Response Surface Methodology

Contents Conclusion Further Study 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

1.1 Concept of Response Surface MethodOriginal SystemRSM : Response Surface Method : Response Surface Model

1.2 Definition of Response Surface MethodA 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.Response surface method is a collection of statistical and mathematical techniques useful for developing, improving, and optimizing processes.Response surface method is a method for constructing global approximations to system behavior based on results calculated at various points in the design space.Box G.E.P. and Draper N.R.,1987Myers R.H., 1995Roux W.J.,1998

1.3 History of Response Surface Method1951 Box and Wilson - CCD1959 Kiefer - Start of D-optimal Design1960 Box and Behnken - Box-Behnken deign 1971 Box and Draper - D-optimal Design1972 Fedorov - exchange algorithm 1974 Mitchell - D-optimal Design1996 Burgee - design HSCT 1997 Ragon and Haftka - optimization of large wing structure 1998 Koch, Mavris, and Mistree - multi-level approximation1999 Choi / Mavris Robut, Reliablity-Based DesignResearch of DOEApplication in OptimizationApp in Optimization & Reduce the Approximation Error

1.4 Introduction - Motivation of RSMHeavy Computation Problem ApproximationWhen Sensitivity is NOT AvailableGlobal BehaviorReal / Numerical ExperimentWhen the Batch Run is ImpossibleFor Any System Which has Inputs and ResponsesEasy to ImplementPart of MDO, Concurrent EngineeringProbabilistic ConceptNoisy Responses or EnvironmentsApproximation ErrorSize of the Approx. Domain is Very DominantAdvantagesDisadvantages

Part I(Classical RSM)

DOE (Design Of Experiments) Experiments (Numerical) & DatabasesOptimization Using RS Model (Meta Model)Construction of RSM (Response Surface Model)

2.1 DOE 1 Factorial Design- 2 / 3 level Factorial Design- Full / Fractional Factorial Design2 level Full Factorial DesignFractional Factorial DesignClassifications

2.1 DOE 2 Central Composite Design(CCD)Factorial PointsAxial PointsCenter Pointsx1x2-1 0 1103 DV2 DVQuadratic RS ModelEffective than Full-Factorial DesignRotatabilityCharacteristics

Quadratic RS ModelEffective 3 Level DesignBalanced Incomplete Block DesignCharacteristicsBlock1Block2Block3Center Point2.1 DOE 3 Box-Behnken Design

Good fittingThe Most Popular DOEArbitrary Number of Experiment PointsPossible to Add PointsSpecified Functional Form of the ResponseCharacteristicsApproximation Function(RSM)Coefficients of RSMVariance of Coefficients2.1 DOE 4 D-Optimal Design

2.1 DOE 5 Latin-Hypercube DesignArbitrary Number of Experiment PointsNo Priori Knowledge of the Functional Form of the ResponseCharacteristics1. No. of Levels = No. of Experiments2. Experiment points in the design space are distributed as regular as possible.No. of Variables : kNo. of Experiments : nInitial InformationMain Principles

DOE (Design Of Experiments) Experiments (Numerical) & DatabasesOptimization Using RS Model (Meta Model)Construction of RSM (Response Surface Model)

2.2 Experiments (Numerical) & DatabasesBlack BoxedSystem(FE Model)InputResponse*.bdf*.cdb*.f06 , *.pch*.rstNASTRANANSYSRewrite Input FilesRead Output Files

DOE (Design Of Experiments) Experiments (Numerical) & DatabasesOptimization Using RS Model (Meta Model)Construction of RSM (Response Surface Model)

2.3 Construction of RSM Least Squares MethodInputResponse- Global Approximation 1 RS Function at all pts Constant Coefficients

- 1. Approximation Function(RSM)- 2. Least Squares Function- 4. The coefficients of the RS model - 3. Minimize Least Squares Function2.3 Construction of RSM Least Squares Method()DOE

2.3 Example Construction Of RSMSoftwareJMP, SAS, SPSSMATLAB Statistics ToolboxVisual-DOCIn-House CodesNumber of Design Variable = 2Number of Experiment(FFD) = 9RS Model = Quadratic ModelOriginal FunctionRSM Function

6math-2dv-table_analysis

NoXlXu

NOreal_x[1]real_x[2]output(1)-510NOreal_x1real_x2output(1)

1-5.0-5.0-194.2RSM11-5.0-5.0-194.2RSM1

22.5-5.0-98.81-30.16RSM1x1x2obj2-5.02.5-102.91-30.16

310.0-5.0-228.7x17.89Xopt2.002674.649950.583453-5.010.0-128.4x17.89

4-5.02.5-102.9x29.82True opt25642.5-5.0-98.8x29.82

52.52.5-4.9x1*x1-2.00Error(%)0.134-7.001-90.27652.52.5-4.9x1*x1-2.00

610.02.5-131.8x1*x20.03XlXu62.510.0-30.3x1*x20.03

7-5.010.0-128.4x2*x2-1.06x1-510710.0-5.0-228.7x2*x2-1.06

82.510.0-30.3x2-510810.02.5-131.8

910.010.0-157.0910.010.0-157.0

1-1.01.6-33.845590

22.01.6-5.560000RSM2

35.01.6-33.8455901-36.9533784593

4-1.04.6-15.985860x110.9315140741RSM2x1x2obj

52.04.65.840000x213.3437499259Xopt1.99855.18.01582

65.04.6-15.985860x1*x1-2.7328785185True opt256

7-1.07.6-24.436270x2*x2-1.307782963Error(%)-0.075233.597

82.07.6-0.760000XlXu

95.07.6-24.436270x1-15

x21.67.6

10.83.9-2.620000

22.03.94.790000RSM3

33.23.9-2.5800001-94.7374309609Xopt

2.3 Construction of RSM Test CriteriaThe model was fitted well. F-Test (ANOVA)

The coefficient of determination t-TestAdjusted R2 Where Cjj diagonal term in (XX)-1 corresponding to bj xj is a dominant term of RS model 1.01.0Prediction Test2.3 Construction of RSM Test Criteria (Continued)

