Computer and Fluids

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www.elsevier.com/locate/compfluid

Computers & Fluids 37 (2008) 705–723

Improving the hydrodynamic performance of diffuser vanesvia shape optimization

Tushar Goel a,*,1, Daniel J. Dorney b,2, Raphael T. Haftka a,3, Wei Shyy c,4

a Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611, United Statesb ER42, NASA Marshall Space Flight Center, AL 35812, United States

c Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109, United States

Received 17 February 2007; received in revised form 1 August 2007; accepted 1 October 2007Available online 10 October 2007

Abstract

The performance of a diffuser in a pump stage depends on its configuration and placement within the stage. The influence of vaneshape on the hydrodynamic performance of a diffuser has been studied. The goal of this effort has been to improve the performanceof a pump stage by optimizing the shape of the diffuser vanes. The shape of the vanes was defined using Bezier curves and circular arcs.Surrogate model based tools were used to identify regions of the vane that have a strong influence on its performance. Optimization ofthe vane shape, in the absence of manufacturing and stress constraints, led to a nearly 8% reduction in the total pressure losses comparedto the baseline design by reducing the base separation.� 2007 Elsevier Ltd. All rights reserved.

1. Introduction

The space shuttle main engine (SSME) is required tooperate over a wide range of flow conditions. This require-ment imposes numerous challenges on the design of turbo-machinery components. One concept that is being exploredis the use of an expander cycle for future vehicles. A sche-matic of a representative expander cycle for a conceptualupper stage engine is shown in Fig. 1. Oxidizer and fuelpumps are used to feed LOX (liquid oxygen) and LH2(liquid hydrogen) to the combustion chamber of the mainengine. The combustion products are discharged from thenozzle. The pumps are driven by turbines, which use thegasified fuel as the working fluid.

There is a continuing effort to develop subsystems liketurbopumps and turbines used in a typical expander cycle

0045-7930/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compfluid.2007.10.002

* Corresponding author. Tel.: +1 925 245 4547.E-mail address: [email protected] (T. Goel).

1 Currently with Livermore Software Technology Corporation.2 Aerospace Engineer.3 Distinguished Professor.4 Clarence L ‘‘Kelly’’ Johnson Professor.

based upper stage engine. The requirements on the designof subsystems are influenced by size, weight, efficiency,and manufacturability of the system. Besides, often therequirements for subsystems are coupled, for example,the above-mentioned design constraints require turbo-pumps to operate at high speeds to have high efficiencywith low weight and compact design. This may requireseeking alternate designs for the turbopumps as the currentdesigns may not prove adequate over a wide range of oper-ating conditions. Mack et al. [1,2] optimized the design of aradial turbine that allows high turbopump speeds, per-forms comparable to an axial turbine at design conditions,and yields good performance at off-design conditions.

Dorney et al. [3] have been exploring different conceptsfor turbopump design. A simplified schematic of a pump isshown in Fig. 2. Oxidizer or fuel enters from the left. Thepressure increases as the fluid passes through the impeller.The fluid emerging from the impeller periphery typicallyhas high tangential velocity, which is partially convertedinto pressure by passing it through a diffuser. Dorneyet al. [3] found that a diffuser with vanes is more efficientthan a vaneless diffuser at off-design conditions. Over arange of operating conditions, the performance of a pump

mailto:[email protected]

Diffuser vane

blade

Splitter blade

Inlet guide vane Main impeller

Fig. 2. Schematic of a pump.

IGV

Main/Splitter blade

Flow path

Fig. 3. Meanline pump flow path (IGV: inlet guide vane).

Fig. 1. A representative expander cycle used in the upper stage engine(courtesy wikipedia).

706 T. Goel et al. / Computers & Fluids 37 (2008) 705–723

is driven by the flow in the diffuser. Generally, the diffuserwill stall before the impeller, and the performance of thediffuser will drop off more rapidly at off-design operatingconditions. The main objectives of the current effort areto improve the hydrodynamic performance of a diffuser,using advanced optimization techniques, and to study thefeatures of the diffuser vane that influences its performance.While numerous aspects such as diffusion factor, efficiency,and pressure increase, can be used to characterize the per-formance of a diffuser, we limit the scope of this study tooutlet-to-inlet pressure ratio at design conditions.

Improvement in performance via shape optimization hasbeen successfully achieved in many areas. For example,there are numerous instances of improvement in lift to dragratio via airfoil or wing shape design [4–10], shape of bladesis optimized to increase the efficiency of turbines [4,11] andpumps [12] in past. Shape design typically involves optimi-zation that requires a significant number of function eval-uations to explore different concepts. If the cost ofevaluating a single design is high, as is usually the case, sur-

rogate models of objectives and constraints are frequentlyused to reduce the computational burden.

Surrogate based optimization approach is widely used inthe design of space propulsion systems, such as radial tur-bines [1,2], supersonic turbines [4], diffusers [13], rocketinjectors [14,15], and combustion chamber geometry [16].Detailed reviews on surrogate based optimization are pro-vided by Li and Padula [17] and Queipo et al. [18]. Thereare many surrogate models, and it is not clear which surro-gate model performs the best for any particular problem.In such a scenario, one possible approach to account foruncertainties in predictions is to use an ensemble of surro-gates [19]. This multiple surrogate approach has been dem-onstrated to work well for several problems includingsystem identification [20], and compressor blade design [12].

Specifically, the objectives of the present study are: (i) toimprove the hydrodynamic performance, characterized byoutlet to inlet pressure ratio, of the diffuser via shapeoptimization of vanes, (ii) to identify important regionsin diffuser vanes that help optimize pressure ratio, and(iii) to demonstrate the application of multiple surrogatesbased strategy for space propulsion systems.

The paper is organized as follows. We define the geome-try of the vanes, and numerical tools to evaluate the diffuservane shapes in Section 2. We describe the relevant details insurrogate model based optimization in Section 3. Theresults obtained for the current optimization problem anda discussion of the physics involved in flow over optimalvane is presented in Section 4. Finally, we conclude by reca-pitulating the major findings of the paper in Section 5.

2. Problem description

Representative radial locations in the meanline pumpflow path (not-to-scale) are shown in Fig. 3 [3]. The fluidenters from the left, and is guided to the unshrouded impel-ler via an inlet guide vane assembly. The flow from theimpeller passes through the diffuser before being collectedin the discharge collector. In this study, we focus our effortson a configuration of 15 inlet guide vanes, seven main and

Fig. 5. Definition of the geometry of the diffuser vane (refer to Table 1 forvariable description).

T. Goel et al. / Computers & Fluids 37 (2008) 705–723 707

seven splitter blades, and 17 diffuser vanes. The length ofthe diffuser vanes is also fixed according to the locationof the collector. Our goal is to maximize the performanceof the diffuser, characterized by the ratio of pressure atthe outlet and the inlet, hereafter called as ‘pressure ratio’.The performance of the diffuser is governed by the shape ofthe diffuser vane.

