Aero-mechanical multidisciplinary optimization of a high speed centrifugal impeller

Aerospace Science and Technology 95 (2019) 105452

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Aero-mechanical multidisciplinary optimization of a high speed centrifugal impeller Chenxi Li a , Jing Wang b , Zhendong Guo c,∗ , Liming Song a , Jun Li a a b c

Institute of Turbomachinery, Xi’an Jiaotong University, Xi’an 710049, China AECC Commercial Aircraft Engine Co. Ltd, Shanghai, 200241, China School of Computer Science and Engineering, Nanyang Technological University, 639798, Singapore

a r t i c l e

i n f o

Article history: Received 21 November 2018 Received in revised form 2 April 2019 Accepted 30 September 2019 Available online 3 October 2019 Keywords: High speed centrifugal impeller Multidisciplinary optimization design SMODE algorithm ANOVA

a b s t r a c t An aero-mechanical multidisciplinary optimization was carried out for a high-speed centrifugal impeller, SRV2-O, by integrating a Self-adaptive Multi-Objective Differential Evolution (SMODE) algorithm, RANS solver technique, Finite Element Method (FEM), and data mining technique of analysis of variance (ANOVA). Specifically, the optimization of the impeller was conducted for the maximization of isentropic efficiency and the minimization of maximum stress. During optimization, constraints were imposed on the total pressure ratio at the optimization point and the mass flow rate at the choked point as well. Where, the former constraint intends to guarantee the working capability of the impeller while the latter tries to fix the working range. After optimization, six optimal Pareto solutions are finally obtained. The isentropic efficiency of the optimal solutions is increased by 2.07% at most while the maximum stress is decreased by 6.36% at most among the Pareto solutions. The better performance of optimal designs was demonstrated through detailed aerodynamic and mechanical analysis. Then, the ANOVA is used to explore the effects of variables on performance function in design space, it is found that, the design variables located at the meridian channel and the leading edge of the full blade have significant effect on aerodynamic performance. These variables are crucial to reduce the loss caused by shock wave at impeller inlet and the leakage flow at blade tip. Meanwhile, the design variables located near the leading edge of full blade root section have great effect on strength performance, as they are effective to decrease the bend of blades and thus reduce the maximum stress. Thereby, the better aeromechanical performance can be achieved by dedicated adjustment on the curves of both the shroud of meridional channel and leading edge of full blade. The results of ANOVA are consistent with the aero-mechanical analysis. Therefore, the effectiveness of aero-mechanical optimization and data mining framework is demonstrated. © 2019 Elsevier Masson SAS. All rights reserved.

1. Introduction Centrifugal compressors are widely applied in small gas turbines, turbochargers and chemical engineering. In recent years, centrifugal compressors with low weight, small size, and high loading care are demanded more than ever. Thus, it is goal for researchers to improve the performance such as aerodynamic performance and strength performance of high speed impellers. Many researchers have studies that flow characteristic inside high speed centrifugal impellers. Krain et al. [1,2] applied experiment and numerical investigation of a high speed centrifugal compressor impeller. The formation of shocks and their effects on the 3D flow and the secondary flow structure are analyzed at differ-

*

Corresponding author. E-mail address: [email protected] (Z. Guo).

https://doi.org/10.1016/j.ast.2019.105452 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.

ent operation conditions, with comparison to low speed machines. Cumpsty et al. [3] analyzed the inside flow of compressor stages. For impellers with tip gaps, the greatest source of loss generation comes from the tip leakage flow. In addition, under the transonic condition, the flow in transonic centrifugal compressors exhibits strong 3D characteristics and the shock waves exist at the inlet. The interaction of the shock wave, the boundary layer, and the tip leakage vortex will further deteriorate the performance of centrifugal impellers [4,5]. Ali et al. [6] gave the three-dimensional steady and unsteady simulations to analyze the flow structure within the impeller and diffuser passages. Su et al. [7] applied quadratic constitute relation on the compressor to improve the prediction of the corner separation. Then, Liu et al. [8] studied the influences of the end wall corner jet with different locations, yaw angles and jet-toinflow total pressure ratios on the aerodynamic performance of a high-speed compressor cascade in detail by numerical simulation. The flow in transonic centrifugal compressors exhibits complexity

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Nomenclature x SMODE ANOVA

Variable vector of the design space Self-adaptive Multi-Objective Differential Evolution Analysis of variance ηis Isentropic efficiency max f stress (x) Maximum stress P 2t / P 1t Total pressure ratio ˙ m Blockage mass flow rate . . . . . . . . . . . . . . . . . . . . . . . . . (kg/s) α Flow angle

