Performance improvement of optimization solutions by POD-based data mining

Ynhui DUAN , Wenhu WU , Peihong ZHANG , Fulin TONG ,Zholin FAN , Guiyu ZHOU , Jiqi LUO

a Computational Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China

b School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China

KEYWORDS Aerodynamic shape optimization;Computational fluid dynamics;Data mining;Particle swarm optimization;Proper Orthogonal Decomposition;Transonic flow;Turbomachinery

Abstract The performance of an optimized aerodynamic shape is further improved by a secondstep optimization using the design knowledge discovered by a data mining technique based on Proper Orthogonal Decomposition (POD) in the present study. Data generated in the first-step optimization by using evolution algorithms is saved as the source data, among which the superior data with improved objectives and maintained constraints is chosen. Only the geometry components of the superior data are picked out and used for constructing the snapshots of POD. Geometry characteristics of the superior data illustrated by POD bases are the design knowledge, by which the second-step optimization can be rapidly achieved.The optimization methods are demonstrated by redesigning a transonic compressor rotor blade, NASA Rotor 37, in the study to maximize the peak adiabatic efficiency, while maintaining the total pressure ratio and mass flow rate.Firstly, the blade is redesigned by using a particle swarm optimization method, and the adiabatic efficiency is increased by 1.29%. Then, the second-step optimization is performed by using the design knowledge, and a 0.25% gain on the adiabatic efficiency is obtained. The results are presented and addressed in detail,demonstrating that geometry variations signif icantly change the pattern and strength of the shock wave in the blade passage. The former reduces the separation loss,while the latter reduces the shock loss, and both favor an increase of the adiabatic efficiency.

1. Introduction

Data mining is one of data analysis techniques, which can extract useful information (named knowledge in this paper)from non-sequence data. For multi-disciplinary optimization,design knowledge obtained from data mining can be used for formulation of design problems, decision making, and design steering.1Three data mining techniques, i.e., Self-Organizing Map(SOM),ANalysis Of VAriance(ANOVA),and rough set theory, were used to help designers to determine the final design from non-dominated multi-objective optimization solutions of a f lyback-booster.2,3Data mining for non-dominated solutions of multi-disciplinary optimization of a transonic regional-jet wing showed that design knowledge had a chance to give a better solution compared with the optimized ones.4

Proper Orthogonal Decomposition(POD)-based data mining technique,which is independent on how a shape is parameterized, was used to analyze the non-dominated solutions of multi-objective optimization of a transonic airfoil, and design knowledge was the geometry characteristics illustrated by the POD bases.5Being different from multi-disciplinary and multi-objective optimization, the source data of data mining is the one generated during the entire optimization procedure.6Guo et al.extracted design knowledge from a design space and then used it to validate the single-objective optimization solution.7From the aforementioned literature,it can be found that design knowledge obtained from data mining was only used for analysis and optimization validations. To the knowledge of the authors, there were relatively few studies devoted to use design knowledge for further improving the aerodynamic performance of optimization solutions.

This study strives to further improve the aerodynamic performance of optimization solutions for a transonic compressor rotor blade, NASA Rotor 37, by a second-step optimization on use of design knowledge. The snapshot POD is adopted as the data mining technique since it can successfully extract the system characteristics of the original problem and then reduce the system order.8,9For Aerodynamic Shape Optimization(ASO), ‘‘the original problem” can be a series of flow fields and a group of geometry shapes.10-13Respectively, the flow characteristics and the geometry characteristics are the design knowledge.However,only design knowledge including geometry characteristics is taken into account in the present study because it is more convenient to be used for reducing the design space rather than constructing a reduced-order model.

In the present paper, the method of POD-based data mining and how to use the design knowledge will be firstly introduced. Then, methods of ASO are described and used for design optimization of NASA Rotor 37.Results are presented in detail, and the effects of geometry variations on the performance improvements are addressed.