2.1. Vane shape definition

The description of geometry is the most important stepin the shape optimization. The current shape of the diffuservanes, referred as baseline design, (shown in Fig. 4a) wascreated and analyzed using meanline and geometry genera-tion codes at NASA [21], which are suitable for preliminarydesign. The baseline design yields a pressure ratio of 1.065when simulated using the standalone turbomachineryanalysis code PHANTOM developed at NASA MarshallSpace Flight Center [22] (details of the code are providedin Section 2.2). It is obvious from Fig. 4b that the existingshape allows the flow to remain attached while passing onthe vanes but causes significant flow separation while leav-ing the vane. The separation induces significant loss ofpressure recovery that is the primary goal of using a dif-fuser. The baseline design was created subject to severalconstraints, including: (i) the number of vanes was set at17 based on the number of bolts used to attach the twosides of the experimental rig, (ii) the vanes had to accom-modate a 3/8-in. diameter bolt, with enough excess mate-rial to allow manufacturing, and (iii) the length of thevanes was set based on the location of the collector. Theseconstraints resulted in an initial optimized design thatlooked quite similar to the baseline design, and yielded sim-ilar performance in both isolated component and full stagesimulations. In an effort to more thoroughly explore thedesign space for diffusers (and allowing for future improve-ments in materials and manufacturing), the constraintsresulting from the bolts (the number and thickness of the

Fig. 4. Baseline diffuser vane sh

vanes) were relaxed. Other factors that must be consideredin the following optimization include: (i) the designs result-ing from the optimization techniques shown in this paperare based on diffuser-alone simulations (different resultsmay be obtained if the optimizations were based on fullstage simulations), and (ii) no effort was made to determineif the proposed designs would meet stress and/or manufac-turing requirements.

To reduce the separation, we represent the geometry of avane by five sections using a circular arc and Bezier curvesas shown in Fig. 5. Sections 1, 3, and 4 are Bezier curves,and Section 2 is a circular arc. To keep the number ofdesign variables manageable, we fixed the parameters thatwere not considered to be critical for flow separation. Theshape of the inlet nose (between points B1 and B2) isobtained from the existing baseline vane shape. The circu-lar arc in Section 2 is described by fixing the radius r

(r = 0.08), the location of the center C1 (�3.9,6.1), the startand end angles. The arc begins at angle p + hr + h/2 (point

ape and time-averaged flow.


A1) and ends at p + hr � h/2 (point A2). Here, we usehr = 22.5� and h is a design variable. We provide the infor-mation about the coordinates and slope of the tangents atendpoints such that different Bezier curves (Appendix A)are generated by varying the length of tangents. The pointsA1 and A2, shown by red dots in Fig. 5, and the tangents tothe arc, serve as the end points location and tangents usedto define the Bezier curves in Sections 1 and 3. Coordinatesand the slopes at points B1 and B2 (obtained using the dataof inlet nose of the baseline design Fig. 5) are used to defineBezier curves in Sections 1 and 4. The Bezier curve in Sec-tion 1 (Fig. 5) is defined using the coordinates and slope oftangents at B2 and A2. The lengths of the tangents t1 and t2

control the shape of the curve. While one end point coor-dinates and slope for Bezier curves in Sections 3 and 4are known at points A1 and B1, the second end point andslope are obtained by defining the location of point P

(Py,Pz). The slope of tangent at point P is taken as five

degrees more than the slope of the line joining points P

and B1. The additional slope is specified to avoid all thepoints in Section 4 fall in a straight line. The values ofthe fixed parameters are decided based on the designrequirements. The lengths of tangents t1–t6 serve as vari-ables to generate Bezier curves in Sections 1, 3 and 4.

The ranges of the design variables, summarized in Table1, are selected such that we obtain practically feasible vanegeometries and the corresponding grids (discussed in nextsection). We note that present Bezier curve and circulararc based definition of diffuser vane shapes provides signif-icantly different vane geometries than the baseline design,particularly near the outlet region. The main rationalebehind this choice of vane shape is the relaxation of man-ufacturing and stress constraints, ease of parameterizationand better control of the curvature of the vane that allowsthe flow to remain attached.

2.2. Mesh generation, boundary conditions, and numerical

simulation

Performance of diffuser vane geometries is analyzedusing the NASA PHANTOM code [22] developed at Mar-

Table 1Design variables and corresponding ranges

Minimum Maximum

h 60� 110�t1 1.25 2.50t2 0.75 1.50t3 0.05 0.30t4 0.50 1.00t5 0.50 1.00t6 1.00 2.00Py �2.50 �2.00Pz 5.70 5.85

Angle h controls the size of circular arc, t1–t6 define the lengths of thetangents of Bezier curves and determine its curvature, and (Py,Pz) are thecoordinates of the point P in the middle of the section. Angle h is given indegrees and the units of other variables are same as used for baseline design.

shall Space Flight Center to analyze turbomachinery flows.This 3D, unsteady, Navier–Stokes code utilizes structured,overset O- and H-grids to discretize and to analyze theunsteady flow resulting from the relative motion of rotatingcomponents. The code is based on the Generalized Equa-tions Set methodology [23] and utilizes a modified Bald-win-Lomax turbulence model [24]. The inviscid andviscous fluxes are discretized using a third-order spatiallyaccurate Roe’s scheme, and second-order central differenc-ing, respectively. The unsteady terms are modeled using asecond-order accurate scheme.

For this problem, we solve three-dimensional, incom-pressible, unsteady, non-rotating, turbulent, single phase,constant-material-property flow through the diffuser vane.The working fluid is water. By taking the advantage ofperiodicity, only a single vane is analyzed here. A combina-tion of H- and O-grids with 13065 grid points has beenused to analyze diffuser vane shapes. A typical grid isshown in Fig. 6. The boundary conditions imposed onthe flow domain are as follows. Mass flux, total tempera-ture, and flow angles (circumferential and radial) are spec-ified at the inlet. Mass flux is fixed at the outlet. All solidboundaries are modeled as no slip, adiabatic walls, withzero normal derivative of pressure. Periodic boundary con-dition is enforced at outer boundaries. With this setup, ittakes approximately 15 min on a single Intel Xeon proces-sor (2.0 GHz, 1.0 GB RAM) to simulate each design.

3. Methodology

As discussed earlier, the surrogate model basedapproach is suitable to reduce the computational cost ofoptimization. In this section, we discuss different stepsand considerations in surrogate modeling and its applica-tion to the design problem(s). A stepwise procedure of sur-

Fig. 6. A planar view of the combination of H- and O-grids used toanalyze diffuser vane. Body-fitted O-grids are shown in grey and algebraicH-grid is shown in black.


rogate based analysis and optimization is explained withthe help of Fig. 7. Firstly, we identify the objectives, con-straints, and design variables. Next, we develop a proce-dure to evaluate different designs. In context of thecurrent problem, we identified design variables, objective,and numerical procedure to evaluate different designs inthe previous section. To reduce the computational expenseinvolved in optimization, we construct multiple surrogatemodels of the objectives and constraints. A brief descrip-tion of popular surrogate models is as follows.

3.1. Surrogate modeling

There are many surrogate models e.g., polynomialresponse surface approximations, kriging, radial basis neu-ral network, support vector regression etc. A detailed dis-cussion of different aspects of surrogate modeling wasreviewed by Li and Padula [17] and Queipo et al. [18].We give a brief description of different surrogate modelsas follows.

