von karman institute for fluid dynamics rto, avt 167, october, 2009 1 r.a. van den braembussche von...

von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 1

R.A. Van den Braembussche

von Karman Institute for Fluid Dynamics

Tuning of Optimization Strategies


Improving Convergence of GA

Performance

Database

Geometry

GA

NSNavier-Stokes

Metafunction

NS, HT, FEA

Predict

Learn

Requirements

Start

FEAStress

analysis

HTHeat

transfer

Parallel computing


Improving Convergence of GA

Performance

Database

Geometry

GA

NSNavier-Stokes

Metafunction

NS, HT, FEA

Predict

Learn

Requirements

Start

FEAStress

analysis

HTHeat

transfer


1. Population size N

2. Substring length l

3. Crossover Probability Pc

4. Mutation Probability Pm

5. Number of children ch

Optimal parameter setting

( to accelerate evolution )

Genetic AlgorithmOptimal parameter setting


%100.minOFOF

OFOFq

REF

GAREF

Genetic Algorithm

Optimal parameter setting

OF defined by test function Tests on 7 and 27

parameter function

GA = non-deterministic

Conclusions based on:5 optimization

Result of given effort

5000 OF evaluations

Six hump camel back test function


Genetic Algorithm(1) Population size# evaluations = 5000 = N * t

t number of generations

N population size

Small populations Premature convergence

Local optimum

Low number of feasible geometries

Large populations Low number of generations

No evolution

10 < N < 20Population size


Genetic Algorithm(2) Substring length

l = # of bits / variable

2l values / variable

ε = desired resolution

min max

2log

i ii

i

x xl

Global minimum

Best possible solution

average

OF

xmin xmax

00 01 10 01


N variables

l bits / variable

L < 3 too low resolution

L > 10 too large design space(slower convergence)

Substring length (# of bits)

Genetic Algorithm (2) Substring length


Uniform crossover

Swap with probability pc

Genetic AlgorithmCross over00000 11111

00011

# of function evaluations

Single crossover

One random swap / individual

00000 11111

01101


pm probability a bit is swapped

Mu

tatio

n p

roba

bili

ty

Total string length ( l .N )

Optimal pm

pm =1/(l.n) pm =2/(l.n)

Genetic AlgorithmMutation00000 11111

00011

01011

mutation

Optimal pm

______ pm = 1/(l.N)_ _ _ _ pm = 2/(l.N)


Genetic AlgorithmNew generation

(n,ch) n best of ch offspring's replace the old population (best individuals can be lost)

(n+ch) n best of (ch offspring's + n old population) replace the old population (elitism)

(n/i+ch) n/i best contribute to new generation

diversity


Genetic AlgorithmOptimal number of children


OptimizationConvergence

Performance

Database

Geometry

GA

NSNavier-Stokes

Metafunction

NS, HT, FEA

Predict

Learn

Requirements

Start

FEAStress

analysis

HTHeat

transfer


MetafunctionType (ANN, RBF) Structure (# hidden l)

RBF 5 hidden neurons

De Jong 2D test function

ANN 2 hidden layers10 hidden neurons

Database # of samplesDistribution of samples


Database

x x

xx

Systematic scanning

2 values /variable

n variables

full factorial 2n evaluations

7 variables 128 NS evaluations

27 variables 107 NS evaluations


Databasemerit function

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


A B C ABC1 + - - +2 - + - +3 - - + +4 + + + +5 + + - -6 + - + -7 - + + -8 - - - -

RunParameters

A B C ABC1 + - - +2 - + - +3 - - + +4 + + + +5 + + - -6 + - + -7 - + + -8 - - - -

RunParameters

Database

Alt: Latin Hypercube – Random selection

3 variables

2 values (+ -) / variable

23 = 8 combinations

1, 2, 3 and 4 : main effect

5, 6, 7 and 8 : interaction

Design Of Experiment

DOE


DatabaseANN's global error for diffrent number of training samples

919

42

104

275

64

105 110

144

0

20

40

60

80

100

120

140

160

180

200

220

240

260

280

64 32 16 8 8 rand_first 16 rand _first 8 rand_second 8 rand_third 8 rand_fourth

Number of samples

64 32 16 8 8 rand_first 16 rand _first 8 rand_second 8 rand_third 8 rand_fourth

))(()(06,0))((002,0)(001,01 23 AECFFABFECDAR

Random

DOE

64 32 16 8 8a 16 8b 8c 8d

6 parameters

Full factorial =

26 = 64

Err

or

in 6

4 p

oin

ts


Statistical analysis

2k factorial =64 2 k-2 factorial =16 ))(()(06,0))((002,0)(001,01 23 AECFFABFECDAR k=6

Database


Statistical analysis

2 k-3 factorial = 8

k=6

2 level variables

(25% and 75% of non dimensional range)

1 central variable

(all variables at 50% of range)

12 to 15 variables 16 runs

16 to 31 variables 32 runs

Database


MetafunctionANN

n

j

ibjpjiWFTia1

1111 ))()(),(()(

)(1

1)(

xexFT

Learning : define W (weight) and b (bias)N

avi

er

Sto

kes

resu

lts

Ge

om

etr

y &

bo

und

. co

nd.


MetafunctionKriging

Linear least square approximation

Predicts value and uncertainty

N

i i

j

k

j jj

xfxwxf

functionGausianxZ

functionsregressiong

xZxgxf

1

1

)().()(~

)(

)()(.)(~

Accurate evaluations in regions of high uncertainty

Very time consuming


MetafunctionRBF

Learning : define W (weight) and b (bias)

Gausian activation function


Multi-objective optimizationConvergence


Genetic AlgorithmGray coding

1 0012 0113 0104 1105 1116 1017 100

Gray coding

Value Code

1 0012 0103 0114 1005 1016 1107 111

Binary coding

Value Code

No real advantage observed

von karman institute for fluid dynamics rto, avt 167, october, 2009 1 r.a. van den braembussche von...

Documents

karman institute

fluid dynamics rto

database slide

x slide

ns evaluations slide

test function slide

population size slide

substring length slide