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A novel global optimization algorithm and data-mining methods for turbomachinery design

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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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References

  1. Aslimani N, Ellaia R (2017) A new hybrid algorithm combining a new chaos optimization approach with gradient descent for high dimensional optimization problems. Comput Appl Math 4:1–29

  2. Barsi D, Perrone A, Ratto L, Simoni D, Zunino P (2015) Radial inflow turbine design through multi-disciplinary optimisation technique. ASME Turbo Expo 2015:GT2015–G42702

  3. Barsi D, Perrone A, Qu Y, Ratto L, Ricci G, Sergeev V, Zunino P (2018) Compressor and turbine multidisciplinary design for highly efficient micro-gas turbine. J Therm Sci 27(3):259–269

  4. Blondet G, Boudaoud N, Duigou J (2015) Simulation data management for adaptive design of experiments. Mech Ind 16(6):611

  5. Chen L, Chen J (2015) Aerodynamic optimization design of multi-stage turbine using the continuous adjoint method. Int J Turbo Jet-Engines 32(2):199–211

  6. Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multi objective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197

  7. Deshmukh A, Allison J (2016) Multidisciplinary dynamic optimization of horizontal axis wind turbine design. Struct Multidiscip Optim 53(1):15–27

  8. Dinh C, Ma S, Kim K (2017) Aerodynamic optimization of a single-stage axial compressor with stator shroud air injection. AIAA J 55(8):2739–2754

  9. Elfert M, Weber A, Wittrock D, Peters A, Voss C, Nicke E (2017) Experimental and numerical verification of an optimization of a fast rotating high-performance radial compressor impeller. J Turbomach 139(10):101007

  10. Gowda K, Prasad S, Nagarajaiah V (2016). Design optimization of t-root geometry of a gas engine HP compressor rotor blade for lifing the blade against fretting failure. ASME 2016, Power 2016–59331

  11. Grosso A, Jamali A, Locatelli M (2009) Finding maximin Latin hypercube designs by iterated local search heuristics. Eur J Oper Res 197(2):541–547

  12. Hehn A, Mosdzien M, Grates D, Jeschke P (2018) Aerodynamic optimization of a transonic centrifugal compressor by using arbitrary blade surfaces. J Turbomach 140(5):051011

  13. Heinrich M, Schwarze R (2016) Genetic algorithm optimization of the volute shape of a centrifugal compressor. Int J Rotating Mach 2016:13

  14. Hu J, Zhou Q, Jiang P, Shao X, Xie T (2017) An adaptive sampling method for variable-fidelity surrogate models using improved hierarchical kriging. Eng Optim 50(1):145–163

  15. Huang D, Allen T, Notz W, Zeng N (2006) Global optimization of stochastic black-box systems via sequential kriging meta-models. J Glob Optim 34(3):441–466

  16. Javed A, Pecnik R, Buijtenen J (2013) Optimization of a centrifugal compressor impeller design for robustness to manufacturing uncertainties. J Eng Gas Turbines Power 138(11):43

  17. Jones D, Schonlau M, Welch W (1998) Efficient global optimization of expensive black-box functions. J Glob Optim 13(4):455–492

  18. Kato H, Funazaki K (2014) POD-driven adaptive sampling for efficient surrogate modeling and its application to supersonic turbine optimization. ASME Turbo Expo 2014:GT2014–G27229

  19. Kobrunov A, Priezzhev I (2016) Hybrid combination genetic algorithm and controlled gradient method to train a neural network. Geophysics 81(4):IM35–IM43

  20. Kristensen J, Ling Y, Asher I, Wang L (2016) Expected-improvement-based methods for adaptive sampling in multi-objective optimization problems. ASME 2016:DETC2016–DET59266

  21. Leborgne M, Lonfils T, Lepot I (2015) Development and application of a multi-disciplinary multi-regime design methodology of a low-noise contra-rotating open-rotor. ASME Turbo Expo 2015:GT2015–G43432

  22. Li X, Zhao Y, Liu Z, Chen H (2016) The optimization of a centrifugal impeller based on a new multi-objective evolutionary strategy. ASME Turbo Expo 2016:GT2016–G56592

  23. Li X, Liu Z, Lin Y (2017) Multi point and multi objective optimization of a centrifugal compressor impeller based on genetic algorithm. Math Probl Eng 2017(1):1–18

  24. Liu H, Xu S, Ma Y, Chen X, Wang X (2015) An adaptive Bayesian sequential sampling approach for global metamodeling. J Mech Des 138(1):011404

  25. Liu Y, Shi Y, Zhou Q, Xiu R (2016) A sequential sampling strategy to improve the global fidelity of metamodels in multi-level system design. Struct Multidiscip Optim 53(6):1295–1313

