the use of artificial neural networks for the optimisation of...

von Karman Institute for Fluid Dynamics

The Use of Artificial Neural Networks for the Optimisation of

Turbomachinery Components

R.A. Van den Braembusschevon Karman Institute for Fluid Dynamics

Ercoftac Introductory course on “ Design Optimisation”, Garching, 1-3 April 2003


Main problems

⇒ Geometry definition

⇒Convergence speed

real optimum

⇒ Target specification

Optimisation

MinimiseF=F(U,Xi) i=1,ND

Xi=ND geometrical parameters

Aerodynamic constraintsRk(U,Xi)=0. k=1, Neq*Npt

Geometric constraintsGj(Xi) ≤ 0 j=1, Ncond


Parameterisationreduced number of unknown

18 geometric parameters to be defined

• continuous curvature

• geometrical constraints

Geometry definition


Allow all possible Blade Geometries !

Geometry definition


Gradient Method

•Define search direction

n+1 calculations

•Perform 1D search

1 calculation

•Respect constraints

•Verify convergence

m steps

n parameters * m iterations ⇒ (n+1)*m N.S. calculations

Convergence


Systematic Search

covering design space

n parameters * 3 values ⇒ 3n N.S. Solutions 38=6561

Convergence

x

x x

x

x x

x

x

x


Convergence

Initial blade selectionInitial blade selectionInitial blade selectionInitial blade selection

Each new geometry is analysed by Navier Stokes

Genetic Algorithms

Use information on previous designs to define new ones


Convergence

Gene design variable digitChromosome design variable Individual blade shapePopulation set of bladesFitness performance of an

individual.

coding

Genetic algorithm


Convergence

Child abcDEF

•Tournament selection

•Uniform crossover (p=0.5)

•Jump Mutation

Parent #1

abcdef

Child abcDEM

Parent #2

ABCDEF

(*) D. Carroll http://www.cuaerospace.com/carroll/ga.html

Genetic algorithm


Convergence

•fast but lower accuracy

⇒ ANN + GA

•slow but high accuracy

⇒ NS

Genetic algorithm

Two level optimisation


Artificial Neural Network (*)

learning: definitions of coefficients Wi(n),bi(n)

predict: performance predictioninput output

(*) http://www-ra.informatik.uni-tuebingen.de/SNNS

Network structure

�

=+=

n

jibipjiWFTia

11111 ))()(),(()(


Artificial Neural Network

Non-dimensionalisation -1 < output < 1

)(11

1)(

xexFT −+

=

-1

11FT

x

∑=

+=n

jibipjiWFTia

11111 ))()(),(()(

model free relation

sigmoid function


Artificial Neural NetworkLearning process

Training by back propagation of errors

≈≈≈≈ function minimalisation by gradient technique (objective = error)

Learning time >< Navier Stokes

1−∆+∂∂=∆ ii WWE

W αγ

∑=

+=n

jibipjiWFTia

11111 ))()(),(()(



local minimum



merit function ≈≈≈≈ objective function m(x) = f(x) -ρm dm(x)


Accuracy:

Noise in data (error in the predictions – measurements)

* same Navier Stokes solver

* on similar grids•Learning process (number of training samples)

(number of validation samples)

•Network structure (number of hidden layers)(number of nodes)

• Database (covering design space)

• Self-learning




training + validation error


Accuracy:







• Self-learning



Σ

Σ

Σ

FT2

FT2

FT2

W2(k)

W2(1)

b2(1)

b2(2)

b2(k)FT1

FT1

FT1

FT1Σ FT3

W3(k)

W3(1)

b3(1)

Σ FT3

b3(2)

Σ FT3

b3(m)

Σ FT3

b3(3)

Σ FT3

b3(m-1)

Po1

ß1

Xn

X1

η

ß2

Mm

M1

Mm-1

Σ

b1(n)W1(n)

W1(1)

Σ

b1(3)

Σ

b1(2)

Σ

b1(1)


one hidden layer - number of nodes ?


Artificial Neural NetworkNetwork structure

Critical number of hidden nodes:

More hidden nodes less learning error

more prediction error

1*

++−=

outin

outouttrainh nn

nnnn


Accuracy:







• Self-learning



Database

Geometry:Blade definition by Bezier Curves

18 parameters

Aerodynamic conditionsInlet and outlet flow conditions

Po1, To1,

�

1, P2

PerformanceEfficiency �

Outlet flow angle

�

Mach number distribution

(40 points)

Mis

s

input & output of Navier Stokes calculation


Test function : 6 parameters 6 parameters

F=1-0,001(A-D)3+0,002(C+E)(F-B)-0,06(A-F)2+(F+C)(E+A)

1<A<5 2<B<3 4<C<5 3<D<4 2<E<3 2<F<6

Full factorial (2 values) 26 =64 samples

Half factorial 25=32 samples

1/4 factorial 24=16 samples

DatabaseDOE


A B C D E F1 1 2 4 3 2 2 19.582 5 2 4 3 2 2 47.583 1 3 4 3 2 2 19.574 5 3 4 3 2 2 47.575 1 2 5 3 2 2 22.586 5 2 5 3 2 2 54.587 1 3 5 3 2 2 22.578 5 3 5 3 2 2 54.579 1 2 4 4 2 2 20.19

