The Use of Artificial Neural Networks for the Optimisation of...

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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

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

jibipjiWFTia

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

Artificial Neural Network

Non-dimensionalisation -1 < output < 1

xexFT −+

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

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

b2(k)FT1

FT1Σ FT3

Σ FT3

b3(m-1)

b1(n)W1(n)

one hidden layer - number of nodes ?

Artificial Neural NetworkNetwork structure

Critical number of hidden nodes:

More hidden nodes less learning error

more prediction error

++−=

outouttrainh nn

Accuracy:

• Self-learning

Database

Geometry:Blade definition by Bezier Curves

18 parameters

Aerodynamic conditionsInlet and outlet flow conditions

Po1, To1,

PerformanceEfficiency �

Outlet flow angle

Mach number distribution

(40 points)

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

effecteffect

p ro b

a bi li

p ro b

a bi li

64 samples 32 samples

DatabaseDOE

16 samples 8 samples

effect effect

a bili

ANN's global error for diffrent number of training samples

104 105110

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

Num ber of samples

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

��

⋅=samplesn

samplesn_

_: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

Ablade

β1blade

used for ANN or NS

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

βββββ

Blade angle variation

Geometry definition

d m )(. βθ =

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

Performance Analysis

Result OK

Navier Stokes solver

FEA Stress

analysis

Database

Stress Analysis

Database

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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