and strong 3D characteristics, which influences the aerodynamic performance importantly. As for the strength performance of impellers, it is beneficial to analyze the main characteristic through simplified model. Rsmamurti et al. [9] proposed a simplified model on the stress analysis of a typical turbocharger compressor impeller using the concept of cyclic symmetry. Luo et al. [10] analyzed the stress of centrifugal impellers by ANSYS. Based on above studies, in recent years, combining optimization algorithm in the design of impeller to improve the overall performance is an efficient way. Various optimization methods have been applied on the design of impeller. Based on Simulated Annealing, Ashihara et al. [11] proposed an optimization approach of impeller to improve the impeller efficiency. Luo et al. [12] conducted the automated design optimization of a typical axial transonic compressor blade by using a modified differential evolutionary algorithm. Then, Guo S et al. [13] also selected evolutionary algorithm as the optimization method on a mini-centrifugal compressor. And They used the weighted metric methods to specific optimal solution in multiple Pareto front solutions. Recently, Liu et al. [14] applied artificial neural network on the method of optimization to improve the performance of centrifugal compressor with splitters. Tang et al. [15] applied an adjoint-response surface method to provide efficient surrogate model in a parametrized design space for aerodynamic optimization of blades. Luo J [16] presented a hybrid model method based on proper orthogonal decomposition (POD). Then, they applied POD on the aerodynamic design optimization of the last stage of a low-speed 4.5-stage compressor. The results indicated that optimization reduced the flow separation zones that appear in the suction-end wall corners. In addition, multi-objective and multidisciplinary optimization method has been attracted attention. In the study of multi-objective optimization, Guo et al. [17] taken isentropic efficiency and total pressure ratio as objective functions, and proposed a multi-objective aerodynamic optimization design of a high pressure ratio centrifugal impeller. They not only obtained Pareto solutions with better efficiency and total press ratio, but also they analyzed the relation between objective functions and parameters. In the study of multidisciplinary optimization, Siller et al. [18] proposed the automated, multidisciplinary optimization of fans and compressors on basis of a highly loaded, transonic axial compressor. This method is divided two steps. The first step is for an aerodynamic performance optimization, and the second step is for the structural side coupled with a finite element analysis. Then, Verstraete et al. [19] presented a multidisciplinary optimization system of a small radial compressor impeller by genetic algorithm and artificial neural network to find a compromise between the conflicting demands of high efficiency and low centrifugal stresses in the blades. Note that the optimization of aerodynamic performance has been studied more, and some researchers have considered structural performance after aerodynamic performance. In fact, the optimization of impellers is demanded by considering the aerodynamic performance as well as strength performance simultaneously. However, the related research is still rare. Therefore, to trade-off aerody-

Superscript s

Static

t

Total

Subscript 1

Inlet

2

Outlet

Table 1 Geometrical parameters of SRV2-O. Geometric parameters

Values

Blade count full/splitter Tip clearance at inlet/mm Tip clearance at inlet/mm Leading edge hub radius/mm Leading edge tip radius/mm Impeller Tip radius/mm Exit blade height/mm Blade angle Trailing edge/◦ Blade angle Leading edge tip/◦

13/13 0.5 0.3 30 78 112 10.2 52 26.5

namic and strength performance in the design of high-speed impellers is still a challenge. It is necessary to study multidisciplinary optimization of high-speed centrifugal impeller. For the design optimization of impellers, the relation between design variables and objective function is complex and indistinct. If single traditional optimization is adopted, there are only optimal designs, but the information of design space and design experience cannot be obtained. Thus, data mining technique is proposed, which can explore the relation between design variables and objective function, identify significant variable and obtain the information of design space. Some researchers have applied data mining technique on turbomachinery. Obayashi et al. [20] and Oyama et al. [21] initially applied data mining techniques such as analysis of variance and self-organizing map to different aerodynamic designs, tradeoff information between contradicting objectives and the effect of design parameters on the objectives are explored. In this paper, an aero-mechanical multidisciplinary optimization is proposed and demonstrated through the redesign of a typical high-speed impeller SRV2-O. First, based on Self-adaptive Multi-Objective Differential Evolution (SMODE) algorithm, aeromechanical multidisciplinary optimization of SRV2-O is carried out for maximizing isentropic efficiency and minimizing the maximum stress of blade. Then, detailed flow analysis and stress analysis between reference design and the optimal designs are provided. Finally, ANOVA based on data mining technique is used to explore information of design space and obtain significant design variables. 2. Numerical method and validation In this paper, a high-pressure-ratio, high specific-speed, transonic centrifugal compressor rotor SRV2-O is as the computation model. The computation model is from the experiment of Krain et al. [1,2]. The detailed geometrical parameters of SRV2-O are given in Table 1. 2.1. Numerical method for aerodynamic performance In the simulation of aerodynamic performance, Reynoldsaveraged Navier–Stokes (RANS) equations are applied based on commercial software NUMECA FINE/Turbo. The Spalart–Allmaras

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Fig. 1. Aerodynamic computation gird of SRV2-O. Table 2 Boundary conditions of SRV2-O. Aerodynamic parameters

Values

Rotating speed/(rpm) Inlet total pressure/Pa Inlet total temperature/K Blade Inlet angle /◦ Design mass flow rate/(kg/s) Working medium

50000 101325 288.15 0 2.55 Air(perfect)

