## Abstract

A new multi-objective, multi-disciplinary global optimization strategy is proposed to address the high-dimensional, computationally expensive black box problem (HEB) in turbomachinery design. The strategy consists of an adaptive sampling hybrid optimization algorithm (ASHOA), two data-mining techniques, a 3D blade parameterization method, and the aerodynamic/mechanical codes. Firstly, the ASHOA is established by integrating a novel adaptive sampling Kriging metamodel and a new hybrid optimization method. Secondly, two data-mining methods (analysis of variance (ANOVA) and self-organizing map (SOM)) are applied to set the initial design space and optimization objectives of the transonic centrifugal compressor. A refined design space and objective parameters of the optimization problem are eventually obtained. Finally, the optimization process of a transonic centrifugal compressor is carried out based on the refined design space and objectives using ASHOA. The results show that the search efficiency of the optimization strategy is 2–10 times higher when compared to other excellent optimization algorithms. For the optimized compressor, both isentropic efficiency and total pressure ratio at design condition are improved by 1.61% and 4.13%, respectively, and the maximum stress decreases by 9.68%.

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## Acknowledgments

The authors would like to acknowledge National Laboratory of Engine Turbocharging Technology, North China Engine Research Institute, for providing experimental data.

### Funding information

This work was supported by the National Natural Science Foundation of China under Grant number 11672206.

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## Appendices

### Appendix 1

### ZDT1

ZDT1 has 30 design variables and multiple Pareto fronts. The mathematical expression is

The Pareto-optimal region for the case ZDT1 corresponds to *x*_{1} ∈ [0, 1], *x*_{i} = 0, *i* = 2, ⋯, 30.

### ZDT2

ZDT2 has 30 design variables and multiple Pareto fronts. The mathematical expression is

The Pareto-optimal region for the case ZDT2 corresponds to *x*_{1} ∈ [0, 1], *x*_{i} = 0, *i* = 2, ⋯, 30.

### ZDT4

ZDT4 has 10 design variables and multiple local Pareto fronts. The mathematical expression is

The Pareto-optimal region for the case ZDT4 corresponds to *x*_{1} ∈ [0, 1], *x*_{i} = 0, *i* = 2, ⋯, 10. There exist 21^{9} local Pareto-optimal solutions and about 100 distinct Pareto fronts.

The basic optimization parameters of ASHOA for ZDT1, ZDT2, and ZDT4 are given in Table 13. Figure 26 shows the optimization results for test cases ZDT1, ZDT2, and ZDT4.

### Appendix 2

The demonstrations of the procedure (see Section 2.3.3) to search the optimal *λ* for test cases ZDT3 and ZDT6 (Figs. 27 and 28).

### ZDT 3

*(1) λ* = 0*, the growth rate Δλ* = 0.05*, λ*_{1} = *λ* + *Δλ* = 0.05*.*

*(2) λ*_{1} < 1*, go next.*

*(3) & (4)*

Fig. 3a*, the 1st iteration,*

*λ* = 0*,λ*_{1} = 0.05*,N*_{λ1} = 15 > *N*_{λ} = 1*, then update λ* = 0.05*, λ*_{1} = 0.1*, go next.*

Fig. 3b*, the 2nd iteration,*

*λ* = 0.05*,λ*_{1} = 0.1*,N*_{λ1} = 6 > *N*_{λ} = 1*, then update λ* = 0.1*,λ*_{1} = 0.15*, go next.*

Fig. 3c*, the 3rd iteration,*

*λ* = 0.1*,λ*_{1} = 0.15*,N*_{λ1} = 6 > *N*_{λ} = 2*, then update λ* = 0.15*,λ*_{1} = 0.2*, go next.*

Fig. 3d*, the 4th iteration,*

*λ* = 0.15*,λ*_{1} = 0.2*,N*_{λ1} = 17 > *N*_{λ} = 1*, then update λ* = 0.2*,λ*_{1} = 0.25*, go next.*

Fig. 3e*, the 5th iteration,*

*λ* = 0.2*,λ*_{1} = 0.25*,N*_{λ1} = 20 > *N*_{λ} = 1*, then update λ* = 0.25*,λ*_{1} = 0.3*, go next.*

Fig. 3f*, the 6th iteration,*

*λ* = 0.25*,λ*_{1} = 0.3*,N*_{λ1} = 22 > *N*_{λ} = 0*, then update λ* = 0.3*,λ*_{1} = 0.35*, go next.*

