Design of Experiments - NTUAvelos0.ltt.mech.ntua.gr/ERCOFTAC/symp03/poloni_doe.pdf · – Eliminates redundant observation. ... are examining. ... (Design Of Experiments) Design computed
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1ERCOFTAC
Munich – 1st April, 20031
Dipartimento di Energetica
DOE
D.O.E.Design of Experiments
Carlo Poloni, Valentino Pediroda, Alberto ClarichDipartimento di Energetica
Universita’ di Trieste
Silvia PolesESTECOTrieste
www.esteco.comTU Munich 1st April 2003
2ERCOFTAC
Munich – 1st April, 20032
Dipartimento di Energetica
Why D.O.E.?
! Get the most relevant qualitative information from a data-base of experiments making the smallest possible number of experiments.
! Look for the best data set to build a simplified model
! Look for robust solution that are not influenced by small variation of the design variables.
3ERCOFTAC
Munich – 1st April, 20033
Dipartimento di Energetica
Why D.O.E.?
! Pro:– Reduced number of experiments, more than one variable is changed
in each new experiment.– Eliminates redundant observation.– Reduce the time and the resources to make the experiments.– Give information on the major interactions between the variables.
! Cons:– The response variables-objectives is pre-defined.– Only “simple” relations are detected (often only linear or quadratic).
4ERCOFTAC
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Dipartimento di Energetica
DOE (Design Of Experiments)
The DOE approach should be used to determine thegeneral behaviour of the objective function that we are examining.
Examples:
! Determining the most important design variables;! Research of the region most favourable for the objective functions;! Creating the data base for the response surface training.
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Dipartimento di Energetica
Classification
A possible classification of DOE techniques can be classified asfollow:
• Random and Quasi-Random sampling (random points are selected in the design space)
• Factorial DOE (systematic sampling on pre-defined variables intervals)
• Orthogonal Arrays (sampling is done according to orthogonal arrays)
• Adaptive sampling ( DOE and RSA are tightly connected and new points are selected using available dataset)
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Dipartimento di Energetica
Base DOE (Design Of Experiments)
! The DOE Random & Sobol Sequences are able to cover sufficiently the dominium of the functions.
! The mathematical theory is the Random Number Generation.
– Sequence Random (function with “many” variables)– Sequence Sobol (function with “less” variables < 6)
! Random sequences of experiments allow the sampling of a configuration space with continuous and discrete variables without pre-defined interactions
! The use of random sequences avoids the risk of “correlated sampling” even in the case of limited sampling
7ERCOFTAC
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Dipartimento di Energetica
Base DOE (Design Of Experiments)
Random sequence
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Sobol sequence
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
The Sobol algorithm covers better the function’s dominium (2 variables case)
Quasi-RandomSuitable for medium-large sampling
Pseudo-RandomSuitable for small sampling
8ERCOFTAC
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Dipartimento di Energetica
Base DOE (Design Of Experiments)
Es. the accuracy of a Monte Carlo Integration is higher and converges more rapidlywith Sobol sequence
9ERCOFTAC
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Dipartimento di Energetica
Factorial DOE (Design Of Experiments)
Factorial methods for the DOE:
– Full Factorial;– Reduced Factorial.
!Pro:
– They show “all” interactions between the design variables;
!Cons:
– Normally the number of the design is too big.
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Dipartimento di Energetica
Factorial DOE (Design Of Experiments)
Full factorial: Number of the designs = mn
m = base of each variablen = number of design variables
It gives all the information related to the influence of each variable at each interaction.The number of experiments is increased of a factor 2 for each added variable.
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Dipartimento di Energetica
n. design x1 x2 x3 fit
1 + + + f12 + + - f23 + - + f3 4 + - - f45 - + + f56 - + - f67 - - + f78 - - - f8
Factorial DOE Example
Full Factorial 2 levels2n Experiments allows the computation of 1nd order interactions
Function with 3 input variables (x1,x2,x3) 0<xi<1
range [0,0.5] ⇒ -
range [0.5,1] ⇒ +
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Munich – 1st April, 200312
Dipartimento di Energetica
Factorial DOE (Design Of Experiments)
Full Factorial 3 levels3n Experiments allows the computation of 2nd order interactions
3 variables27 experiments
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Dipartimento di Energetica
Factorial DOE (Design Of Experiments)
Full Factorial advantages:– For every variable we have the same number of designs in
the range + and in the range -;– The DOE will be very large in the space of function’s
definition;– Good reaching of the variables interactions.