2. Data mining and knowledge application

2.1. Procedure of data mining and knowledge application

Compared with other data mining methods used for aerodynamic shape optimization,knowledge of POD-based data mining can signif icantly reduce the order of the optimization problem involving a decreased number of design variables.In such cases,the POD-based data mining method is employed in the present study. There are mainly three steps for present POD-based data mining,which are data generation,data mining, and knowledge application. Details of each step are described as follows.

Step 1. Data generation.Data generated during the first optimization procedure is saved as source data. Then, the superior data with improved objectives and maintained constraints is chosen from source data. Meanwhile, only the coordinates of the geometry shape of each superior data are kept, because present data mining concerns only geometry characteristics.

Step 2. Data mining.Data obtained from the first step is regarded as the snapshots of POD.Bases can then be obtained by using a singular value decomposition.The geometry characteristics described by the POD bases are the knowledge of data mining.

Step 3. Knowledge application.POD bases with the geometry knowledge are used to represent the geometry shape of the first-step optimization solutions.Finally, the second-step optimization can be accomplished with a higher efficiency due to the following two reasons:

(1) The order of the optimization problem is reduced. Suppose that the order of the optimization problem equals to the number of design variables. When using POD bases to represent the aerodynamic shape, only the first several primary POD bases are needed,meaning that the number of design variables is dramatically reduced. In most cases, the number of design variables is much lower than that of the original problem.

(2) The design space of the optimization problem is reduced. POD bases include only superior geometry characteristics rather than arbitrary geometry characteristics, which limits the variation of the aerodynamic shape. Therefore, the design space can be reduced.

2.2. POD-based data mining

2.2.1. Principle of snapshot POD

POD is a procedure that provides a set of optimal bases for construction of multidimensional data, which allows a reduction in the order of the original system under consideration from very large numbers to very small ones. Basic POD can provide a series of bases by solving an eigenvalue problem of an auto-relation matrix, and the order of the matrix is always very large, which is hard and sometimes impossible to solve.14Snapshot POD is a more eff icient way to determine POD bases, since the order of the autocorrelation matrix is equal to the number of snapshots, which is much smaller than that of basic POD.8

Here, only snapshot POD is introduced shortly, and to make the description simple, ‘‘POD”is used to replace ‘‘snapshot POD”. Assume that the original system is described by a set of vectors:s1,s2,...,sm,which are also named as snapshots.To calculate their POD bases:φ1,φ2,...,φm,the first step is to solve the following eigenvalue problem:

where R is the autocorrelation matrix defined as

After solving Eq. (1), POD bases can be calculated by

where φkdenotes the k th POD basis, anddenotes the thelement of the corresponding eigenvector of the k th eigenvalue.

2.2.2. Data mining using POD

As mentioned above, snapshots consist of the coordinates of the geometry shape of the superior data generated from the first-step optimization. All geometry shapes are composed of surface grid points. To describe geometry shapes, def ine a set of vectors:g1,g2,...,gm,each of which consists of the coordinates of grid points arranged in a certain sequence,e.g.,gican be denoted as

where X, Y, and Z are the coordinates of X, Y, and Z axes,respectively, which are arranged in a sequence according to the index of grid points. If the number of grid points of each geometry shape is n, the dimension of each vector is 3n.

Then, the snapshot of POD is calculated as follows:

where g0is the vector of the original geometry shape. Eq. (5)means that the snapshots of POD only contain variations of each geometry shape. Then, POD bases can be calculated by Eqs. (1)-(3), in which the design knowledge of data mining is implicitly contained.

The energy ratio of POD bases can help to analyze the knowledge,5which is defined as follows:

where ERidenotes the energy ratio of the i th POD base,and Eidenotes the energy of the i th POD base, which can be defined as

where φi,jdenotes the j th element of the i th POD base. As shown in Eq. (7), the energy represents the sum of the geometry shape variations of the corresponding POD bases.