  26. Liu B, Grout V, Nikolaeva A (2018) Efficient global optimization of actuator based on a surrogate model assisted hybrid algorithm. IEEE Trans Ind Electron 65(7):5712–5721

  27. Long Q, Wu C (2014) A hybrid method combining genetic algorithm and Hooke-Jeeves method for constrained global optimization. J Ind Manag Optim 10(4):1279–1296

  28. Lophaven S, Nielsen H, Sondergaard J (2002) Dace—a MATLAB kriging toolbox (version 2) informatics and mathematical modeling. Technical University of Denmark, Copenhagen

  29. Lu H, Li Q, Pan T (2016) Optimization of cantilevered stators in an industrial multistage compressor to improve efficiency. Energy 106:590–601

  30. Luo L, Hou X, Zhong J, Cai W, Ma J (2017) Sampling-based adaptive bounding evolutionary algorithm for continuous optimization problems. Inf Sci 382–383:216–233

  31. Ma C, Su X, Yuan X (2017) An efficient unsteady adjoint optimization system for multistage turbomachinery. J Turbomach 139(1):011003

  32. Ma S, Afzal A, Kim K (2018) Optimization of ring cavity in a centrifugal compressor based on comparative analysis of optimization algorithms. Appl Therm Eng 138:633–647

  33. Mahmood G, Acharya S (2006) Experimental investigation of secondary flow structure in a blade passage with and without leading edge fillets. J Fluids Eng 129(3):253–262

  34. Ning F, Xu L (2001) Numerical investigation of transonic compressor rotor flow using an implicit 3D flow solver with one-equation Spalart-Allmaras turbulence model, ASME Turbo Expo 2001-GT-0359

  35. Pellegrini R, Iemma U, Leotardi C, Campana E, Diez M (2016) Multi-fidelity adaptive global metamodel of expensive computer simulations. IEEE World Congress Comput Intell https://doi.org/10.1109/CEC.2016.7744355

  36. Pholdee N, Bureerat S (2015) An efficient optimum Latin hypercube sampling technique based on sequencing optimisation using simulated annealing. Int J Syst Sci 46(10):1780–1789

  37. Picheny V, Ginsbourger D, Roustant O, Haftka R, Kim N (2010) Adaptive designs of experiments for accurate approximation of a target region. J Mech Des 132(7):071008

  38. Salnikov A, Danilov M (2017) A centrifugal compressor impeller: a multidisciplinary optimization to improve its mass, strength, and gas-dynamic characteristics. ASME Turbo Expo 2017:GT2017–G64123

  39. Sevastyanov V (2010) Hybrid multi-gradient explorer algorithm for global multi-objective optimization. AIAA/ISSMO Multidisciplinary Analysis Optimization Conference

  40. Song L, Guo Z, Li J, Feng Z (2016) Research on meta-model based global design optimization and data mining methods. J Eng Gas Turbines Power 138(9):092604

  41. Verstraete T, Mueller L, Mueller J (2017) Multidisciplinary adjoint optimization of turbomachinery components including aerodynamic and stress performance. 35th AIAA, Applied Aerodynamics Conference, https://doi.org/10.2514/6.2017-4083

  42. Viana F, Venter G, Balabanov V (2010) An algorithm for fast optimal Latin hypercube design of experiments. Int J Numer Methods Eng 82(2):135–156

  43. Viana F, Haftka R, Watson L (2013) Efficient global optimization algorithm assisted by multiple surrogate techniques. J Glob Optim 56(2):669–689

  44. Walther B, Nadarajah S (2015) Optimum shape design for multi-row turbomachinery configurations using a discrete adjoint approach and an efficient RBF deformation scheme for complex multi-block grids. J Turbomach 137(8):33

  45. Wang L, Han R, Wang T, Ke S (2018) Uniform decomposition and positive-gradient differential evolution for multi-objective design of wind turbine blade. Energies 11(5):1–19

  46. Wu J, Xia J, Chen J, Cui Z (2011) Moving object classification method based on SOM and K-means. J Comput 6(8):1654–1661

  47. Wu G, Qiu D, Yu Y, Pedrycz W, Ma M, Li H (2014) Superior solution guided particle swarm optimization combined with local search techniques. Expert Syst Appl 41(16):7536–7548

  48. Xu S, Liu H, Wang X, Jiang X (2014) A robust error-pursuing sequential sampling approach for global metamodeling based on Voronoi diagram and cross validation. J Mech Des 136(7):69–74

  49. Zhu H, Liu L, Long T, Peng L (2012) A novel algorithm of maximin Latin hypercube design using successive local enumeration. Eng Optim 44(5):551–564

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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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Correspondence to Zhengxian Liu.

