The Optimal Design for Low Noise Intake System Using Kriging Method with Robust Design∗

873

Kyung-Joon CHA∗∗, Chung-Un CHIN∗∗∗,Je-SeonRYU∗∗ and Jae-Eung OH∗∗∗∗

This paper proposes an optimal design scheme to improve an intake’s capacity of noise reduction of the exhaust system by combining the Taguchi and Kriging method. As a measur- ing tool for the performance of the intake system, the performance prediction software which is developed by Oh, Lee and Lee (1996) is used. In the first stage, the length and radius of each component of the current intake system are selected as control factors. Then, the L18 table of orthogonal arrays is adapted to extract the effective main factors. In the second stage, we use the Kriging method with the robust design to solve the non-linear problem and find the optimal levels of the significant factors in intake system. The L18 table of orthogonal arrays with main effects is proposed and the Kriging method is adapted for more efficient results. We notice that the Kriging method gives noticeable results and another way to analyze the intake system. Therefore, an optimal design of the intake system by reducing the noise of its system is proposed.

Key Words: Intake System, Sound Reduction, Kriging Model, Robust Design

to reduce the intake noise has been applied by the method 1. Introduction of trial and error after the design of the engine room is The noise from the exhaust system and the intake sys- finished. In addition methods of the excessive noise re- tem of an automotive vehicle has a uncomfortable effect duction cause a bad effect rather than reduce the intake on the riding quality of the passenger well as causes the noise. environmental noise. In addition, as the number of an au- It is difficult to design an optimal automotive intake tomotive vehicle is increasing, the quietness in a vehicle’s system because the design of the intake system affects the passenger compartment becomes one of the most impor- engine performance and the space of the engine room is tant performances of high quality vehicle, and recently the limited. Recently, various analysis methods (the trans- intake noise is being considered the important object of fer matrix method, the acoustic finite element method and the research. etc.) and the experimental method using the simulator are In general, the intake noise is the low frequency noise proposed(9), (25) – (28), (33). below 500 Hz. The booming noise generated by the intake However, it requires a lot of time and cost so noise transferring to the interior of the vehicle has a un- far to design the intake system optimally because vari- comfortable effect on the riding quality. However, it is a ous analysis methods and the experimental method de- difficult to reduce the time and the cost for the develop- pend on the method of trial and error. Therefore, the ment of the low noise intake system because the method Taguchi method which can improve the performance of the system with a low cost and time is frequently ∗ Received 21st November, 2003 (No. 03-5145) applied(16), (18), (23), (37) – (39), (43). ∗∗ Department of Mathematics, Hanyang University College In this study, the characteristics of the noise reduction of Natural Sciences, 17 Haengdang-Dong, Seongdong- are evaluated using the intake system (1 500 cc, DOHC en- Gu, Seoul 133–791, Korea. gine) as shown Fig. 1. Also, in order to solve the non- E-mail: [email protected], [email protected] linear optimization problem which is inefficient in the ro- ∗∗∗ BK21 Division in Mechanical Engineering, Hanyang Uni- bust design for the intake system, the Kriging method is versity, 17 Haengdang-Dong, Seongdong-Gu, Seoul 133– ff 791, Korea. E-mail: lada @hanmail.net applied. That is, after the most e ective factor is selected ∗∗∗∗ Division of Mechanical Engineering, Hanyang Univer- using the robust design, the level of the significant param- sity, 17 Haengdang-Dong, Seongdong-Gu, Seoul 133– eter is subdivided and the non-linear optimal condition of 791, Korea. E-mail: [email protected] parameters is obtained in the limit condition by applying

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one conﬁdence that the design is “inﬁlterating” the design space well and important in the robust design. Nomenclature