Full Factorial disadvantages:– If the number of the variables is high, the number of the
requested designs becomes huge.
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Dipartimento di Energetica
Factorial DOE (Design Of Experiments)
Reduced Factorial: number of requested design = 2m
m<nn = number of design variables
Example: function with 4 variables
(x1,x2,x3,x4) ⇒ n=4, m=3
Requested design = 8
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Dipartimento di Energetica
Factorial DOE (Design Of Experiments)
Design computed with Reduced Factorial DOE
n. design x1 x2 x3 x4 (=x1*x2)1 + + + +2 + + - +3 + - + -4 + - - -5 - + + -6 - + - -7 - - + +8 - - - +
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Dipartimento di Energetica
Factorial DOE(Design Of Experiments)
Experiment with 3 variables (A,B,C) 2 levelsExperiments A B C Y 1 - - - 33 2 + - - 63 3 - + - 41 4 + + - 57 5 - - + 57 6 + - + 51 7 - + + 59 8 + + + 53
A BTot + 224 210Tot - 190 204Diff 34 6
Effect 8,5 1,5
! there is half number of experiments
! the information related to binary interaction between variables is kept and main effects are visible
! the reduction is higher when the number of variables is increased: 10 variables 2 levels, from 1024 to 64 experiments.
A BTot + 108 116Tot - 92 84Diff 16 32
Effect 8 16
Full factorial Reduced factorial
! Compute the mean value of one factor: A- A+Diff=Tot+ - Tot-Effect=Diff / 4
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Dipartimento di Energetica
Factorial DOE (Design Of Experiments)
Reduced Factorial Advantages:
– The number of requested designs is smaller than the Full Factorial DOE;
Reduced Factorial Disadvantages:
– It is impossible to get all the interactions between the variables;
– There is a limit in the number of variables (with m=3, max 6 variables (x4=x1*x2, x5=x1*x3,x6=x2*x3) ⇒ saturated factorial.
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Dipartimento di Energetica
Factorial DOE (Design Of Experiments)
Latin Square:– It computes a DOE with more variable levels (not only + and –);– The Requested Design Number is m2 where m is the number of levels.
Example:– Latin Square for 3 variables (X1,X2,X3)
with 3 levels :
– X1(1,2,3), – X2(A,B,C),– X3(a,b,c)
213132
321ACBBAC
CBAbacacb
cba
b2Aa1Cc3B
a1Bc3Ab2C
c3Cb2Ba1A
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Dipartimento di Energetica
Factorial DOE (Design Of Experiments)
Latin Square advantages:– The number of computed designs does not depend on the
variables number;– It can be used in the Significance Analysis (e.g. t-Student);
Latin Square disadvantages:– The DOE is not representative of the entire design space.
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Dipartimento di Energetica
Cubic Face Centred (Design Of Experiments)
Cubic Face Centred
! 2n + 2*n +1 Experiments! allows the computation of
2nd order interactions! Less expensive than a 3
levels full factorial
Full Factorial 3 levels
!3n Experiments!allows the computation of 2nd order interactions
3 variables15 experiments
3 variables27 experiments
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Dipartimento di Energetica
Box-Behnken (Design Of Experiments)
The Box-Behnken algorithm is similar in intent to a Cubic Face Centered algorithm, but with the difference that no corners or extreme points are used. The Box-Behnken experiments fill out a polyhedron, approximating a sphere.The experiments are placed in the design variables hyper-cube as follows:
On the mid-points of each edge On the hyper-cube's centre.
