The design of high efficiency, high pressure ratio, and wide flow range centrifugal impellers is a challenging task. The paper describes the application of a multiobjective, multipoint optimization methodology to the redesign of a transonic compressor impeller for this purpose. The aerodynamic optimization method integrates an improved nondominated sorting genetic algorithm II (NSGAII), blade geometry parameterization based on NURBS, a 3D RANS solver, a selforganization map (SOM) based data mining technique, and a time series based surge detection method. The optimization results indicate a considerable improvement to the total pressure ratio and isentropic efficiency of the compressor over the whole design speed line and by 5.3% and 1.9% at design point, respectively. Meanwhile, surge margin and choke mass flow increase by 6.8% and 1.4%, respectively. The mechanism behind the performance improvement is further extracted by combining the geometry changes with detailed flow analysis.
The reliance on numerical methods in the aerodynamic design process of turbomachinery components has considerably increased in the last decades. Nowadays, Computational Fluid Dynamics (CFD) codes have matured to a level where they are capable of not only providing a substantial insight into the threedimensional flow field in turbomachines, but also calculating aerodynamic performances of the machines [
In order to obtain better designs and reduce design cost, automated design optimization of centrifugal impeller has received a widespread attention in recent years. Guo et al. [
The optimization for the whole speed line is even more challenging, especially near surge condition (certain impellers require good efficiency near the surge and most compressors need good surge margin), as CFD convergence is often not guaranteed under or near such a condition. Demeulenaere et al. [
The purpose of this paper is to develop a multipoint and multiobjective design optimization method for high pressure ratio impellers to achieve better aerodynamic performances at both design and offdesign conditions and with a wider operating range. The remaining of this paper is organized as follows: first, a time series autoregressive (AR) model is developed to predict surge point from the CFD simulation of a single impeller flow passage and validated by experimental results. A selforganization map (SOM) is then carried out on samples of CFD results to explore the relation between objective functions and design parameters. A total of 27 key design variables selected by the SOM are then employed in the followup multiobjective optimization. Finally, the results from the optimization are shown and discussed, and some conclusions and remarks are drawn.
Surge is instability of the centrifugal compressors and is associated with strong unsteadiness of the inlet and outlet pressures and temperatures of the compressor. Therefore, monitoring the time based signal by adopting a Fast Fourier Transform analysis makes it possible to state the instant when the compressor starts to surge and to highlight the typical frequency peak related to surge occurrence [
Here we propose a new method for surge detection in CFD; it is based on autoregressive (AR) statistical pattern recognition algorithms [
The slope of pressure rise is a reliable indicator of compressor stability. If the slope is positive the compressor will be unstable. The maximum or peak compressor pressure ratio thus defines the stability limit. Numerical studies of axial compressors [
In the surge detection, one may use the frequencies or amplitude of the unsteady signal. These two parameters are however dimensional so their values depend directly on compressor size and speed, making them unsuitable as universal thresholds in surge detection. By contrast, the time series model methods, such as AR and ARMA, are based on monitoring model residual variances, which is independent of specified compressor impellers and speed.
Autoregression is a data processing technique that is commonly used in constructing a model from time sequence data for extracting underline trends. For a stationary time series
Autocovariance
Coefficients
For the outlet boundary condition, mass flow rate condition is not appropriate as the convergence of CFD cannot be guaranteed near surge condition, resulting in premature sinusoidal waves of flow parameters. Thus, a static pressure condition is adopted and the signal of mass flow rate at LE is monitored. It tends to be a periodic wave but not sinusoidal.
The rise of static pressure at outlet boundary is flattened out near surge condition; a dichotomy or sectional method is applied to the determination of the maximum outlet static pressure near surge condition. This method will be demonstrated in the next section with an example. The flow chart of surge detection is presented in Figure
The flow chart of surge detection method.
This surge detection approach is applied to a higher pressure centrifugal impeller, SRV2AB impeller [
The surge detection validation is carried out by the comparison of surge line between CFD prediction and experimental results of the SRV2AB impeller with vaneless diffuser, which serves as the baseline for the subsequent optimization. The basic design parameters of this compressor are given in Table
SRV2AB rotor design parameters.
Shaft speed  50000 rpm 
Design mass flow rate  2.55 kg/s 
Impeller tip radius  112 mm 
Diffuser outlet radius  212.8 mm 
Rotor tip speed  586 m/s 
Rotor pressure ratio  6.2 : 1 
Blade number full/splitter  13/13 
LE hub radius  30 mm 
LE tip radius  78 mm 
Blade angle LE tip  26.5 deg (from tangential) 
Blade angle TE  52 deg (from tangential) 
Exit blade height  8.7 mm 
Diffuser inclination against radial  13 deg 
Tip clearance  0.5 mm at inlet to 0.3 mm at exit 
A mesh independence investigation was firstly conducted by coarse, medium and fine meshes with total mesh number of 0.3, 0.9, and 1.7 million, respectively (for one blade channel). The mesh has a structured HO topology and the minimum value of
Mesh independence investigation.
Aerodynamic computational domain and grid of SRV2AB (single passage).
The sketch of dichotomy to reach the highest outlet static pressure at surge point is shown in Figure
The sketch of dichotomy to reach the highest static pressure at surge point.
The time based mass flow rate signals at LE, monitored at the last stable point and the first unstable point for design speed (50000 rpm), are presented in Figure
Time based mass flow rate signal at LE.
Autocovariance and partial autocovariance versus order of AR model.
Variance versus number of time steps.
Figure
Comparison of predicted and measured total pressure ratio of SRV2AB impeller with vaneless diffuser.
In the turbomachinery design using CFDbased optimization, it is important to determine a small number of key design variables from design space to simplify design problem. The information about the design space, such as tradeoff between objective functions, the relations between design variables and objective functions, and why performance of the optimized designs has been improved will be useful for this purpose by eliminating the design variables which do not have a large influence on the objective functions, thereby the efficiency as well as the reliability of optimization process may be greatly improved. Furthermore, it is preferable for a designer to provide some alternative or suboptimum solutions for the decision making of the final design. The process to extract information from the design space by detecting the features of the optimization results is called “data mining.” This paper deals with the data mining technique based on SOM.
In terms of 3D parameterization of impeller geometry, refer to authors’ previous work [
Control points of endwalls and blade camber curves.
Endwalls
Root section
Tip section
Table
Variable names and corresponding numbers of control points.
Variable name  Variable number 