2.3 Construction of RSM Variable SelectionAll Possible Regression Stepwise Regression Original SystemRS ModelConcept Forward regression Backward regression Stepwise regression (Backward + Forward )MinimizeUnnecessary Term

DOE (Design Of Experiments) Experiments (Numerical) & DatabasesOptimization Using RS Model (Meta Model)Construction of RSM (Response Surface Model)

2.4 Optimization Using RSM - Whole SequencesApproximation DomainDOE & ExperimentsConstruct RSMOptimization Using RSMEstimated Opt ResponseFinal Optimal SolutionOptimization ProblemVariable Selection

2.5 Example 1 System / Problem SetupInitial variablesMin : weights.t : Problem SetupSystem(FE Model)

2.5 Example 1 Optimization Using RSM

2.5 Example 2 - Induction Motor FE Model UpdateModel : WM0F3A-S Induction motor Upper HousingRotorStatorLower HousingReal SystemFE Model (NASTRAN)LMS CADA-XReliable FE Model ?? (close to Real Model )

2.5 Example 2 - Modal Analysis(1/2)Rotor12Lower Housing12Upper Housing21Stator21

42-4dvdot_results

DVPhysicalXLXUX_OptexperiInitialerror(%)predictedpredic errrealerror(%)

x1E of shaft1.50E+112.30E+112.251087E+11f19739851.23973.01-0.56978.46680.56

x2E of mass around the shaft2.00E+101.50E+112.748357E+10f23305371112.283305.33-1.143343.4171.16

x3density of shaft5.50E+031.02E+049.103201E+03

x4density of mass5.50E+031.02E+045.728883E+03

RSMf1f2

1946.573506.15

X197.29334.41

X224.98194.90

X3-116.17-505.70

X4-31.67-34.08

X1*X1-4.58-14.65

X2*X14.9437.91

X2*X2-13.52-105.41

X3*X1-11.60-45.47

X3*X2-3.25-30.68

X3*X326.16110.51

X4*X1-3.45-6.72

X4*X20.054.38

X4*X3-1.451.23

X4*X48.376.08

table_analysis2

============< EXPERIMENT RESULTS>===========

--------------------------------------------------------------------------------------------------------------------------------------------------------------------

Nonorm_x(1)norm_x(2)norm_x(3)norm_x(4)norm_x(5)output(1)output(2)output(3)output(4)output(5)output(6)

--------------------------------------------------------------------------------------------------------------------------------------------------------------------

11.00000-1.000001.00000-1.00000-1.00000534.77170725.75910750.93360788.439801559.255001553.62100RSMf1f2f3f4f5f6

2-1.000001.000001.000001.00000-1.00000479.76860645.12040654.03290702.097701328.839001323.767001548.0439743.2894772.9851834.60161504.37491495.6679

31.00000-1.000001.000001.000001.00000518.39940692.32230725.75910787.576801456.457001446.40800X188.9624112.0083121.9068132.3060279.0827276.2355

4-0.483561.00000-1.000001.00000-1.00000617.51400843.59510877.17900943.010201674.687001666.22800X254.441276.985980.124784.2272149.4518148.9536

5-1.00000-0.41306-1.000001.000001.00000454.71750672.16830700.37220749.328401179.787001174.04000X3-41.2243-68.3565-76.6626-84.0448-86.8328-85.5759

6-0.512971.00000-1.00000-1.000001.00000573.36260834.87080872.77910936.409101503.949001495.76500X40.6236-1.8266-1.87180.30812.17041.7945

71.00000-1.00000-1.00000-1.000000.20087614.59280843.43440888.86860963.682401684.116001671.69000X5-13.4569-8.1772-3.68360.0015-63.4594-64.2435

81.00000-0.10189-0.03064-1.000001.00000616.26260838.07030882.47330957.018801699.889001687.51100X1*X1-2.9050-1.44552.0571-1.8728-3.4211-2.9685

90.000380.009110.01237-1.00000-0.20345550.70090741.45690773.45880834.014401516.832001508.26800X2*X18.233110.040911.682312.039726.277026.1001

101.00000-1.00000-1.000001.00000-1.00000634.90690847.91830888.86980964.580601783.788001771.48100X2*X2-2.7138-6.1382-5.6414-4.7563-7.0117-7.1538

11-0.08905-1.00000-0.069540.115201.00000472.94490657.50590687.35150740.084701273.567001266.16900X3*X1-8.2340-8.0816-9.7955-12.3413-18.5078-17.9511

12-1.000001.00000-0.06239-0.085730.16044499.60630695.72540720.96730772.731001333.066001327.20000X3*X2-3.4086-6.8931-10.0031-7.7431-6.5885-6.9472