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Dipartimento di Energetica
Orthogonal DOE (Design Of Experiments) <<<
Taguchi• Taguchi experiments are controlled by
published orthogonal arrays.• very efficient in experimental testing• similar to other methods in case of
deterministic numerical analysis
• Example :
• effect of three design variables with two levels FRONTIER uses the L4 orthogonal array as follows:
• DOE ID Columns • 1 0 0 0 • 2 0 1 1 • 3 1 0 1 • 4 1 1 0
!L(n) is a (n)x(n-1) matrix containing integer between 0 and (levels-1)!If L(n) is the right orthogonal array for the problem, n experiments will be generated.
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Dipartimento di Energetica
FRONTIER OPT-ADVANCED MACK (1/3)
MACK® (Multivariate Adaptive Crossvalidating Kriging) algorithm that automatically sample the design space where the interpolation is less accurate.
Motivation:• In many circumstances the designer is initially more interested in the exploration of the design space more than in the search for the optimum.• None of traditional DOE algorithms have an iterative behaviour while it would be desirable to sample the design space in order to maximize the extraction of information.
Idea:• Starting from an initial set of points each new experiment is placed in the design space region where the interpolation error of a Kriging Geographic Model is larger for each of the responses being analysed.
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Dipartimento di Energetica
FRONTIER OPT-ADVANCED MACK (2/3)
The performance of this algorithm are shown in the following with the help of the mathematical function:
[ ]a b=
=
− −− −
=
0 5 1 01 5 2 0
2 0 1 51 0 0 5
1 0 2 0. .. .
. .
. .. .α
F x y A B A B1 1 12
2 221( , ) [ ( ) ( ) ]= − + + + +
A a sin b
B a sin b
i i j j i j jj
i i j j i j jj
= ⋅ + ⋅
= ⋅ + ⋅
=
=
∑
∑
( ( ) cos( ))
( ( ) cos( ))
, ,
, ,
α α
β β
1
2
1
2
Big absolute values of the function and therefore where the absoluteerrors of an interpolator are higher.
Small absolute values of the function and therefore where the relative errors of an interpolator are higher.
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Dipartimento di Energetica
FRONTIER OPT-ADVANCED MACK (3/3)
Plain Crossvalidation Relative Error Crossvalidation
Absolute Error Crossvalidation
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Statistical analysis
Statistical analysis
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Statistical Analysis
After the DOE table is evaluated, we can post-process the results extracting important information about problem:
! Which are the most important design variables?! Can we reduce the variables space?! What is the best design space region to address for the
optimisation process?! What is the reasonable number of objectives or
constraints to define?
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Munich – 1st April, 200328
Dipartimento di Energetica
Statistical Analysis
With the DOE’s design:• Medium value of the function for
every variables (range + or -): A- A+
• The same for the interactions between the variables: AB++ - - AB+- -+
Diff=Tot+ - Tot-Effect=Diff / 4
A B C D OBJ1 - - - - 65,62 - - + + 79,33 - + - + 51,34 - + + - 69,65 + - - + 59,86 + - + - 77,77 + + - - 74,28 + + + + 87,9
A B C DTot + 300 283 315 278Tot - 266 282 251 287Diff 34 0,6 64 -9
Effect 8,5 0,2 16 -2
AB307
258,448,612,5
AC231,2282,9-51,7
-12,925
AD282,9282,50,40,1
Simple statistical analysis :
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Dipartimento di Energetica
Statistical Analysis
!To obtain better information from the DOE table we can use different statistical methods like thet-Student parameter.
!The t-Student theory shows how to calculate acorrelation index between a design variable and a design objective.
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Dipartimento di Energetica
Statistical Analysis
• T-Student parameter:1
11
1x
xx
x
yyt
σ−+
−=
−
−
−
−
+
+
+
+
∑
∑
=
=
=
=
1
1
1
1
1
1
1
1
1,
1,
x
n
ini
x
x
n
ini
x
n
yy
n
yy
x
x
x
x
Objective’s mean values in the two ranges + and -
( ) ( )( )( ) ( )
−+
−+−+
+ −
−−++
+⋅−+
−+−=∑ ∑
= =
11
1111
1,1
1,1
1111
1 ,1,1,1,1,1,1
1 1
2,
2,
2 xxxxxx
n
i
n
ixixxix
x nnnnnn
yyyyx x
σ
Mean standard deviation:
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Dipartimento di Energetica
Statistical Analysis
High t-Student Low t-Student
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Dipartimento di Energetica
Statistical Analysis
!High t-Student value:
" This variable is probably important (there is a large difference between the range + and - in the objective values );
" The design variable’s range can be limited to either the + or – range, reducing the searching path for the optimisation phase.
