Response Surface Methodology

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Response Surface Methodology
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Contents

• Basic Concept, Definition, History of RSM
• Introduction Motivation

• DOE (Design Of Experiments)
• Experiments (Numerical) Databases
• Construction of RSM (Response Surface Model)
• Optimization Using RS Model (Meta Model)
• Examples

• Efficient RS Modeling Using MLSM and Sensitivity
• Design Optimization Using RSM and Sensitivity

• Conclusion
• Further Study

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1.1 Concept of Response Surface Method
Original System
RSM Response Surface Method Response
Surface Model
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1.2 Definition of Response Surface Method
Box G.E.P. and Draper N.R.,1987
A simple function, such as linear or quadratic
polynomial, fitted to the data obtained from the
experiments is called a response surface, and the
approach is called the response surface method.
Myers R.H., 1995
Response surface method is a collection of
statistical and mathematical techniques useful
for developing, improving, and optimizing
processes.
Roux W.J.,1998
Response surface method is a method for
constructing global approximations to system
behavior based on results calculated at various
points in the design space.
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1.3 History of Response Surface Method
Research of DOE
1951 Box and Wilson - CCD 1959 Kiefer - Start of
D-optimal Design 1960 Box and Behnken -
Box-Behnken deign 1971 Box and Draper -
D-optimal Design 1972 Fedorov - exchange
algorithm 1974 Mitchell - D-optimal Design
Application in Optimization
1996 Burgee - design HSCT 1997 Ragon and Haftka
- optimization of large wing structure 1998
Koch, Mavris, and Mistree - multi-level
approximation 1999 Choi / Mavris Robut,
Reliablity-Based Design
App in Optimization Reduce the Approximation
Error
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1.4 Introduction - Motivation of RSM
Heavy Computation Problem ? Approximation When
Sensitivity is NOT Available Global Behavior Real
/ Numerical Experiment When the Batch Run is
Impossible For Any System Which has Inputs and
Responses Easy to Implement Part of MDO,
Concurrent Engineering Probabilistic
Concept Noisy Responses or Environments
Advantages
Approximation Error Size of the Approx. Domain is
Very Dominant
Disadvantages
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Part I(Classical RSM)
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DOE (Design Of Experiments)
Experiments (Numerical) Databases
Construction of RSM (Response Surface Model)
Optimization Using RS Model (Meta Model)
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2.1 DOE 1 Factorial Design
Classifications
- 2 / 3 level Factorial Design - Full /
Fractional Factorial Design
2 level Full Factorial Design
Fractional Factorial Design
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2.1 DOE 2 Central Composite Design(CCD)
Characteristics
3 DV
Quadratic RS Model Effective than Full-Factorial
Design Rotatability
Factorial Points
2 DV
1
Axial Points
0
x2
-1 0 1
Center Points
x1
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2.1 DOE 3 Box-Behnken Design
Characteristics
Quadratic RS Model Effective 3 Level
Design Balanced Incomplete Block Design
Block1
Block2
Block3
Center Point
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2.1 DOE 4 D-Optimal Design
Characteristics
The Most Popular DOE Arbitrary Number of
Experiment Points Possible to Add
Points Specified Functional Form of the Response
Approximation Function (RSM)
Coefficients of RSM
Variance of Coefficients
Good fitting
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2.1 DOE 5 Latin-Hypercube Design
Characteristics
Arbitrary Number of Experiment Points No Priori
Knowledge of the Functional Form of the Response
Initial Information
No. of Variables k No. of Experiments n
Main Principles
1. No. of Levels No. of Experiments 2.
Experiment points in the design space are
distributed as regular as possible.
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DOE (Design Of Experiments)
Experiments (Numerical) Databases
Construction of RSM (Response Surface Model)
Optimization Using RS Model (Meta Model)
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2.2 Experiments (Numerical) Databases
Input
Response
Black Boxed System(FE Model)
.bdf .cdb
.f06 , .pch .rst
NASTRAN ANSYS
Rewrite Input Files
Read Output Files
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DOE (Design Of Experiments)
Experiments (Numerical) Databases
Construction of RSM (Response Surface Model)
Optimization Using RS Model (Meta Model)
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2.3 Construction of RSM Least Squares Method
Response

• - Global Approximation
• 1 RS Function at all pts
• Constant Coefficients

Input
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2.3 Construction of RSM Least Squares Method()
- 1. Approximation Function (RSM)
- 2. Least Squares Function
DOE
- 3. Minimize Least Squares Function
- 4. The coefficients of the RS model
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2.3 Example Construction Of RSM
Original Function
Number of Design Variable 2 Number of
Experiment(FFD) 9 RS Model Quadratic Model
RSM Function
JMP, SAS, SPSS MATLAB Statistics
Toolbox Visual-DOC In-House Codes
Software
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2.3 Construction of RSM Test Criteria
F-Test (ANOVA)
The model was fitted well.
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2.3 Construction of RSM Test Criteria
(Continued)
The coefficient of determination
1.0
Adjusted R2
1.0
t-Test
Where Cjj diagonal term in (XX)-1
corresponding to bj
xj is a dominant term of RS model
Prediction Test
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2.3 Construction of RSM Variable Selection
Concept
Unnecessary Term
Original System
RS Model
All Possible Regression
Minimize
Stepwise Regression

• Forward regression
• Backward regression
• Stepwise regression (Backward Forward )