2.3. Knowledge application

The most crucial issue for knowledge application is the reconstruction of the geometry shape of the first-step optimization solutions by the POD bases. Then, the POD bases are used to parameterize the geometry shape in the second-step optimization. The following equation shows how to reconstruct a geometry shape (denoted by g) by the POD bases:

where nbdenotes the number of primary POD bases, and αndenotes the coefficient of the n th POD base. Error analysis is used to determine nb.The geometry shape of the first-step optimization solutions is reconstructed by different numbers of POD bases, and a series of error is calculated between each reconstructed and exact geometry shapes. Both the average and maximum errors versus the number of POD bases are plotted to help determine nb. When the error is within the tolerance, the corresponding nbis selected.

Def ine the error as

Once nbis determined, the number of design variables is specif ied to be equal to nb.As shown in Eq.(8),the coefficients of POD bases are regarded as the design variables. All snapshots can be denoted by POD bases. The maximum and minimum values of each coefficient are used to determine the range of the corresponding design variable.

3. Methods for ASO

A complete ASO package usually contains four essential modules, i.e., flow field calculation, optimization, parameterization, and grid generation. In the flow field calculation module, a parallel CFD program, named PMB3D-Turbine before but now ASPAC,is used to calculate the cost function.In the optimization module, a Particle Swarm Optimization(PSO) method is adopted, by which the cost function of each particle in the swarm is calculated parallelly.15In the parameterization module, the 3D surface is divided into a series of characteristic curves, which are finally parameterized by Hicks-Henne shape functions. In the grid generation module,the algebraic deformation method is used, which is robust and easy to code. The method used in each module will be introduced hereinafter.

3.1. Method for flow field calculation

Reynolds-Averaged Navier-Stokes (RANS) equations are solved in the rotating reference frame.16The one-equation Spalart-Allmaras turbulence model,17widely used for calculating a turbomachine flow field, is solved to capture the turbulence flow. Inviscid fluxes are calculated by the Roe scheme18with second-order precision by MUSCL interpolation,19and the minimum modulus (min-mod) limiter is used. Viscous fluxes are calculated by the second-order central differentiation scheme.

3.2. Particle swarm optimization

PSO is a global optimization algorithm imitating biological behaviors introduced by Kennedy and Eberhart in 1995.20PSO simulates the hunting process of a bird f lock (or f ish):each particle in a population updates its position in a design space based on its own information and feedback from other particles, until f inding the global optimal solution. Here, only some core parts of the method are introduced.

Initialization:Positions and speeds of particles are initialized by a random initialization method, as shown in the following equations:

where r1and r2are random numbers within the range [0, 1],xminand xmaxare the upper and lower boundaries of the design space,andandare the initial position and velocity of the i th particle, respectively.

Update of the positions:The positions of the particles are updated by

Update of the speeds:For particles satisfying constraints,the method in Ref. 21 is adopted, as shown in the following equation:

where w is an inertial parameter of particles, and c1=c2=2 according to Ref. 20. pirepresents the best particle when the i th particle evolves to the k th generation, andrepresents the best particle of all at the k th generation.

For particles violating the constraints, the method in Ref.22 is adopted. Assume that the i th particle does not satisfy the constraints, the method for updating the speeds is shown in the following equation:

Eq. (14) shows that an unsatisfying particle results from unreasonable speed, so remove this item from Eq. (14), and keep only information of two best particles.

Update of the inertial parameter:The attenuation method based on the Coeff icient Of Variation(COV)is used to update the inertial parameter.22COV is calculated by

where σ and fa, respectively indicate the standard deviation and average value of cost functions of a certain population.The population here can be either the whole or a part of it.COVupis a certain threshold value. When COV is less than COVup, w is reduced by

where fwis a value between 0 and 1.In this paper,fwis set to be 0.975, and COVupis set to be 0.5.

3.3. Parameterization method

In this paper, geometry variations are imposed on the original blade to perturb the aerodynamic shape. The parameterization method is used to produce only the geometry variations. To parameterize a 3D blade, the blade is divided into a series of characteristic sections, where geometry variations are imposed firstly. Then the perturbed blade can be determined by an interpolation method based on the characteristic sections.