Fig. 3g*, the 7th iteration,*

*λ* = 0.3*,λ*_{1} = 0.35*,N*_{λ1} = 2 < *N*_{λ} = 24*, end loop, go next.*

*(5) The current λ* = 0.3 *is the optimal value, and it is used in the optimization algorithm (ASHOA), and the procedure stops.*

### ZDT 6

*(1) λ* = 0*, the growth rate Δλ* = 0.05*, λ*_{1} = *λ* + *Δλ* = 0.05*.*

*(2) λ*_{1} < 1*, go next.*

*(3) & (4)*

Fig. 3a*, the 1st iteration,*

*λ* = 0*,λ*_{1} = 0.05*,N*_{λ1} = 6 > *N*_{λ} = 0*, then update λ* = 0.05 *λ*_{1} = 0.1*, go next.*

Fig. 3b*, the 2nd iteration,*

*λ* = 0.05*,λ*_{1} = 0.1*,N*_{λ1} = 10 > *N*_{λ} = 1*, then update λ* = 0.1*,λ*_{1} = 0.15*, go next.*

Fig. 3c*, the 3rd iteration,*

*λ* = 0.1*,λ*_{1} = 0.15*,N*_{λ1} = 1 < *N*_{λ} = 8*, end loop, go next.*

*(5) The current λ* = 0.1 *is the optimal value, and it is used in the optimization algorithm (ASHOA), and the procedure stops.*

### Appendix 3

The optimal *λ* found by the procedure (see Section 2.3.3) for test cases ZDT1, ZDT2, and ZDT4 are shown in Fig. 29.

### Appendix 4

In Section 2.1, the proportion of individuals to carry out gradient mutation is specified to be 1/*q* × 100% in *step (5)* of the new gradient mutation algorithm. This threshold value is an important parameter which may influence the global optimization performance and convergence efficiency of the optimization algorithm. Therefore, sensitivity analysis of the proportion is investigated with the same test cases (ZDT1 to ZDT6, except for ZDT5).

In this paper, the proportion is 1/*q* × 100%, where *q* is the number of objectives in optimization. For multi-objective optimization problems, *q* usually equals 2–4. In the study, the five mathematical test cases all have two objectives (1/*q* = 0.50). Four types of proportion (0.25, 0.50, 0.75, 1.00) are imposed for the optimization of the five test cases. The Pareto front solutions are searched over the same period with different proportions, and the results are plotted in Fig. 30. Overall, the global optimization performance and convergence efficiency are not very sensitive to the proportion between 0.25 to 1.00 for all test cases, particularly for ZDT2, ZDT3, and ZDT5. If the Pareto front solutions are compared carefully, it is found that both convergence efficiency and global optimization performance decrease slightly under small or large proportions.

This can be explained by the following facts. When a small proportion is imposed, few individuals are selected to carry out gradient mutation, and some individuals with large gradient values are ignored. Therefore, the convergence efficiency may suffer. On the other hand, if the proportion is large, more individuals are chosen to implement gradient mutation. Those individuals which have very small gradient (near zero) are also selected to conducted gradient mutation. It will increase the computational resources and further decrease convergence efficiency. Additionally, the gradient mutation of small gradient individuals has tiny contributions to convergence. By contrast, it cannot protect excellent individuals, leading to the poorer global optimization performance and convergence efficiency.

As shown in Section 2.3.3, the normalized gradient mutation step size *λ* is selected adaptively during the optimization process. When the proportion of individuals for gradient mutation is specified, the ASHOA will determine a suitable *λ* to balance the high convergence efficiency and good global optimization performance. Taking ZDT1 as an example, when the proportion is 0.25, the optimal *λ* = 0.9 while when the proportion is 0.50 the optimal *λ* = 0.8. The effect of the varying proportion on the optimization algorithm is counteracted by adaptive *λ*. Therefore, the specific proportion (1/*q*) for the proposed algorithm ASHOA is reasonable and robust.

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Li, X., Zhao, Y. & Liu, Z. A novel global optimization algorithm and data-mining methods for turbomachinery design.
*Struct Multidisc Optim* **60, **581–612 (2019). https://doi.org/10.1007/s00158-019-02227-5

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### Keywords

- Adaptive sampling
- Hybrid optimization
- Data-mining
- Turbomachinery