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Appendices

Appendix 1

ZDT1

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

$$ {\displaystyle \begin{array}{l}\operatorname{Minimize}\kern0.36em {f}_1={x}_1\kern0.36em \mathrm{and}\kern0.36em {f}_2=g\left(1-\sqrt{\frac{f_1}{g}}\right)\\ {}\mathrm{where},g=1+\frac{9}{n-1}\sum \limits_{i=2}^n{x}_i;\kern0.84em 0\le {x}_i\le 1,i=1,\cdots, n;n=30\end{array}} $$
(23)

The Pareto-optimal region for the case ZDT1 corresponds to x1 ∈ [0, 1], xi = 0, i = 2, ⋯, 30.

ZDT2

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

$$ {\displaystyle \begin{array}{l}\operatorname{Minimize}\kern0.36em {f}_1={x}_1\kern0.36em \mathrm{and}\kern0.36em {f}_2=g\left[1-{\left(\frac{f_1}{g}\right)}^2\right]\\ {}\mathrm{where},g=1+\frac{9}{n-1}\sum \limits_{i=2}^n{x}_i;\kern0.84em 0\le {x}_i\le 1,i=1,\cdots, n;n=30\end{array}} $$
(24)

The Pareto-optimal region for the case ZDT2 corresponds to x1 ∈ [0, 1], xi = 0, i = 2, ⋯, 30.

ZDT4

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

$$ {\displaystyle \begin{array}{l}\operatorname{Minimize}\kern0.36em {f}_1={x}_1\kern0.36em \mathrm{and}\kern0.36em {f}_2=g\left(1-\sqrt{\frac{f_1}{g}}\right)\\ {}\mathrm{where},g=1+10\left(n-1\right)+\sum \limits_{i=2}^n\left({x}_i^2-10\cos \left(4\pi {x}_i\right)\right);\\ {}0\le {x}_1\le 1,\kern0.48em -5\le {x}_i\le 5,i=2,\cdots, n;n=10\end{array}} $$
(25)

The Pareto-optimal region for the case ZDT4 corresponds to x1 ∈ [0, 1], xi = 0, i = 2, ⋯, 10. There exist 219 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.

Table 13 Parameters of ASHOA for ZDT1, ZDT2, and ZDT4
Fig. 26
figure26

Results for test cases ZDT1, ZDT2, and ZDT4 optimized by ASHOA and five other state-of-the-art algorithms (AME, CV-Voronoi, EGO, ADE, and MBOE) at the same search time

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).

Fig. 27
figure27

The process of searching the optimal λ for ZDT3

Fig. 28
figure28

The process of searching the optimal λ for ZDT6

Fig. 29
figure29

Effect of different values of λ on optimization efficiency and solutions diversity for ZDT1, ZDT2, and ZDT4

ZDT 3

(1) λ = 0, the growth rate Δλ = 0.05, λ1 = λ + Δλ = 0.05.

(2) λ1 < 1, go next.

(3) & (4)

Fig. 3a, the 1st iteration,

λ = 01 = 0.05,Nλ1 = 15 > Nλ = 1, then update λ = 0.05, λ1 = 0.1, go next.

Fig. 3b, the 2nd iteration,

λ = 0.051 = 0.1,Nλ1 = 6 > Nλ = 1, then update λ = 0.11 = 0.15, go next.

Fig. 3c, the 3rd iteration,

λ = 0.11 = 0.15,Nλ1 = 6 > Nλ = 2, then update λ = 0.151 = 0.2, go next.

Fig. 3d, the 4th iteration,

λ = 0.151 = 0.2,Nλ1 = 17 > Nλ = 1, then update λ = 0.21 = 0.25, go next.

Fig. 3e, the 5th iteration,

λ = 0.21 = 0.25,Nλ1 = 20 > Nλ = 1, then update λ = 0.251 = 0.3, go next.

Fig. 3f, the 6th iteration,

λ = 0.251 = 0.3,Nλ1 = 22 > Nλ = 0, then update λ = 0.31 = 0.35, go next.

Fig. 3g, the 7th iteration,

λ = 0.31 = 0.35,Nλ1 = 2 < Nλ = 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, λ1 = λ + Δλ = 0.05.

(2) λ1 < 1, go next.

(3) & (4)

Fig. 3a, the 1st iteration,

λ = 01 = 0.05,Nλ1 = 6 > Nλ = 0, then update λ = 0.05 λ1 = 0.1, go next.

Fig. 3b, the 2nd iteration,

λ = 0.051 = 0.1,Nλ1 = 10 > Nλ = 1, then update λ = 0.11 = 0.15, go next.

Fig. 3c, the 3rd iteration,

λ = 0.11 = 0.15,Nλ1 = 1 < Nλ = 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.

Fig. 30
figure30

The comparison of Pareto front solutions searched with different individuals’ proportion over the same period

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