β : the unknown vector and should be estimated from n observed values βˆ : the usual generalized least squared estimate of β C12 : a real part of the cross spectrum for inlet f (x) : a known linear function of input x k : wave number T {pr vr} : the state vector at the upstream point r T {pr−1 vr−1} : the state vector at the downstream point r−1 Fig. 1 Overview of an intake system + p1 : an inlet sound pressure − p2 : an outlet sound pressure the Kriging method, which can estimate the mutually cor- Q12 : an imaginary part of cross spectrum for inlet related data. r(x) : the correlation vector between the response val- Kriging is based on the field of geostatistics, in ex- ues at the observed points x1,···, xn and the re- amples, hybrid discipline of mining engineering, geol- sponse at a given location x ogy, mathematics and statistics. The approach to pre- R : correlation matrix of input x diction advocated in this paper is known as Kriging af- R(xi, x j) : correlation function between any two points xi ter Dr. D.G. Krige’s work(17) on the Rand gold deposit, and x j of n sampled points in southern Africa. He developed an empirical method S aa : an incident spectrum for inlet for determining a true ore grade distribution from distri- S bb : a reflected spectrum for inlet butions based on sampled ore grade in the 1950’s. Since S cc : an incident spectrum for outlet (22) 1963, Matheron have developed this Kriging technique S dd : a reflected spectrum for outlet in France. This performs well in predicting the value of wi : the energy of inlet a possible but actually not taken observation of a spatially wt : the energy of outlet (17) distributed variable such as a mine grade , a soil char- yij : the transmission loss acteristic(41), rain fall(1), gene frequency(31),orimagese- y(x) : the unknown function of interest quence coding(11). Z(x) : the realization with mean 0, variance σ2, nonzero Recently, Kriging goes by a variety of names in- covariance cluding DACE (Design and Anlaysis of Computer Ex- periments) model, which is the title of the inaugural pa- 2. Backgrounds of Simulator and Experiments (34) per by Sacks, et al. This is derived from geostatistics 2. 1 Analysis of simulator and used for fitting the model of the deterministic output As a method of modeling the transfer characteristic of ffi as the realization of random process for e cient predict- acoustics, the transfer matrix method which introduces the ing. There have been some studies for DACE model at concept of impedance is used. It is widely used for acous- AIAA (American Institute of Aeronautics and Astronau- tic systems for its computational simplicity. This method tics, inc.), in particular. Sacks, et al., first introduced Krig- makes a design easy since modeling by each factor makes ing as a tool of interpolation of deterministic computer up the whole system. experiments in 1989. They proposed to use the Kriging Adopting acoustic pressure p and mass velocity v method with space filling design such as minimizing inte- as the two state variables in the transfer matrix method, grated mean squared error (IMSE), minimizing maximum we could find the four-pole parameters from the condi- mean squared error (MMSE) and maximizing entropy. tions of both sides which can be written as Eq. (1), where Giunta(14) and Giunta, et al.(15), performed a preliminary T {pr vr} is called the state vector at the upstream point r investigation into the use of Kriging for the multidisci- T and {pr−1 vr−1} is called the state vector at the downstream plinary design optimization of a high speed civil transport point r −1. aircraft. Booker(3) and Booker, et al.(5), use the Kriging method to study the aeroelastic and dynamic response of pr = Transfer matrix pr−1 v × v (1) a helicopter rotor with orthogonal arrays. Simpson(36) and r 2 2 r−1 Ryu, et al.(32), perform the Kriging method with evenly The transmission loss is an energy loss of acoustic spaced samples. elements, so the ratio of sound pressure between the in- The experimental design we used in this paper is let and outlet of acoustic elements can be expressed in dB orthogonal arrays and the Taguchi method. These give scale. Equation (2) shows a ratio between incident and re-

