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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 shu@mbaib.gsbs.tsukuba.ac.jp shu@mbaib.gsbs.tsukuba.ac.jp and Hiroe TSUBAKI tsubaki@mbaib.gsbs.tsukuba.ac.jp tsubaki@mbaib.gsbs.tsukuba.ac.jp Supported by Grant-in-Aid for Scientific Research 16200021 (Representative: Hiroe Tsubaki), Ministry of Education, Culture, Science and Technology

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 2 Introduction Application of computer simulation: - Computer Aided Engineering in manufacturing - R & D stage in pharmaceutical industry Advantages of computer simulation: - reduction of time - better solution by examining many possibilities, - sharing knowledge by describing a model

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 3 Computer simulation Simulation study of a phenomena by computer calculation in stead of physical experiments such as finite element method Sometimes called Digital engineering CAE (computer aided engineering) Simulation study

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 4 1. Interpretation of requirements 2. Developing a computer simulation model 3. Application of the developed simulation model Optimization in terms of various viewpoints Output Input Define Validation: Comparison of simulation results to reality Application of DOE depending on the situation DOE helps well Validation Specialist knowledge Screening Approximation Application of DOE in computer simulation study

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 5 Outline of this talk 1. 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

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 6 1. Validation of the developing model Validation of the developing model by comparing with the reality Physical experiments Simulation result compare

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 7 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

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 8 (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 Simulation y y y x1

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 9 (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

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 10 Validation of the developing computer simulation model punch depth (x1) protrusion height Simulation experiments Physical 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.

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 11 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

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 12 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

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 13 No. z1z2z3z4Discrepancy 1 000021 2 100011 3 010019 4 001020 5 000113 6 110011 7 101010 100111 Computational parameters Sequential experiments The optimum level of the computational parameters z1, …, z4 are found by analyzing the relation between “discrepancy” and z1, …, z4

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 14 punch depth (x1) protrusion height Original simulation results Physical experiment Adjusted simulation results

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 15 Future problems at validation stage (1) Statistical tests to judge the appropriateness of the simulation model (2) Identification of the trend (3) Design to examine the simulation model efficiently

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 16 2. Screening problem Example: cantilever Ex. FEM measures the maximum stress Factors x 1, x 2, …, x p Theoretical equations (x 1, x 2, …, x p ) are applied to calculate the response variables under given factor level Design a beam in which one side is fixed to the wall

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 17 Design a beam (Iwata and Yamada (2004)) Stepping cantilever Factors Hight x1 ～ x15 LevelsNo.130(mm) No.235(mm) No.340(mm) Response Maximum stress Variation of the maximum stress Weight of the beam

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 18 An application of three-level supersaturated design Requirements The relation between the responses and their factors are complicated It takes several hours to calculate the response in a design There are many factor effects Linear effects15 Interaction effects105 Quadratic effects15 Impossible to estimate all factor effects simultaneously.

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 19 Application of three-level supersaturated design (Yamada and Lin (1999))

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 20 Screening procedure More than There are many factor effects Linear effects15 Interaction effects105 Quadratic effects15 (0) Impossible to assess all possibilities (1) Stepwise selection of F value (2) Stepwise selection of F value with order principle and effect heredity

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 21 Analysis strategy Order 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.

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 22 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

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 23 Applied design YAMADA, S. and LIN, D. K. J., (1999), Three- level 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. n=9 n=18

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 24 What is a right choice Consistency with the knowledge in the mechanical engineering Physical property 1 ． The factors closing to the wall are important such as x1,x2, x3 2. Sometimes, the edge side are important such as x14, x15 3. Interaction can be considered at a connected two factors such as x1×x2 ， x2×x3 x1x4 x3 x2 x5x15

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 25 EO select a reasonable selection in terms of the physical property of the cantilever

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 26 3. Approximation and optimization Example: wire bonding in IC Yamazaki, Masuda and Yoshino,(2005), Analysis of wire-loop resonance during al wire bonding, 11 th 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.

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 27 Background (1)The shape of the wire is shown in the right (2)Material: AL x1: Width 2mm - 4mm x2: 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

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 28 Output of FEM software x1: 3.14 x2: 1.28 x3: 0.03 x1: 3.00 x2: 0.78 x3: 0.03 FEMAP v8.2.1+ CAFEM v8.0 The peaks will be determined by x1: width, x2: height and x3: diameter Complex function

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 29 Requirements (1)Outline of the factors in process x1: WidthSome 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: FrequencyControllable 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.

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 30 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.

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 31 Uniform design Three dimensional uniform design may be the best choice. However is applied because of computational restriction,

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 32 http://www.math.hkbu.edu.hk/~ktfang/

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 33 frequency (210 levels) x1: 3.14 x2:1.28 x3: 003 x1: 3.00 x2: 0.78 x3: 0.03

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 34 Approximation (1) It is beneficial to find levels of x1, x2, x3 and f with resonance precisely rather than an well fitted model uniformly in the space of x1, x2, x3 and f. (2) The main aim is to find levels x1, x2, x3 and f with resonance. The estimation of the moment is not the major aim. (3) A RBF (radial basis function) like model is applied for the approximation. ＊ RBF is sometimes applied to describe the impulse response in neural network.

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 35 Model x1=3.00, x2=0.78 X3=0.03

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 36 x1: 3.14 x2:1.28 x3: 003

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 37 (4) Fit a model for each run (No. 1~30), i.e. estimate the following parameters (5) The estimates are treated as response variables whose factors are x1, x2, x3 and f. An approximation of M=g(x1, x2, x3, f) is derived as follows:

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 38 (4) Fit a model for each run (No. 1~30), i.e. estimate the parameters

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 39 (5) The estimates are treated as response variables whose factors are x1, x2, x3 and f.

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 40 Using the approximation (x1=2.143, x2=1.357, x3=0.03)

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 41 (x1=3.000, x2=0.786, x3=0.03)

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 42 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.

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 43 Application of the approximation x1: width 3mm, x3 diameter 0.3 ｘ 2 height frequency f Good choice

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University of Tsukuba MBA in International Business July 11, 2006 Shu YAMADA and iroe TSUBAKI 44 4. Grammar of DOE in computer simulation study 1. Interpreting requirements 2. Developing a simulator 3. Applying the simulator Optimization from various viewpoints Output Input Define Validation knowledge in the field Validation: Comparison of simulation results to reality Application of DOE depending on the situation

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