Skip to main content

Advertisement

SpringerLink
Log in

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

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

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

References

  1. Robinson TD (2007) Surrogate based optimization using multifidelity models with variable parameterization, PhD Thesis, Massachusetts Institute of Technology, USA

  2. Samad A (2008) Numerical optimization of turbomachinery blade using surrogate models, PhD Thesis, Inha University, Republic of Korea

  3. Houlin L, Yong W, Shouqi Y, Minggao T, Kai W (2010) Effects of blade number on characteristics of centrifugal pumps. Chin J Mech Eng 23(6):742–747

    Article  Google Scholar 

  4. Chakraborty S, Pandey KM (2011) Numerical studies on effects of blade number variations on performance of centrifugal pumps at 4000 RPM. IACSIT Int J Eng Technol 3(4):410–416

    Article  Google Scholar 

  5. Kamimoto G, Matsuoka Y (1956) On the flow in the impeller of centrifugal type hydraulic machinery (The 2nd report). Trans JSME Series 22(113):55–59

    Article  Google Scholar 

  6. Varley FA (1961) Effects of impeller design and surface roughness on the performance of centrifugal pumps. Proc Inst Mech Eng 175(21):955–969

    Article  Google Scholar 

  7. Li WG (2002) The influence of number of blades on the performance of centrifugal oil pumps. World Pumps 427:32–35

    Google Scholar 

  8. Li WG (2008) Numerical study on behavior of a centrifugal pump when delivering viscous oils—part 1: performance. Int J Turbo Jet Engines 25:61–79

    Google Scholar 

  9. Luo X, Zhang Y, Peng J, Xu H, Yu W (2008) Impeller inlet geometry effect on performance improvement for centrifugal pumps. J Mech Sci Technol 22:1971–1976

    Article  Google Scholar 

  10. Sanda B, Daniela CV (2012) The influence of the inlet angle over the radial impeller geometry design approach with Ansys. J Eng Stud Res 18(4):32–39

    Google Scholar 

  11. Ohta H, Aoki K (1996) Effect of impeller angle on performance and internal flow of centrifugal pump for high viscosity liquids. Proc Sch Eng Tokai Univ 36(1):159–168

    Google Scholar 

  12. Samad A, Kim KY, Goel T, Haftka RT, Shyy W (2008) Multiple surrogate modeling for axial compressor blade shape optimization. J Propuls Power 24(2):302–310

    Article  Google Scholar 

  13. Viana FAC, Simpson TW, Balabanov V, Toropov V (2014) Metamodeling in multidisciplinary design optimization: how far have we really come? AIAA J 52(4):1–21

    Google Scholar 

  14. Peter J, Marcelet M (2008) Comparison of surrogate models for turbomachinery design. WSEAS Trans Fl Mech 3(1):10–17

    Google Scholar 

  15. Jung IS, Jung WH, Baek SH, Kang S (2012) Shape optimization of impeller blades for a bidirectional axial flow pump using polynomial surrogate model. World Acad Sci Eng Technol 66:775–781

    Google Scholar 

  16. Derakhshan S, Pourmahdavi M, Abdolahnejad E, Reihani A, Ojaghi A (2013) Numerical shape optimization of a centrifugal pump impeller using artificial bee colony algorithm. Int J Comput Fluids 81:145–151

    Article  Google Scholar 

  17. Joseph VR, Hung Y, Sudjianto A (2008) Blind kriging: a new method for developing metamodels. J Mech Des Trans ASME 130:1–8

    Article  Google Scholar 

  18. Peter J, Marcelet M (2008) Assessment of surrogate models for the global optimization of turbomachinery flows, WCCM8, 5th ECCOMAS, Venice, 30 June–5 July 2008

  19. Toal D, Keane A, Benito D, Dixon J, Yang J, Price M, Robinson T, Remouchamps A, Kill N (2014) Multifidelity multidisciplinary whole-engine thermomechanical design optimization. J Propuls Power 30(6):1654–1666