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Fig. 2 Schematics of transmission loss measurement

flective pressure through acoustic elements. Also, a two- microphone method is used at the end of the acoustic element to remove the influence of reflected waves. + w p = i = 1 TL[dB] 10log10 w 20log10 − (2) t p2 where wi is the energy of inlet, wt is the energy of outlet, (a) Block diagram of the experimental set-up + − p1 is an inlet sound pressure, and p2 is an outlet sound + − pressure. Here, p1 and p2 are derived from Eq. (1). Fig- ure 2 shows schematics of transmission loss measurement. Transmission loss obtained from Eq. (2) is used to inter- pret a intake system. 2. 2 Analysis of experimental results The two-microphone method separates the incident wave and the reflected wave in the pipe. The transmission loss can be written as Eqs. (3) and (4) using two- (b) Transmission loss measurement using the two microphone microphones. method S = aa Fig. 3 Experimental setup TL[dB] 10log10 (3) S cc S aa( f ) = [S 11( f )+S 22( f )−2C12( f )cosk(x1 − x2) 2 +2Q12 sink(x1 − x2)]/4sin k(x1 − x2)

S bb( f ) = [S 11( f )+S 22( f )−2C12( f )cosk(x1 − x2) 2 +2Q12 sink(x1 − x2)]/4sin k(x1 − x2)

S cc( f ) = [S 33( f )+S 44( f )−2C34( f )cosk(x3 − x4) 2 +2Q34 sink(x3 − x4)]/4sin k(x3 − x4)

S dd( f ) = [S 33( f )+S 44( f )−2C34( f )cosk(x3 − x4) 2 +2Q34 sink(x3 − x4)]/4sin k(x3 − x4)(4)

Where S aa is an incident spectrum for inlet, S bb is a reflected spectrum for inlet, S cc is an incident spectrum Fig. 4 Simplified model for outlet, and S dd is a reflected spectrum for outlet. Also, C12 is a real part of the cross spectrum for inlet, Q12 is an imaginary part of cross spectrum for inlet, and k is wave air cleaner, pipe and resonator. The manifold and plenum number. are not considered as design parameters because the man- Figure 3 (a) is a block diagram of the experimental ifold and plenum are designed by considering the engine setup and Fig. 3 (b) shows the experimental setup of a performance in the early stage. Also, the resonator is not transmission loss measurement using Eqs. (2) and (3) in considered as the design parameter because the resonator detail. We install a non-reflected part (Anechoic termina- is designed after the intake system is designed. tor) in the outlet for complete separation of the reflected The simplified intake system is represented in Fig. 4, wave in Fig. 3 (b). Because Eq. (3) is constructed exclud- and, control factors and levels are represented in Table 1. ing the information in reflected spectrum, the experiment Table 1 shows 8 factors that are chosen as the control fac- is performed to get S aa and S cc as shown in Fig. 3 (b). This tors to apply to the preliminary experiment, and the bold experimental value is used to evaluate the performance of faced numbers are values of the current level. These con- aintakesystem. trol factors, which are expected to contribute to the characteristic value, are chosen by experience so that the ex- 3. Design of Experiments periment can be easily modified and the total size of ex- 3. 1 Object of experiment and design periments can be kept to a minimum in the design process The intake system consists of the manifold, plenum, as well.

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1 7 Table 1 Control factors and levels Table 2 Parameter design using L18(2 ×3 ) orthogonal array

