Model Validation and Uncertainty Quantification, Volume 3

Preface	6
Contents	8
1 Calibration of System Parameters Under Model Uncertainty	11
1.1 Introduction	11
1.2 Multi-fidelity Calibration Method	12
1.2.1 Surrogate Model: Polynomial Chaos Expansion	12
1.2.2 Uncertainty Quantification	13
1.2.3 Bayesian Calibration	13
1.2.4 Validation	14
1.2.5 Multi-fidelity Implementation	14
1.3 Numerical Example	15
1.3.1 Problem Description	15
1.3.2 Results	17
1.3.3 Discussion	17
1.4 Conclusion	19
References	19
2 On the Aggregation and Extrapolation of Uncertainty from Component to System Level Models	21
2.1 Introduction	21
2.2 Example Problem Description	22
2.2.1 Dumbbell Configuration (Level 1)	22
2.2.2 Three-Leg Configuration (Level 2 and Application Space)	23
2.2.3 Energy Dissipation Model: Iwan Model	23
2.2.4 Quantity of Interest: Energy Dissipated (Ed)	25
2.3 Uncertainty Quantification, Aggregation and Propagation	25
2.3.1 Training Data and Partial Least Squares Regression	25
2.3.2 Choice of the Number of Latent Components	26
2.3.3 Unit-to-Unit Variability	28
2.3.4 Bias and Uncertainty in the Simulation and Corrected Simulation Results	29
2.3.5 Results	29
2.4 Summary	30
References	32
3 Validation of Strongly Coupled Models: A Frameworkfor Resource Allocation	34
3.1 Introduction	34
3.2 The Problem of Resource Allocation in Model Validation	35
3.3 Model Validation Framework	36
3.4 Resource Allocation in Strongly Coupled Models	37
3.4.1 Code Prioritization	37
3.4.2 Experiment Prioritization	38
3.5 Conclusions	39
References	39
4 Fatigue Monitoring in Metallic Structures using Vibration Measurements	42
4.1 Introduction	42
4.2 Fatigue Monitoring using Operational Vibrations	43
4.2.1 Deterministic Fatigue Damage Accumulation	43
4.2.2 Stochastic Fatigue Damage Accumulation	43
4.3 Strain Monitoring using Output-Only Vibration Measurements	44
4.3.1 Stationary Stochastic Excitations	45
4.3.2 Non-stationary Deterministic Excitations	45
4.4 Applications	46
4.4.1 N -DOF Spring-Mass Chain-like Model	46
4.4.2 Small Scale Vehicle-like Frame Structure	47
4.5 Conclusions	48
References	48
5 Uncertainty propagation in Experimental Modal Analysis	50
5.1 Introduction	50
5.2 Estimation of FRF Variance	51
5.3 The Polymax Plus Method	53
5.3.1 First Stage: Maximum Likelihood Estimator	53
5.3.2 Second Stage: PolyMAX Applied to the MLE Synthesized Model	54
5.3.3 Third Stage: Estimation of the Confidence Bounds from an MLE Modal Model Formulation	54
5.4 Numerical and Measurement Examples	55
5.4.1 Acoustic Modal Analysis Measurement Example	55
5.4.2 Numerical Example: Lightly-Damped Structure with Closely-Spaced Modes	55
5.4.3 Multiple-Input In-Flight Measurement Data	57
5.4.4 Comparison of Strain Gauges and Accelerometers	59
5.5 Conclusions	59
References	60
6 Quantification of Prediction Bounds Caused by Model Form Uncertainty	61
6.1 Introduction	62
6.2 Objectives	63
6.3 Design Specifications and Experimental Setup	64
6.4 Experimental Campaigns of Vibration Testing	65
6.4.1 Effect on the Measured Response of Varying the Bolt Torque Level	65
6.4.2 Effect on the Measured Response of Changing How the Bolt Is Torqued	65
6.4.3 Effect on the Measured Response of Changing the Bolt Tightening Sequence	66
6.4.4 Overall Assessment of Experimental Variability	66
6.5 Finite Element Models of the Portal Frame	68
6.5.1 Development of the “Bounding-Case” Finite Element Models	68
6.5.2 Quantification of Numerical Uncertainty for the Finite Element Simulations	68
6.5.3 Parameter Study of the Contact Stiffness Representation	69
6.5.4 Parameter Study of the Tied Node Representation	70
6.6 Test-Analysis Correlation and Assessment of Bounding Predictions	71
6.7 Conclusion	72
References	74
7 Composite Fuselage Impact Testing and Simulation: A Model Calibration Exercise	75
7.1 Introduction	75
7.2 Composite Subfloor Test Article	76
7.3 Time Domain Calibration	76
7.3.1 Metric 1: Displacement Norm	76
7.3.2 Metric 2: Orthogonality	77
7.4 Pretest Analysis for Modal Test	78
7.4.1 Modal Testing	78
7.4.2 Comparison of Modal Test Results with Analysis	80
7.5 Impact Testing	82
7.5.1 Photogrammetry Data Processing	83
7.6 Multi-dimensional Model Calibration	83
7.7 Concluding Remarks	86
References	86
8 Noise Sensitivity Evaluation of Autoregressive Features Extracted from Structure Vibration	87
8.1 Introduction	87
8.2 Measurement Noise Sensitivity for the AR Damage Features	88
8.2.1 Theoretical Expression of Structural Response Autocovariance Function (ACF)	88
8.2.2 Sensitivity of the AR Coefficients/Spectra to the Increase in the Noise Level	90
8.2.3 Sensitivity of the Distance Measures to Changes in AR Coefficients/Spectra	91
8.3 Simulation Example: Sensitivity Analysis for a 10-DOF Structure	92
8.4 Conclusion	94
References	94
9 Uncertainty Quantification and Integration in Multi-level Problems	96
9.1 Introduction	96
9.2 Model Calibration	97