1–13 

14–26 

27–34 

35–42 

43–50 

51–58 

59–63 

64–68 
Active variable numbers, range of variations, and constraints in initial design space.
Active variable number  Range of variation  

Hub  7, 8, 9, 10, 11 

20, 21, 22, 23, 24 


Shroud  29, 30, 31, 32, 33, 34 

37, 38, 39, 40 


Full root  44, 45, 46, 47, 48, 49, 50 

Full tip  51, 52, 53, 54, 55, 56, 57, 58 

Splitter root  59, 60, 61, 62, 63 

Splitter tip  64, 65, 66, 67, 68 

The impeller will be optimized at 50000 rpm for maximizing isentropic efficiency and total pressure ratio at design operating condition (2.55 kg/s), while striving for smaller surge and larger choke mass flows. The corresponding mathematical expression of aerodynamic optimization of SRV2AB is as follows:
SOM expresses the information in a qualitative way and visualizes not only the relation between design variables and objective functions but also the tradeoff between the objective functions. It employs a nonlinear projection algorithm from high to lowdimensions and a clustering technique. This projection is based on selforganization of a lowdimensional array of neurons. In the projection algorithm, the weights between the input vector and the array of neurons are adjusted to represent features of the highdimensional data on the lowdimensional map. Figure
Schematic map of SOM.
The learning algorithm of SOM starts with finding the bestmatching unit (winning neuron) which is closest to the input vector
Adjustment of the bestmatching unit and its neighbors.
Repeating this learning algorithm, the weight vectors become smooth not only locally but also globally. Thus, the sequence of the vectors in the original space results in a sequence of the corresponding neighboring neurons in the twodimensional map.
Once the highdimensional data projected on the twodimensional regular grid, the map can be used for visualization and data mining. The same location on each component map corresponds to the same SOM neuron, which is colored according to its related neutron component values. By comparing the behavior of the color pattern in the same region, one can analyze the correlations among parameters. Parameters are correlated if there exist similar color patterns in the same region of the corresponding component maps.
The SOM was carried out by an inhouse code on initial design space to select key design parameters in the design space for further optimization. 100 impellers generated in random were simulated, in which 49 parameters (4 objectives and 45 active design variables) were analyzed. All the employed parameters are normalized (variance is normalized to one). 25 SOM neutrons were used in component map. Figure
SOMs colored by objective functions of initial design space.
Figure
SOMs colored by active design variables of initial design space.
Variable 21
Variable 22
Variable 33
Variable 40
Variable 44
Variable 52
Variable 58
Variable 68
Based on data mining results, one may derive the following conclusions.
Table
Active variable numbers and range of variations in final design space.
Active variable number  Range of variation  

Hub  9, 10 

20, 22, 23 


Shroud  31, 32, 33 

39, 40 


Full root  44, 45, 46, 48, 50 

Full tip  52, 53, 54, 55, 58 

Splitter root  59, 61, 63 

Splitter tip  64, 65, 67, 68 

Active variables of endwalls and blade camber curves in final design space.
Endwalls
Root section
Tip section
A multipoint and multiobjective design optimization method based on an improved NSGAII genetic algorithm [
The flow chart of this multipoint and multiobjective design optimization method is shown in Figure
Settings of the genetic algorithm & CFD.
Population size  30 
Maximum iteration number  20 
Crossover probability 
0.9 
Crossover probability 
0.3 
Mutation probability 
0.2 
Mutation probability 
0.05 
Convergence criterion of CFD  10^{−5} 
Flow chart of multipoint, multiobjective optimization method.
The overall performances of baseline impeller and optimal impeller are presented in Table
The overall performances of baseline and optimal impellers at 50000 rpm.
Parameters  Baseline  Optimal  Improvement 