13-0.318631.000001.000001.000001.00000519.23240706.82120735.89190791.873401413.817001406.35600X3*X34.038614.549514.405313.08966.40986.7105

14-1.00000-1.00000-1.00000-1.000001.00000424.53500627.69020654.02940699.745701101.463001096.09900X4*X10.4942-0.6353-1.81050.76501.60981.2187

151.000001.000001.00000-1.000000.68970637.84310854.58180888.86890964.737201799.815001787.70700X4*X20.6120-0.24791.30990.73410.73290.7866

161.000001.00000-0.106261.000000.00507701.17350936.32150984.222201067.977001965.080001951.40100X4*X3-0.82920.83632.0438-0.7567-1.8627-1.4214

17-1.00000-0.71783-1.00000-0.98631-1.00000472.23090653.38680676.70670725.424501264.415001258.90900X4*X40.0736-3.8829-5.3938-1.19710.6660-0.0535

19-1.00000-1.000001.000001.00000-1.00000391.77350526.94090534.01570573.260801085.255001081.11300X5*X11.3416-3.3939-5.31431.4236-3.8684-4.9619

200.08853-0.31342-0.051140.05249-1.00000553.68380744.58720763.20000823.844901552.609001544.23400X5*X2-0.9562-1.1874-0.94530.6149-5.1926-5.4441

211.000000.386941.000000.21867-1.00000620.58900842.23540871.37160914.975801809.421001802.87400X5*X33.2141-3.8777-0.8033-0.85439.12459.0700

22-0.26917-1.000001.00000-0.153480.12740435.91310582.53520605.70320652.324301205.054001198.88600X5*X4-0.33191.40291.7532-0.0788-1.2220-0.9251

23-1.00000-0.182631.00000-1.000001.00000408.92570565.84500586.04080628.231101094.878001090.11200X5*X5-0.02685.71234.44870.52423.24793.9552

241.000000.56748-1.000000.296431.00000709.19790993.304701048.665001136.249001918.790001904.38900

250.03894-0.13044-0.024950.11185-0.10011546.74130737.79880770.10830830.521901503.750001495.11200

26-0.514791.000001.00000-1.00000-1.00000524.16660711.01910712.62540766.975801471.670001465.37100

27-0.01132-0.08150-1.000000.02262-0.07203588.93910818.85550856.65240922.779101586.984001577.57100

28-1.000000.15386-0.033060.086420.00098464.44350643.20930666.22360714.166901242.913001237.49100

290.19101-0.071631.000001.00000-0.12982523.42620704.79840721.37420779.163001471.012001462.92200

f1~f5

opt ptgoodDVPhysicalXLXUX_Opt

x1thickness1.00E-031.40E-031.15E-03

x2E1(body)8.00E+101.20E+119.12E+10

x3density12.40E+033.60E+032.40E+03

x4E2(solid)1.60E+112.40E+112.28E+11

x5density26.29E+039.43E+038.75E+03

at opt ptexperiInitialerror(%)predictedpredic errrealerror(%)

f1567548-3.35533.15640.35531.29922-6.30

f27167413.49754.1939-0.46757.67895.82

f3790773-2.15785.656-0.87792.54790.32

f4904834-7.74852.83870.11851.9276-5.76

f51339150312.251414.3050.331409.7025.28

f6144614953.391404.909-0.201407.674-2.65

f1~f6bad

opt ptthicknessE1(body)density1E2(solid)density2

x1x2x3x4x5

1.12E-031.01E+112.56E+032.00E+118.90E+03

at opt ptexperiInitialerror(%)predictedpredic errrealerror(%)

f1567548-3.3509700176535.328-7.4163366251578.211.9770723104

f27167413.4916201117759.821-6.4110017614811.8713.3896648045

f3790773-2.1518987342793.457-6.7157704155850.587.6683544304

f4904834-7.7433628319853.5425-6.8490123322916.31.360619469

f51339150312.24794622851420.99231-8.35352705391550.51515.7964899178

f6144614953.38865836791412.9608-8.25334985211540.0686.5053941909

12table_doe-normalized

Nonorm_x[1]norm_x[2]norm_x[3]norm_x[4]output(1)output(2)output(3)output(4)

1111-1861.37991119.88201262.22002127.5100

2-11-11593.9438772.7447876.77491455.3170

310.063-1-1692.2947886.3386932.30091581.0520

411-11594.4263772.7935886.57991475.9830

51-0.057811664.2859847.9056899.98641508.0560

61-1-11533.0735647.6674658.67111097.1680RSMf1f2f3f4

710.01540.0017-1756.4104967.29331013.34301719.39801675.136238866.561642903.5014471519.143373

