A Dynamic Aggregation Strategy Enhanced Efficient Global Optimization Algorithm for Solving High-Dimensional Turbomachinery Design Problems
Authors: Qineng Wang, Zhendong Guo, Yun Chen, Guangjian Ma, Liming Song, Jun Li
Organizations: Institute of Turbomachinery, Xi’an Jiaotong University, No.28, West Xianning Road, 710049, Xi’an, China · AVIC Shenyang Engine Design Institute, Fangjialan Road, Wanlian Street, 110066, Shenyang, China
In order to solve the high-dimensional (d≥30) expensive black-box problems within budget, an efficient global optimization (EGO) algorithm with a dynamic aggregation strategy is proposed, labeled as DA-EGO. Specifically, the DA-EGO decomposes the original high-dimensional design space into a set of low-dimensional subspaces for efficient surrogate-based optimization search, and the optimal solutions of subspaces are combined as an elite point for the global search. Most importantly, the subspaces are not fixed. Instead, the subspace variables are updated in each iteration, according to the variable interaction analyses in the sub- and full-spaces. The perturbation method and the analysis of variance are used to detect variable interactions. To further accelerate the optimization progress, the searching ranges of subspaces are also adaptively adjusted according to the analyses of subspace optimization results of the previous iteration. Tests on 21 benchmark instances, comprising seven functions at 30, 60, and 90 dimensions, show that DA-EGO is effective on separable and partially separable problems under a budget of 1500 function evaluations. Its advantage is case-dependent: on the non-separable shifted Rosenbrock function, GSGA performs better at 60 and 90 dimensions, while the 30-dimensional results are statistically comparable to IKAEA and GSGA. Moreover, the advantage of DA-EGO is also seen in the aerodynamic optimization of a transonic rotor blade with 28 variables as well as the compressor stage optimization with 60 variables. With the above, the effectiveness of the proposed DA-EGO has been well demonstrated.
Figures & tables
Figure 1: Framework of the variable interaction based decomposition algorithm
Figure 2: The framework of the proposed DA-EGO algorithm.
Figure 3: The data flow of the proposed DA-EGO algorithm.
Index
Dim
Name
Property
Range
F1
30/60/90
Shifted Elliptic
separable
[−10,10]D
F2
30/60/90
Shifted Rastrigin
separable
[−5,5]D
F3
30/60/90
Shifted Ackley
separable
[−32,32]D
F4
30/60/90
Rotated Elliptic
partially-separable
[−10,10]D
F5
30/60/90
Rotated Rastrigin
partially-separable
[−5,5]D
F6
30/60/90
Rotated Ackley
partially-separable
[−32,32]D
Table 1: Benchmark functions
Index
Description
F1
f(x)=Felliptic (Zsft(x))
F2
f(x)=Frastrigin (Zsft(x))
F3
f(x)=Fackley (Zsft(x))
F4
f(x)=Felliptic (Zrot(x))
F5
f(x)=Frastrigin (Zrot(x))
F6
f(x)=Fackley (Zrot(x))
Table 2: The equations of benchmark function
Func.
Algorithm
Best
Worst
Mean
Std
Wilcoxon
F1
DA-EGO
2.226E-03
4.311E-02
1.730E-02
1.263E-02
N/A
RG-EGO
1.170E-01
6.607E-01
3.295E-01
1.591E-01
+
DE
7.722E+05
2.090E+06
1.336E+06
3.541E+05
+
GA
2.339E+05
5.043E+06
1.814E+06
1.289E+06
+
GSK
1.758E+05
1.470E+06
4.590E+05
2.633E+05
+
Nash-EGO
1.359E+02
3.958E+02
2.442E+02
1.032E+02
+
Table 3: Optimization results of 8 algorithms on the seven 30-D benchmark functions.
Figure 4: Average convergence curves on F1–F6 at 30 dimensions.
Func.
Algorithm
Best
Worst
Mean
Std
Wilcoxon
F1
DA-EGO
1.658E-01
1.109E+00
4.646E-01
2.419E-01
N/A
RG-EGO
6.170E-01
5.215E+00
1.849E+00
1.252E+00
+
DE
5.749E+06
1.293E+07
1.017E+07
1.741E+06
+
GA
2.912E+06
1.165E+07
6.508E+06
2.431E+06
+
GSK
1.985E+06
5.321E+06
3.867E+06
9.036E+05
+
IKAEA
2.219E+05
2.002E+06
6.489E+05
4.406E+05
+
Table 4: Optimization results of 7 algorithms on the seven 60-D benchmark functions.