Assume that there is a 3D blade, the suction and pressure sides are respectively represented by a block of the surface grid,which is extracted from the volume grid used for CFD. The I,J, and K directions of the grid are respectively in the chordwise, pitchwise, and spanwise directions, as shown in Fig. 1. Details of the parameterization method are presented in the following.

Step 1.Select characteristic sections.Characteristic sections are selected along the spanwise (K) direction of the blade. In fact, a characteristic section is the grid line with constant K,and all the points of the section are located on an S2stream surface by carefully generating the volume grid.

Step 2.Calculate geometry variations of each selected section. In this step, each section is divided into two curves respectively on suction and pressure surfaces. The geometry variations of the curve on the suction or pressure surface can be calculated as follows:

where ks is the index of the section in the K direction,jw is the index of the blade surface in the J direction, i means the i th point on the grid line, nvaris the number of design variables,anis the n th design variable, and bnis the n th Hicks-Henne function,23as shown in the following equation:

where arci,jw,ksdenotes the normalized arc length along the grid line, which can be calculated by

Step 3.Interpolate the geometry variations from the selected sections to the others. Cubic spline is used to interpolate the geometry variations from the points at the selected sections to the others on each grid line along the K direction.For interpolation, dependent variables are the geometry variations, and independent variables are the arc length.Interpolation is preceded on each grid line along the K direction one by one until geometry variations of all points are obtained.

Fig. 1 I-K directions and selected spanwise sections on the suction surface of NASA Rotor 37.

Step 4.Add the geometry variations to the initial geometry shape. The geometry variation of each point is added to the initial geometry shape along certain direction. The normal direction of the blade surface is used to keep new points on the same S2stream surface when the geometry variations are not too large. A new geometry shape is obtained as follows:

where

3.4. Grid deformation method

During optimization design, a grid has to be generated after the geometry is changed. The grid deformation method is a highly eff icient method, and the topology of a grid is always kept. Simple algebraic interpolation is adopted in the study.Assuming that the blade span direction is along the K index of the grid,and the normal outward direction of the blade surface is along the J index, then algebraic interpolation is proceeded on the grid line with the same I and K indices. For a certain point on the grid line, its coordinates in the new grid can be calculated as follows:

where arci,j,kdenotes the normalized arc length along the grid line. Its calculation method is the same as that of arci,jw,ks,except that the loop index is changed from I to J. It should be pointed out that,when the variations of the blade geometry are not large, grid deformation processing can only be conducted to the grid block enclosing the blade surface.

4. Case of NASA Rotor 37

4.1. CFD validation

The validation case is based on NASA Rotor 37, which is always used for the validation case of CFD programs due to its detailed test data.Aerodynamic design parameters,summarized in Table 1,are given by Reid and Moore.24The flow field is considered to be periodic before flow is stalled, so it isreasonable to choose only one passage of the rotor as the computation domain. A grid is generated by the AutoGRID5 module in NUMECA. Topology of the grid is O4H type: the blade is enclosed with an O-grid, and surrounded with a 4Hgrid.There are 56 cells distributed along the spanwise direction of the blade,16 cells at the top clearance of the blade,152 cells in the flow direction, and totally about 900000 cells in the entire passage.

The choking mass flow calculated by ASPAC is 20.95 kg/s within the error band of experimental results: (20.93±0.14)kg/s. Fig. 2 shows performance map comparisons between numerical results calculated by ASPAC and experimental results. The comparison of total pressure ratios shown in Fig.2(a)indicates that numerical results of ASPAC are consistent with experimental results. From the comparison of adiabatic eff iciencies shown in Fig.2(b),it can be seen that the results calculated by ASPAC are a little lower than experimental ones,where the peak adiabatic efficiency of ASPAC differs from that of experiments by about 3%.Nevertheless,flow field trends toward geometry variations are the most important point for ASO, and this of ASPAC has been validated.25

Fig.3 Comparisons of the geometry shapes at 25%,50%,and 75%spans(the original geometry shapes and optimal geometry shapes).

Fig. 4 Comparison of pressure coefficient contours between original and first optimization solutions.