8-11-1-1727.4170946.41381055.03201772.8130X10.4884543.1539654.44446517.71174

9-1-111629.9476749.6078779.28511255.0930X237.33801284.728206130.236345234.481056

1011-1-0.1022657.9166855.3408973.55301629.8210X357.33865471.27228578.080769128.985523

11-0.0661-11-1771.8763907.7280954.45741549.3810X4-69.093909-83.177185-88.319683-150.983724

12-1-1-11532.4053633.6316658.61611060.8180X1*X10.295007-0.0664531.5242853.100661

131-1-1-1652.8362776.6903806.70101335.0060X2*X10.087048-4.101323.339613-4.169078

141-11-1772.4413918.8292954.50151579.4790X2*X20.76397-21.57188415.1659410.717691

15-0.02361-0.0931-1790.90321028.69501152.26001938.0650X3*X10.030551-0.8720181.4557261.454112

16-0.0004111703.0032914.35051042.24201731.9390X3*X23.0267967.1939710.49383219.276498

17-1-1-1-1652.0067759.5267806.63351290.4260X3*X3-2.830685-3.866636-3.94694-6.71652

18-1110.1231760.5209989.48431115.47501859.8020X4*X1-0.07608-0.825291-0.170855-2.048003

191-10.15510.1714637.0615768.6475787.17711308.4180X4*X2-3.765359-12.562131-8.445428-23.303408

200.0346-0.11831-0.0434728.0571928.3646964.56701619.8500X4*X3-5.720911-8.275237-5.949173-12.394537

21-1-10.106-1720.5182839.2650891.39531425.9700X4*X410.8452611.93175314.28822623.632609

22-10.22051-1825.26671061.11901112.58301874.6340

opt ptPhysicalxLXUX Opt

x1height of stiffener1.00E-031.40E-031.29E-03

x2thickness of shell1.00E-031.40E-031.37E-03

x3E of body5.00E+107.00E+105.00E+10

x4density of body2.40E+033.60E+032.98E+03

at opt ptexperiInitialerror(%)predictedpredic errrealerror(%)

f1628675.5357.57647.149719-0.11647.88643.17

f2863863.99930.12844.8399660.47840.9149-2.56

f3919906.6369-1.35938.8532-0.48943.33612.65

f416361520.64-7.051576.9646-0.051577.774-3.56

1st RSMb1h1b2h2tE1density1E2density2

1X1X2X3X4X5X6X7X8X9

675.5350.0140.070.0870.3837.4454.123-58.8520.003-0.009

Sheet1

3statorx_lowerx_upperx_normalizedx_realobjfunc call

x1(E)1.80E+102.20E+100.9514752.19E+101.79E+0431RSMf1f2

x2(density)77008020-0.3085227.81E+0311462.22333333331474.2063333333

x173.210573.8105

x2-14.874-14.996

DVPhysicalXLXUX_Optx1*x1-1.8335-1.8485

x1(E)1.80E+102.20E+102.192E+10x1*x2-0.74525-0.75125

x2(density)770080207.815E+03x2*x20.2270.229

experiInitialerror(%)predictedpredic errrealerror(%)

f1144014621.531535.09480.001535.0796.60

f216421474-10.231547.6750.001547.659-5.75

Sheet2

Sheet3

2.5 Example 2 - Model Update Using RSM : Rotor

2.5 Example 2 - Model Update Using RSM: Other Parts

2.5 Example 2 - Model Assemble & Analysis4312Mode ShapeNatural FrequenciesThese good Results are from the good part models Sensitivities of all design variables w.r.t. the each frequencies5 Design Variables are selected

Sheet1

1_22

1D.V.Physical PropertyxLXUX OptexperiInitialerror(%)predictedpredic errrealerror(%)RSMf1f2f3f4RSMf1f2f3f4f5f6

x1height of stiffener1.00E-031.40E-031.29E-03f1628675.5357.57647.149719-0.11647.88643.171675.136866.562903.5011519.1431548.044743.289772.985834.6021504.3751495.668

x2thickness of shell1.00E-031.40E-031.37E-03f2863863.99930.12844.8399660.47840.9149-2.56X10.4883.1544.44417.712X188.962112.008121.907132.306279.083276.236

x3E of body5.00E+107.00E+105.00E+10f3919906.6369-1.35938.8532-0.48943.33612.65X237.33884.728130.236234.481X254.44176.98680.12584.227149.452148.954

x4density of body2.40E+033.60E+032.98E+03f416361520.64-7.051576.9646-0.051577.774-3.56X357.33971.27278.081128.986X3-41.224-68.357-76.663-84.045-86.833-85.576

X4-69.094-83.177-88.320-150.984X40.624-1.827-1.8720.3082.1701.795

X1*X10.295-0.0661.5243.101X5-13.457-8.177-3.6840.001-63.459-64.243

2DVPhysical PropertyXLXUX_OptexperiInitialerror(%)predictedpredic errrealerror(%)X2*X10.087-4.1013.340-4.169X1*X1-2.905-1.4462.057-1.873-3.421-2.969

x1thickness of Shell1.00E-031.40E-031.1465E-03f1567548-3.35533.15640.35531.29922-6.30X2*X20.764-21.57215.1660.718X2*X18.23310.04111.68212.04026.27726.100

x2E1(body) of Body8.00E+101.20E+119.1167E+10f27167413.49754.1939-0.46757.67895.82X3*X10.031-0.8721.4561.454X2*X2-2.714-6.138-5.641-4.756-7.012-7.154

x3density1 of body2.40E+033.60E+032.4000E+03f3790773-2.15785.656-0.87792.54790.32X3*X23.0277.19410.49419.276X3*X1-8.234-8.082-9.795-12.341-18.508-17.951

x4E2(solid) of bearing1.60E+112.40E+112.2756E+11f4904834-7.74852.83870.11851.9276-5.76X3*X3-2.831-3.867-3.947-6.717X3*X2-3.409-6.893-10.003-7.743-6.589-6.947

x5density2 of bearing6.29E+039.43E+038.7550E+03f51339150312.251414.3050.331409.7025.28X4*X1-0.076-0.825-0.171-2.048X3*X34.03914.54914.40513.0906.4106.710

f6144614953.391404.909-0.201407.674-2.65X4*X2-3.765-12.562-8.445-23.303X4*X10.494-0.635-1.8100.7651.6101.219