3.1.1. Polynomial response surface approximation (PRS,

[25])

The observed response y(x) of a function at point x isrepresented as a linear combination of basis functionsfi(x) (mostly monomials are selected as basis functions)and coefficients bi. Error in approximation e is assumedto be uncorrelated and normally distributed with zeromean and r2 variance. That is,

yðxÞ ¼X

i

bifiðxÞ þ e; EðeÞ ¼ 0; V ðeÞ ¼ r2: ð1Þ

The polynomial response surface approximation of y(x) is,

Fig. 7. Surrogate based design a

yðxÞ ¼X

i

bifiðxÞ; ð2Þ

where bi is the estimated value of the coefficient associatedwith the ith basis function fi(x). The coefficient vector b isobtained by minimizing the error in approximationðeðxÞ ¼ yðxÞ � yðxÞÞ at Ns sampled design points in a leastsquare sense as,

b ¼ ðX TX Þ�1X Ty; ð3Þ

where X is the matrix of basis functions and y is the vectorof responses at Ns design points. The quality of approxima-tion is measured by computing the coefficient of multipledetermination R2

adj defined as,

R2adj ¼ 1� r2

aðN s � 1ÞXNs

i¼1

ðyi � �yÞ2,

; ð4Þ

where �y ¼ ðNsÞ�1PNsi¼1yi and RMS error at sampling points

is given as ra ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðN s � N bÞ�1PNs

i¼1ðyðxiÞ � yðxiÞÞ2q

. For a

good fit, R2adj should be close to 1. For more details on poly-

nomial response surface approximation, refer to Myers andMontgomery [25].

3.1.2. Kriging (KRG, [26])

Kriging is named after the pioneering work of D.G. Kri-ge (a South African mining engineer) and was formallydeveloped by Matheron [26]. Kriging estimates the valueof an objective function y(x) at design point x as the sumof a linear polynomial trend model

PNvi¼1bifiðxÞ and a sys-

tematic departure Z(x) representing low (large scale) andhigh frequency (small scale) variations around the trendmodel.

nd optimization procedure.


yðxÞ ¼ yðxÞ ¼X

i

bifiðxÞ þ ZðxÞ: ð5Þ

The systematic departure components are assumed to becorrelated as a function of distance between the locationsunder consideration. The Gaussian function is the mostcommonly used correlation function.

CðZðxÞ; ZðsÞ; hÞ ¼YNv

i¼1

expð�hiðxi � siÞ2Þ: ð6Þ

The parameters bi, hi are obtained by maximizing a like-lihood function that is a measure of the probability of thesample data being drawn from it. We use an implementa-tion by Lophaven et al. [27].

3.1.3. Radial basis neural network (RBNN, [28])

The objective function is approximated as a weightedcombination of responses from radial basis functions (alsoknown as neurons).

yðxÞ ¼XNRBF

i¼1

wiaiðxÞ; ð7Þ

where ai(x) is the response of the ith radial basis function atdesign point x and wi is the weight associated with ai(x).Mostly, the Gaussian function is used for radial basis func-tion a(x) as

a ¼ radbasðks� xkbÞ; radbasðnÞ ¼ e�n2

: ð8Þ

The parameter b in the above equation is inverselyrelated to a user defined parameter ‘‘spread constant’’ thatcontrols the response of the radial basis function. A highspread constant would cause the response of neurons tobe very smooth and small spread constant would result intoa highly non-linear response function. Typically, spreadconstant is selected between zero and one. The number ofradial basis functions (neurons) and associated weightsare determined by satisfying the user defined error ‘‘goal’’on the mean squared error in approximation. The usualvalue of goal is the square of 5% of the mean response.

As discussed here, we have many surrogate models andit is unknown a priori, which surrogate would be most suit-able for a given problem. Besides, the choice of best surro-gate model changes with sampling density and nature ofthe problem [19]. It has been shown by Goel et al. [19] thatin such scenario, simultaneously using multiple surrogatemodels protects us from choosing wrong surrogates. Theyproposed using a weighted averaged surrogate that isdescribed as follows.

3.1.4. PRESS-based weighted average surrogate model

(PWS, [19])

We develop a weighted average surrogate model as

ypwsðxÞ ¼XNSM

i

wiyiðxÞ; ð9Þ

where ypwsðxÞ is the predicted response by the weightedaverage of NSM surrogate models, yiðxÞ is the predicted re-sponse by the ith surrogate model and wi is the weight asso-ciated with the ith surrogate model at design point x.Furthermore, the sum of the weights must be oneðPNSM

i¼1 wi ¼ 1Þ so that if all the surrogates agree, ypwsðxÞ willalso be the same. Weights are determined as follows:

w�i ¼ ðEi=Eavg þ aÞb; wi ¼ w�iX

i

w�i ;

,

Eavg ¼XNSM

i¼1

Ei=NSM; b < 0; a < 1;

ð10Þ

where Ei is the global data-based error measure for ith sur-rogate model. In this study, generalized mean squaredcross-validation error (GMSE) (leave-one-out cross valida-tion or PRESS in polynomial response surface approxima-tion terminology), defined in Appendix B, is used as globaldata-based error measure, by replacing Ei by

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiGMSEi

p

(RMS PRESS error). We use a = 0.05 and b = �1. Theabove mentioned formulation of weighting schemes is usedwith polynomial response surface approximation (PRS),kriging (KRG) and radial basis neural networks (RBNN)such that,

ypws ¼ wprsyprs þ wkrgykrg þ wrbnnyrbnn: ð11Þ

For more details about the weighted average surrogatemodel, we refer the reader to Goel et al. [19].

Next, we use these surrogate models to characterize theimportance of different variables and to identify the mostand the least important variables for different objectivesand constraints. This information is vital to understandthe role of the variables. Also, we can fix the least impor-tant variables to reduce the dimensionality of the problem.We use global sensitivity analysis method proposed bySobol [29].

3.2. Global sensitivity analysis (GSA, [29])

Sobol [29] presented a variance based, non-parametricapproach to perform global sensitivity analysis. In thisapproach, the response function is decomposed into uniqueadditive functions of variables and their interactions suchthat the mean of each additive function is zero. Thisdecomposition allows the variance (V) to be computed asa sum of individual partial variance of each variable (Vi)and partial variance of interactions (Vij) of different vari-ables. The sensitivity of the response function with respectto each variable is assessed by comparing the sensitivityindices (Si,Sij) that is the relative magnitude of partialand total variance of each variable. The details of themethod are as follows.

A function f(x) of a square integrable objective as afunction of a vector of independent uniformly distributedrandom input variables, x in domain [0,1] is assumed.The function can be decomposed as the sum of functionsof increasing dimensionality as


f ðxÞ ¼ f0 þX

i

fiðxiÞ þXi<j

fijðxi; xjÞ þ � � �

þ f12...Nvðx1; x2; . . . ; xNvÞ; ð12Þ

where f0 ¼R 1

x¼0f dx. If the following condition:Z 1

0

fi1...is dxk ¼ 0; ð13Þ

is imposed for k = i1, . . . , is, then the decomposition de-scribed in Eq. (12) is unique. In context of global sensitivityanalysis, the total variance denoted as V(f) can be shownequal to

V ðf Þ ¼XNv

i¼1

V i þX

16i;j6N v

V ij þ � � � þ V 1...Nv ; ð14Þ

where V(f) = E((f � f0)2), and each of the terms in Eq. (14)represents the partial contribution or partial variance ofthe independent variables (Vi) or set of variables to the to-tal variance and provides an indication of their relativeimportance. The partial variances can be calculated usingthe following expressions:

V i ¼ V ðE½f jxi�Þ;V ij ¼ V ðE½f jxi; xj�Þ � V i � V j;

V ijk ¼ V ðE½f jxi; xj; xj�Þ � V ij � V ik � V jk � V i � V j � V k;

ð15Þand so on, where V and E denote variance and expectedvalue, respectively. Note that E½f jxi� ¼

R 1

0fi dxi, and

V ðE½f jxi�Þ ¼R 1

0f 2

i dxi. This formulation facilitates the com-putation of the sensitivity indices corresponding to theindependent variables and set of variables. For example,the first- and second-order sensitivity indices are computedas