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DOE (Design Of Experiments)
Experiments (Numerical) Databases
Construction of RSM (Response Surface Model)
Optimization Using RS Model (Meta Model)
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2.4 Optimization Using RSM - Whole Sequences
Optimization Problem
Variable Selection
Approximation Domain
DOE Experiments
Construct RSM
Optimization Using RSM
Estimated Opt Response
Final Optimal Solution
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2.5 Example 1 System / Problem Setup
Problem Setup
System(FE Model)
Min weight s.t
Initial variables
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2.5 Example 1 Optimization Using RSM
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2.5 Example 2 - Induction Motor FE Model Update
Model WM0F3A-S Induction motor
Real System
LMS CADA-X
Upper Housing
Stator
Lower Housing
Rotor
Reliable FE Model ?? (close to Real Model )
FE Model (NASTRAN)
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2.5 Example 2 - Modal Analysis(1/2)
Lower Housing
Upper Housing
2
1
1
2
Rotor
Stator
2
1
2
1
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2.5 Example 2 - Model Update Using RSM Rotor
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2.5 Example 2 - Model Update Using RSM Other
Parts
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2.5 Example 2 - Model Assemble Analysis
Mode Shape
Natural Frequencies
1
2
4
3
These good Results are from the good part models
Sensitivities of all design variables w.r.t. the
each frequencies
5 Design Variables are selected
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2.5 Example 2 - Model Update Whole Motor
Optimization

• Gradient-based Optimization
• Hybrid(RSMGRAD) Optimization

Final Results Using Hybrid Method
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2.5 Example 3 - AUTOMOTIVE SIDE IMPACT
Example by k.k.choi, U. of Iowa, Moving Least
Square Method for Reliability-Based Design
Optimization, WCSMO4, 2001
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References
Myers, R. H., and Montgomery, D. C. Response
Surface Methodology Process and Product
Optimization Using Designed Experiments. John
Wiley Sons. Inc., New York, 1995
???, ????, ???, 1998
???, ???????, ???, 1996
???, Efficient Response Surface Modeling and
Design Optimization Using Sensitivity, ???? ??,
???????, 2001
Nguyen, N. K., and Miller, F. L. A Review of
Some Exchange Algorithms for Constructing
discrete D-optimal Designs, Computational
Statistics Data Analysis, 14, 1992, pp.489-49
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Part II(Advanced RSM)
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3.1 Introduction-Motivation
Function Test
Efficient Construction of RSM using Sensitivity
Reduce Approximation Errors ? Local Global
Approximation (MLSM)
Reduce the Computation Time ? Effect of
Function Sensitivity
Restriction -Available Cheap Sensitivity
Optimization using RSM and Sensitivity-based
Method
Induction Motor FE Model Update
RSM Optimization ? Global Behavior / Large
Approximation Error
Sensitivity-based Optimization ? Accurate
Fast Convergence / local Behavior
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3.2 Moving Least Squares Method
Response

• - Global Approximation
• 1 RS Function at all pts
• Constant Coefficients

Input
Response

• Local Approximation
• 1 RS Function at 1 pt
• Various Coefficients

Input
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3.2 Numerical Derivation (1/2) Moving Least
Squares Method
- Response Function
- Least Squares Function
- The coefficients of the RS model
Function of location x
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3.2 Numerical Derivation (2/2) MLSM with
Sensitivity
- Gradient Function
- New Least Squares Function
- The coefficients of the RS model
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3.2 Numerical Examples (Graphical Analysis)
Rosenbrock Function

• Function Characteristics
• Banana Function
• V-shaped Valley

Basis Model Quadratic Weight Function of Resp
4th order polynomials Weight Function of Grad
4th order polynomials
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3.2 Numerical Examples (Error Analysis)
SSE/n Sum of Squared Errors / No of Sampling
Pts SSE/nt Sum of Squared Errors / No of Test
Pts
Error Table
Resp Error
Grad Error
Global Error
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3.2 Numerical Examples (Graphical Analysis)
2D six-hump camel back function
Basis Model Quadratic Weight Function of Resp
4th order polynomials Weight Function of Grad
Exponential
4 local optimums and 2 global optimums within
the bounded region
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3.2 Numerical Examples (Error Analysis)
SSE/n Sum of Squared Errors / No of Sampling
Pts SSE/nt Sum of Squared Errors / No of Test
Pts
Error Table
Resp Error
Grad Error
Global Error
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3.2 Numerical Examples (Efficiency Test)
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3.3 Concept of Hybrid Optimization of RSM
gradient-based optimization
Hybrid Optimization (Function Plot)
Using Response Surface Method
Original Response
Use the approximated Function instead of the
original system
RSM Response
Optimum By RSM

• (Adv) Global Behavior
• (Dis) Large Approximation Error

True Optimum by Gradient-based optimization
Hybrid Optimization (Contour Plot)
Using Gradient-Based Method
Search the direction s.t. improve the
objective Use the original system

• (Adv) Accurate Fast Convergence
• (Dis) local Behavior

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3.3 Sequences of the optimization
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3.3 Numerical Example
Optimization Problem
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3.4 Conclusion
Efficient Construction of RSM using Sensitivity
Local Global Approximation (MLSM) ? Reduce
the Approximation Errors
Function Tests ? Accuracy Efficiency
Effect of Function Sensitivity ? Reduce the
Calculation Time
Optimization using RSM and Sensitivity-based
Method
RSM Optimization ? Global Behavior
Function Test Induction Motor FE Model Update
Sensitivity-based Optimization ? Accurate
Fast Convergence
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3.4 Further Study

• Apply to Real Optimization Problems Using these
  Methods
• Reliability-Based Design Optimization Using This
  RSM
• Proper Selection of The Weight Factor of
  Gradient Error (SWg)
• Use of Design Of Experiments

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3.5 Other Approximation Methods
Kriging Model Neural Network