July 11, 2006 1 University of Tsukuba MBA in International Business Application of design of experiments in computer simulation study Shu YAMADA [email protected]and Hiroe TSUBAKI [email protected]Supported by Grant-in-Aid for Scientific Research 16200021 (Repr esentative: Hiroe Tsubaki), Ministry of Education, Culture, Science and Technology
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University of Tsukuba MBA in International Business July 11, 2006 1 Application of design of experiments in computer simulation study Shu YAMADA [email protected].
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July 11, 20061
University of TsukubaMBA in International Business
Application of design of experiments in computer simulation study
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI5
University of TsukubaMBA in International Business
Outline of this talk1. Validation of the developing model
Example: Forging of an automobile parts
Technique: Sequential experiments
2. Screening of many factors
Example: Cantilever
Technique: Supersaturated design and F statistic
3. Approximation of the response
Example: Wire bonding
Technique: Non-linear model and uniform design
July 11, 2006
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI6
University of TsukubaMBA in International Business
1. Validation of the developing model
Validation of the developing model by comparing with the reality
pxxxy ,,, 21
Validation
knowledge in the field
pxxxy ,,, 21
Validation
knowledge in the field
Validation
knowledge in the field
Physical experiments
Simulation result
compare
July 11, 2006
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI7
University of TsukubaMBA in International Business
Forging example
protrusion
Mr. Taomoto (Aisin Seiki Co.)
height (response)
x1
chamfer
x1: punch depth x2: punch width x3: shape of corner (w)x4: shape of corner (h)y: height
depth
x2
July 11, 2006
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI8
University of TsukubaMBA in International Business
(a) Judgment:
Appropriate or not?
(b) Adjustable by changing computational parameters?
(Young ratio, mechanical property,… )
(c) Revise the simulation model
What should be done?
punch depth (d)
Simulation experiments
Physical experiment
Simulation
Physical
Physical
Simulation
y
y
y
x1
x1
x1
July 11, 2006
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI9
University of TsukubaMBA in International Business
(a) Judgment: Appropriate or not?
Application of statistical tests
(b) Adjustable by changing computational parameters?
How to determine the level of the computational parameters systematically
(c) Revise the simulation model
Not a statistical problem
Simulation experiments
Physical experiment
July 11, 2006
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI10
University of TsukubaMBA in International Business
Validation of the developing computer simulation model
punch depth (x1)
prot
rusi
on h
eigh
t
Simulation experimentsPhysical experiment
Adjusted simulation results
Find appropriate levels of computational parameters to fit the simulation results to physical experimental results
Computational parameters
Yong ratio, poison ratio, etc.
July 11, 2006
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI11
University of TsukubaMBA in International Business
Problem formulation of adjusting computational parameters
Requirement: Small run number is better
Aim:
To determine the level of computation parameters z1, z2, …, zq to minimize the difference between the physical experiment results and computer simulation results over the interested region of x1, x2, …, xp.
Simulation results
Physical experiments
Minimization of
by computational parameters
dxxxYzzxxRx pqp
2
111 ,,,,,,,
qp zzzxxxy ,,,,,,, 2121
pxxxY ,,, 21
qzz ,,1
July 11, 2006
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI12
University of TsukubaMBA in International Business
An approach by “easy to change” factor
Punch depth (x1) : Easy to change factor levels because it does not require remaking of mold. Physical experiments can be performed by using the mold by several levels.
At each combination of computational parameters, the discrepancy between physical and experimental experiments are calculated by
5
1
244221141442211 ,...,,,,,,...,,
iii axaxxxYzzaxaxxx
D
July 11, 2006
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI13
University of TsukubaMBA in International Business
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI20
University of TsukubaMBA in International Business
Screening procedure
More than There are many factor effects
Linear effects 15
Interaction effects 105
Quadratic effects 15
(0) Impossible to assess all possibilities
(1) Stepwise selection of F value
(2) Stepwise selection of F value with order principle and effect heredity
31135 1005648.42
July 11, 2006
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI21
University of TsukubaMBA in International Business
Analysis strategyOrder principle
Lower order terms are more important higher order terms
(Linear effect, interaction and quadratic,...)
Effect heredity
When two-factor interaction is detected,
(i) at least one factor effect of the two factors
(ii) both of the linear effects
should be included in the model. The strategy (ii) is implemented.
July 11, 2006
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI22
University of TsukubaMBA in International Business
Procedure to ensure effect heredity and order principle (EO)
Step 1 Candidate set
Step 2
Quadratic term of the selected effect is added to the candidate set
Ex 1
Interaction term of the selected two factors is added
Ex 2
151,..., xx
2221151 ,,..., xxxxx
22151,..., xxx
July 11, 2006
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI23
University of TsukubaMBA in International Business
Applied designYAMADA, S. and LIN, D. K. J., (1999), Three-le
vel supersaturated design, Statistics and Probability Letters, 45, 31-39. etc
Yamada, S., Ikebe, Y., Hashiguchi, H. and Niki, N., (1999), Construction of three-level supersaturated design, Journal of Statistical Planning and Inference, 81, 183-193.