Si ¼V i

V ðf Þ ; Sij ¼V ij

V ðf Þ : ð16Þ

Under the independent model inputs assumption, thesum of all the sensitivity indices is equal to one. The first-order sensitivity index for a given variable represents themain effect of the variable, but it does not take into accountthe effect of the interaction of the variables. The total con-tribution of a variable to the total variance is given as thesum of all the interactions and the main effect of the vari-able. The total sensitivity index of a variable is then definedas

Stotali ¼

V i þPj;j 6¼i

V ij þPj;j 6¼i

Pk;k 6¼i

V ijk þ � � �

V ðf Þ : ð17Þ

Note that the above referenced expressions can be easilyevaluated using surrogate models of the objective func-tions. Sobol [29] has proposed a variance-based non-para-metric approach to estimate the global sensitivity for anycombination of design variables using Monte Carlo meth-ods. To calculate the total sensitivity of any design variablexi, the design variable set is divided into two complemen-

tary subsets of xi and Z (Z = xj, "j = 1, Nv; j 5 i). The pur-pose of using these subsets is to isolate the influence of xi

from the influence of the remaining design variablesincluded in Z. The total sensitivity index for xi is thendefined as

Stotali ¼ V total

i =V ðf Þ; ð18Þ

where

V totali ¼ V i þ V i;Z ; ð19Þ

where Vi is the partial variance of the objective with respectto xi and Vi,Z is the measure of the objective variance thatis dependent on interactions between xi and Z. Similarly,the partial variance for Z can be defined as Vz. Thereforethe total objective variability can be written as

V ¼ V i þ V Z þ V i;Z : ð20Þ

While Sobol [29] had used Monte Carlo simulations toconduct the global sensitivity analysis, we use Gauss-quad-rature numerical integration of different partial varianceterms in Eq. (15) to calculate sensitivity indices.

Surrogate models, which represent objectives and con-straints, are used to optimize the performance of the sys-tem. If we use multiple surrogates, we will find more thanone candidate optimal solutions. The performance of suchpredicted optimal designs is validated using numerical sim-ulation. If we are satisfied with the performance of the sub-system, we terminate the search procedure. Otherwise, werefine the design space in the region of interest. The designspace refinement can be done in multiple ways, (i) we canfix the least important variables at the optimal values (asrealized by optimization) or mean values to reduce thedimensionality of the problem, (ii) we sample more pointsin the design space, and (iii) we identify the region wherewe expect potential improvement and concentrate on thatregion. We repeat this optimization procedure tillconvergence.

4. Results and discussion

We present application of the above-mentioned surro-gate based optimization framework to optimize the perfor-mance of diffuser vanes in this section. We also present adetailed analysis of optimal design.

4.1. Surrogate model construction

First step in surrogate modeling is to sample data indesign variable space. Different design of experiment(DOE) techniques are effective in reducing the computa-tional expense of generating high-fidelity surrogate models.The most popular DOE methods, are Latin hypercubesampling (LHS) and face-centered central compositedesigns (FCCD). The choice of DOE depends on the nat-ure of problem and the number of samples required.

For this problem, we have nine design variables hencewe selected 110 design points to allow adequate data to


evaluate 55 coefficients of a quadratic polynomial responsesurface. Since face-centered central composite designrequires unreasonably large number of samples (531 pointsto approximate 55 coefficients), we use Latin hypercubesampling (LHS) to construct surrogate models. We gener-ated LHS designs using Matlab routine ‘lhsdesign’ with100 iterations to maximize the minimum distance betweenpoints. We evaluated each diffuser vane shape usingPHANTOM. This dataset is referred as ‘Set A’. The rangeof the data, given in Table 2, indicated the potential ofimprovement in the performance of the diffuser by shapeoptimization. We constructed four surrogate models, poly-nomial response surface approximation (PRS), kriging(KRG), radial basis neural network (RBNN), andPRESS-based weighted average surrogate (PWS) of theobjective pressure ratio. We used quadratic polynomialfor PRS, and linear trend model with Gaussian correlationfunction for kriging. For RBNN, the spread coefficient wastaken as 0.5 and the error goal was the square of 5% of themean value of the response at data points. The parametersa, and b for PWS model were 0.05 and �1, respectively.The summary of quality indicators for different surrogatemodels is given in Table 2.

All error indicators are desired to be low compared tothe response data, except R2

adj, which is desired to be closeto one. The RMS PRESS error (Appendix B) and RMSerror (�1.0e�2) were very high compared to the range ofdata. This indicated that all surrogate models poorlyapproximated the actual response, and were likely to yieldinaccurate results if used for global sensitivity analysis andoptimization. To identify the cause of poor surrogate mod-

Table 2Summary of pressure ratio on data points and performance metrics fordifferent surrogate models fitted to Set A

Surrogate Parameter Value Weight

Pressure ratio # of points 110Minimum of data 1.001Mean of data 1.041Maximum of data 1.093

PRS R2adj 0.863 0.32

RMSE 8.00e�3RMS PRESS error 1.33e�2Max error 2.62e�2Mean absolute error 4.40e�3

KRG sqrt Process variance 1.04e�2 0.41RMS PRESS error 1.02e�2

RBNN RMS PRESS error 1.58e�2 0.27Max error 2.03e�2Mean absolute error 3.33e�3

PWS Max error 1.39e�2Mean absolute error 1.95e�3

We tabulate the weights associated with the surrogates used to constructPRESS-based weighted surrogate (PWS). PRS: Quadratic Polynomialresponse surface, KRG: kriging, RBNN: Radial basis neural networks,RMSE: Root mean square error, RMS PRESS error: Root mean squareof predicted residual sum of squares error.

eling, we conducted a lack-of-fit test (refer to Appendix C)for PRS. A low p-value (�0.017) indicated that the chosenorder of the polynomial was inadequate in the selecteddesign space. Since, the data available at 110 points isinsufficient to estimate 220 coefficients in a cubic polyno-mial, this issue of model inadequacy also reflected lack ofdata.

4.2. Design space refinement

We addressed the issue of model accuracy or data inad-equacy using two parallel approaches. Firstly, we addedmore data in the design space to improve the quality offit. We sampled 330 additional points using Latin hyper-cube sampling such that we had 440 design points to fit acubic polynomial (220 coefficients). We call this datasetas ‘Set B’. Secondly, the low mean value of the responsedata (1.041) at 110 points compared to the baseline design(1.065) indicated that large portion of the current designspace was undesirable due to inferior performance. Hence,it might be appropriate to identify the region where weexpect improvements in the performance of the designs,and construct surrogate models by sampling additionaldesign points in that region (reasonable design spaceapproach [31]). To identify the region of interest, we usedthe surrogate models, constructed with Set A data (110design points), to evaluate response at a large number ofpoints in design space. Specifically, we evaluated responsesat a grid of four Gaussian points in each direction (total49 = 262,144 points). We chose Gaussian points insteadof usual uniform grids because Gaussian points lie insidethe design domain, and are less susceptible to extrapolationerrors than corners of uniform grids that might fall outsidethe convex hull of LHS design points used to construct sur-rogate models. Any point with a predicted performance(due to any surrogate model) of 1.08 units or better(1.5% improvement over the baseline design) was consid-ered to belong to the potential region of interest. This pro-cess identified 29,681 unique points (�11%) in the potentialgood region. We selected 110 points from this data setusing D-optimality criterion. D-optimal designs were gen-erated using Matlab routine ‘candexch’ with a maximumof 100 iterations to maximize D-efficiency [25, pp. 93]. This110 points dataset is called ‘Set C’.