4.2. Data generation - the first-step optimization

This paper is intended to maximize the peak adiabatic efficiency of NASA Rotor 37 under the design rotational speed,meanwhile limit changes of both total pressure ratio and mass flow rate at the design point.The optimization objective can be expressed as

where η denotes the peak adiabatic efficiency, ˙m denotes the flow mass, π denotes the total pressure ratio, and subscript‘‘0” denotes the value of the original geometry shape.

Fig. 5 Comparisons of limiting streamlines of relative velocity on the suction surface between original and first optimization solutions.

Fig.6 Comparisons of relative Mach number contours between original and first optimization solutions at 25%,50%,and 75%spans.

In Ref. 26, the objective is converted to f ind a minimum of entropy production, and constraints are converted to penalty functions. However, sometimes penalty functions cannot limit constraints strictly,so the most direct method is used to def ine a cost function as the reciprocal of the adiabatic efficiency,and when the constraints are not met,amplifying the reciprocal for five times, as shown in the following equation:

Five spanwise sections at approximately 100%,75%,50%,25%, and 0% blade spans are selected to parameterize the blade, as shown in Fig. 1. Ten Hicks-Henne shape functions are imposed on the suction surface of the spanwise sections at 75%,50%,and 25%blade spans,while the pressure surface maintains unchanged. Therefore, the total number of design variables is 30. The design space of each design variable is(-0.01, 0.01). 80 particles are used to evolve 25 generations,and 8 CPUs are used to calculate the objective function of each particle.

Table 2 compares the performances between original and first optimized solutions (marked by Optimal solutions-1st).The first optimized blade increases the adiabatic efficiency by 1.29% with a 0.42% reduction in the total pressure ratio and a 0.48%increase in the mass flow rate.This can be considered as a fairly good optimal design in terms of efficiency gain and satisfaction of constraints.

Original and optimal geometries at 25%, 50%, and 75%spans are depicted in Fig. 3. Maximum thicknesses at 75%and 50% spans are increased, and their positions are moved toward the trailing edge.At 25%span,the maximum thickness is decreased, but its position is also moved toward the trailing edge.

Fig. 4 shows comparisons of pressure coefficient Cpcontours between original and first optimization solutions. There is a slight difference upstream from the mid-chord on the pressure surface,and the pattern of the shock wave on the suction surface (dashed red lines) is changed obviously. Changes on the pressure surface will be discussed together with Fig. 5.

To gain more insight into the changes on the suction surface, Fig. 5 depicts limiting streamlines of relative velocity on the suction surface, where the flow moves radially outwards immediately after the shock, so radial limiting streamlines show the position of the shock wave. Comparisons of limiting streamlines of relative velocity on the suction surface are shown in Fig.5,from which it can be seen clearly that the position of the shock wave over 25% span moves downstream, so the separation region induced by the shock wave is shrunk,which can reduce the loss and then increase the adiabatic efficiency.

Fig. 6 shows relative Mach number contours of original and first optimization solutions at 25%,50%,and 75%spans.As can be seen,shock losses of Rotor 37 are generated by two shocks, one is a bow shock, for one passage which can be divided into two parts,its oblique extension ahead of the leading edge (marked by S1) and the first passage shock (marked by S2),and the other is a weak passage normal shock(marked by S3). For all three spans, there are almost no changes in S1,so only changes in the other two shocks will be discussed below.

(1) At 75% span (Fig. 6(a)): S2near the suction surface is weaker than that in the original flow field; S3is moved downstream a little, and slightly stronger than that in the original flow field. In addition, very different from that in the original flow field, flow near the suction surface and before S2is reaccelerated, as indicated by a dashed blue circle.

(2) At 50% span (Fig. 6(b)): S2near the suction surface is much weaker, and changes of S3are little. There is another shock at this span, which is a ref lection shock of S2(marked by Smr2). Smris moved far away from the leading edge and much weaker.

