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Application of Kriging Model
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Kriging for Multi-Objective Problem
ParEGO: converts the k different objective function into a single objective function via a parameterized scalarizing weight vector
Utility Function
weighting vector
EGOMOP: converts all objective functions into EI of objective functions and these values are directly used as fitness value in the multi-objective problem
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Optimization Problem
Transonic Airfoil Design
Minimize obj1: Drag at fixed lift of 0.75 (Mach=0.70)
obj2: Drag at fixed lift of 0.67 (Mach=0.74)
subject to t/c > 11%
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Definition of Geometry and Design Variables (NURBS)
Total 26 parameters for airfoil definition
Number of design variable :26 2010년 1월 15일 금요일
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Initial Sample Point Selection
Latin Hypercube Sampling with Constraint Evaluation
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Pareto Front of EI & Update
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Initial and Additional Sample Points
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Designed Airfoil
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Selection of Sample from Pareto Front of EIs
EI1
EI2
2 objectives n objectives3 objectives
?
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High Dimensional Pareto solu2ons about EI
able to control the number of addi2onal sample points in high dimensional problem
EGOMOP - Clustering for additional sample points
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High Dimensional Pareto solu2ons about EI
able to control the number of addi2onal sample points in high dimensional problem
EGOMOP - Clustering for additional sample points
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Summary of Design Procedure
12
Construction of Kriging Modelswith N sample points
N=N+m: Sample points
Explora2on of Pareto solu2ons about
Clustering analysis to
select the promising points for design
EI1 EI2 EI3
Good Bad Bad
Good Good Good
Good Bad Normal
Good Normal Good
Normal Good Good
Good Bad Good
Good Good Bad
Bad Normal Bad
Ini2al sample pointsselec2on by
La2n Hypercube sampling
High-‐FidelityAnalysis
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Definition of Design Problem
Objective functions Soot
Thermal NO
CO
Thermal efficiency
Design variables
(dv2) dv11 : Injection angle (100~180[deg])
lip
RL1
L2
(dv1)
(dv3)
(dv10)(dv4, y1)
(dv5,y2)
(dv6,dv7)
(x4,dv8)(x5,dv9)
Between and is interpolated by spline curve is defined to keep constant compression ratio
(volume) Total 11 design variables are used
Minimization
Maximzation
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Result (6 iterations, 43 additional sample points)
optimum direction
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Result
Baseline Design A Design B
NO
SOOT
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Reduction of Dimension
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High-dimensional Problem
Reason of failure to find Pareto optimal front
Lack of computational resource Limited number of individuals and generation lead to insufficient convergence
Existence of non-conflicting objective functionsRedundant objective function prevents from converging to pareto optimal front
Remedy: Use more computational resources
Remedy: eliminate redundant objective functions
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Example: DTLZ5
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Example of High-dimensional Case
DTLZ(I, M) I: dimension of Pareto-frontM: number of objective functions
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Example of High-dimensional Case
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Difficulties with Many Objectives
High-Dimensionality of Pareto-optimal frontierMany Objectives are in trade-offLack of point
If N points are needed for adequately representing a one-dimensional Pareto-optimal front, O(NM) points will be required an M-dimensional Pareto-optimal front
Difficulty in visualizing
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Principle Component Analysis
Initial data set: is i-th objective and n×1 vector
Standardized data set:
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Principle Component Analysis Covariance Matrix
Correlation Matrix
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Principle Component Analysis
i-th component of standardized data set X: (Yi)
is i-th eigenvector of R
which maximize Var(Yi)= e`i R ei
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Application to DTLZ(2,10)
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Application to DTLZ(3,10)
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Definition of Design ProblemObjective functions : Natural vibration frequencies
Design variables
Side to sidetranslation
(L0)
Rotational torsion(C0)
First-orderradial(f1)
Second-orderside to side
(L2)
Second-ordercross-sectional
(f2)
Elasticity of 43 elements of tire
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Result
Initial sample points selection Latin Hypercube Sampling (LHS)
Construction of Kriging Model Cross-Validation
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Result
Comparison of optimized solutions with baseline
After 6 design iterations with 64 additional samples
obj1 obj2 obj3 obj4 obj5
opt1 × ○ × ○ ○
opt2 ○ ○ × ○ ○
opt3 ○ ○ ○ × ○
○: better performance than baseline
×: worse performance than baseline
There is no solution which shows better performance than baseline in term ofall objective functions.
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Result: Self-Organizing Map (SOM)
0
0
0 1
1 0 1
101
L0
f1
f2
C0
L2
Obj1=-1.*L0 Obj2=C0
Obj3=-1.*f1 Obj4=L2
Obj5=f2
Severe Trade-off exist among objective functions
Thus, it was difficult to improve all objective functions at the same time
Domination relation should be investigated
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Principal Component Analysis (PCA)
Correlation Matrix (R)Correlation Matrix (R)Correlation Matrix (R)Correlation Matrix (R)Correlation Matrix (R)
1 0.7084 0.7872 -0.8804 -0.7386
0.7084 1 -0.9856 0.9255 0.6029
0.7872 -0.9856 1 -0.9749 -0.6910
-0.8804 0.9255 -0.9749 1 0.7811
-0.7386 0.6029 -0.6910 0.7811 1
EigenvaluesEigenvaluesEigenvaluesEigenvaluesEigenvalues
4.247 0.4935 0.2400 0.0185 0.0004
Proportional EigenvaluesProportional EigenvaluesProportional EigenvaluesProportional EigenvaluesProportional Eigenvalues
0.8495 0.0987 0.0480 0.003 0.0008
EigenvectorsEigenvectorsEigenvectorsEigenvectorsEigenvectors
0.4329 0.3160 0.7995 0.2658 0.0537
-0.4483 0.5141 0.2011 -0.5691 0.4129
0.4703 -0.3372 -0.1265 -0.1448 0.7928
-0.4820 0.0700 -0.0460 0.7499 0.4454
-0.3977 -0.7192 0.5498 -0.1491 -0.0095
PCA1 PCA2 PCA3 PCA4 PCA5
From the first principle component, the most positive and the most negative elements are selected.