As before, we conducted simulations at data points inthe Sets B and C using PHANTOM. One point in eachset failed to provide an appropriate mesh. The mean, min-imum, and maximum values of the pressure ratio for thetwo datasets are summarized in Table 3. We observed onlyminor differences in the mean pressure ratio of the Set Bcompared to the Set A (Table 2) but the responses in theSet C data clearly indicated high potential of improvement.This demonstrates the effectiveness of reasonable designspace approach used to identify the region of interest.

We approximated the data in the Set B and the Set Cusing four surrogates. We employed a reduced cubic, anda reduced quadratic polynomial for PRS approximation

Table 3Range of data, quality indicators in approximation of pressure ratio fordifferent surrogate models, and weights associated with the components ofPWS

Surrogate Parameter Set B Set C

Value Weights Value Weights

# of points 439 109Minimum of data 1.000 1.052Mean of data 1.040 1.075Maximum of data 1.097 1.105

PRS Order of thepolynomial

3 0.44 2 0.36

R2adj 0.959 0.978

RMSE 4.07e�3 1.65e�3RMS PRESS error 4.84e�3 8.74e�3Max error 1.02e�2 3.84e�3Mean absoluteerror

2.67e�3 1.05e�3

KRG sqrt Processvariance

1.05e–2 0.37 1.03e–2 0.34

RMS PRESS error 5.89e�3 9.47e�3

RBNN RMS PRESS error 1.17e�2 0.19 1.07e�2 0.30Max error 1.53e�2 2.57e�2Mean absoluteerror

1.47e�3 2.89e�3

PWS Max error 6.85e�3 7.91e�3Mean absoluteerror

1.30e�3 1.04e�3

PRS: polynomial response surface, KRG: kriging, RBNN: radial basisneural networks, PWS: PRESS-based weighted surrogate, RMSE: rootmean square error, RMS PRESS error: root mean square of predictedresidual sum of squares error, we used a reduced cubic and a reducedquadratic polynomial to approximate the Set B and Set C data,respectively.


of the Set B and the Set C data, respectively. As can be seenfrom different error measures in Table 3, the quality of sur-rogate models fitted to the Set B and the Set C data was sig-nificantly better than the surrogate models fitted to the SetA data. This improvement in surrogate approximation wasattributed to the increase in the sampling density (Set B)allowing a cubic model and the reduction of the designspace (Set C). Both PRS models did not fail the lack-of-fit test (p-value �0.90+) indicating the adequacy of fittingsurrogate model in the respective design spaces. TheRMS PRESS error metric and weights associated with dif-ferent surrogates suggested that the PRS was the best sur-rogate model for the Set B and the Set C data, unlikekriging for the Set A data.

We used the surrogate models fit to the Set B data forglobal sensitivity analysis and the surrogate models fittedto the Set C data for optimization. The optimization ofthe performance using the surrogate models fitted to theSet B data was inferior to the optimal design obtainedusing the Set C data based surrogates. Some of the optimaldesigns from the Set B data based surrogates could not beanalyzed. This anomaly arose because large design spacewas sampled with limited data such that large regionsremained unsampled; and hence, susceptible to significant

errors, particularly near the corners where optima werefound. The same issue restricted the use of surrogate mod-els constructed using Set C data for global sensitivity anal-ysis as there is large extrapolation error outside the regionof interest where no point was sampled. Hence, surrogatemodels constructed using Set B data were more suited forconducting global sensitivity analysis.

4.3. Global sensitivity assessment

We used global sensitivity analysis (GSA) to identify themost and the least important design variables. We usedGauss quadrature numerical integration scheme with fourGauss points along each direction (total 49 = 262,144points) to evaluate different integrals in GSA. The responseat each point was evaluated using surrogate models fit tothe Set B data. Corresponding sensitivity indices of maineffects of different variables are shown in Fig. 8. Thoughthere were differences in the exact magnitude of sensitivityindices from various surrogates, all surrogates indicatedthat the pressure ratio was most influenced by three vari-ables, Pz, t2, and Py. A comparison of sensitivity indicesof total and main effect of design variables (using PWS)in Fig. 9 suggested that the interactions between variableswere small but non-trivial.

4.4. Validation of global sensitivity analysis results

To validate the findings of the global sensitivity analysis,we evaluated the variation in the response function (pres-sure ratio) by varying one variable at a time while keepingremaining variables at the mean values. We specified fiveequi-spaced levels for each design variable and used trape-zoidal rule to compute the actual variance. The responsesat design points were evaluated by performing actualnumerical simulations. The results of actual variance com-putations are shown in Fig. 10.

The one-dimensional variance computation results alsoindicated that variables Pz, t2, and Py were more importantthan all other variables. This validated the findings of theglobal sensitivity analysis. The differences in the results ofone-dimensional variance computation and global sensitiv-ity analysis can be explained as follows: (i) the number ofpoints used to compute one-dimensional variance is small,(ii) one-dimensional variance computation does notaccount for interactions between variables, and (iii) thereare approximation errors in using surrogate models for glo-bal sensitivity analysis. Nevertheless, the main implicationsof the finding clearly show that the performance of the dif-fuser vane was pre-dominantly affected by the location ofpoint P and the length of tangent t2 in Section 1 (Fig. 5).

4.5. Preliminary optimization of diffuser vane performance

Next, we used surrogate models, fitted to the Set C data,to maximize the pressure ratio by exploring different diffuservane shapes. To avoid the danger of large extrapolation

t30.5%

t40.1% t5

0.1%

t211.7%

t60.1%

Py

6.8%Pz

79.8%

0.4%

t10.4%

t10.2%

0.8%

Pz

79.1%

Py

6.0%

t60.0%

t213.1%

t5

0.0%

t4

0.0%

t30.9%

(a) Polynomial response surface (b) Kriging

t31.5% t4

0.9%

t51.0%

t212.9%

t61.3%

Py

7.5%

Pz

72.6%

1.3%

t11.0%

t10.3%

0.6%

Pz

79.2%

Py

6.5%

t60.1%

t212.5%

t5

0.1%

t40.1%

t30.6%

(c) Radial basis neural network (d) PRESS-based weighted average

θ θ

θθ

Fig. 8. Sensitivity indices of main effect using various surrogate models (Set B).

0.00 0.20 0.40 0.60 0.80 1.00

t1

t2

t3

t4

t5

t6

Py

Pz

Var

iabl

es

Sensitivity index

Total effect

Main effect

Fig. 9. Sensitivity indices of main and total effects of different variablesusing PWS (Set B).

Pz

63.2%

t5

0.0%

t3

0.8%

t2

20.3%

t6

0.2%

t4

0.4%

0.6%

t1

0.7%

Py

13.7%

Fig. 10. Actual partial variance of different design variables (no interac-tions are considered).


errors in the unsampled region, we employed the surrogatemodel (PRS surrogate) from the Set A as a constraint (allpoints in the feasible region had predicted response greaterthan a threshold value of 1.08). Our use of PRS from theSet A as the constraint was motivated by the simplicity ofthe constraint function, and the fact that PRS contributedto the most number of points in the potential region ofinterest.