(3) At 25% span (Fig. 6(c)): S2near the suction surface is slightly weaker, and S3near the pressure surface is moved downstream and a little stronger.

In summary, although parts of S3are strengthened a little,S2near the suction surface is weakened more signif icantly,hence it can be concluded that the loss of shock wave in the flow field of the optimized blade is reduced.

By the analyses of changes in both geometry shapes and the flow field, it can be known that variations of the geometry shapes change both the pattern and strength of the shock waves, the former decreases the loss of the separation region induced by the shock waves, and the latter decreases the loss of the shock waves. The loss reduction from these two parts finally makes the adiabatic efficiency increase.

4.3. Knowledge discovery

Fig. 7 Ratios of different kinds of data.

Fig. 8 Energy ratios of the first ten POD modes (bases).

The number of particles generated during 25 generations is 2000.Adding 80 particles of initial generation,there are totally 2080 particles.Particles,objective function values of which are larger than that of NASA Rotor 37,and which also satisfy the constraints, are selected as the superior data. There are 1734 particles meeting the requirements,which are 83%of the total particles. Fig. 7 shows ratios of different kinds of data, in which the data is classif ied according to how much the adiabatic efficiency is increased, and the data out of the rule is denoted as ＜0. The superior data is transferred to snapshots according to Eqs. (4) and (5), from which POD bases (knowledge) are extracted.

Fig. 9 Variations of each grid point transferred from the first three modes (ranges of the three axes are the same but contour levels are different).

Fig. 8 shows energy ratios of the first ten POD modes(bases). The first mode is dominating (more than 94%), and the first three modes represent more than 99% of the total energy, in which the main knowledge is saved. However, the original data in POD bases (X, Y, and Z) is not convenient to f ind the knowledge, so the variation of each grid point(marked by VG) is defined as follows to show rules of shape changes of the blade:

Fig.10 Variations of grid points of the first three POD modes at 25%, 50%, and 75% spans.

Fig. 11 Mean and maximum errors with the number of POD bases.

where ni,jw,kis the unit surface normal vector of NASA Rotor 37, andis the original data in POD bases rearranged on each grid point.In fact,the meaning of VGi,jw,kis the same as that of Δxi,jw,kin Eq. (22).

Fig.9 shows variations of each grid point transferred from the first three POD modes. This f igure indicates that the first mode contributes primarily to the most part of the suction surface change, and contributions of the second and third modes are decreased in a proper order. Another characteristic is that almost all variations are located between the mid-chord and the trailing edge for all three modes. To gain more insight,variations of each grid point of all three modes at 25%,50%, and 75% spans are depicted in Fig. 10 respectively.Distributions rules (absolute amplitudes and locations of VG) of each POD mode are consistent with the observation from Fig. 9. Comparing the first modes of the three spans,the maximum VG is located at 50% span.

Fig. 12 Comparisons of performance maps between exact and POD-based geometry shapes.

All of these characteristics of the POD modes are the knowledge implied in the superior data, and undoubtedly, all geometry shapes represented by POD bases according to Eq.(8) will contain the characteristics of the superior data. In other words, variation of the geometry shape represented by the POD bases is limited, which is very useful to reduce the design space and thus increase the optimization efficiency. In addition, to start a new optimization design of NASA Rotor 37 or other similar 3D blades, the knowledge proposes that design variables should be located between the mid-chord and the trailing edge, the range of which at 50% span should be larger than those at other spans.

4.4. Knowledge application - the second-step optimization

The geometry shape of the first optimization solutions is represented by one to twenty POD bases separately according to Eq. (8). Mean and maximum errors with the number of POD bases are shown in Fig. 11. Both average and maximum errors are reduced when the number of POD bases is increased,except with some fluctuations. When using eight POD bases,there are local minimums on both curves(marked by a dashed blue circle), and the mean of error is less than 10-5while the maximum is less than 10-2. Therefore, the geometry shape of the first optimization solutions is represented by the first eight POD bases.To check the accuracy of the geometry shape represented by the POD bases, performance maps of both exact and POD-based geometry shapes are calculated and compared,which are shown in Fig.12.All points are closed except that two points near stalling are with a little deviation.