3. 2 Design of Taguchi method In order to design the parameter, tables of orthogonal arrays and Taguchi method which are important in the robust design are applied. In Table 2, the design parameters using the model of mixed tables of orthogonal arrays 1 7 L18(2 ×3 ) which is proper for control factors and levels shown in Table 1 are represented(2), (6), (8), (20). The eight control factors are arranged on a L18 table of orthogonal arrays which is given in Table 2. For an intake performance analysis, the intake performance prediction software that is developed by one of the authors and his associates is used. Figure 5 shows the comparison between the simulation and the experimental results. In general, we would use 0 – 300 Hz for frequency range of interest. Note that the booming noise, which overwhelms the intake noise, hardly occurs in the range over 200 Hz. Fig. 5 Comparison between simulation and Even though an intake has a frequency characteristic over experimental results 1 000 Hz, it can be neglected by the absorbing characteristics of the car interior. We can see that there is no signifi- 3. 3 Kriging model cant difference between the simulation and the experimen- The Kriging model is formulated as a combination of tal results in the interesting region (0 – 300 Hz). Hence, we a linear regression model plus departure, i.e., would say that the performance prediction software can be used to convert simulation results into the transmission y(x) = β f (x)+Z(x), (5) loss of a system. where y(x) is the unknown function of interest, f (x)isa Control factors (A, B, C, D, E, F, G, H) shown in Ta- known linear function of input x and Z(x) is the realiza- ble 1 are placed in the inner array and uncontrollable fac- tion with mean 0, variance σ2, nonzero covariance. Also, tors (U: temperature, V: humidity, W: noise) are placed β is the unknown vector and should be estimated from n in the outer array. When the analysis of variance is pro- observed values(1), (3), (5), (13) – (15), (19), (21), (32), (34) – (36). ceeded, the SN ratio is used as the characteristic value. The covariance matrix of Z(x) is represented by The characteristic value y is the transmission loss which ij i , j = σ2 i, j , , = ,···, , is used to evaluate the acoustic characteristic and the per- Cov[Z(x ) Z(x )] R[R(x x )] i j 1 n formance of the reduction of acoustic elements. The trans- where R is correlation matrix of input x and R(xi, x j)is mission loss is an energy loss of acoustic elements, so the correlation function between any two points xi and x j of ratio of sound pressure between the inlet and outlet acous- n sampled points. So, R(xi, x j) is specified by the users tic elements can be expressed in dB scale. as in Table 3, and reflects the association of the outputs The overall value, which is the average value of TL generated by computer code. in the frequency region of interest, is used as the char- In this paper, a unique θ value for each dimension is acteristics value and the larger-the-better characteristic is considered on the design space to [0,1]d. It is worth not- also applied because the larger value implies better perfor- ing that in some cases using a single correlation parameter mance. gives sufficiently good results(4), (29). So the exponential

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Table 3 Summary of correlation functions 0 F −λ(x) f (x) = . (9) FR c(x) r(x) Then, by inverting the partitioned matrix, the BLUP can be written as −1 yˆ(x) = f (x)βˆ +r (x)R (Yx − Fβˆ), (10) −1 −1 −1 where βˆ = (F R F) F R YX is the usual generalized least squared estimate of β. Assuming the Gaussian process, the likelihood is a function of the β’s, variance σ2 and correlation parameter θ. Given the correlation parameters, MLE of σ2 is 1 σˆ 2 = (y − Fβˆ)R−1(y − Fβˆ). (11) n x x With the definitions of βˆ andσ ˆ 2, the problem is to correlation function, which is used in this paper, may be maximize 1 rewritten as φ(θ) = −(detR) n σˆ 2, (12) d i j R(x , x ) = exp −θ |dk| . (6) which is a function of only the correlation parameter and k=1 the data. While any values for the θ in Eq. (6) create an in- Another term of interest is the correlation vector, terpolative approximation model, the best Kriging model r(x), between the response values at the observed points is found by solving the unconstrained nonlinear optimiza- x1,···, xn and the response at a given location x. This can tion problem given by Eq. (12) with respect to R andσ ˆ 2. be expressed as 4. Statistical Analysis of a Design Process r(x) = [R(x, x1),···,R(x, xn)]. (7) 4. 1 Design and analysis using L18 tables of orthog- Consider the linear predictor onal arrays yˆ(x) = c (x)yX In order to evaluate the above design factors and lev- 1 × 7 of y(x) at an untried x. We can replace yX by the cor- els, the experiment satisfying the condition of 2 3 are 1 n responding random quantity YX = {Y(x ),···,Y(x )}, treat done. The characteristic values of each experiment are ob- yˆ(x) as random, and compute the mean squared error of tained from the experiment explained in Table 2. this predictor averaged over the random process. The best Here, the characteristic values which are obtained linear unbiased predictor (BLUP) is obtained by choos- from the SN ratio are the overall value of the transmission ing the n × 1 vector c(x) to minimize mean squared er- loss from 0 Hz to 500 Hz. The result which is achieved 2 ror (MSE), MSE[ˆy(x)] = E[c (x)YX −Y(x)] , subject to the by the analysis of variance for the SN is represented in (20), (30), (39), (40) unbiasedness constraint E[c (x)YX] = E[Y(x)]. To give Table 4 . some technical details connected with the implementing From Table 4, it is noticed that the reliability is gener- 2 the BLUP of the response at an untried input we use the ally high since R is 0.998 647. Also, it is verified that the p-value for B factor is below 0.01 in detail and the p-values notation f (x) = [ f1(x),···, fd(x)] for the d functions in the regression, of A, B, C, D, E, F factors are below 0.05. Therefore, it is   1 noticed that B factor is the most significant.  f (x )   .  From the previous study, the interaction of the intake F =  .   .  system is not considered since interaction between factors   f (xn) can be ignored and even can be assumed independent(27). ff for the n×d design matrix, R = {R(xi, x j) | 1 ≤ i, j ≤ n},for The e ectiveness for each factor and level by using the n×n matrix of correlations, and r(x) for the vector of the mean square is represented in Fig. 6. It is verified that correlations with this definition. Then, MSE ofy ˆ(x)is optimal level is A2B3C1D3E1F1G2H1 and sensitive design parameter to the SN ratio is A (the diameter of a/c), B E[c(x)Y −Y(x)]2 x (the length of the outlet pipe), C (the diameter of the outlet = σ2 , − Rr(x) c(x) . pipe), D (the length of a/c), E (the length of the inlet pipe), [c (x) 1] − (8) r (x)1 1 and F (the diameter of the inlet pipe). The BLUP of Y(x) is obtained by minimizing The pareto diagram for the mean square is repre- MSE[ˆy(x)] subject to Fc(x) = f (x). Using the Lagrange sented in Fig. 7, and it shows that B factor is a more sig- multiplier for the constrained minimization of the MSE nificant than other factors. The result of a optimization for produces Rc(x) = Fλ(x) + r(x) and the coefficient c(x)of the main factors except for insignificant G and H factors the BLUP must satisfy is represented in Table 5.