5.7  6.0  5.3% 

80.0  81.5  1.9% 

2.37  2.21  6.8% 

2.87  2.91  1.4% 
Figure
Comparison of offdesign performance between baseline and optimal impellers.
Total pressure ratio
Isentropic totaltototal efficiency
These results demonstrate the power of this multipoint and multiobjective optimization method. A singlepoint optimization is less likely to get similar results. Note that this multipoint and multiobjective optimization is only conducted at design shaft speed (50000 rpm), and it could further improve offdesign performance if the same technique is applied to both design and offdesign speeds. This will significantly increase computational time and resources (about 3 times of this study). A tradeoff is needed between the effort and the gains.
Figure
Comparison of geometry between baseline and optimal impeller.
Endwalls
Blade camber curves at root section
Blade camber curves at tip section
The backsweep angles (variables 50 and 63) are slightly reduced in the optimal impeller at its hub for both full blade and splitter blade, and this increases impeller work and pressure ratios. Because the same angle is not reduced at the shroud, the reduction at the hub decreases the diffusion imbalance between the shroud and the hub, resulting in more uniform impeller outflow. This compensates the higher diffuser inlet velocity caused by the reduced hub backsweep angle.
It is concluded that the design variables’ optimization for a better performance corresponds to the analysis in SOM. It proves that SOM analysis is effective in extracting the key design variables and their effects, and the proposed optimization method was able to find a proper tradeoff between all the objectives.
The flow field at design operating condition is analyzed. Figure
Entropy distributions at streamwise sections and streamlines.
Baseline
Optimal
Relative Mach number contour at 60% and 95% spans.
Baseline 60% span
Optimal 60% span
Baseline 95% span
Optimal 95% span
From Figure
Figure
Spanwise performance 10 mm downstream of impeller.
Total pressure ratio
Isentropic efficiency
A multipoint and multiobjective design optimization strategy of centrifugal impeller is proposed by integrating a genetic algorithm, 3D geometry parameterization, CFD tools, time series based surge detection method, and SOMs based data mining technique. This approach was successfully applied to high pressure ratio centrifugal impeller SRV2AB for higher total pressure ratio and better efficiency at design operating condition and smaller surge mass flow and larger choke mass flow at design speed. The main conclusions are drawn as follows:
(1) A time series based surge detection method was introduced. It uses autoregression to detect compressor instability and is independent of the impeller geometry, shaft speed, and boundary condition. This method was successfully applied to SRV2AB impeller surge prediction.
(2) By SOMbased data mining on initial design space, tradeoff relations between objective functions and correlations among design variables and objective functions were visualized and analyzed. The key design variables were then identified and kept in the final design space.
(3) The optimization improves the overall performance of the impeller at whole design speed line and widens the compressor flow range. At offdesign speeds and compressor efficiency, surge margin is also enhanced. The mechanism behind the performance improvement is further explained by combining geometry changes with detailed flow analysis.
The surge detection method still needs to be checked by experiment and by more applications. It is currently computational intensive. It essentially detects surge by the macroscopic time based signal instead of local flow features such as the development of stall cells, which may be more efficient. The method needs further refinement.
Another future work concerns with the method of selecting the data for data mining. Results of data mining techniques may depend on the used data. For the consistency of information obtained from data mining, a robust data selection method is necessary.
Autoregressive processes
Autoregressive moving average processes
Amplitude of time based signal
White noise series
Degree
Genetic algorithm
Leading edge
Moving average processes
Mass flow rate
Choke mass flow
Weight vector
Surge mass flow (last stable mass flow near surge condition)
Normally and independently distributed
Shaft speed (rpm)
Probability
Upper border of crossover probability
Lower border of crossover probability
Upper border of mutation probability
Lower border of mutation probability
Pressure
Order of AR model
Radial coordination (mm)
Autocovariance function
Selforganization map
Trailing edge
Original time series signal
Zeromean time series
Axial coordination (mm).
Upper border of variance in AR model
Total pressure ratio
Isentropic efficiency
Azimuthal angle (rad)
Mean value
Autocorrelation function
Variance
Coefficients of AR model
Partial autocovariance function.
Crossover
Choke condition
Design condition
Full blade
Isentropic
Mutation
Root section
Splitter blade
Surge condition
Total conditions
Tip section.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research work is supported by National Natural Science Foundation of China (Grant no. 11672206) and State Key Laboratory of Internal Combustion Engine Burning (Grant no. K201607).