3DVPhysical PropertyXLXUX_OptX4*X3-5.721-8.275-5.949-12.395X4*X20.612-0.2481.3100.7340.7330.787

x1(E) of Stator1.80E+102.20E+102.1916E+10experiInitialerror(%)predictedpredic errrealerror(%)X4*X410.84511.93214.28823.633X4*X3-0.8290.8362.044-0.757-1.863-1.421

x2(density) of Stator770080207.8147E+03f1144014621.531535.09480.001535.0796.60X4*X40.074-3.883-5.394-1.1970.666-0.053

f216421474-10.231547.6750.001547.659-5.754_2X5*X11.342-3.394-5.3141.424-3.868-4.962

RSMf1f2X5*X2-0.956-1.187-0.9450.615-5.193-5.444

4DVPhysical PropertyXLXUX_OptexperiInitialerror(%)predictedpredic errrealerror(%)1946.5713506.149X5*X33.214-3.878-0.803-0.8549.1259.070

x1E of shaft1.50E+112.30E+112.2511E+11f19739851.23973.01-0.56978.46680.56X197.292334.412X5*X4-0.3321.4031.753-0.079-1.222-0.925

x2E of mass around the shaft2.00E+101.50E+112.7484E+10f23305371112.283305.33-1.143343.4171.16X224.977194.899X5*X5-0.0275.7124.4490.5243.2483.955

x3density of shaft5.50E+031.02E+049.1032E+03X3-116.175-505.701

x4density of mass5.50E+031.02E+045.7289E+03X4-31.671-34.082

X1*X1-4.576-14.6463

experiInitialerror(%)predictedpredic errrealerror(%)X2*X14.94237.912RSMf1f2

f1451474.415.19474.410.00474.415.19X2*X2-13.524-105.41211462.2231474.206

52DVPhysicalDirectionXLXUX_Optf2502492.22-1.95492.220.00492.21-1.95X3*X1-11.604-45.470x173.21173.811

x1spring k(bearing)z2.00E+033.80E+041.743098E+04f3613609.48-0.57601.680.05601.41-1.89X3*X2-3.247-30.679x2-14.874-14.996

x2spring k(connection)z2.00E+093.80E+102.030161E+10f4624613.79-1.64613.250.44610.58-2.15X3*X326.162110.512x1*x1-1.833-1.848

x3spring k(bearing)y/x2.00E+093.80E+102.032543E+10f5763789.073.42771.160.01771.111.06X4*X1-3.455-6.716x1*x2-0.745-0.751

x4spring k(bearing)y/x1.41E+092.68E+101.433662E+10f611021031.06-6.441031.070.001031.06-6.44X4*X20.0464.380x2*x20.2270.229

f711101063.21-4.221063.210.001063.21-4.22X4*X3-1.4471.227

f815381378.92-10.341378.950.001378.92-10.34X4*X48.3736.079

5_8

58DVPhysicalXLXUX_OptexperiInitialerror(%)predictedpredic errrealerror(%)RSMf1f2f3f4f5f6f7f8

x1Elasticity of shaft1.58E+112.93E+112.461909E+11f1451474.415.19472.22-0.11472.754.821472.249675490.226377584.827227611.073596765.8308151022.6827271050.6131541392.179764

x2Density of shaft6.37E+031.18E+046.372241E+03f2502492.21-1.95496.740.24495.55-1.28X138.21598441.387634.346765.786113-0.96907995.766765109.63835375.462943

x3Density of mass on shaft4.01E+037.45E+036.186602E+03f3613601.41-1.89601.460.20600.27-2.08X2-3.412597-4.69781-2.182844-0.928528-3.409813-32.953164-36.426914-108.367014

x4Elasticity of lower body6.38E+101.19E+119.852380E+10f4624610.58-2.15632.94-0.23634.381.66X3-46.622506-46.592922-4.211997-5.124542-3.081245-79.837016-90.868693-23.791534

x5Elasticity of upper body3.50E+106.50E+104.437427E+10f5763771.111.06753.622.35736.35-3.49X410.36216715.36587460.90499583.10845620.09743731.68125335.28472682.790203

f611021031.06-6.441067.40.311064.08-3.44X515.6186188.2781021.1785141.78604156.09910813.5761784.3911821.901104

f711101063.21-4.221102.431.031091.15-1.70X1*X1-9.249617-10.7555413.7892493.850869-3.114455-7.573029-11.547198-38.483735

f815381378.92-10.341554.530.991539.360.09X2*X12.8374742.764556-2.560242-2.023085-1.031511-1.2215992.84187120.535164