! Low t-Student value:
" This variable is probably NOT so important (the difference between the objective values in the two ranges + and – is small );
" The optimisation phase could ignore the variable.
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Dipartimento di Energetica
Statistical Analysis
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Statistical Analysis
!An accurate assessment of the DOE data (t-Student, ANOVA, etc.) speeds up the optimisation phase reducing the complexity order of our problem limiting the number of variables and the variables definition range.
!Be aware: the statistical tools need DOE tables able to represent correctly all the design space.
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DOE
D.O.E.Examples
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DOE
Examples 1
How to use modeFRONTIER to get the most relevant qualitativeinformation from a data-base of experiments making the
smallest possible number of experiments.
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Dipartimento di Energetica
Mathematical functions
Two different mathematical
functions
[ ]a b=
=
− −− −
=
0 5 1015 20
2 0 1510 0 5
10 2 0. .. .
. .
. .. .α
F x y A B A B1 1 12
2 221( , ) [ ( ) ( ) ]= − + + + +
A a sin b
B a sin b
i i j j i j jj
i i j j i j jj
= ⋅ + ⋅
= ⋅ + ⋅
=
=
∑
∑
( ( ) cos( ))
( ( ) cos( ))
, ,
, ,
α α
β β
1
2
1
2x y, [ , ]∈ −π π
F x y x y22 23 1( , ) [( ) ( ) ]= − + + +
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Dipartimento di Energetica
Factorial DOE (Design Of Experiments)
16 Designs computed with Full Factorial<ID> y x dummy1 dummy2 out1 out2
0 -3.14 -3.14 -100.0 -10.0 -9.458044 -4.59921 -3.14 -3.14 -100.0 10.00 -9.458044 -4.59922 -3.14 -3.14 100.00.00 -10.0 -9.458044 -4.59923 -3.14 -3.14 100.00.00 10.00 -9.458044 -4.59924 -3.14 3.14 -100.0 -10.0 -9.454443 -42.27925 -3.14 3.14 -100.0 10.00 -9.454443 -42.27926 -3.14 3.14 100.00.00 -10.0 -9.454443 -42.27927 -3.14 3.14 100.00.00 10.00 -9.454443 -42.27928 3.14 -3.14 -100.0 -10.0 -9.458832 -17.15929 3.14 -3.14 -100.0 10.00 -9.458832 -17.1592
10 3.14 -3.14 100.00.00 -10.0 -9.458832 -17.159211 3.14 -3.14 100.00.00 10.00 -9.458832 -17.159212 3.14 3.14 -100.0 -10.0 -9.455302 -54.839213 3.14 3.14 -100.0 10.00 -9.455302 -54.839214 3.14 3.14 100.00.00 -10.0 -9.455302 -54.839215 3.14 3.14 100.00.00 10.00 -9.455302 -54.8392
2 more variables are added
Initial inputvariables
Added inputvariables
Results
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Dipartimento di Energetica
Factorial DOE
dummy1 and dummy2 have significance 0 in both functions.
Hint: “The number of design variables can be reduced.”
Full Factorial gives all theinformation related to theinfluence of each variable.
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Dipartimento di Energetica
Reduced Factorial DOE
Reduced Factorial provides reasonable coverage of the experiments space, while requiring fewer experiments
dummy1 and dummy2 have significance close to 0 in both functions
Hint: “The number of design variables can probably be reduced.”