Figure 5: Average convergence curves on F1–F6 at 60 dimensions.
Func.
Algorithm
Best
Worst
Mean
Std
Wilcoxon
F1
DA-EGO
9.531E-01
4.996E+00
2.096E+00
1.044E+00
N/A
RG-EGO
3.997E+00
5.521E+01
1.128E+01
9.817E+00
+
DE
2.184E+07
3.440E+07
2.886E+07
4.476E+06
+
GA
1.490E+07
2.899E+07
2.191E+07
4.544E+06
+
GSK
5.962E+06
1.461E+07
9.845E+06
2.433E+06
+
IKAEA
8.137E+05
1.954E+06
1.535E+06
4.015E+05
+
Table 5: Optimization results of 7 algorithms on the seven 90-D benchmark functions.
Figure 6: Average convergence curves on F1–F6 at 90 dimensions.
Figure 7: Average convergence curves on shifted Rosenbrock (F7), from left to right: 30D, 60D, and 90D. The horizontal axis is the number of function evaluations (NFE), and the vertical axis is the objective value.
Dimension
Algorithm
Average time
Mean optimum
30D
DA-EGO
35 min
1.73E-02
RG-EGO
36 min
3.30E-01
DE
0.741 s
1.34E+06
GA
3.2 s
1.81E+06
GSK
0.124 s
4.59E+05
Nash-EGO
46 min
2.44E+02
Table 6: Average runtime and mean optimum on shifted elliptic (F1) at 30, 60, and 90 dimensions. Time units are shown explicitly.
Figure 8: Average convergence curves for sub-task iteration multipliers titer=6,9,12 on (a) F1, (b) F4, and (c) F7. Numbers in the DA-EGO legend indicate titer ; GSGA is included for comparison. NFE denotes the number of function evaluations.
Geometric definition
variable index
control coefficient in 0% span
x1,⋯,x5
control coefficient in 25% span
x6,⋯,x10
control coefficient in 50% span
x11,⋯,x15
control coefficient in 75% span
x16,⋯,x20
control coefficient in 100% span
x21,⋯,x25
circumferential translation parameter
x26
Table 7: Design variables in engineering test case
Figure 9: The sketch map of the 3D parameterization method
Condition name
Value
equivalent rotational speed[rpm]
17188.7
number of rotor blades
36
rotor blade aspect ratio
1.19
tip clearance gap[mm]
0.356
inlet total temperature[K]
288.15
inlet total pressure[Pa]
101325
Table 8: Design conditions of the Rotor 37 blade
Figure 10: The comparison of feature curves of CFD simulation and experimrnt result
Algorithm
Efficiency
Pressure ratio
Mass flow(kg/s)
Efficiency improvement
Baseline
0.8539
2.040
20.62
—
DA-EGO
0.8707
2.055
20.96
1.65%
GSGA
0.8696
2.051
20.89
1.57%
IKAEA
0.8665
2.050
20.88
1.26%
GA
0.8673
2.049
20.88
1.34%
DE
0.8645
2.049
20.83
1.06%
Table 9: The optimization results of the engineering test case 1 in detail.
Figure 11: The limit streamlines the suction surface of the DA-EGO algorithms and the baseline design of the Rotor 37 blade.
Figure 12: The geometry model and mesh of the multistage compressor.
Condition name
Value
equivalent rotational speed[rpm]
8614.2
number of rotor blades
51/47/67/57/97
tip clearance gap[mm]
0.3
inlet total temperature[K]
288.15
inlet total pressure[Pa]
101325
design average outlet static pressure[Pa]
220000
Table 10: Design conditions of the multistage compressor
Algorithm
Efficiency
Pressure ratio
Mass flow(kg/s)
Efficiency improvement
Baseline
88.072%
2.4604
53.6465
—
DA-EGO
89.142%
2.4644
53.9005
1.070%
GSGA
88.973%
2.4630
53.7780
0.901%
IKAEA
88.905%
2.4623
53.8325
0.833%
GA
88.804%
2.4634
53.8110
0.732%
DE
88.495%
2.4575
53.5025
0.423%
Table 11: The optimization results of the engineering test case 2 of the multistage axial compressor in detail.
Figure 13: total-efficiency distribution along the blade span.
Figure 14: The comparison of entropy function contour of DA-EGO’s optimization result and the baseline design in engineering case 2.
Shanghai Institute of AI for Education, and School of Computer Science and Technology, East China Normal University, Shanghai 200062, China · Ant Group, Hangzhou 310013, China · Nanjing University, Nanjing 210023, China