X2*X2-1.220529-0.7510141.0052111.4796997.2193083.6225433.34337624.606745

X3*X10.8322932.633564-5.859659-5.932582-4.753774-31.582848-32.605836-3.010238

X3*X20.2286281.3308231.3739890.8027880.01053922.96310332.587405-28.675277

X3*X37.3431757.6770754.3504644.042839-1.10160619.90503218.88109119.123374

X4*X17.2361189.457747-5.313963-6.804532-2.426263-22.41176-13.093192-16.831496

X4*X2-2.164786-1.2533221.482595-0.0093030.0634258.0294945.19078223.421585

X4*X3-5.461929-7.5770333.658897.521769-3.12607926.02382419.09131-1.343106

X4*X4-2.652697-4.232089-8.332874-4.145229-11.08825210.9248518.591516-27.327853

X5*X13.4315262.460812-0.11362-4.020144.02793114.27747515.136914-3.991974

X5*X23.6925444.183426-2.864545-4.451858-0.6697589.60669412.5632082.543733

X5*X3-2.519702-2.1257850.6987481.726843-0.392715-8.572588-9.436356-17.375372

X5*X4-0.916958-1.0474031.3190663.54382719.799911-10.837183-10.9395824.281333

X5*X54.6950384.605444-6.422384-8.499246-12.0468759.02672412.536395-6.587147

Sheet2

Sheet3

Sheet1

domain3RSM+DOTdomain4RSM+DOT

obj=34.1obj=30.6

DVPhysicalXLXUX_OptexperiInitialerror(%)predictedpredic errrealerror(%)realerror(%)realerror(%)

x1Elasticityshaft1.58E+112.93E+112.461909E+11f1451474.415.19472.22-0.11472.754.82+471.1594.47469.434.09

x2Densityshaft6.37E+031.18E+046.372241E+03f2502492.21-1.95496.740.24495.55-1.28497.048-0.99499.31-0.54

x3Densitymass around shaft4.01E+037.45E+036.186602E+03f3613601.41-1.89601.460.20600.27-2.08+600.532-2.03600.06-2.11

x4Elasticitylower body6.38E+101.19E+119.852380E+10f4624610.58-2.15632.94-0.23634.381.66+633.9411.59633.371.50

x5Elasticityupper body3.50E+106.50E+104.437427E+10f5763771.111.06753.622.35736.35-3.49753.059-1.30752.72-1.35

f611021031.06-6.441067.40.311064.08-3.44+1085.994-1.451080.91-1.91

f711101063.21-4.221102.431.031091.15-1.70+1091.740-1.651096.86-1.18

f815381378.92-10.341554.530.991539.360.09+1538.5360.031538.530.03

206.2667340752sumsq58.927214949334.1405460532

RSM+DOT

0experiInitialerror(%)predictedpredic errrealerror(%)realerror(%)

f1451474.415.19468.9472-0.79472.674.81471.1594.47

f2502492.21-1.95496.4845-0.88500.87-0.23497.048-0.99

f3613601.41-1.89603.2610.61599.61-2.18600.532-2.03

f4624610.58-2.15634.03820.26632.421.35633.9411.59

f5763771.111.06749.31882.70729.64-4.37753.059-1.30

f611021031.06-6.441088.52-0.831097.65-0.391085.994-1.45

f711101063.21-4.221093.472-1.341108.28-0.161091.740-1.65

f815381378.92-10.341527.535-0.711538.410.031538.5360.03

sumsq49.03193233634.1405460532

-0.5experiInitialerror(%)predictedpredic errrealerror(%)Finalerror(%)

f1451474.415.19469.1766-0.61472.074.67471.0234.44

DVPhysicalReal_xl(i)Real_xu(i)f2502492.21-1.95495.8851-0.45498.14-0.77After497.065-0.98

x1Elasticityshaft2.00E+113.50E+11f3613601.41-1.89604.10760.53600.95-1.97Gradient600.424-2.05

x2Densityshaft2.50E+038.00E+03f4624610.58-2.15633.6846-0.10634.351.66Based633.7731.57

x3Densitymass around shaft6.00E+031.20E+04f5763771.111.06748.35523.06726.16-4.83Method751.891-1.46

x4Elasticitylower body6.38E+101.19E+11f611021031.06-6.441087.693-0.271090.66-1.031086.141-1.44

x5Elasticityupper body3.50E+106.50E+10f711101063.21-4.221091.187-1.081103.06-0.621092.040-1.62

f815381378.92-10.341534.986-0.221538.400.031538.5290.03

Objective206.26753.78734.149

objrsmobj obj

DVPropertyPartReal_xl(i)Real_xu(i)

x1Elasticityshaft1.00E+113.50E+11

x2Densityshaft2.50E+031.20E+04

x3Densitymass around shaft2.50E+031.20E+04

x4Elasticitylower body6.38E+101.19E+11

x5Elasticityupper body3.50E+106.50E+10

Sheet2

Sheet3

2.5 Example 2 - Model Update : Whole MotorFinal Results Using Hybrid MethodOptimizationGradient-based OptimizationHybrid(RSM+GRAD) Optimization