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Table 4 ANOVA table for SN data

Fig. 8 Comparison between current and optimal designs (simulation)

Fig. 6 SN ratio versus control factors (optimal value : )

Fig. 9 Maximum likelihood function

factors because these factors are verified to be the most significant two factors from L18 tables of orthogonal arrays. So, C, D, E, F, G and H factors are kept in current levels because these are not significant. Firstly, we find Kriging estimate in section 3.3. In Eq. (10), the Kriging estimate is only dependent on correlation matrix R, so we have to solve the unconstrained nonlinear optimization problem in Eq. (12). The scalar Fig. 7 Pareto diagram for mean square coefficient of correlation θ in Eq. (6) has to be estimated from a one-dimensional analytic function in Eq. (12). Table 5 Optimum specification of an intake system In fact, the determination of θ requires another optimization process. The golden section refinement is used to solve it. As in Fig. 9, the optimal coefficient of correlation θ is determined as θ = 38.196 66 with 2 levels of A and 3 levels of B factors. To find the Kriging prediction values of A and B fac- The re-analysis result using the software for the per- tors, each factor is sampled at 30×30 = 900 equally spaced formance evaluation of the intake system (Oh, J.E., Han, levels between the minimum and maximum of it. Al- K.H., Son, D.Y., 1996) applying the above condition though there are many optimization methods, in this paper, (A2B3C1D3E1F1) is represented in Fig. 8. The overall we only use many equally spaced samples. The Kriging level of TL is 7.049 dB that is larger than current speci- prediction surface of SN ratio for A and B factors is rep- fication. resented in Fig. 10. From this, it is verified that the max- 4. 2 The optimization using the Kriging method imum value is determined in the neighborhood of the last In this section, we find the optimal level of A and B level of the B factor, but accurately the last level is not the

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Table 6 Comparing before and after