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Dipartimento di Energetica
Random DOE (Design Of Experiments)
16 Designs computed with Random DOE<ID> y x dummy1 dummy2 out1 out2
0 1.44995 -0.5647 -59.0 -3.34566 -11.181237 -11.9329411 2.93755 -3.1016 93.0 8.79731 -9.543931 -15.5146232 2.8084 2.7449 -21.0 -3.04964 -9.475473 -47.5077873 -1.29335 0.0407 -77.0 5.41072 -51.874394 -9.3319114 1.00415 -2.15565 -24.0 -7.20475 -5.135367 -4.7295445 1.2243 1.91685 -99.0 0.4627 -2.007196 -29.1229246 1.53225 -2.2481 -4.0 0.8911 -3.587914 -6.9776447 0.4842 -1.85315 25.0 -6.30586 -15.461887 -3.5181158 -3.07295 -2.12865 -65.0 0.80794 -12.21101 -5.0563739 2.9757 -1.59875 -21.0 -5.64796 -12.64405 -17.769692
10 -0.427 -1.6758 78.0 -9.23347 -39.012283 -2.08183511 0.58015 0.9745 -76.0 3.04954 -8.176557 -18.29352412 3.04155 -1.8417 -25.0 -0.733 -12.022765 -17.67578513 -1.04495 -0.35665 1.0 9.97962 -59.930812 -6.9893214 0.81895 2.5722 2.0 -0.17098 -1.618762 -34.35799215 -0.44725 -1.2052 44.0 9.2483 -49.654233 -3.52684
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Dipartimento di Energetica
Random DOE
Random DOE does not provide reasonable coverage of the experiments space.
The variable significances are not correct.
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Conclusion <<<
!The use of experiments plan allows the analysis of macro-effects
!An accurate assessment of the DOE data (t-Student, ANOVA, etc.) speeds up the optimisation phase reducing the complexity order of our problem limiting the number of variables and the variables definition range.
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Dipartimento di Energetica
DOE
Examples 2
How to use modeFRONTIER to get the most relevant qualitativeinformation from a data-base of experiments making the
smallest possible number of experiments.
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Dipartimento di Energetica
j
outlet
i
i
j
Deflector 1
Deflector 2
Deflector 4
T=791 KV=40 m/s
T=591 KV=40 m/s
δTδV
Example: Fluid Dynamic Mixing
• Simplified problem:– 2D geometry– adiabatic mixing of two gases
• 2 objectives:– min temperature variation at
outlet δT – min velocity variation at
outlet δV
• 6 variables:– position and height of three
deflectors
• 1 constraint:– pressure losses ∆p<3000 Pa
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Dipartimento di Energetica
Example: Fluid Dynamic Mixing
• DOE• Full factorial 2 levels (64 experiments)
• Results analysis• Computation of influences using a 2nd order
interpolation polynomial• Elimination of one deflector
• New DOE• Box Behnken 3 levels (57 experiments) on 4 variables
remaining• New interpolation and minimisation
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Example: Fluid Dynamic Mixing
The Deflector 4 has a lower influence both in size and position
Can Be Eliminated ?
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Example: Fluid Dynamic Mixing
RMS T 1.9 DP 2700
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Conclusion <<<
!The use of experiments plan allows the analysis of macro-effects
! It is very efficient in case of linear phenomenaor when the linear component is dominant
!Not suitable for non-linear phenomena (Ex. Kinematics/dynamic problems, multimodalfunctions)
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DOE
Examples 3
How to use modeFRONTIER to research the most favourable region for the objective functions.
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Maximise a Mathematical function
[ ]a b=
=− −− −
=0 5 1015 2 0
2 0 1510 0 5
10 2 0. .. .
. .