turbulence model is used to solve. The fourth-order Runge–Kutta scheme is applied for temporal discretization. To ensure the convergence and the precision of numerical simulation in all situations, the computational outlet is selected 35 mm away from the real impeller exit [17]. After grid independence verification, the number of grid cells is selected to be 680,000. Wherein, the gird number in I, J K direction is 65, 53 and 121, respectively. The grid number in impeller clearance is 13. In addition, the wall-adjacent grid distance decreases to 0.003 mm, and the value of Y+ is set to be range from 1–5. The aerodynamic computation grid is shown in Fig. 1. The H-O-H topology is used to generate hexahedral structured grid The computational boundary conditions were brought into correspondence with Krain’s experiment [1]. Uniform total pressure and temperature with axial flow direction are imposed at the inlet boundary. An averaged static pressure is imposed at the outlet. Stationary and adiabatic boundary condition is adopted at the casing wall. The detailed flow boundary conditions of SRV2-O are given in Table 2. Our work is based on paper [17]. In paper [17], the overall performance at off-design conditions between the results of our numerical method and Krain’s experiment have been shown. In addition, the relative Mach number at different streamwise sections and the location of streamwise sections are compared. All the results indicated that the numerical results and the experimental results were in well agreement. Therefore, the numerical method adopted here can be effectively used for the optimization process of SRV2-O. To ensure the reliability of design optimization, the same grid template is employed for the performance evaluation of different designs during the optimization process. 2.2. Numerical method for strength performance The strength performance of SRV2-O blades is evaluated using Finite Element Analysis (FEA) based on software ANSYS Mechanical APDL. In order to save computation resources and simplify model, the centrifugal stress of full blade and splitter blade in

static strength performance simulation are considered in simulation. In computational model, the type of element is SOLID45 element with 8 nodes is used to generate blade mesh. The displacement at the root of blade in axial, circumferential, and radial directions is zero. The rotating speed which generates centrifugal load is as same as that in aerodynamic analysis, which is 50000 rpm. The material of blades is structure steel. The material properties are as follows: the mass density equals 7800 kg/m3 , the elasticity modulus is 200000 Pa; and the Poisson’s ratio is 0.3. In addition, the yield stress of the material is 1029 MPa. The finite element model and the computational domain of strength simulation are shown as Fig. 2. The number of computational grids in strength performance numerical computation is set to be 195026. After computation, Fig. 3 shows the computation results. The maximum equivalent stress of blade is 906.471 MPa, which is less than the yield stress of the material. The result indicates that the numerical computation meets the requirement of the material. Meanwhile, noted that the maximum equivalent stress of blade is close to the yield stress of the material. Thus, it is necessary to decrease the maximum equivalent stress of blade by the aeromechanical optimization in the present study. 3. Aero-mechanical multidisciplinary optimization of SRV2-O 3.1. Optimization platform By integrating data mining techniques, RANS solver, FEM and the SMODE algorithm, a design optimization method for a multidisciplinary and multi-objective design optimization method of centrifugal impeller is proposed and shown in Fig. 4. This platform includes two parts: (1) the optimization design part, in which the optimal solutions are provided. This part contains four modules, namely, optimization algorithm module, variable module, constraint handing module and performance analysis module; this part can obtain the optimal result in design space; (2) the data mining part, in which the significant variables in design space are explored through the analysis of variance (ANOVA). 3.2. Optimization method The optimization algorithm controls the design process of impeller. In this optimization, SMODE algorithm is applied, which is based on the self-adaptive mechanism from evolution strategies, Pareto based ranking and crowding distance sorting [22]. As for the performance of SMODE, a typical multi-constrained and multiobjective problem as SNR [23] is used to test the effectiveness of the algorithm. The expression of SNR is as follows.

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Fig. 2. Strength simulation grid of SRV2-O.

Fig. 3. The overall stress distribution of blades.

min s.t .

f 1 = (x1 − 2)2 + (x2 − 1)2 + 2 f 2 = 9x1 − (x2 − 1)2 x21 + x22 − 255 ≤ 0 x1 − 3x2 + 10 ≤ 0

(1)

Fig. 5 shows the optimization results of SNR. The red circle represents the Pareto solutions of SNR. As shown in Fig. 5, all obtained non-dominated solutions converge to the Pareto-optimal frontiers of the problem; and the obtained non-dominated solutions are distributed uniformly. These results indicate that the SMODE algorithm has a good global search performance. It is effective to apply the SMODE algorithm to solve multi-objective problems. Therefore, SMODE algorithm is considered as the optimization algorithm in the optimization of SRV2-O in this paper. 3.3. Parameterization method In optimization, the parametrization method determines the design space of optimization problem. The number and the selection of design variables also depend on parameterization method. The parameterization method based on paper [17] is adapted in this paper, including 3D blade profile parameterization and meridional channel parameterization. The specific method is presented in Fig. 6 and Fig. 7. In Fig. 6, the 3D blade profile parameterization mainly adjusts the load of blade through changing 2D section profile in order to improve aerodynamic performance. The steps are as follows: First, 2D sections which have large effects on the aerodynamic perfor-

mance are selected. There, the tip and root sections of full blade and splitter blade are selected. Second, the non-uniformed B-spline is used to generate these section profiles. The mean camber and the thickness distribution of 2D section profiles are fitted by nonuniformed B-spline. In this optimization, the mean camber will be modified by active controlling points which are described as design variables. In addition, the controlling points near the leading and trailing edge are fixed. Thus, new 2D section profile will be obtained by new mean camber and the thickness distribution. Third, after finishing the parameterization of 2D section profile, the reshaped 3D blade is obtained by the radial stacking line of blade. As for the parameterization of the meridional channel, the nonuniformed B-spline is also applied as Fig. 7. First, the meridional channel of reference design is fitted by the non-uniformed Bspline. Then, some active points at B-spline are selected to adjust the channel profiles. In Fig. 7, hollow dots denote the control points of B-spline. The adjust direction of these active points at the meridional channel is different. If active points are located at solid line, the axial direction is the adjust direction. While, if active points are located at dashed line, the radial direction is the adjust direction. Combining the parameterization of 3D blade and meridional channel, the reshaped full blade, splitter blade and meridional channel are obtained. In optimization process, the model of every design is will established by this parameterization method.

