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A comparative study of kriging variants for the optimization of a turbomachinery system

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Abstract

Kriging is a well-established approximation technique for deterministic computer experiments. There are several Kriging variants and a comparative study is warranted to evaluate the different performance characteristics of the Kriging models in the computational fluid dynamics area, specifically in turbomachinery design where the most complex flow situations can be observed. Sufficiently accurate flow simulations can take a long time to converge. Hence, this type of simulation can benefit hugely from the computational cheap Kriging models to reduce the computational burden. The Kriging variants such as ordinary Kriging, universal Kriging and blind Kriging along with the commonly used response surface approximation (RSA) model were used to optimize the performance of a centrifugal impeller using CFD analysis. A Reynolds-averaged Navier–Stokes equation solver was utilized to compute the objective function responses. The responses along with the design variables were used to construct the Kriging variants and RSA functions. A hybrid genetic algorithm was used to find the optimal point in the design space. It was found that the best optimal design was produced by blind Kriging, while the RSA identified the worst optimal design. By changing the shape of the impeller, a reduction in inlet recirculation was observed, which resulted into an increase in efficiency.

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Abbreviations

BKR:

Blind kriging

BLUP:

Best linear unbiased predictor

CFD:

Computational fluid dynamics

CKR:

co-Kriging

CVE:

Cross validation error

DOE:

Design of experiments

GA:

Genetic algorithm

KR:

Kriging

LSPR:

Least square polynomial regression

MLE:

Maximum likelihood estimation

MSE:

Mean square error

OKR:

Ordinary kriging

Opt:

Optimal

PR2:

Polynomial regression with degree 2

PS:

Pressure side

RANS:

Reynolds-averaged Navier–Stokes

RMSE:

Root mean square error

RSA:

Response surface approximation

SKR:

Simple kriging

SQP:

Sequential quadratic programming

SRF:

Spatial random field

SS:

Suction side

SST:

Shear stress transport

UKR:

Universal kriging

b :

Blade width, mm

c :

Absolute fluid flow velocity, m/s

D :

Diameter, mm

e :

Error

F :

Objective function

f(x):

Polynomial function

G :

Number of blades

g :

Acceleration due to gravity, m/s2

H :

Head generated, m

ΔH :

Hydraulic losses, m

i, j :

Integer value 1, 2, 3,

k :

Turbulence kinetic energy, J

N :

Impeller speed, rpm

n s :

Number of sampled data points

n dv :

Number of design variables

P :

Power consumed by pump, kW

Q :

Volume flow rate, m3/s

R :

Correlation matrix

R’ :

Elements of spatial correlation matrix

Re :

Reynolds number

Ref:

Reference

r(x):

Correlation between n s and f(x)

t :

Blade thickness, mm

u :

Peripheral velocity, m/s

w :

Relative velocity, m/s

Z(x):

Localized deviation

α :

Co-efficient of regression function

β :

Blade angle, °

γ:

Flow angle, °

ε :

Rate of kinetic energy dissipation, J/s

η :

Hydraulic efficiency,  %

θ :

Correlation parameter

μ :

Unknown constant regression function

ρ :

Density of fluid, kg/m3

σ 2 :

Process variance

ω :

Turbulence frequency, Hz

1:

Inlet

2:

Outlet

a:

Actual

design:

Design

h:

Hydraulic

krg:

Kriging

m:

Meridional component

opt:

Optimal

CFD:

Computational fluid dynamics

ref:

Reference

sp:

Specific

T:

Total

th:

Theoretical

U:

Peripheral component

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Acknowledgments

A. Samad would like to acknowledge Indian Institute of Technology Madras for an NFSC grant (Grant code: OEC/10-11/529/NFSC/ABDU) and Inha University for conducting this research. Also IC acknowledges the support of the Department of Information Technology (INTEC), Ghent University-iMinds, Ghent, Belgium for conducting this research. Ivo Couckuyt is a post-doctoral research fellow of FWO-Vlaanderen. The research has (partially) been funded by the Interuniversity Attraction Poles Program BESTCOM initiated by the Belgian Science Policy Office.

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Authors and Affiliations

  1. Department of Ocean Engineering, IIT Madras, Chennai, 600036, India

    Sayed Ahmed Imran Bellary & Abdus Samad

  2. Department of Information Technology (INTEC), Ghent University-iMinds, Ghent, Belgium

    Ivo Couckuyt & Tom Dhaene

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  1. Sayed Ahmed Imran Bellary
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  2. Abdus Samad
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  3. Ivo Couckuyt
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  4. Tom Dhaene
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Correspondence to Abdus Samad.

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Bellary, S.A.I., Samad, A., Couckuyt, I. et al. A comparative study of kriging variants for the optimization of a turbomachinery system. Engineering with Computers 32, 49–59 (2016). https://doi.org/10.1007/s00366-015-0398-x

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