We used sequential quadratic programming optimizer tofind the optimal shapes. The optimal configuration of blade

shapes obtained using different surrogate models as func-tion evaluators is shown in Fig. 11, and the correspondingoptimal design variables are given in Table 4. The optimaldesigns obtained from all surrogates were close to eachother in both function and design space. A few minor dif-ferences were observed in relatively insignificant designvariables (refer to the results of global sensitivity analysis).Notably all design variables touched the bounds for PRSand were close to the corner for other surrogate models.The small value of h (near its lower bound) indicated shar-per nose, as was used in the baseline design. Also, most tan-

Fig. 11. Baseline and optimal diffuser vane shape obtained using differentsurrogate models. PRS indicates function evaluator is quadratic polyno-mial response surface, KRG stands for kriging, RBNN is radial basisneural network, and PWS is PRESS-based weighted surrogate model asfunction evaluator.

Table 5Comparison of actual and predicted pressure ratio of optimal designsobtained from multiple surrogate models (Set C)

Design by Actual P-ratio Surrogate-prediction using

PRS KRG RBNN PWS

PRS 1.117 1.120 1.103 1.084 1.103KRG 1.113 1.112 1.111 1.085 1.104RBNN 1.106 1.106 1.104 1.105 1.105PWS 1.114 1.116 1.108 1.102 1.109

Average absolute error 0.0014 0.0060 0.0186 0.0072

Each row shows the results of optimal design using a particular surrogateand different columns show the prediction of different surrogate models ateach optimal design. PRS is quadratic polynomial response surface, KRGis kriging, RBNN is radial basis neural network, and PWS is PRESS-based weighted surrogate.


gents were at their lower bounds that resulted in low curva-ture sections. Near the point P, the tangents were on theirupper limits to facilitate gradual transition in the slope.The optimal vane was thinner in the middle section andwas longer compared to the baseline design (Fig. 11). Thereduction in the thickness reduced the incidence angle ofthe vane and contributed to improvement in diffuser per-formance. The central region of the optimal design wasnon-convex compared to the convex section for the base-line design, which helped in streamlining the flow. Theresulting shape of the optimized vane indicated potentialbenefits of using an airfoil-like shape to improve diffuserperformance.

We simulated all four candidate optimal designs fromdifferent surrogate models to evaluate the improvements.The actual and predicted performances from different sur-rogates are compared in Table 5. We observed that theerror in approximation for different surrogates was compa-rable to their respective RMS PRESS errors. Nevertheless,PRS was the most accurate surrogate model and furnishedthe best performance shape. RBNN was the worst surro-gate model. PRESS-based weighted average surrogatemodel performed significantly better than the worst surro-gate (RBNN). The best predicted diffuser vane yielded sig-nificant improvements in the performance (1.117)compared to the baseline design (1.065). We refer to thisdesign as ‘intermediate optimal’ design.

Table 4Optimal design variables and pressure ratio (P-ratio) obtained using different

Surrogate h t1 t2 t3 t4

PRS 60.00 1.25 0.75 0.05 1.00KRG 63.58 1.25 0.80 0.05 0.81RBNN 64.49 1.32 0.78 0.07 0.94PWS 63.65 1.25 0.75 0.05 1.00

PRS is quadratic polynomial response surface, KRG is kriging, RBNN is rad

It is obvious that the design space refinement in theregion of interest based on multiple surrogates has pay-offsin the improved performance of surrogates and the identi-fication of optimal design. The high confidence in the opti-mal predictions was also derived from the similarperformance of all surrogate models. The results alsoshowed the incentives (protection against the worst design,proper identification of the reasonable design space) ofinvesting a small amount of computational resources inconstructing multiple surrogate models (less than the costof a single simulation) for computationally expensive prob-lems, and then the extra cost of evaluating multiple optima.

We compared time-averaged flow fields from the inter-mediate optimal design (from PRS) with the baselinedesign in Fig. 12. The intermediate optimal design allowedsmoother turning of the flow compared to the baselinedesign, and reduced the losses due to separation of theflow, which were significantly high in the baseline design.Consequently, the pressure at the outlet was higher for thisintermediate optimal design. We also noted the increase inpressure on the vane for the intermediate optimal design.

4.6. Design space refinement via dimensionality reduction

We used first design space refinement by identifying theregion of interest. This helped in identifying the intermedi-ate optimal design. Since most design variables in the opti-mal design were at the boundary of the design space (Table4), further improvements in the performance of the diffusermight be obtained by refining the design space. To reducethe computational expense, we reduce the design space byutilizing the findings of the global sensitivity analysis. We

surrogates constructed using Set C data

t5 t6 Py Pz Predicted P-ratio

1.00 1.00 �2.00 5.85 1.1200.88 1.43 �2.00 5.85 1.1110.54 1.08 �2.04 5.84 1.1050.53 1.09 �2.00 5.85 1.109

ial basis neural network, and PWS is PRESS-based weighted surrogate.

Fig. 12. Comparison of time-averaged flow fields of baseline and intermediate optimal (PRS) designs.

Table 7Range of data, summary of performance indicators in approximation ofpressure ratio, and weights associated with different surrogate models inthe refined design space

Surrogate Parameter Value Weights

# of points 20Minimum of data 1.117Mean of data 1.136Maximum of data 1.151

PRS R2adj 0.956 0.59

RMSE 1.75e�3RMS PRESS error 2.74e�3Max error 2.53e�3Mean absolute error 1.11e�3

KRG sqrt Process variance 1.10e–2 0.25RMS PRESS error 7.14e�3

RBNN RMS PRESS error 1.11e�2 0.16Max error <1.0e�6Mean absolute error <1.0e�6

PWS PRESS 4.21e�3Max error 1.51e�3Mean absolute error 6.62e�4

PRS: (cubic) polynomial response surface, KRG: kriging, RBNN: radialbasis neural networks, PWS: PRESS-based weighted surrogate, RMSE:root mean square error, RMS PRESS error: root mean square of pre-dicted residual sum of squares error.


fixed six relatively insignificant design variables at the opti-mal design (predicted using PRS), and expanded the rangeof the three most important design variables. The modifiedranges of the design variables and the fixed parameters aregiven in Table 6. We selected 20 design points using a com-bination of face-centered central composite design (15points FCCD), and LHS designs (5 points) of experiments.The range of pressure ratio at 20 design points is given inTable 7. Note that all the tested designs in the refineddesign space perform better than the intermediate optimaldesign.

As before, we constructed four surrogate models in therefined design space. The performance metrics, specifiedin Table 7, indicated that all surrogate models approxi-mated the response function very well. The weights associ-ated with different surrogate models, suggested that aquadratic PRS approximation represents the data the best.This result is not unexpected since any smooth function canbe represented by a second-order polynomial if the domainof application is small enough. As before, PWS model wascomparable to the best surrogate model. While PWS is notthe best surrogate for this problem, we note that PWSmodel is significantly better than the worst surrogate(RBNN) and performs at par with the best surrogate forall datasets. So when the constraints on resources limitthe number of simulations, the PWS model offers a safeapproach in constructing surrogate models.