. .. .α
F x y A B A B1 1 12
2 221( , ) [ ( ) ( ) ]= − + + + +
A a sin b
B a sin b
i i j j i j jj
i i j j i j jj
= ⋅ + ⋅
= ⋅ + ⋅
=
=
∑
∑
( ( ) cos( ))
( ( ) cos( ))
, ,
, ,
α α
β β
1
2
1
2
x y, [ , ]∈ −π π
Maximise:
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Dipartimento di Energetica
Factorial DOE (Design Of Experiments)
16 Designs computed with Full Factorial(4 levels)
<ID> x y out1 obj10 -3.14 -3.14 -9.458044 -9.4580441 -3.14 -1.0467 -15.174235 -15.1742352 -3.14 1.04665 -2.611807 -2.6118073 -3.14 3.14 -9.458832 -9.4588324 -1.0467 -3.14 -18.063888 -18.0638885 -1.0467 -1.0467 -59.029914 -59.0299146 -1.0467 1.04665 -15.209256 -15.2092567 -1.0467 3.14 -18.007172 -18.0071728 1.04665 -3.14 -3.116181 -3.1161819 1.04665 -1.0467 -25.814344 -25.814344
10 1.04665 1.04665 -2.980187 -2.98018711 1.04665 3.14 -3.098072 -3.09807212 3.14 -3.14 -9.454443 -9.45444313 3.14 -1.0467 -15.137027 -15.13702714 3.14 1.04665 -2.613207 -2.61320715 3.14 3.14 -9.455302 -9.455302
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Factorial DOE
Variable XThe design variable’s range can be limited to either the + or – range, reducing the searching path for the optimisation phase.
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Factorial DOE
Variable YThe design variable’s range can be limited to either the + or – range, reducing the searching path for the optimisation phase.
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Conclusion <<<
!The use of a factorial experiments plan allows the analysis of macro-effects
!The design variable’s range can be limited to either the + or – range
!The searching path can be reduced for the optimisation phase.
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DOE
Examples 4
How to use modeFRONTIER to create a good data base for the response surface training
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Mathematical functions
x y, [ , ]∈ −π π
F x y x y22 23 1( , ) [( ) ( ) ]= − + + +
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Sobol versus Random
Random sequence
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Sobol sequence
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
The Random sequence can generate clusters.The Sobol algorithm covers better the dominium of the function The experiments in Sobol sequence are maximally avoiding of each other, filling in a uniform way the design space.
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Sobol DOE (Design Of Experiments)
10 Designs computed with Sobol DOE(well distributed in the range [-π,π]x[-π,π])
<ID> x y out20 0 0 -101 -1.57 -1.57 -2.36982 1.57 1.57 -27.48983 -0.785 0.785 -8.092454 2.355 -2.355 -30.512055 -2.355 2.355 -11.672056 0.785 -0.785 -14.372457 -1.1775 2.7475 -17.36526258 1.9625 -0.3925 -24.99546259 -2.7475 1.1775 -4.8052625
60ERCOFTAC
Munich – 1st April, 200360
Dipartimento di Energetica
Residual Chart
Residual ChartLeast sum of square technique has been used to fit the points. The residual errors are very low. Any other point in the range can be estimate with low error.
61ERCOFTAC
Munich – 1st April, 200361
Dipartimento di Energetica
Random DOE (Design Of Experiments)
11 Designs computed Randomly(badly distributed in the range [-π,π]x[-π,π])
<ID> x y out20 -0.9153 1.35405 -9.8875254931 -0.7303 0.215 -6.627763092 0.4435 -0.55075 -12.059517813 -1.26485 1.1714 -7.7257234834 0.1629 0.9111 -13.656239625 0.4287 0.8602 -15.216327736 0.35735 0.83125 -14.625275597 2.5042 -1.68275 -30.76236528 1.8801 -1.1282 -23.831811259 -1.41465 -0.41745 -2.852699125
10 -2.9272 -1.3828 -0.15183568
Points used for training
62ERCOFTAC
Munich – 1st April, 200362
Dipartimento di Energetica
Residual Chart
Residual ChartLeast sum of square technique has been used to fit the first 10 points (all in the range [-1,1]x[-1,1]). The residual errors seem very low.
63ERCOFTAC
Munich – 1st April, 200363
Dipartimento di Energetica
Residual Chart
Residual ChartResidual error for the 11th point (outside of the range [-1,1]x[-1,1]).The residual error is high.
64ERCOFTAC
Munich – 1st April, 200364
Dipartimento di Energetica
Conclusion <<<
!DOE can be used to create the data base for theresponse surfacetraining.
!The use of a correct DOE minimises the interpolation errors.