Fig. 13 Adiabatic efficiency evaluations with the number of function evaluations of the twice optimization.

Table 3 Comparison of performance at the design point between the first and second optimization solutions.

For the second optimization, twenty particles are used.After ten generations, optimization solutions (marked by Optimal solutions-2nd) are shown in Table 3. On the basis of the first optimized blade,the second optimized blade further increases the adiabatic efficiency to 1.54%with a 0.50%reduction in the total pressure ratio and a 0.43% increase in the mass flow rate. There is a rise of 0.25% compared with that of the first optimized blade.

Adiabatic efficiency evaluations with the number of function evaluations of the twice optimization are depicted in Fig. 13. It can be seen that the performance improvement of the second optimization is obvious, while the computational cost, totally calculating 220 times objective functions, is very low compared with that of the first optimization. To further explore this, object function changes per calculating function(denoted by OFC) is defined as below:

where ηkis the adiabatic efficiency at the k th step of optimization,η0is the adiabatic efficiency of NASA Rotor 37,and npis the number of calculating functions per step. For PSO, npis equal to be the number of particles.

Table 4 shows comparisons of mean and maximum of OFCs between the first and second optimizations. The mean of OFCs of the second optimization is about four times larger than that of the first optimization, and the maximum is about two times larger than that of the first optimization. Comparisons in Fig.12 and Table 4 show that the second optimization based on the geometry represented by POD bases can optimize the 3D blade with a higher efficiency, which results from the application of the knowledge.

Comparisons of shapes between the first and second optimization solutions at 25%, 50%, and 75% spans are shownin Fig. 14. It can be seen that all changes appear between the mid-chord and the trailing edge. At 75% span, the maximum thickness is further increased, and the thickness near the trailing edge is decreased signif icantly. At 50%span,the thickness near the trailing edge is slightly decreased. At 25% span, the thickness near the trailing edge is slightly increased.

Table 4 Comparisons of mean and maximum of OFCs between the first and second optimizations.

Fig. 15 shows comparisons of streamlines on the suction surface between the first and second optimized solutions.The shock wave between 25% span and 75% span, indicated by radial limiting streamlines, moves downstream, so the region of the induced separation is shrunk further, which reduces the loss of the separation.

Fig. 16 shows relative Mach number contours of the first and second optimized solutions at 25%,50%,and 75%spans.There are almost no changes in S1and S3at all three spans,and at 25% span, even S2is not changed. At 75% and 50%spans, S2near the suction surface are further slightly

Fig. 15 Comparisons of limiting streamlines of relative velocity on the suction surface between the first and second optimization solutions.

weakened.The most obvious change is that, at 50%span,is banished in the second optimization solutions.

By the analysis of changes in both the geometry and the flow field, it can be concluded that variations of the geometry further reduce the loss of the separation region and the shock waves and thus increase the adiabatic efficiency.

5. Conclusions

(1) It is effective and eff icient to use the knowledge of PODbased data mining to improve the performance of first optimization solutions.

(2) The knowledge of POD-based data mining can reduce the order and design space of optimization problems and thus increase the optimal efficiency; in addition,the knowledge gives some useful advice on setting the locations and ranges of design variables when designing NASA Rotor 37 or other similar blades.

(3) The methods of ASO,including the flow field calculation module, optimization module, parameterization module, and grid deformation module, do a very effective optimization design for the 3D blade.

(4) Analyses of the changes of the geometric shape and the flow field show that geometry variations change the pattern and strength of the shock waves in the passage of NASA Rotor 37,the former reduces the loss of the separation and the latter reduces the loss of the shock.Moreover, both of them finally increase the adiabatic efficiency.

(5) The second optimization and POD-based data mining technique described in this paper can be used to solve other ASO problems, such as airfoil, wing, wing-body,and so on.

Acknowledgement

This study was supported by the National Natural Science Foundation of China (Nos. 51676003, 51206003, and 11702305).