design of experiments - ntuavelos0.ltt.mech.ntua.gr/ercoftac/symp03/poloni_doe.pdf · –...

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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

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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.

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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).

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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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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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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

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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

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Es. the accuracy of a Monte Carlo Integration is higher and converges more rapidlywith Sobol sequence

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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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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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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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Full Factorial 3 levels3n Experiments allows the computation of 2nd order interactions

3 variables27 experiments

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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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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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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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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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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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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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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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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



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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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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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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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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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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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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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!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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• 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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High t-Student Low t-Student

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!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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!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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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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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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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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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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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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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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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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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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• 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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The Deflector 4 has a lower influence both in size and position

Can Be Eliminated ?

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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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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

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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.

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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

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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.

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Residual Chart

Residual ChartResidual error for the 11th point (outside of the range [-1,1]x[-1,1]).The residual error is high.

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Conclusion <<<

!DOE can be used to create the data base for theresponse surfacetraining.

!The use of a correct DOE minimises the interpolation errors.

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DOE

Examples 5

How to use modeFRONTIER to create a good data base for optimisation

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[ ]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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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)

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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)

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Example Conclusion

Starting from different initial population, Genetic Algorithm evolutions are different.

MOGA with Sobol initial population reaches faster the best point.

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DOE

Examples 6

How to use modeFRONTIER for robust design

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Maximise:

( )

( ) ( )[ ]0.5,0.5,

5.25.2

4.020),(

22

222 2

−∈−++=

=+−⋅= ⋅

−

yxyx

yxeyxF

ασ

σα

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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)

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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

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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)

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Monte Carlo DOE

Frequency Histogram. The output variable is not influenced by small variation of the design variables.

Monte Carlo Perturbation for input variables.

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Monte Carlo DOE

Frequency Histogram. The output variable isinfluenced by small variation of the design variables.

Monte Carlo Perturbation for input variables.

77ERCOFTAC



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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design of experiments - ntuavelos0.ltt.mech.ntua.gr/ercoftac/symp03/poloni_doe.pdf · –...

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