10 5 2 4 4 2 2 49.7511 1 3 4 4 2 2 20.1812 5 3 4 4 2 2 49.7413 1 2 5 4 2 2 23.1914 5 2 5 4 2 2 56.7515 1 3 5 4 2 2 23.1816 5 3 5 4 2 2 56.7417 1 2 4 3 3 2 25.5818 5 2 4 3 3 2 53.5819 1 3 4 3 3 2 25.5720 5 3 4 3 3 2 53.5721 1 2 5 3 3 2 29.5822 5 2 5 3 3 2 61.5823 1 3 5 3 3 2 29.5624 5 3 5 3 3 2 61.5625 1 2 4 4 3 2 26.1926 5 2 4 4 3 2 55.7527 1 3 4 4 3 2 26.1828 5 3 4 4 3 2 55.7429 1 2 5 4 3 2 30.1930 5 2 5 4 3 2 63.7531 1 3 5 4 3 2 30.1732 5 3 5 4 3 2 63.7333 1 2 4 3 2 6 30.19

Sample Parmeters Response

Full factorial64 calculations

Low influence of variable C

Large influence of variable A

DatabaseDOE


A B C AB AC BC ABC

a + - - - - + +b - + - - + - +c - - + + - - +adc + + + + + + +ab + + - + - - -ac + - + - + - -bc - + + - - + -(1) - - - + + + -

Factorial effectTreatment Combination

)(5,0 abccbalA +−−= )(5,0 abccbalBC +−−=

)(5,0 abccbalB +−+−= )(5,0 abccbalAC +−+−=

)(5,0 abccbalC ++−−= )(5,0 abccbalAB ++−−=

ABC = + C = C*ABC = ACABC = + C = C*ABC = AC22B = ABB = AB

DatabaseDOE


DatabaseDOE

effecteffect

p ro b

a bi li

ty

p ro b

a bi li

ty

64 samples 32 samples


DatabaseDOE

16 samples 8 samples

effect effect

prob

abili

ty

prob

a bili

ty


ANN's global error for diffrent number of training samples

104 105110

144

42

64

19

9

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

8 8.1 rand 01 8.2 rand 01 8.3 rand 01 16 16 rand 01 32 64

Num ber of samples

Glo

bal e

rror

[%]

8 8.1 rand 01 8.2 rand 01 8.3 rand 01 16 16 rand 01 32 64

∑=

��

�

��

��

��

⋅=samplesn

i

samplesn_

1

_:100eExact_valu

valuePredicted_ -eExact_valuyDescrepanc

DatabaseDOE



35 sample Database

von Karman Institute for Fluid Dynamicsbased on 25 samples



Objective function

OF2D = Pgeom + Pmeca + PAeroBC + Pmach + P Manuf + P cost

β2

ξ

...

Ablade

Imin

Imax

α

Rle

Rte

RLE

β1blade

Buri

dsdM

sdMd2

2ds

dRC

used for ANN or NS

m&


Turbine blade design

Table 1. Imposed parameters, mechanical and aerodynamic requirements

β1flow(ο) 18.0 Imposed After

M2is 0.9 Min. Max. 18 modif.

Re 5.8 10.5 surface 5.2 10.-4 6.8 10.-4 5.36 10.-4

γ=Cp/Cv 1.4 Imin(m4) 7.5 10.-9 1.2 10.-8 7.45 10.-9

Τu(%) 4 Imax(m4) 1.25 10.-7 2.2 10.-7 1.28 10.-7

Cax(m) 0.052 αImax -50.00 -30.00 -37.50

Pitch/Cax 1.0393 β2flow(ο) -57.80 57.80 -57.62

TE thick (m) 1.2 10.-3 loss coef.(%) 0.0 0.0 1.9

Requirements


Evolution of Mach number and Geometry

Design ConvergenceTurbine blade design


Design Convergence Turbine blade design

self learning process


Quasi 3D Design


Geometry definition

Full 3D Radial impeller design

variable

dependent variable


Design space limitation




01

22

33

.

..)(

βββββ

+++=

u

uuu

Blade angle variation


Geometry definition

Geometry definition

Rtgdm

d m )(. βθ =

mβ




Cost function evolution

von Karman Institute for Fluid DynamicsMach number evolution

shroud hub


von Karman Institute for Fluid DynamicsGeometry evolution

meridional blade to blade



Concurrent designRequirements

Optimiser =

Geometry generator

+

Search Vehicle

ANN

Performance Analysis

Result OK

Navier Stokes solver

FEA Stress

analysis

Database

ANN

Stress Analysis

Database

no

yes


Conclusions•ANN accuracy depends on quality of Database

•number and quality of “independent” samples

•architecture of ANN

•learning process

•Efficient design system

•two level optimisation

•self learning

•fully automated

•Realistic designs (geometry model)

•Accurate: based on Navier Stokes solver

•Easy off-design analysis

•2D and 3D compressors and turbines, axial and radial

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