application of kriging model - ifs.tohoku.ac.jp

Application of Kriging Model

2010년 1월 15일 금요일

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

2010년 1월 15일 금요일

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

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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Summary of Design Procedure

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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

(497 Pareto solutions on Kriging model)2010년 1월 15일 금요일

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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)

2010년 1월 15일 금요일

Robust Design

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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

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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

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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)

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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

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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

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Condition attributesCondition attributesCondition attributesCondition attributes Decision attribute

No

1 Level 1 Level 2 Level 5 Level 4 Level 2



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)

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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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2010년 1월 15일 금요일

Cubic Spline Interpolation

From Condition 1,

From Condition 2,

set and

From Condition 3,

From Condition 4,

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(Eq.1)

(Eq.2)

(Eq.3)

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Put into Eq.1 and Eq.2

Put Eq.6 and Eq.7 into Eq. 8

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(Eq.4)

(Eq.5)

(Eq.6)

(Eq.7)

(Eq.8)

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Solving tridiagonal system of equation

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B-Spline Curve

With n control points and n+p knot, B-spline curve can be defined by following equation

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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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application of kriging model - ifs.tohoku.ac.jp

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