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Fig. 7. Parameterization of meridional channel.

Fig. 4. The multidisciplinary design optimization platform of impeller.

Fig. 5. The test results of SMODE.

4. Optimization results and discussion 4.1. Design variables and objective function Based on parameterization method, the reshaped model of SRV2-O are obtained by non-uniformed B-spline and active points.

These active points denote design variables of SRV2-O. Fig. 8 shows the distribution of design variables. The red circles represent design variables. For meridional channel, eight active points are selected as design variables (x1 ∼ x8 ) at hub and shroud profiles. The adjust direction of x1 ∼ x6 is the axial direction, While, the adjust direction of x7 and x8 is the radial direction. For full blade, four active points (x9 ∼ x12 ) at root section and four active points (x13 ∼ x16 ) at tip section are selected to control the 2D section profiles. In addition, one circumference translation parameter (x17 ) is selected to control blade folding in space. For splitter blade, the distribution of design variables is similar as that in full blade. There are also four active points (x18 ∼ x21 ) at root section and four active points (x22 ∼ x25 ) at tip section. In addition, design variable (x26 ) is the circumference translation parameter for splitter blade. Table 3 shows the range of every design variable. In all, twenty-six design variables are selected to rebuild the SRV2-O blade. In aero-mechanical multidisciplinary optimization, the isentropic efficiency and the maximum stress of blades are selected as two objective functions. The isentropic efficiency represents the aerodynamic performance and the maximum stress of blades represents the structural performance. The optimization tries to maximize isentropic efficiency and minimize maximum stress of impeller with constraints. In this paper, the original design point of impeller is under the relative mass flow condition 0.89mre f ,chocked . To accelerate the optimization process, we select the optimization

Fig. 6. Three-dimensional blade parameterization of SRV2-O [17].

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Table 3 The range of every design variable. Parameters

x1 ,x3 ,x4 ,x6

x2

x5

x7 ,x8

Range

[−0.01s, 0.01s]

[−0.012s, 0.012s]

[−0.015s, 0.015s]

[−1mm, 0.5mm]

Parameters

x9 ,x13 ,x18 ,x22

x10 ,x11 ,x14 ,x15 ,x19 ,x20 ,x23 ,x24

x12 ,x16 ,x21 ,x25

x17 ,x26

Range

[−0.01l, 0.01l]

[−0.015l, 0.015l]

[−0.015l, 0.02l]

[−2deg, 2deg]

s – the arc length of hub or shroud line. l – the chord length of a section profile. Table 4 Performance comparison of references and optimal designs.

Reference design Design A Change Design B Change Design C Change

Isentropic efficiency (%)

Maximum stress (MPa)

Mass flow at the chocked point (kg·s−1 )

Total pressure ratio at the optimization point

80.458 82.532 2.07% (Increment) 81.871 1.401% (Increment) 82.460 2.00% (Increment)

906.471 947.231 4.50% 848.795 −6.36% 887.578 −2.08%

3.053 3.058 0.16% 3.047 0.20% 3.038 0.42%

6.3393 6.5399 3.16% 6.5717 3.67% 6.6150 4.35%

Fig. 8. The distribution of design variables.

point under the relative mass flow condition 0.96mre f ,chocked of SRV2-O. More importantly, in order to improve the behavior of the impellers with vary mass flow mates and retain the working range, the aerodynamic performance of every design at the design point and chocked point is all computed. The constraints are imposed on the total pressure ratio at the design point and the mass flow rate under the choked condition as well. In fact, the constraints are also considered as objective functions. During the all optimization process, the penalty function is used to hand the constraint problem. Our objective functions include the information of no mass flow rate deviation at the choked point actually. Therefore, two working conditions are taken consideration into the optimization; this optimization is the overall performance optimization of SRV2-O at design and off-design conditions. The corresponding mathematical expression for aero-mechanical optimization of SRV2-O is as follows:

max s.t .

F (x) = [ηis (x), − max f stress (x)] ˙ (x)chocked≤ 1.005mre f ,chocked 0.995m  re f ,chocked  ≤m 0.995 P 2t / P 1t re f ,optimization ≤ P 2t / P 1t (x) optimization



 t

≤ 1.05 P 2t / P 1

(2)

re f ,optimization

where, F (x) denotes objective function; ηis (x) denotes isentropic efficiency at the optimization point; max f stress (x) denotes the maximum stress; mre f ,chocked denotes the mass flow rate of reference design under the chocked mass flow condition; ˙ (x)chocked denotes the mass flow rate of optimal design under m the chocked mass flow condition; ( P 2t / P 1t )re f ,optimization denotes total pressure ratio of reference design at the optimization point; ( P 2t / P 1t )optimization denotes total pressure ratio of optimal design at the optimization point.

Fig. 9. Optimal Pareto solutions.

