method to generate the new geometry of the impeller and radial diffuser for all the samples obtained from Step (2). Additionally, the mass of blades can be obtained during this process. The parametric modeling method will be introduced in Section 3.2. Figure 1: 3D view of the original centrifugal compressor. 4. Conduct high-fidelity simulations such as computational fluid dynamics (CFD) and finite element analysis (FEA) and obtain the relevant responses (objectives and constraints) for all the sample points. 5. Generate the initial sample database by combing the sample points and their relevant responses. 6. Remove some of the samples (unfeasible or poor designs) and the samples are then sorted according to the satisfaction of the objectives and constraints. In other words, the best point in the samples is placed in the first place, and the suboptimal point is placed in the second place, and so on. 7. Generate a snapshot matrix. The data were standardized to have a zero mean and unit variance. In this study, 120 sample points were chosen to generate the snapshot matrix. 8. Conduct a POD analysis of the snapshot matrix, and determine the number of principal components to be retained. In this study, only four principal components were retained. The POD method will be introduced in Section 3.4.1. 9. Check whether the contribution rate is satisfied or not. If not, increase the sample points using an adaptive sampling method and then update the snapshot matrix. If yes, the cycle will continue. In this study, only one cycle of the sample point adding process is performed to satisfy the requirement of the set value for the contribution rate (85%), and 150 new sample points are added. The sample point adding strategy will be introduced in Section 3.4.2. 10. Select POD coefficients to act as new design variables and a modified geometry in the reduced order space by changing the POD coefficients. Then, project the data back to the original space. 11. Utilize the parametric modeling method to generate the new geometry of the impeller and radial diffuser as Step (3). 12. Conduct a multidisciplinary analysis in sequence. The analysis includes aerodynamics analysis, strength analysis, and vibration analysis, which will be introduced in Section 3.3. 13. Optimize the POD coefficients using an appropriate algorithm. The formulation of the optimization problem will be introduced in Section 3.5. In this study, a hybrid optimization searching algorithm was adopted. The genetic algorithm (GA) was firstly used to search the optimized point and then sequential quadratic programming (SQP) was employed for local optimization. The whole process was integrated and optimized by commercial software iSIGHT (Version 5.9.4, Dassault Systems, Velizy-Villacoublay, France, 2015) [28]. 14. Check whether a convergence or termination criterion is satisfied or not. If not, repeat Steps (10)-(13). If yes, the cycle will be finished, and an optimal design will be obtained. 15. The feedbacks are set up in the process to reduce searching time, which is stated as follows:  
* Parameterization part: Check the requirements of the exit lean angle.