von karman institute for fluid dynamics rto, avt 167, october, 2009 1 r.a. van den braembussche von...
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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
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 2
Improving Convergence of GA
Performance
Database
Geometry
GA
NSNavier-Stokes
Metafunction
NS, HT, FEA
Predict
Learn
Requirements
Start
FEAStress
analysis
HTHeat
transfer
Parallel computing
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 3
Improving Convergence of GA
Performance
Database
Geometry
GA
NSNavier-Stokes
Metafunction
NS, HT, FEA
Predict
Learn
Requirements
Start
FEAStress
analysis
HTHeat
transfer
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 4
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
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 5
%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
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 6
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
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 7
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
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 8
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
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 9
Uniform crossover
Swap with probability pc
Genetic AlgorithmCross over00000 11111
00011
# of function evaluations
Single crossover
One random swap / individual
00000 11111
01101
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 10
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)
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 11
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
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 12
Genetic AlgorithmOptimal number of children
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 13
OptimizationConvergence
Performance
Database
Geometry
GA
NSNavier-Stokes
Metafunction
NS, HT, FEA
Predict
Learn
Requirements
Start
FEAStress
analysis
HTHeat
transfer
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 14
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
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 15
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
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 16
Databasemerit function
merit function objective function m(x) = f(x) -m dm(x)
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 17
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
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 18
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
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 19
Statistical analysis
2k factorial =64 2 k-2 factorial =16 ))(()(06,0))((002,0)(001,01 23 AECFFABFECDAR k=6
Database
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 20
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
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 21
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.
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 22
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
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 23
MetafunctionRBF
Learning : define W (weight) and b (bias)
Gausian activation function
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 24
Multi-objective optimizationConvergence
von Karman Institute for Fluid DynamicsRTO, AVT 167, October, 2009 25
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