4.2. Optimization results and analysis Based on optimization platform, a SMODE algorithm is applied in optimization process. The initial population size is set to be 61, which represents 61 initial design samples. These design samples are obtained by uniform design method [24]. In addition, the maximum generation is set to 150, while the number of total samples is 9150. During optimization process, 22 central processing units are used and the total computation time is about 60 days. Fig. 9 presents the optimal Pareto solutions of the optimization process. In Fig. 9, square denotes reference design. The blue dots denote the optimal Pareto solutions of the last optimization generations. The green arrow denotes the optimization direction. Noted that the aerodynamic performance of final Pareto solutions are all better than reference design. Finally, six Pareto optimal solutions are obtained. Then, three typical Pareto solutions, Design A, Design B, and Design C are selected to analyzed the performance of optimization solutions in detail. Table 4 compares the overall performance of impeller among reference design and three Pareto optimal solutions. Design A has the best aerodynamic performance. The isentropic efficiency of Design A is increased by 2.07%, but the maximum stress is increased by 4.50%. Design B has the best strength performance. The maximum stress of Design B is decreased by 6.36%, but the isentropic efficiency is increased a little. Design C has the optimal overall performance. The isentropic efficiency is increased by 2.00%, while the maximum stress is decreased by 2.08%. The total pressure ratio at the optimization point of the three optimal designs has been

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Fig. 10. Overall performance at off-design conditions.

Fig. 11. Relative Mach number contours at meridional surface.

increased. Meanwhile, the optimal Pareto solutions all meets the constraints, especially for the mass flow rate at chocked point. Then, the detailed aerodynamic performance and strength performance of SRV2-O among reference design and three optimal Pareto solutions are analyzed in follow. For aerodynamic performance, Fig. 10 shows the behavior of impellers before and after optimization under varying mass flow rates. From Fig. 10, the working range of Design A is slightly increased; the maximum increase of isentropic efficiency during the overall working range is 2.3%, and the total pressure ratio is all improved. For Design B, the maximum increase of isentropic efficiency during the overall working range is 1.6%, and the maximum increase of total pressure ratio during the overall working range is about 3.2%. For Design C, although the working range is a little decreased, the maximum increase of isentropic efficiency during the

overall working range is 2.1%, and the maximum increase of total pressure ratio during the overall working range is about 4.4%. Especially, at chocked point, the mass flow rate of three designs is similar with the reference design. At the original design point and optimization point, the aerodynamic performance of the three optimal design has been improved. Therefore, after optimization, the aerodynamic performance of optimal Pareto solutions is all improved at both design points and off-design conditions. Fig. 11 shows the relative Mach number at meridian plane of reference design and the three optimal designs. Compared with reference design, the supersonic region at the tip of splitter blade becomes almost disappeared in three optimal designs, especially in Design A. Meanwhile, in optimal designs, the radial distribution of supersonic region at the full blade leading edge reduced a little, although the Mach number near the inlet of full blade tip increases

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Fig. 12. Entropy contours at meridional surface.

a little. The results indicate that the optimal designs reduce shock wave at inlet of splitter. In addition, the interaction between the separation of boundary layer and tip clearance near downstream are reduced. Therefore, the isentropic efficiency in optimal Pareto solutions is improved. Then, a comparison of entropy distribution at meridian surface is shown in Fig. 12. It is found that the entropy distribution of Design B is similar with the reference design. However, in Design A and Design C, the value of entropy at the tip of splitter blade decreased, it indicated that the tip leakage loss are reduced after optimization. The analysis is consistent with the result of Fig. 11. In addition, the value of entropy at the diffuser decrease obviously in Design A and Design C. Thus, the flow loss at the diffuser is also reduced after optimization. Based on the analysis of Fig. 11 and Fig. 12, the aerodynamic performance of Design A has improved most, while the aerodynamic performance of Design B has improved. While the aerodynamic performance of Design B has improved least among optimal designs. These results are consistent with the overall performance of Table 4. In order to analyze the flow characteristic in reference design and optimal designs in detail, Fig. 13 and Fig. 14 show the flow characteristics of B2B section at 80% span. Fig. 13 represents the distribution of relative Mach number at 80% span. The great changes exist in Region 1 and Region 2. Region 1 presents the flow passage between the pressure surface of the full blade and the suction surface of the splitter blade, while Region 2 presents the flow passage between the suction surface of the full blade and the pressure surface of the splitter blade. In reference design, Region 1 is a low-energy fluid gathering area for low relative Mach number. While there is a little low–energy fluid in Region 2. Thus, the flow loss is mainly caused by low velocity flow in Region 1. After optimization, the relative Mach number of Region 1 is all increased significantly in three optimal designs although the Mach number of Region 2 is reduced a little. In all, the overall flow loss of flow

passages caused by the low velocity flow in tip leakage is obviously reduced and the distribution of flow loss has been improved after optimization. Fig. 14 represents the distribution of entropy at 80% span. Corresponding to Fig. 13, Fig. 14 is much clear to show the improvement of the optimal designs in aerodynamic performance. A low-energy fluid gathering area for large flow loss and entropy is mainly in Region 1. From Fig. 14, it is found that the great change exists only in Region 1 after optimization. Compared with reference design, the value of entropy in Region 1 is decreased more in the three optimal designs. Meanwhile, the value of entropy in other regions has nearly unchanged. The optimization improves the low-energy fluid flow at the exit and reduce the gathering of low energy flow. Therefore, the uniformity of flow at the exit of impeller has improved. Then, in Fig. 15, the entropy distribution at the full blade and the splitter blade of the optimal designs is significantly improved. For the full blade, the high flow loss region at the leading edge in optimal designs become smaller than that in reference design, while the largest entropy value of high loss region in optimal designs is also reduced. The entropy distribution of the full blade is more uniform. For the splitter blade, the entropy values of tip near the leading edge in optimal designs are greatly reduced. So the supersonic area at the tip of splitter blade nearing the leading edge become almost invisible in the optimal designs. In addition, Fig. 15 also shows the limiting streamlines of the blades. Note that the location of the separate region near the full blade leading edge is move toward the blade root. All the results in Fig. 15 indicate that optimal designs reduce the shock wave at the inlet of impeller and the leakage flow interactions between neighboring flow passages. As for strength performance, Fig. 16 and Fig. 17 show the overall and local distribution of static stress through Von Mises contour. The maximum stress exists the root of full blade. Fig. 16 shows the overall distribution of static stress of blades. In reference

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Fig. 13. Relative Mach number at 80% span.