65ERCOFTAC
Munich – 1st April, 200365
Dipartimento di Energetica
DOE
Examples 5
How to use modeFRONTIER to create a good data base for optimisation
66ERCOFTAC
Munich – 1st April, 200366
Dipartimento di Energetica
Maximise a Mathematical function
[ ]a b=
=− −− −
=0 5 1015 2 0
2 0 1510 0 5
10 2 0. .. .
. .
. .. .α
F x y A B A B1 1 12
2 221( , ) [ ( ) ( ) ]= − + + + +
A a sin b
B a sin b
i i j j i j jj
i i j j i j jj
= ⋅ + ⋅
= ⋅ + ⋅
=
=
∑
∑
( ( ) cos( ))
( ( ) cos( ))
, ,
, ,
α α
β β
1
2
1
2
x y, [ , ]∈ −π π
Maximise:
67ERCOFTAC
Munich – 1st April, 200367
Dipartimento di Energetica
Example 5:
• DOE Algorithm• Latin Square of 4 levels (16 experiments)
• Optimisation Algorithm• Multi Objective Genetic Algorithm (MOGA)
• 10 Generations• Probability of Directional Cross-Over 70%• Probability of Selection 5%• Probability of Mutation 10%
• Results• 160 designs• Maximum value reached –1.080 (x=1.018 y=2.281)
68ERCOFTAC
Munich – 1st April, 200368
Dipartimento di Energetica
Example 5:
• DOE Algorithm• Sobol sequence (16 experiments)
• Optimisation Algorithm• Multi Objective Genetic Algorithm (MOGA)
• 10 Generations• Probability of Directional Cross-Over 70%• Probability of Selection 5%• Probability of Mutation 10%
• Results• 160 designs• Maximum value reached –1.001 (x=2.031 y=7.172e-1)
69ERCOFTAC
Munich – 1st April, 200369
Dipartimento di Energetica
Example Conclusion
Starting from different initial population, Genetic Algorithm evolutions are different.
MOGA with Sobol initial population reaches faster the best point.
70ERCOFTAC
Munich – 1st April, 200370
Dipartimento di Energetica
DOE
Examples 6
How to use modeFRONTIER for robust design
71ERCOFTAC
Munich – 1st April, 200371
Dipartimento di Energetica
Maximise a Mathematical function
Maximise:
( )
( ) ( )[ ]0.5,0.5,
5.25.2
4.020),(
22
222 2
−∈−++=
=+−⋅= ⋅
−
yxyx
yxeyxF
ασ
σα
72ERCOFTAC
Munich – 1st April, 200372
Dipartimento di Energetica
Robust Design
• In many real world optimisation problems, the design parameters are not fixed, normally we identify the mean value and the standard deviation of those parameters.
0 5 10 15 20X
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
P(X
) Mean Value
Standard deviationExample:
X=10 mm ± σ (=1.25 mm)
73ERCOFTAC
Munich – 1st April, 200373
Dipartimento di Energetica
Robust Design
• Maximisation problem where the design parameters are defined by the mean and the deviation.
x
F(x)
x1 x2
Point Value
Average Value
x1 x2
74ERCOFTAC
Munich – 1st April, 200374
Dipartimento di Energetica
Monte Carlo D.O.E.
Monte Carlo perturbations around a point look for robust solution that are not influenced by small variation of the design variables.
Example:Var1=2.5 mm ± σ (=0.1 mm)Var2=-2.5 mm ± σ (=0.1 mm)
75ERCOFTAC
Munich – 1st April, 200375
Dipartimento di Energetica
Monte Carlo DOE
Frequency Histogram. The output variable is not influenced by small variation of the design variables.
Monte Carlo Perturbation for input variables.
76ERCOFTAC
Munich – 1st April, 200376
Dipartimento di Energetica
Monte Carlo DOE
Frequency Histogram. The output variable isinfluenced by small variation of the design variables.
Monte Carlo Perturbation for input variables.
77ERCOFTAC
Munich – 1st April, 200377
Dipartimento di Energetica
There is no single solution to design optimisation tasks. Many techniques are available and most of them have pro and cons. While in the well established world of “simulation” the principles are clear (build a model able to reproduce numerically the physics of a phenomena), in the optimisation arena the driving force should become “improve your available design”
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