Sheet1

domain3RSM+DOTdomain4RSM+DOT

obj=34.1obj=30.6

DVPhysicalXLXUX_OptexperiInitialerror(%)predictedpredic errrealerror(%)realerror(%)realerror(%)

x1Elasticityshaft1.58E+112.93E+112.461909E+11f1451474.415.19472.22-0.11472.754.82+471.1594.47469.434.09

x2Densityshaft6.37E+031.18E+046.372241E+03f2502492.21-1.95496.740.24495.55-1.28497.048-0.99499.31-0.54

x3Densitymass around shaft4.01E+037.45E+036.186602E+03f3613601.41-1.89601.460.20600.27-2.08+600.532-2.03600.06-2.11

x4Elasticitylower body6.38E+101.19E+119.852380E+10f4624610.58-2.15632.94-0.23634.381.66+633.9411.59633.371.50

x5Elasticityupper body3.50E+106.50E+104.437427E+10f5763771.111.06753.622.35736.35-3.49753.059-1.30752.72-1.35

f611021031.06-6.441067.40.311064.08-3.44+1085.994-1.451080.91-1.91

f711101063.21-4.221102.431.031091.15-1.70+1091.740-1.651096.86-1.18

f815381378.92-10.341554.530.991539.360.09+1538.5360.031538.530.03

206.2667340752sumsq58.927214949334.1405460532

RSM+DOT

0experiInitialerror(%)predictedpredic errrealerror(%)realerror(%)

f1451474.415.19468.9472-0.79472.674.81471.1594.47

f2502492.21-1.95496.4845-0.88500.87-0.23497.048-0.99

f3613601.41-1.89603.2610.61599.61-2.18600.532-2.03

f4624610.58-2.15634.03820.26632.421.35633.9411.59

f5763771.111.06749.31882.70729.64-4.37753.059-1.30

f611021031.06-6.441088.52-0.831097.65-0.391085.994-1.45

f711101063.21-4.221093.472-1.341108.28-0.161091.740-1.65

f815381378.92-10.341527.535-0.711538.410.031538.5360.03

sumsq49.03193233634.1405460532

-0.5ExperiInitialerror(%)Predictedpredic errRealerror(%)Finalerror(%)

f1451474.415.19469.1766-0.61472.074.67471.0234.44

DVPhysicalReal_xl(i)Real_xu(i)f2502492.21-1.95495.8851-0.45498.14-0.77After497.065-0.98

x1Elasticityshaft2.00E+113.50E+11f3613601.41-1.89604.10760.53600.95-1.97Gradient600.424-2.05

x2Densityshaft2.50E+038.00E+03f4624610.58-2.15633.6846-0.10634.351.66Based633.7731.57

x3Densitymass around shaft6.00E+031.20E+04f5763771.111.06748.35523.06726.16-4.83Method751.891-1.46

x4Elasticitylower body6.38E+101.19E+11f611021031.06-6.441087.693-0.271090.66-1.031086.141-1.44

x5Elasticityupper body3.50E+106.50E+10f711101063.21-4.221091.187-1.081103.06-0.621092.040-1.62

f815381378.92-10.341534.986-0.221538.400.031538.5290.03

Objective206.26753.78734.149

objrsmobj obj

DVPhysicalReal_xl(i)Real_xu(i)

x1Elasticityshaft1.00E+113.50E+11

x2Densityshaft2.50E+031.20E+04

x3Densitymass around shaft2.50E+031.20E+04

x4Elasticitylower body6.38E+101.19E+11

x5Elasticityupper body3.50E+106.50E+10

Sheet2

Sheet3

Example by k.k.choi, U. of Iowa, Moving Least Square Method for Reliability-Based Design Optimization, WCSMO4, 2001 2.5 Example 3 - AUTOMOTIVE SIDE IMPACT

ReferencesNguyen, 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 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

Part II(Advanced RSM)

3.1 Introduction-MotivationEfficient Construction of RSM using SensitivityOptimization using RSM and Sensitivity-based MethodReduce the Computation Time Effect of Function & SensitivityReduce Approximation Errors Local & Global Approximation (MLSM)RSM Optimization Global Behavior / Large Approximation ErrorSensitivity-based Optimization Accurate & Fast Convergence / local Behavior Function TestInduction Motor FE Model UpdateRestriction-Available Cheap Sensitivity

InputResponse- Global Approximation 1 RS Function at all pts Constant Coefficients3.2 Moving Least Squares MethodInputResponse Local Approximation 1 RS Function at 1 pt Various Coefficients

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

3.2 Numerical Derivation (2/2) MLSM with Sensitivity- New Least Squares Function- The coefficients of the RS model - Gradient Function

Rosenbrock Function 3.2 Numerical Examples (Graphical Analysis)Basis Model : QuadraticWeight Function of Resp : 4th order polynomialsWeight Function of Grad : 4th order polynomialsFunction Characteristics Banana Function V-shaped Valley

Sheet1

SSE/n

PbrunSensitivityMLSMX2Bi- X2

TF316XX2.809E+051.870E-10

1.837E+061.836E+06

1.911E+053.391E+04

X1.57.346E+02X

1.927E+06

3.424E+04

opt1opt11.005E+038.687E+02

1.655E+051.511E+05

2.744E+042.712E+04

Use ofUse ofBasisSSE/n of RespSSE/n of GradSSE/n of Resp

NoSensitivityMLSMModelfor 16 Experi ptsfor 16 Experi ptsfor 100 Test ptsDataClassical LSMMoving LSMMLSM with Sensitivity

1XXQuadratic2.809E+051.837E+061.911E+05SSE/n of Resp for 16 Experi pts2.809E+057.346E+021.005E+03

2Bi-Quadratic1.870E-101.836E+063.391E+04SSE/n of Grad for 16 Experi pts1.837E+061.927E+061.655E+05