Fig. 14. Entropy at 80% span.

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Fig. 15. The entropy distribution and limiting streamlines at the full blade and the splitter blade.

Fig. 16. The overall stress distribution of blades.

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Fig. 17. The local stress distribution at the root of full blade.

Fig. 18. Root section profile of full blade.

design, the stress of blades at the blade tip is small and the stress at the blade root is large. The stress on blade surface increases gradually from blade tip to blade root. The smaller the radius, the stronger the cumulative effect of centrifugal stress generated by rotation. The maximum stress of blade surface occurs at the root of full blade near leading edge and pressure surface. That result is caused by the rotation characteristics of impeller. The radius at the root is the smallest, and the root endures the centrifugal force of the whole blade. The maximum stress exists the root of blade. Usually, the life of blades depends on the maximum stress of blades. Thus, the optimization needs to reduce the maximum stress. Form Fig. 16, the overall distribution of stress in optimal designs and reference design are similar although the stress of the blade middle is increase a little. In addition, after optimization, the distribution of static stress of blade in optimal designs is more uniform than that in reference design. Then, Fig. 17 gives the detailed the comparison of stress local distribution. For Design B and Design C design, although the stress of middle blade increased, but the maximum stress at the root of blade near the leading edge reduced. It is obvious that the large stress area of the blade root is reduced and the value of maximum stress is decreased. However, for Design A, the maximum stress is increased and the large stress area at the blade root is also increased. As for reason, Fig. 18 shows the comparison of the blade profile at the root section. The profile of leading edge at the root section has changed obviously in Design A. The leading edge moves toward pressure surface, and the bend of blade in Design A is increased, so the maximum stress of blade in Design A increases. For Design B and Design C, the profile of blade at the root near the leading edge is similar with the reference design, and the bend of blade is decreased compared with reference design. Thus, the maximum stress of the blade is decreased.

In all, after optimization, the flow loss caused by tip leakage in the flow passage, the flow loss of inlet shock wave and the flow loss at the exit are reduced. The optimization has improved aerodynamic performance. In addition, by adjusting the bend of blades, the maximum stress at the root of full blade is reduced. Therefore, the effectiveness of our proposed optimization method is demonstrated. 5. Knowledge discovery based on ANOVA The analysis of variance (ANOVA) is a quantitative analysis based variance, which can analyze the effects of the design variable for objective functions [25,26]. Through ANOVA, the main effect and variance proportion of every design variable are obtained [27]. The variance proportion of every design variable denotes the influence of every design variable on objective function. Detailed expressions of ANOVA are shown in Appendix. The larger the variance proportion of design variable is, the more significant influence the corresponding design variable has. Before the analysis of ANOVA, the reliability of ANOVA is validated. In this paper, the ANOVA is based on Kriging model. Thus, cross validation is used to validate the reliability of ANOVA. The detail expression of cross validation is as follow.

ei =

y (x(i ) ) − yˆ −i (x(i ) ) s2−i (x(i ) )

(3)

where, e i denotes probability; the y (x(i ) ) denotes the true values; yˆ −i (x(i ) ) denotes the predict values of Kriging model; s2−i (x(i ) ) denotes the mean square error of Kriging model [28]. The confidence probability of model is set as 99.75%, so the range of e i is [−3, 3]. If the value of e i for all design samples is

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Fig. 21. Variance proportion of significant design variables about isentropic efficiency.

Fig. 19. Cross validation of isentropic efficiency.

Fig. 22. Cross validation of maximum stress.

Fig. 20. Variance proportion of significant design variables about isentropic efficiency.

belong to [−3, 3], the uncertainty of the model is within confidence interval, and the model is accurate. Especially, if the value of e i for all design samples is belong to [−2, 2], the Kriging model has high accuracy and ANOVA has high reliability. Analysis samples are initial samples obtained by uniform design. Firstly, the reliability of ANOVA has been validated. Fig. 19 gives the results of cross validation plot, which shows that the e i of samples belong to [−3,3]. In addition, the average value of the relative error between the predicted value and true value is 1.19%, and the mean square root of the relative error is 1.96%. It indicates that the kriging model have good accuracy and the analysis of ANOVA is reliable. Fig. 20 shows the variance proportion of significant design variables about isentropic efficiency. In this paper, when variance proportion is larger than 1% or nearly 1%, the corresponding design variables are regarded as significant variables. From Fig. 19, it is quite clear that the variance proportion of x5 is 42.30%, and the variance proportion between x5 and other design variables is 14.25%. That result indicated that x5 has most important influence for isentropic efficiency. In addition, x6 , x7 , x9 , x12 and x13 have certain effects, while the variance proportion is 14.96%. Then, the effects of these design variables are analyzed in Fig. 21. According to the distribution of design variables in Fig. 8, these significant design variables are divided two groups. The first group including design variables (x5 , x6 , x7 ) controls the shroud of meridional channel design, while the variance proportion of the first group design variables is up to 50.12%. The second group in-