Table 7 Optimum speciﬁcation of an intake system

Fig. 10 Interpolation graph using Kriging method

Fig. 13 Simulation results of transmission loss from Kriging method and current exp. Fig. 11 SN ratio versus B factor

all value and SN ratio. As these results, L18 and Kriging method increase respectively 7.049 dB and 14.259 dB as compared to current condition. An optimal level of factor A and B is presented in Table 7. We can see that the optimal level can be obtained. Figure 13 shows comparison of values between current experiment and Kriging method. As shown in Fig. 13, the Kriging method gives better results. 5. Conclusions The Taguchi method can be easily applied to find significant main effects. Thus, for the preliminary experiment, the L18 design is first performed for the 8 control factors. The L18 design could not take interactions of all Fig. 12 SN ratio versus A factor combinations into consideration, and it is possible to miss significant interactions to the optimal condition. The inter- maximum value. action of the intake system is not considered since interac- Figure 11 shows the predicted value of SN ratio for tion between factors can be ignored. Also, the Kriging B factor. As shown in this, it does not have the maximum method is adapted to analyze the significant main factor to value at last level but has the maximum value within level, get the optimal condition. which is similar to the result of Taguchi method. Also In this experiment, there is the large reduction of time Fig. 12 shows the SN ratio for A factor. It shows the same by using Taguchi method. Also, in the case of the non- result as the Taguchi method, that is, the maximum value linear optimization problem, the Kriging method searches is obtained at the last level. provided a more accurate optimal condition for the inter- Table 6 is a comparison between current condition, esting factor. orthogonal arrays L18 and the Kriging method for over- In this paper, the conclusions of noise reduction of