Table 6Modified ranges of design variables, and fixed parameters in refined designspace

Variables Min Max Fixedparameters

Value Fixedparameters

Value

t2 0.60 0.75 h 60� t4 1.00Py �2.00 �1.50 t1 1.25 t5 1.00Pz 5.85 6.00 t3 0.05 t6 1.00

4.7. Final optimization

The design variables for the four optimal designs of thediffuser vane obtained using different surrogate models andcorresponding surrogate-predicted and actual (CFD simu-lation) pressure ratio are listed in Table 8. The error in pre-dictions of surrogate models compared well with thequality indicators and all surrogate models had only minordifferences in the performance. Also, the optimal vaneshapes from different surrogates were similar. In this casealso, polynomial response surface approximation conceded

Table 8Design variables and pressure ratio at the optimal designs predicted by different surrogates

t2 Py Pz Actual P-ratio Surrogate-predictions using

PRS KRG RBNN PWS

PRS 0.60 �1.99 6.00 1.150 1.151 1.151 1.155 1.152KRG 0.60 �1.87 6.00 1.149 1.150 1.152 1.152 1.151RBNN 0.61 �1.97 5.99 1.150 1.150 1.150 1.155 1.151PWS 0.60 �1.96 6.00 1.150 1.151 1.152 1.154 1.152

Average absolute error 7.5e�04 1.50e�03 4.25e�03 1.75e�03

PRS is cubic polynomial response surface, KRG is kriging, RBNN is radial basis neural network, and PWS is PRESS-based weighted surrogate.


the smallest errors in prediction. While the performance ofthe optimal diffuser vane has improved compared to theintermediate optimal design (compare to Table 5), the con-tribution of the optimization process was insignificant. Oneof the data points (t2 = 0.60, Py = �2.00, Pz = 6.00)resulted in a better performance (P-ratio = 1.151) thanthe predicted optimal. This result was not surprisingbecause the optimal design existed at a corner that wasalready sampled leaving little scope for further improve-ment. Nevertheless, the optimizers correctly concentratedon the best region. As expected, the optimized design isthinner and streamlined to further reduce the losses, andto improve the pressure recovery. We analyzed the optimaldiffuser vane shape according to the flow structure and theother considerations as follows.

4.8. Flow structure

The time-averaged pressure contours for the optimalvane shape (best data point) are shown in Fig. 13. Weobserved further reduction in separation losses, andsmoother turning of the flow compared to the intermediateoptimal design obtained before design space refinement(Table 5). Consequently, the pressure rise in the diffuser

Fig. 13. Time-averaged pressure for the final optimal diffuser vane shape.

was higher. The main reasons of this improved perfor-mance can be summarized as follows.

1. The optimal vane shape had a notable curvature in themiddle section on the lower side of the vane (near thepoint P). This curvature decelerated the flow and ledto faster increase in pressure (notice the shift of higherpressure region towards the inlet in Figs. 12 and 13).

2. As noted earlier, the performance of the vane isimproved by correction in the incidence angle achievedby thinning of the vane.

3. Since the diffuser vanes convert the kinetic energy of theflow into the pressure energy, turning of the flow overthe vane plays a significant role on the performance.We showed the variation in the pressure ratio withchange in turning angle of the vane in Fig. 14, andobserved an elevation in pressure ratio with increase inturning angles (higher pressure ratios were obtainedfor intermediate and optimal vane shapes). The flowfrom the lower side of the vane turned more than theupper side of the vane (as can also be seen in Fig. 13),such that the flow from two sides became increasinglyparallel (baseline vane results into a diverging flow) atthe exit from the vane.

4.9. Vane loadings

The shapes and pressure loads on the baseline, interme-diate optimal, and final optimal vanes are shown inFig. 15. We noted the increase in the mean pressure onthe diffuser vane for the optimal design. The pressureloads near the inlet tip, and the pressure loading on thediffuser vane, given by the area bound by the pressureprofile on the two sides, had reduced by optimization.However, the optimized diffuser vane might be susceptibleto high stresses as the optimal design was thinner com-pared to the baseline vane. The intermediate optimaldesign served as a compromise design with relativelyhigher pressure ratio (�5%) compared to the baselinedesign and lower pressure loading on the vanes comparedto the optimal design.

In the future, this problem would be studied by account-ing for manufacturing and structural considerations like,stresses in the diffuser vanes. One can either specify a con-

Turning angle vs. Pressure ratio

1.07

1.08

1.09

1.1

1.11

1.12

1.13

1.14

1.15

1.16

0 5 10 15 20 25Turning angle

Pres

sure

ratio

AverageTurnUpperSideLowerSide

Fig. 14. Turning angle versus pressure ratio. UpperSide: turning angle on the upper side of the vane, LowerSide: turning angle on the lower side of thevane, and AverageTurn: average of turning on the upper and lower sides.

Fig. 15. Different vane shapes and corresponding pressure loadings.

1

1.1

1.2

1.3

1.4

40 60 80 100 120 140 160

% flow rate

Pre

ssu

re r

atio Baseline Optimal

Fig. 16. Comparison of pressure ratio for baseline and optimal vaneshapes for variable flow rate flows.


straint to limit stress to be less than the feasible value oralternatively, one can solve a multi-objective optimizationproblem with two competing objectives, minimization of

stress or pressure loading in the vane, and maximizationof pressure ratio.

4.10. Performance at off-design conditions

To further probe the utility of optimal vane shape, wecompared the baseline and optimal vane shapes for differ-ent mass flow rates. The resulting pressure ratio for twodesigns is shown in Fig. 16. We observed that the optimalvane shape consistently performed better than the baselinedesign. The improvement in performance increased as theflow rate increased. The time-averaged flow structures forthe baseline and the optimal vane shapes for varying massflow rates, (a few samples) shown in Fig. 17, revealed thatthe baseline design yielded noticeably high flow separationcompared to the optimal design. The extent of flow separa-tion increased with increase in mass flow rate resulting intoa significant improvement in the performance of optimalvane.

Fig. 17. Time-averaged flow structures of baseline and optimal vane shapes for variable mass flow rates.


4.11. Empirical considerations

Typically, the vane shape design is carried out usingempirical considerations on the gaps between adjacent dif-

fuser vanes as shown in Fig. 18. The empirical suggestionson the ratio of different gaps [32] and actual valuesobtained for different vanes are given in Table 9. Contraryto the empirical relations, the ratio of length to width gap

Fig. 18. Gap between adjacent diffuser vanes.

Table 9Actual and empirical ratios of gaps between adjacent diffuser vanes

W1 W2 L L/W1 W2/W1 P-ratio

Empirical relations 4.00 1.60Baseline vane 0.66 0.94 2.09 3.15 1.42 1.065Intermediate optimal 0.79 1.07 2.03 2.55 1.35 1.117Optimal vane 0.94 1.11 2.02 2.15 1.19 1.150


(L/W1), and ratio of width gaps (W2/W1) decreased as thepressure ratio increases, though the actual magnitude oflength and width of the gaps increase. The discrepanciesbetween the optimal design and the empirical optimalratios [32] are explained by multiple design considerationsused for the empirical optimal. Firstly, we note that theoptimization was carried out for a 2D diffuser vane notfor the combination of vanes and flow for which empiricalratios are provided. This allowed a variable height of thevane for optimization. However, the empirical ratios areobtained by assuming a constant channel height so thatthe area ratio of the channel reduces to the ratio of widthgaps (W2/W1). Nevertheless, this requires further investiga-tion to understand the cause of discrepancies betweenempirical ratios and that obtained for optimal design.

5. Summary

We used surrogate model based optimization strategy tomaximize the hydrodynamic performance of a diffuser,characterized by the increase in pressure ratio, by modify-ing the shape of the vanes. The shape of the diffuser vaneswas defined by a combination of Bezier curves and a circu-lar arc. Firstly, we defined the shape of the vane using ninedesign variables and used surrogate models to represent thepressure ratio. We used lack-of-fit test to identify the issuesof model inadequacy, and insufficiency of the data to rep-resent the pressure ratio. We addressed these issues by, (i)adding more data points, and (ii) identifying the region

of interest using the less-accurate surrogate models. Moresamples were added in the region of interest using the infor-mation from multiple surrogate models.