cluding design variables (x9 , x13 , x12 ) controls the root and the leading edge of the full blade, while the variance proportion of the first group design variables is 7.14%. In first group, x5 , x6 and x7 can affect the blade height through adjusting the shroud of meridional channel design. Therefore, the changes of these three design variables will influence the flow capacity of impeller and the chocked mass flow ratio. Meanwhile, x5 , x6 and x7 also influence the tip leakage flow of splitter blade and flow loss at the exit. In second group, x9 and x13 are located at the leading edge of full blade. These two design variables will influence shock wave at inlet. x12 is located at the root of full blade near trailing edge. The region of flow passage between the pressure surface of the full blade and the suction surface of the splitter blade is changed with the value of x12 . It will influence the flow characteristic at exit of impeller. Therefore, the design variables of the shroud of meridional channel near full blade leading edge, the design variables of the full blade root are needed more attention in design optimization. Fig. 22 and Fig. 23 are used to analyze the effects of design variables for strength performance. Fig. 22 gives the results of cross validation plot, which shows that the e i of samples belong to [−2,2]. In addition, the average value of the relative error between the predicted value and true value is 2.43%, and the mean square root of the relative error is 3.04%. It indicates that the analysis of strength performance by ANOVA has good reliability. Fig. 23 shows the variance proportion of significant design variables for maximum stress. The variance proportion of x13 and x9 is up to 40.31%. These two design variables has most significant effect among design variables. In addition, the variance proportion of x2 , x5 , x21 and x24 is up to 24.76%. They also have certain effect for maximum stress.

C. Li et al. / Aerospace Science and Technology 95 (2019) 105452

Fig. 23. Variance proportion of significant design variables about maximum stress.

Fig. 24. Variance proportion of significant design variables about maximum stress.

In order to analyze the most important design variables for strength performance, four design variables (x9 , x13 , x2 , x5 ) are selected. Then, according to the location of these four design variables, these four significant design variables are also divided two groups. Fig. 24 gives the variance proportion of two groups. The

13

first group includes x9 and x13 , which locate at the leading edge of full blade. The other group significant design variables (x2 and x5 ) are distributed at the meridional channel near the inlet of full blade and the inlet of splitter blade. In all, this result indicates that the design variables located at full blade root near leading edge have most important effect on strength performance. Moreover, from the analysis of strength performance in Fig. 17, the maximum stress of blade occurs the full blade root and the stress will increase with the reduce of radius. Thus, the design of full blade root and the full blade leading edge is most important for the strength performance of SRV2-O. Finally, Fig. 25 shows the comparison of significant design variables and performance in reference design and optimal designs. In Fig. 25, the values of performance in reference design are all 1. For aerodynamic performance, three optimal designs have significant improvement compared with the reference design. Especially, the chocked mass flow of optimal designs retain nearly unchanged. For strength performance, except for Design A, Design B and Design C reduce the maximum stress. Thus, Design A has best aerodynamic performance; Design B has best strength performance; Design C has best overall performance. In addition, according to the analysis of ANOVA, there are seven significant design variables in twenty-six dimension design space. In order to carry out a detailed comparison of the significant design variable value between optimal designs and the reference design, Fig. 25 shows the normalized values of these seven design variables. In reference design, the values of design variables are all 0.5. While the change of the value of design variables is inconsistent in optimal designs. For the design variables at the meridional channel (x2 ), the value is less than 0.5 in Design A but the value is more than 0.5 in Design B. The result indicates that reducing the value of x2 can improve aerodynamic performance, while the maximum stress will be increased. For the design variables at the shroud of meridional channel (x5 , x7 ) and the design variable at the root of full blade (x9 ), the values are more than 0.5 in Design A, but the values are less than 0.5 in Design B. The result indicates that increasing the value ofx5 , x7 and x9 can improve aerodynamic performance, but the maximum stress of blade will be increased. For the design variable at the root of full blade (x6 , x12 , x13 ), in-

Fig. 25. Comparison of significant design variables and performance in reference design and optimal designs.

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C. Li et al. / Aerospace Science and Technology 95 (2019) 105452