From the second principle component, the element having the largest absolute value is selected.
The rest objective functions are eliminated.
495 non-dominated solutions on Kriging model is used
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Result: SOM for PCA case
0 1 0 1
0 1
f1 L2
f2
Obj3=1/f1 Obj4=L2
Obj5=f2
Sweet Spot for Design
(After 5 iterations are over, 100 Pareto solutions on Kriging model was used)
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Robust Design
33
性 能
変 数-Δx +Δx -Δx +Δx
設計 A
設計 B
絶対値最適化設計: 解 A > 解 Bロバスト最適化設計: 解 A < 解 B
変動下での設計A 変動下での
設計B
急激な性能性低下
わずかな性能性低下
解Bのような設計が必要
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Application- Centrifugal fan
34
humid air suction
dry air injection
dehumidifier
heater
mass production120,000 sales / month
washer-dryer
140 mm
uncertainty in design-conditions• dimensions• material properties• environments• aged deteriorations
centrifugal fan
design variables
probability density functionwhen profileis known
when profileis unknown
Multi-objective robust optimization based on statistics Design rule extraction of specific solution
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Robust Optimization Based on Statistic
Efficient statistic-based optimization using Kriging model
35
prediction by Kriging models
evaluation of objective functions
OF
DV2DV1
Kriging model
probability density function
DV
specified mean value
definition of design variables
simulation samples
OF
probability density function
multi-objective genetic algorithm(MOGA)
uncertainty
minimize standard deviation
minimize mean
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Design ProblemDesign variables(8): NURBS control points/uniform uncertainty profile
Objective function(4): mean and std. dev. of fan efficiency and noise level
Constraints (2) : mean and std. dev. of axial input power
CFD: RANS using Star-CD (steady state, k-e turbulence model)
36
CFD mesh145920 cells149292 vertices
blade-to-blade
CFD model Fan efficiency (maximize)
Turbulent noise level (minimize)
79 samples for Kriging model Population 100 / Generation 162 1000 samples for statistical calculations
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Pareto Solutions1268 Pareto solutions/ 17,000,000 evaluations
Turn around design time : 2 weeks
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Designer’s Preference
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1
1
10
S
DNon-dominated solution
Non-dominated front
aspiration vector
data vector
Solution with small is close to the designer’s preference
Designer’s preference
Deviation from preference direction
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Weighted Solutions
39
µ(ηs) weightedw1:w2:w3:w4= 1 : 0 : 0 : 0
µ() weightedw1:w2:w3:w4= 1 : 0 : 1 : 0
σ() wighted w1:w2:w3:w4= 0 : 1 : 0 : 1
compromisedw1:w2:w3:w4= 1 : 1 : 1 : 1
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Association Rule
Analysis of transaction data such as POS data
Rule extraction based on covering search for combination of attributes
Judgement of importance of rule
40
Condition attributesCondition attributesCondition attributesCondition attributes Decision attribute
No
1 Level 1 Level 2 Level 5 Level 4 Level 2
2 Level 5 Level 4 Level 1 Level 3 Level 1
3 Level 3 Level 4 Level 2 Level 2 Level 5
4 ... ... ... ... ...
"if A then B"
・・・ ,, ,
combination of any attributes
Apriori algorithm
(= Commodity of a rule)
(= Accuracy of a rule)
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Design Rules from Association Rule
Design rules for robust weighted solution (w1:w2:w3:w4=0:1:0:1)
41
Low aspect ratio of velocity triangle for robust designs
non-robust U
W
Key rules to achieve the specific solution can be determined
large Beta3
small b2
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Curve Definition
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Cubic Spline InterpolationIn [a, b] ,Spline of interval are of the form
should satisfy following conditions
For all j,
For all j,
For all j,
For all j,
At boundary, or
and
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Cubic Spline Interpolation
From Condition 1,
From Condition 2,
set and
From Condition 3,
From Condition 4,
44
(Eq.1)
(Eq.2)
(Eq.3)
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Cubic Spline Interpolation
Put into Eq.1 and Eq.2
Put Eq.6 and Eq.7 into Eq. 8
45
(Eq.4)
(Eq.5)
(Eq.6)
(Eq.7)
(Eq.8)
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Cubic Spline Interpolation
Solving tridiagonal system of equation
46
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B-Spline Curve
With n control points and n+p knot, B-spline curve can be defined by following equation
47
Blending function
Control Point
order of b-spline curve
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B-Spline Curve Basis of B-Spline
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knot intervals between knot should be same!
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Non-Uniform Rational B-Spline (NURBS)
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NURBSB-Spline
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NURBS
Basis of NURBS
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knot intervals between knot do not need to be same!
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Bezier Curve
With n+1 control points, n degree of Bezier Curve can be defined by following “Bernstein polynomial equation”
Bernstein Basis Polynomial
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Bezier Curve
Cubic Bezier Curve With two end points and two control points
De Castljau Algorithm
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Example 1 Example 1
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Bezier Curve
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