JSME International Journal Series C, Vol. 47, No. 3, 2004 880 intake system are as follows. Astronautics, 8th AIAA/USAF/NASA/ISSMO Sym- 1 ) The new method called the Kriging method is in- posium on Multidisciplinary Analysis and Optimiza- troduced in the design. This method based on statistical tion, 2000/Long Beach, CA, AIAA-2000-4754, (2000), theory can be used in order to solve highly correlated and pp.381–391. ( 8 ) Cochran, W.G. and Cox, G.M., Experimental Designs, nonlinear problem. Therefore the new method for predic- (1957), John Wiley & Sons, New York. tion is proposed that has more systematic and theoretical ( 9 ) Craggs, A., A Finite Element Approach for Damped backgrounds. Acoustic Systems: An Application to Evaluate the Per- 2 ) It can be validated that application of SN ratio at formance of Reactive Mufflers, Journal of Sound and the optimum design procedure obtained a better result than Vibration, Vol.48 (1976), pp.377–392. the application of experiment value directly at the opti- (10) Cressie, N., Statistics for Spatial Data, (1991), pp.1– mization. 143, John Wiley & Sons, New York. 3 ) By applying the Taguchi method to the optimiza- (11) Deceneiere, E., Fouquet, C. and Meyer, F., Appli- cations of Kriging to Image Sequence Cooling, Sig- tion problem, there is the large reduction of time. Also, the nal Processing; Image communication, Vol.13 (1998), nonlinear problem could be solved by Kriging method. pp.227–249. / 4 ) Design parameters such as A (diameter of a c), (12) Dubrule, O., Two Methods with Different Objections, B (length of outlet pipe), C (diameter of outlet pipe), E Splines and Kriging, Mathematical Geology, Vol.15 (length of inlet pipe) and F (diameter of inlet pipe) are (1983), pp.245–257. sensitive in the effect on the transmission loss of intake (13) Fedorov, V.V., Design of Spatial Experiment: Model system. Also, B (length of outlet pipe) is the most sensi- Fitting and Prediction, ORNL/TM-13152, (1996), Oak tive in the effect on parameter of the transmission loss of Ridge National Laboratory. (14) Giunta, A.A., Aircraft Multidisciplinary Design Op- intake system. timization Using Design of Experiments Theory and 5 ) The overall level of transmission loss by optimum Response Surface Modeling, Ph.D. Dissertation and design using L18 is increased by 7.049 dB as compared MAD Center Report No.97-05-01, Department of with current design and optimum design by the Kriging Aerospace and Ocean Engineering, Virginia, Polytech- method is improved the transmission loss up to 14.259 dB. nic Institute and State University, Blacksburg, VA, (1997). Acknowledgement (15) Giunta, A.A., Watson, L.T. and Koehler, J., A This work was supported by the research fund of Comparison of Approximation Modeling Tech- Hanyang University (HY-2003-BT-003). nique: Polynomial Versus Interpolating Models, 7th AIAA/USAF/NASA/ISSMO Symposium on Multi- References disciplinary Analysis & Optimization, St. Louis, MI, (1998), AIAA-98-4758, AIAA. ( 1 ) Bacchi, B. and Kottegoda, N.T., Identification and Cal- (16) Kim, J.K., Optimum Design of a Heat-Exchanger-Fan ibration of Spatial Correlation Patterns of Rain Fall, Casing of Clothe Using the Taguchi Method, KSME Journal of Hydrology, Vol.165 (1995), pp.311–348. International Journal, Vol.13, No.12 (1999), pp.962– ( 2 ) Bendell, A., Disney, J. and Pridmore, A., Taguchi 972. Methods: Applications in World Industry, (1989), (17) Krige, D.G., A Statistical Approach to Some Basic Springer-Verlag, New York. Mine Valuation Problems on the Witwatersrand, Jour- ( 3 ) Booker, A.J., Case Studies in Design and Analysis of nal of the Chemical, Metallurgical and Mining Society Computer Experiments, Proceedings of the Section on of South Africa, Vol.52 (1951), pp.119–139. Physical and Engineering Sciences, American Statisti- (18) Lee, K.H., Eom, I.S., Park, G.J. and Lee, W.I., Robust cal Association, (1996). Design for Unconstrained Optimization Using Taguchi ( 4 ) Booker, A.J., Conn, A.R., Dennis, J.E., Frank, P.D., Method, AIAA Journal, Vol.34, No.5 (1995), pp.1059– Trosset, M. and Torczon, V., Global Modeling for 1063. / / Optimization: Boeing IBM Rice Collaborative Project, (19) Mardia, K.V., Kent, J.T., Goodall, C.R. and Little, J.A., 1995 Final Report, ISSTECH-95-032, (1995), The Kriging and Splines with Derivative Information, Bio- Boeing Company, Seattle, WA. metrica, Vol.83 (1996), pp.207–221. ( 5 ) Booker, A.J., Conn, A.R., Dennis, J.E., Frank, P.D., (20) Mason, R.L., Gunst, R.F. and Hess, J.L., Statistical De- Serafini, D. and Torczon, V., Multi-Level Design Opti- sign and Analysis of Experiments, (1989), John Wiley / / mization: A Boeing IBM Rice Collaborative Project, & Sons, New York. 1996 Final Report, ISSTECH-96-031, (1996), The (21) Matheron, G., Traite de Geotatistique Appliquee, Tome Boeing Company, Seattle, WA. I. Memoires du Burean de Recherches Geologigues et ( 6 ) Box, G.E.P., Hunter, W.G. and Hunter, J.S., Statis- Minieres, No.14 (1962). tics for Experiments, (1978), John Wiley & Sons, New (22) Matheron, G., Principles of Geostatistics, Economic York. Geology, Vol.58 (1963), pp.1246–1266. ( 7 ) Chung, H.S. and Alons, J.J., Comparison of Ap- (23) Mishima, N., Ishii, K. and Mori, K., Robustness Es- proximation Models with Merit Functions for Design timation for Machine Tool Designs Using Taguchi Optimization, American Institute of Aeronautics and