creasing the values of x6 , x12 and x13 will improve the overall performance, and there are the optimum values of these design variables in Design C. All the results indicate that there is the restriction relationship between the improvement of aerodynamic performance and the improvement of strength performance. In all, the design of meridional channel near full blade leading edge, the splitter blade tip and the full blade root has significant effects on the aerodynamic performance and the strength performance. The analysis of ANOVA and the conclusion of optimization are consistent, that verified the effectiveness of the aeromechanical multidisciplinary optimization framework of SRV2-O. 6. Conclusions An aero-mechanical multidisciplinary optimization and knowledge mining framework is proposed in this paper. The framework is established by integrating the SMODE algorithm, 3D parameterization method for impeller blade profile and meridional channel, and the data mining technique of ANOVA. The framework is applied on the optimization and data mining for a typical high speed centrifugal impeller, SRV2-O. Some conclusions are obtained. Based on SMODE algorithm, the impeller is optimized for maximizing isentropic efficiency and minimizing maximum stress. In addition, in order to guarantee the working capability and the working range of the impeller, constraints are imposed on the total pressure ratio at the optimization point and the mass flow rate at the choked point as well. Six optimal Pareto solutions are obtained finally after optimization. Among six Pareto solutions, three typical Pareto solutions are selected to analyze performance with reference design. Design A has the optimal isentropic efficiency, and the isentropic efficiency of Design A is increased by 2.07%. Design B has the smallest maximum stress, and the maximum stress of Design B is decreased by 6.36%. Design C has the best overall performance, in which the isentropic efficiency is increased by 2.00%, while the maximum stress of that is decreased by 2.8%. Detailed flow and mechanical analysis of three typical Pareto solutions indicate that the performance of the optimal Pareto solutions is significantly improved. For aerodynamic performance, the optimal designs reduce the loss of shock wave at inlet, the tip leakage flow and the flow loss of flow passage in impeller. In addition, the working range of optimal Pareto solutions nearly retain constant. More importantly, the overall performance of the three optimal Pareto solutions is all improved at both design point and off-design conditions. For strength performance, expect for Design A, the optimal designs reduce the bend of full blade and retain the profile of full blade root section to reduce the maximum stress. AONVA as the data mining technique is used to explore the relations between design variables and objective functions. Meanwhile, significant design variables are obtained by ANOVA. For aerodynamic performance, the design variables located at the shroud of meridional channel, the full blade tip and the full blade root have significant effect on aerodynamic performance. These variables are crucial to reduce the loss of shock wave at impeller inlet, the leakage flow at blade tip and the flow loss in flow passage. For strength performance, the design variables located near the leading edge of full blade root have great effect on strength performance, as they are effective to decrease the bend of blades and thus reduce the maximum stress. Thereby, the better aeromechanical performance can be achieved by dedicated adjustment on the curves of both the shroud of channel and leading edge of full blade. The results of ANOVA are consistent with the optimization results. The effectiveness of aero-mechanical optimization and data mining framework is demonstrated.

Declaration of competing interest We confirm that there are no known conflicts of interest associated with this publication. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 51676149). Appendix ANOVA is a quantitative analysis based variance, which can explore the effects of the design variable for objective function. The detail expression of ANOVA is as follows. Let yˆ (x) denote the estimated function value of the response surface obtained by Kriging model, x = [x1 , x2 , · · · xn ] T denotes design variables, the total mean a0 and the total variance δ 2 of yˆ (x) in the whole design space can be defined as:



a0 =



yˆ (x)dx1 · · · dxk

··· 

δ2 =

 ···



yˆ (x) − a0

2

dx1 · · · dxk

(4)

The effect of xi on yˆ (x) is usually broken down as follows:



a i ( xi ) =



··· 



[ yˆ (x) − a0 ]dx1 · · · dxi −1 dxi +1 · · · dxk

· · · [ yˆ (x) − a0 ]2 dx1 · · · dxi −1 dxi +1 · · · dxk   ai j (xi , x j ) = · · · [ yˆ (x) − ai − a j − a0 ] δi2 (xi ) =

(5)

dx1 · · · dxi −1 dxi +1 · · · dx j −1 dx j +1 · · · dxk



δi2j (xi , x j ) =



···

[ yˆ (x) − ai − a j − a0 ]2

dx1 · · · dxi −1 dxi +1 · · · dx j −1 dx j +1 · · · dxk

(6)

where, ai (xi ) denotes the main effect of xi on yˆ (x), δi2 (xi ) denotes the related variance caused by xi . Similarly, ai j (xi , x j ) and δ2i j (xi , x j ) denote the joint effect of xi /x j and the related variance, respectively. Usually, δi2 (xi )/δ 2 or δi2j (xi , x j )/δ 2 can quantitatively

indicate the contributions of xi or xi /x j on yˆ (x), are regarded as variance proportion. Thereby the significant variables are detected by variance proportion. Furthermore, the effect of design variable xi on yˆ (x) in a range can be represented according to the expressions about ai (xi ). References [1] H. Krain, Swirling impeller flow, J. Turbomach. 110 (1) (1988) 122–128. [2] G. Eisenlohr, H. Krain, F. Richter, V. Tiede, Investigations of the Flow Through a High Pressure Ratio Centrifugal Impeller, 2002, Asme Paper GT2002-30394. [3] N.A. Cumpsty, Compressor Aerodynamics, Longman Scientific and Technical, Harlow, U.K., 1989. [4] S. Ibaraki, M. Furukawa, K. Iwakiri, K. Takahashi, Vortical Flow Structure and Loss Generation Process in a Transonic Centrifugal Compressor Impeller, 2007, ASME Paper GT2007-27791. [5] R. Hunziker, H.P. Dickmann, R. Emmrich, Numerical and experimental investigation of a centrifugal compressor with an inducer casing bleed system, Pro. Inst. Mech. Eng. A, J. Power Energy. 215 (2) (2001) 783–791. [6] A. Zamiri, B.J. Lee, J. Taek Chung, Numerical evaluation of transient flow characteristics in a transonic centrifugal compressor with vaned diffuser, Aerosp. Sci. Technol. 70 (2017) 244–256. [7] X. Su, X. Yuan, Improved compressor corner separation prediction using the quadratic constitute relation, Pro. Inst. Mech. Eng. A, J. Power Energy. 231 (7) (2017) 618–630.

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