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI26
University of TsukubaMBA in International Business
3. Approximation and optimizationExample: wire bonding in IC
Yamazaki, Masuda and Yoshino,(2005), Analysis of wire-loop resonance during al wire bonding, 11th Symposium on Microjoining in Electrics, February 3-4, 2005, Yokohama
Outline:Recent years AL wire bonding by microjoining is widely applied in many types of IC.
Finite Element Method obtained that it sometimes occurs resonance problem at the mircojoining.
July 11, 2006
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI27
University of TsukubaMBA in International Business
Background(1)The shape of the wire is shown in the right(2)Material: AL
x1: Width 2mm - 4mmx2: Height 0.5mm-1.5mm
x3: Diameter 0.03 - 0.04 mm (3) FEM obtains the moment along with the
f: frequency at the joining(4) The connected point will be broken when the moment is higher than
certain level.(5) The amplitude and its frequency is determined by x1, x2, x3.(6) Given the level of x1, x2 and x3, FEM analysis requires time for
calculation to analyze the frequency and response analysis.(7) The response, moment, is a multi-modulus because of the resonance at
several frequencies.
x1: Width
x2: Height
VibrationFix
July 11, 2006
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI28
University of TsukubaMBA in International Business
Output of FEM software
x1: 3.14x2: 1.28x3: 0.03
x1: 3.00x2: 0.78x3: 0.03
FEMAP v8.2.1+ CAFEM v8.0
The peaks will be determined by x1: width, x2: height and x3: diameter
Complex function
July 11, 2006
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI29
University of TsukubaMBA in International Business
Requirements(1)Outline of the factors in process
x1: Width Some restrictions because of the location with other parts
x2: Height controllable in the range (0.5-1.5mm)
x3: Diameter Specified in the priori process
f: Frequency Controllable by selecting the bonder
(2)The tentative levels of x1, x3 are determined in the priori process. Based on the tentative levels, optimum levels of x2 and f is explored. There is a need to consider the robustness against the difference of f from the specified value.
(3) It takes a long time to evaluate the frequency under a set of levels of x1, x2 and x3. Re-calculation is inefficient when the levels are slightly revised.
July 11, 2006
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI30
University of TsukubaMBA in International Business
Strategy(1) The final goal is to find a good approximated function of M:
moment at the fixed point by x1: width, x2: height, x3: diameter and f: frequency such that
M=g(x1, x2, x3, f)
(2) In the future, various types of wires are applied in the IC design. Thus, the above approximation is helpful.
(3) To find a smooth function, 210-level design is utilized.
(4) Because of the complex relation, uniform design will be beneficial.
July 11, 2006
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI31
University of TsukubaMBA in International Business
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI40
University of TsukubaMBA in International Business
Using the approximation(x1=2.143, x2=1.357, x3=0.03)
5 10 15 20
02
00
40
06
00
80
01
00
0
fr
y
5 10 15 20
02
00
40
06
00
80
01
00
0
fr
pre
dic
t.c
July 11, 2006
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI41
University of TsukubaMBA in International Business
(x1=3.000, x2=0.786, x3=0.03)
5 10 15 20
02
040
60
80
fr
y
5 10 15 20
02
040
60
80
fr
pre
dic
t.c
July 11, 2006
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI42
University of TsukubaMBA in International Business
Comments(1) The essence of the case study is exploring an approximation of
multi-modulus function by uniform design and RBF.
(2) Fitting by RBF brings a good fitting. It is suggested that RBF is beneficial to fit response to frequency.
(3) It is concerned the over fitting in the case study. The fitness should be validated.
(4) The parameters a1, m1, a2, m2,… are estimated precisely, for example the adjusted R^2 is more than 90%. On the other hand, there is a need to estimate s1, s2, … more precisely.
July 11, 2006
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI43
University of TsukubaMBA in International Business
5
10
15
20 0.5
0.6
0.7
0.8
0.9
1
0
5
10
15
5
10
15
205 10 15 20
0.5
0.6
0.7
0.8
0.9
1
Application of the approximation
x1: width 3mm, x3 diameter 0.3
x 2 height
x 2 height
frequency
f
Good choice
July 11, 2006
Shu YAMADA and iroe TSUBAKIShu YAMADA and iroe TSUBAKI44
University of TsukubaMBA in International Business
4. Grammar of DOE in computer simulation study
1. Interpreting requirements
2. Developing a simulator
3. Applying the simulator
pxxxy ,,, 21
3333
2111311332110
31
ˆˆˆˆˆˆ
,ˆ
xxxxxx
xxy
Optimization from various viewpoints
Output
Input
yDefine
pxxx ,,, 21
Validation
knowledge in the field
Validation: Comparison of simulation results to reality