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Method, SOURCE Journal of the Japan Society for in the 3-D Analysis of Expansion Chamber Muffler, Precision Engineering, Vol.64 (1998), pp.1502–1506. Japan of Sound and Vibration, Vol.143, No.7 (1991), (24) Mitchell, T.J. and Morris, M.D., The Spatially Corre- pp.371–394. lation Function Approach to Response Surface Estima- (34) Sacks, J., Welch, W.J., Mitchell, T.J. and Wynn, N.P., tion, Proceedings of the 1992 Winter Simulation Con- Design and Analysis of Computer Experiments, Statis- ference, Arlington, VA, (1992), pp.565–571, IEEE. tical Science, Vol.4, No.4 (1989), pp.409–435. (25) Munjal, N.L., Acoustics of Ducts and Mufflers, (1987), (35) Sasena, M.J., Optimization of Computer Simulations John Wiley & Sons, New York. via Smoothing Splines and Kriging Metamodels, M.S. (26) Oh, J.E., Han, K.H. and Lee, J.C., Analysis and Ad- Thesis, Department of Mechanical Engineering, Uni- vance for Performance of the Muffler of a Passenger versity of Michigan, Ann Arbor, MI, (1998). Car, Proc. of Spring Conference, (in Korea), (1995), (36) Simpson, T.W., A Concept Exploration Method for pp.726–732. Product Family Design, Ph.D. Dissertation, Depart- (27) Oh, J.E., Han, K.H. and Son, D.Y., The Design Tech- ment of Mechanical Engineering, Georgia Institute of nique for Low-Noise Intake System of Vehicle, Proc. Technology, (1998). of Autumn Conference, Korean Society of Automotive (37) Taguchi, G., Quality Engineering of Development and Engineering, (in Korea), (1996), pp.317–322. Design, Korea Standard Association, Translated to Ko- (28) Oh, J.E., Lee, K.T. and Lee, J.C., Development of the rean, (1991). Software for Analysis and Improvement of a Passenger (38) Taguchi, G., Experimental Designs for Quality Engi- Car’s Muffler, Journal of Automotive Engineering, Ko- neering, Korea Standard Association, Translated to Ko- rean Association of Automotive Engineering, (in Ko- rean, (1991). rea), Vol.4, No.6 (1996), pp.133–143. (39) Taguchi, G., Case Studies of Quality Engineering- (29) Osio, I.G. and Amon, C.H., An Engineering Design America and Europe, Korea Standard Association, Methodology with Multistage Bayesian Surrogates and Translated to Korean, (1991). Optimal Sampling, Research in Engineering Design, (40) Wang, H.T., Liu, Z.J., Chen, S.X. and Yang, J.P., Appli- Vol.8, No.4 (1996), pp.189–206. cation of Taguchi Method to Robust Design of BLDC (30) Park, S.K., Application of Quality Engineering for Motor Performance, SOURCE IEEE Transactions on Energy Efficiency of Compressor, The 12th Annual Magnetics, Vol.35, No.5 (1999), pp.3700–3702. Taguchi Symposium, Rochester, New York, (1994), (41) Webster, R., Quantitative Spatial Analysis of Soil in the pp.199–211. Field, Advances in Soil science, 3, edited by Stewart, (31) Piazza, A., Menozzi, P. and Cavalli-Sforza, L., The B.A., (1985), pp.1–70, Springer-Verlag, New York. Making and Testing of Geographic Gene-Frequency (42) Young, C.J. and Crocker, M.J., Prediction of Transmis- Maps, Biometrics, Vol.37 (1983), pp.635–659. sion Loss in Mufflers by the Finite Element Method, (32) Ryu, J.S., Kim, M.S., Cha, K.J., Lee, T.H. and The Acoustical Society of America, Vol.57, No.1 Choi, D.H., Kriging Interpolation Methods in Geo- (1975), pp.144–148. statistics and DACE Model, KSME International Jour- (43) Hwang, W.J., Park, G.J. and Lee, W.I., Structural nal, Vol.16, No.5 (2002), pp.619–632. Optimization Post Processing Using Taguchi Method, (33) Sahasrabudhe, A.D., Anantha, R.S. and Munjal, M.L., JSME Int. J., Ser. A, Vol.37, No.2 (1994), pp.166–172. Matrix Condensation and Transfer Matrix Techniques

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