3XOQuadratic7.346E+021.927E+063.424E+04SSE/n of Resp for 100 Test pts1.911E+053.424E+042.744E+04

4Bi-QuadraticXXX

5OOQuadratic1.005E+031.655E+052.744E+04Original FunctionClassical LSMMoving LSMMLSM with Sensitivity

6Bi-Quadratic8.687E+021.511E+052.712E+04

Sheet2

Sheet3

3.2 Numerical Examples (Error Analysis)Global ErrorSSE/n = Sum of Squared Errors / No of Sampling PtsSSE/nt = Sum of Squared Errors / No of Test PtsGrad ErrorResp ErrorError Table

Sheet1

SSE/n

PbrunSensitivityMLSMX2Bi- X2

TF316XX2.809E+051.870E-10W X

1.837E+061.836E+06

1.911E+053.391E+04

X1.57.346E+02X

1.927E+06

3.424E+04

opt1opt11.005E+038.687E+02

1.655E+051.511E+05

2.744E+042.712E+04

Use ofUse ofBasisSSE/n of RespSSE/n of GradSSE/n of Resp

NoSensitivityMLSMModelfor 16 Experi ptsfor 16 Experi ptsfor 100 Test ptsDataClassical LSMMoving LSMMLSM with Sensitivity

1XXQuadratic2.809E+051.837E+061.911E+05SSE/n of Resp at 16 Experi pts2.81E+057.35E+023.47E+03

2Bi-Quadratic1.870E-101.836E+063.391E+04SSE/n of Grad at 16 Experi pts1.84E+061.93E+063.81E+05

3XOQuadratic7.346E+021.927E+063.424E+04SSE/nt of Resp at 100 Test pts1.91E+053.42E+042.41E+04

4Bi-QuadraticXXX

5OOQuadratic1.005E+031.655E+052.744E+04Original FunctionClassical LSMMoving LSMMLSM with Sensitivity

6Bi-Quadratic8.687E+021.511E+052.712E+04

Sheet2

Sheet3

3.2 Numerical Examples (Graphical Analysis)2D six-hump camel back function 4 local optimums and 2 global optimums within the bounded regionBasis Model : QuadraticWeight Function of Resp : 4th order polynomialsWeight Function of Grad : Exponential

Sheet1

SSE/n

PbrunSensitivityMLSMX2Bi- X2

TF316XX2.809E+051.870E-10W X

1.837E+061.836E+06

1.911E+053.391E+04

X1.57.346E+02X

1.927E+06

3.424E+04

opt1opt11.005E+038.687E+02

1.655E+051.511E+05

2.744E+042.712E+04

Use ofUse ofBasisSSE/n of RespSSE/n of GradSSE/n of Resp

NoSensitivityMLSMModelfor 16 Experi ptsfor 16 Experi ptsfor 100 Test ptsDataClassical LSMMoving LSMMLSM with Sensitivity

1XXQuadratic2.809E+051.837E+061.911E+05SSE/n of Resp for 16 Experi pts2.809E+057.346E+021.005E+03

2Bi-Quadratic1.870E-101.836E+063.391E+04SSE/n of Grad for 16 Experi pts1.837E+061.927E+061.655E+05

3XOQuadratic7.346E+021.927E+063.424E+04SSE/n of Resp for 100 Test pts1.911E+053.424E+042.744E+04

4Bi-QuadraticXXX

5OOQuadratic1.005E+031.655E+052.744E+04Original FunctionClassical LSM/Moving LSMMLSM with SensitivityMLSM with Sensitivity

6Bi-Quadratic8.687E+021.511E+052.712E+04

Sheet2

Sheet3

3.2 Numerical Examples (Error Analysis)Error TableGlobal ErrorGrad ErrorResp ErrorSSE/n = Sum of Squared Errors / No of Sampling PtsSSE/nt = Sum of Squared Errors / No of Test Pts

Sheet1

DataClassical LSMMoving LSMMLSM with Sensitivity

SSE/n of Resp at 16 Experi pts4.23E-164.23E-163.02E-04

SSE/n of Grad at 16 Experi pts5.80E+015.80E+012.26E-01

SSE/nt of Resp at 100 Test pts5.79E-015.79E-011.85E-01

Sheet2

Sheet3

3.2 Numerical Examples (Efficiency Test)

3.3 Concept of Hybrid Optimization of RSM & gradient-based optimization Original ResponseRSM ResponseOptimum By RSMTrue Optimum by Gradient-based optimizationHybrid Optimization (Function Plot)Hybrid Optimization (Contour Plot)Using Response Surface Method (Adv) Global Behavior (Dis) Large Approximation ErrorUsing Gradient-Based Method (Adv) Accurate & Fast Convergence (Dis) local Behavior Use the approximated Function instead of the original systemSearch the direction s.t. improve the objectiveUse the original system

3.3 Sequences of the optimization

3.3 Numerical ExampleOptimization Problem

3.4 ConclusionEfficient Construction of RSM using SensitivityLocal & Global Approximation (MLSM) Reduce the Approximation ErrorsEffect of Function & Sensitivity Reduce the Calculation TimeOptimization using RSM and Sensitivity-based MethodRSM Optimization Global BehaviorSensitivity-based Optimization Accurate & Fast ConvergenceFunction Tests Accuracy & Efficiency Function Test & Induction Motor FE Model Update

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

3.5 Other Approximation MethodsKriging ModelNeural Network