    Article  Google Scholar 

  20. Hung Y (2011) Penalized blind kriging in computer experiments. Stat Sinica 21:1171–1190

    Article  MATH  Google Scholar 

  21. Van Beers WCM, Kleijnen JPC (2015) Kriging interpolation in simulation: a survey, proceedings of the 2004 winter simulation conference. In: Ingalls RG, Rossetti MD, Smith JS, Peter BA (eds) Institute of Electrical and Electronics Engineers, Piscataway, pp 113–121

  22. Hung Y (2008) Contributions to computer experiments and binary time series, PhD Thesis, school of mechanical engineering, school of industrial and systems engineering, Georgia Institute of Technology, USA

  23. Lazarkiewicz S, Troskolanski AT (1965) Impeller pumps, 1st edn. Pergamon Press Ltd, Oxford

    Google Scholar 

  24. Stepanoff AJ (1964) Centrifugal and axial flow pumps, 2nd edn. Krieger Publishing Company, Florida

    Google Scholar 

  25. Srinivasan KM (2008) Rotodynamic pumps. New Age International (P) Ltd, New Delhi

    Google Scholar 

  26. Samad A, Kim KY (2008) Shape optimization of an axial compressor blade by multi-objective genetic algorithm. Proc IMechE Part A J Power Energy 222:599–611

    Article  Google Scholar 

  27. Myers RH, Montgomery DC (1995) Response surface methodology-process and product optimization using designed experiments. Wiley, New York

    MATH  Google Scholar 

  28. McKay MD, Beckman RJ, Conover WJ (1979) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21:239–245

    MATH  MathSciNet  Google Scholar 

  29. Sugimura K (2009) Design optimization and knowledge mining for turbomachinery, PhD Thesis, Tohoku University, Japan

  30. Martin JD, Simpson TW (2005) Use of Kriging models to approximate deterministic computer models. AIAA J 43(4):853–863

    Article  Google Scholar 

  31. Stein ML (1999) Statistical interpolation of spatial data: some theory for kriging. Springer, New York

    Book  Google Scholar 

  32. Lophaven SN, Nielsen HB, Sondergaard J (2002) Aspects of the Matlab toolbox DACE. Tech. rep. Informatics and mathematical modeling. Technical University of Denmark, Lyngby, Denmark

    Google Scholar 

  33. Couckuyt I, Forrester A, Gorissen D, Turck FD, Dhaene T (2012) Blind Kriging: implementation and performance analysis. Int J Adv Eng Softw 49:1–13

    Article  Google Scholar 

  34. Cressie N (1993) Statistics of spatial data. Wiley, New York

    Google Scholar 

  35. He D, Wang F, Mao Z (2008) Hybrid genetic algorithm for economic dispatch with valve-point effect. Electr Power Syst Res 78:626–633

    Article  Google Scholar 

  36. Bellary SAI, Husain A, Samad A (2014) An effectiveness of meta-models for multi-objective optimization of centrifugal impeller. J Mech Sci Technol 28(12):4947–4957

    Article  Google Scholar 

  37. Bradshaw P (1996) Turbulence modeling with application to turbomachinery. Prog Aerosp Sci 32(6):575–624

    Article  Google Scholar 

  38. Gulich JF (2010) Centrifugal pumps, 2nd edn. Springer Publications, Berlin

    Book  Google Scholar 

  39. Gupta M, Kumar S, Kumar A (2011) Numerical study of pressure and velocity distribution analysis of centrifugal pump. Int J Therm Technol 1(1):117–121

    MathSciNet  Google Scholar 

  40. Patel K, Ramakrishnan N (2006) CFD analysis of mixed flow pump. Int ANSYS Conf Proc. http://www.ansys.com/staticassets/ANSYS/.../2006-Int-ANSYS-Conf-255.pdf

  41. Zhou W, Zhao Z, Lee TS, Winoto SH (2003) Investigation of flow through centrifugal pump impellers using computational fluid dynamics. Int J Rotating Mach 9(1):49–61

    Article  Google Scholar 

  42. Tuzson J (2000) Centrifugal pump design, 1st edn. Wiley, New York

    Google Scholar 

Download references

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.

Author information

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

Authors
  1. Sayed Ahmed Imran Bellary
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abdus Samad
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ivo Couckuyt
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tom Dhaene
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abdus Samad.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-015-0398-x

Keywords

Search

Navigation