The surrogate models, constructed with increased dataand/or in smaller design space, were significantly more accu-rate than the initial surrogate models. Also, during thecourse of design space refinement, the best surrogate modelchanged from kriging (initial data) to polynomial responsesurface approximation (all subsequent results). Had we fol-lowed the conventional approach of identifying the best sur-rogate model with the first design of experiments, and thenusing that surrogate model for optimization, we might havenot captured the best design. Thus, we can say that the resultsreflect the improvements in the performance using the designspace refinement approach, and using multiple surrogatesconstructed by incurring a low computational cost.

We conducted a surrogate model based sensitivity anal-ysis to identify the most important design variables in theentire design space. Three design variables controlling theshape of the upper and lower side of the vane were foundto be most influential. We used surrogate model in thereduced design space to identify the optimal design in ninevariable design space. This intermediate optimal designimproved the pressure ratio by more than 5% comparedto the baseline design.

Since all the design variables for intermediate optimaldesign hit the bounds, we further refined the design spaceby fixing the least important variables on optimal valuesto reduce the design space, and relaxing the bounds onthe most important design variables. The optimal designobtained using the surrogate models in the refined designspace further improved the performance of the diffuser bymore than 8% compared to the baseline design. The pres-sure losses in the flow were reduced, and a more uniformpressure increase on the vane was obtained. However, theoptimal vane shape might be susceptible to fail due to highstresses. This behavior was attributed to the absence ofstress constraint that allowed using thinner vanes to maxi-mize the performance. The performance of the optimaldesign was also found to be better than the baseline designin off-design flow conditions. In future, the optimizationwould be carried out by considering the multi-disciplinaryanalysis accounting for stress constraint, manufacturabili-ty, and pressure increase. We also intend to study the per-formance of diffuser vanes coupled with differentsubsystems, and operating conditions.

In terms of the vane shapes, thin vanes helped improvethe hydrodynamic performance significantly. The interest-ing aspect was the change in the sign of curvature of thevane on the suction side that allows an initial speeding ofthe flow followed by a continuous pressure recovery with-out flow separation.

Acknowledgements

The present efforts have been supported by the Institutefor Future Space Transport, under the NASA Constella-


tion University Institute Program (CUIP), Ms. ClaudiaMeyer program monitor. This material is based upon worksupported by the National Science Foundation underGrant No. 0423280.

Appendix A. Bezier curve

A typical parametric Bezier curve f(x), shown in Fig. 19,is defined with the help of two end points P0 and P3, andtwo control points P1 and P2 as follows.

f ðxÞ ¼ P 0ð1� xÞ3 þ 3P 1xð1� xÞ2 þ 3P 2x2ð1� xÞþ P 3x3; x 2 ½0; 1�: ð21Þ

The co-ordinate of any point on the Bezier curve isobtained by substituting the value of x and appropriateco-ordinate of points, accordingly. In this study, the loca-tion of control points is obtained by using the informationabout the slope and the length of the tangents at end points[4].

Appendix B. Generalized mean square cross-validation error

(GMSE or PRESS)

In general, the data is divided into k subsets (k-foldcross-validation) of approximately equal size. A surro-gate model is constructed k times, each time leavingout one of the subsets from training, and using the omit-ted subset to compute the error measure of interest. Thegeneralization error estimate is computed using the k

error measures obtained (e.g., average). If k equals thesample size, this approach is called leave-one-out cross-validation (also known as PRESS in the polynomialresponse surface approximation terminology). Eq. (22)represents a leave-one-out calculation when the general-ization error is described by the mean square error(GMSE).

GMSE ¼ 1

k

Xk

i¼1

ðyi � yð�iÞi Þ

2; ð22Þ

where yð�iÞi represents the prediction at x(i) using the sur-

rogate constructed using all sample points except (x(i),yi).Analytical expressions are available for that case for theGMSE without actually performing the repeated con-struction of the surrogates for both polynomial responsesurface approximation [25, Section 2.7] and kriging [30]

Fig. 19. Parametric Bezier curve.

however here we used brute-force. The advantage ofcross-validation is that it provides nearly unbiased esti-mate of the generalization error and the correspondingvariance is reduced (when compared to split-sample) con-sidering that every point gets to be in a test set once, andin a training set k � 1 times (regardless of how the datais divided).

Appendix C. Lack-of-fit test with non-replicate data for

polynomial response surface approximation

A standard lack-of-fit test is a statistical tool to deter-mine the influence of bias error (order of polynomial) onthe predictions [25]. This test compares the estimated mag-nitudes of the error variance and the residuals unaccountedfor by the fitted model. Lets say, we have M unique loca-tions of the data and at jth location, we repeat the experi-ment nj times, such that total number of points used toconstruct surrogate model is Ns ¼

PMj¼1nj. The sum of

squares due to pure error is given by

SSpe ¼XM

j¼1

Xnj

k¼1

ðyjk � �yjÞ2; ð23Þ

where �yj is the mean response at the jth sample location,given as, �yj ¼ 1

nj

Pnj

k¼1yk.The sum of square of residuals due to lack-of-fit of the

polynomial response surface model is,

SSlof ¼XM

j¼1

njðyðxjÞ � �yjÞ2; ð24Þ

where yðxjÞ is the predicted response at the sampled loca-tion xj. In matrix form, the above expressions are given as

SSpe¼XM

j¼1

yTnj

Inj �1

nj1nj 1

Tnj

� �ynj; ð25Þ

SSlof ¼ yTðINs �X ðX TX Þ�1X TÞy�XM

j¼1

yTnj

Inj �1

nj1nj 1

Tnj

� �ynj;

ð26Þ

where 1nj is the (nj · 1) vector of ones, Inj is (nj · nj) identitymatrix, INs is (Ns · Ns) identity matrix.

We formulate F-ratio using the two residual sum ofsquares as,

F ¼ ðSSlof=d lofÞ=ðSSpe=dpeÞ; ð27Þ

where dlof = Ns � Nb, and dpe = Ns �M, are the degrees offreedom associated with SSlof and SSpe, respectively. Thelack-of-fit in the surrogate model is detected with a-levelof significance, if the value of F in Eq. (27), exceeds the tab-ulated F a;d lof ;dpe value, where the latter quantity is the upper100a percentile of the central F-distribution.

When the data is obtained from the numerical simula-tions, the replication of simulations does not provide anestimate of noise (SSpe), since all replications returnexactly the same value. In such scenario, the variance of


noise can be estimated by treating the observations atneighboring designs as ‘‘near’’ replicates [33, pp. 123].We adopt the method proposed by Neill and Johnson[34], and Papila [35] to estimate the lack-of-fit for non-replicate simulation. In this method, we denote a near-replicate design point xjk (as the kth replicate of the jthpoint �xjÞ such that

xjk ¼ �xj þ djk; ð28Þwhere djk represents the disturbance vector. Then, the Gra-mian matrix is written as

X ¼ X þ D; ð29Þwhere X matrix is constructed using �xj for near replicatepoints and matrix D ¼ X � X . Now the estimated responseat the design points (including near-replicates) is given asb�y ¼ y� Db; ð30Þwhere b is given in Eq. (3). Now, we compute SSpe andSSlof by replacing y, ynj

, and X in Eqs. (25) and (26) withb�y, �ynj , and X .

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