9.3 Model Validation	98
9.4 Uncertainty Integration	100
9.4.1 Physical Relevance	100
9.4.2 Roll-up Methodology	101
9.5 Numerical Example	102
9.6 Summary	103
References	105
10 Reliability Quantification of High-Speed Naval Vessels Based on SHM Data	106
10.1 Introduction	106
10.2 Structural Reliability	108
10.3 Analysis of SHM Data	109
10.4 Illustrative Example	109
10.5 Conclusions	111
References	112
11 Structural Identification Using Response Measurements Under Base Excitation	114
11.1 Introduction	114
11.2 Formulation of Identification Methodology	115
11.2.1 Estimation of the Mass Normalizing Factors with Masses Known at Np Locations	116
11.2.2 Estimation of the Proportional Mass Normalizing Factors with Known Total Mass (MT)	117
11.2.3 Mode Shape Expansion to Unobserved DOFs	118
11.3 Numerical and Experimental Evaluation	119
11.3.1 Numerical Validation	120
11.3.2 Performance with Experimental Data and Application to Structural Damage Detection	121
11.4 Conclusions	124
References	124
12 Bayesian FE Model Updating in the Presence of Modeling Errors	126
12.1 Introduction	126
12.2 SAC Nine-Story Building	127
12.3 Bayesian FE Model Updating	131
12.3.1 Likelihood A	131
12.3.2 Likelihood B	132
12.3.3 Likelihood C	132
12.4 Damage Identification	134
12.4.1 Effects of Modeling Errors and Number of Modes	134
12.4.2 Effects of Noisy Measurements	136
12.4.3 Effects of Number of Sensors	136
12.4.4 Effects of Weight Factors	136
12.5 Conclusions	138
References	139
13 Maintenance Planning Under Uncertainties Using a Continuous-State POMDP Framework	141
13.1 Introduction	141
13.2 The POMDP Framework	142
13.3 Non-linear Stochastic Action Models	144
13.4 Application: Bridge System Maintenance	145
13.5 Conclusion	148
References	149
14 Achieving Robust Design through Statistical Effect Screening	150
14.1 Introduction	150
14.2 Discussion of Performance-Optimal and Robust-Optimal Designs for Multi-Criteria Simulations	152
14.3 Sensitivity Analysis Using the Robustness Performance	155
14.3.1 Info-Gap Decision Theory for Robustness Analysis	155
14.3.2 Morris One-at-a-Time (OAT) Screening for Sensitivity Analysis	157
14.3.3 Integration of IGDT Robustness and Morris OAT Sensitivity Analysis	158
14.4 Application to the NASA Multidisciplinary Uncertainty Quantification Challenge Problem	159
14.4.1 The NASA Langley MUQC Problem	160
14.4.2 Parameter Ranking Using the Morris Elemental Effect Statistics	161
14.4.3 Development of Performance-Optimal and Robust-Optimal Designs	163
14.5 Conclusion	164
References	165
15 Automated Modal Parameter Extraction and Statistical Analysis of the New Carquinez Bridge Response to Ambient Excitations	166
15.1 Introduction	166
15.2 System Identification Using Stochastic Subspace Identification	167
15.3 Long-Term Wireless Structural Monitoring of the New Carquinez Bridge	169
15.4 Automated Extraction of Modal Parameters	170
15.4.1 Knowledge-Based Extraction Method	170
15.4.2 Triangulation Based Extraction Method	170
15.4.3 Results Comparison	171
15.5 Statistical Analysis on Automated Extracted Modal Frequency	173
15.6 Conclusions	174
References	175
16 Evaluation of a Time Reversal Method with Dynamic Time Warping Matching Function for Human Fall Detection Using Structural Vibrations	176
16.1 Introduction	176
16.2 Dynamic Time Warping	177
16.3 Experiment Setup	177
16.4 Fall Signals	179
16.5 Results	179
16.6 Post processing of Results	180
16.7 Conclusion	180
References	181
17 Uncertainty Quantification of Identified Modal Parameters Using the Fisher Information Criterion	182
17.1 Introduction	182
17.2 Fisher Information	183
17.3 Cramer-Rao Lower Bound	183
17.4 Maximum Achievable Accuracy Given Free Vibration Measurements	183
17.5 Forced Vibration	185
17.6 Numerical Illustration	185
17.7 Verification	187
17.8 Experiments and Validation	187
17.9 Conclusions	188
References	189
18 Modal Parameter Uncertainty Quantification Using PCR Implementation with SMIT	190
18.1 Introduction	190
18.2 Modal Identification Using ERA-Based Methods	191
18.2.1 ERA-NExT	192
18.2.2 ERA-NExT-AVG	192
18.2.3 ERA-OKID-OO	192
18.3 Physical Contribution Ratio	193
18.4 Validation Using Measured Data	193
18.4.1 Simulated Simply Supported Beam	194
18.4.2 Ambient Vibration Data from Golden Gate Bridge (GGB)	195
18.5 Conclusion	196
References	197
19 Excitation Related Uncertainty in Ambient Vibration Testing of Bridges	199
19.1 Introduction	199
19.2 Experimental Program	200
19.2.1 Bridge Description	201
19.2.2 Instrumentation, Data Acquisition and Signal Generation	201
19.2.3 Excitation Cases	202
19.3 Data Analysis	203
19.4 Results and Discussion	203
19.5 Concluding Remarks	205
References	205
20 Experiment-Based Validation and Uncertainty Quantification of Coupled Multi-Scale Plasticity Models	206
20.1 Introduction	206
20.2 Meso- and Macro-Scale Coupling of VPSC and ABAQUS FE Codes	208
20.2.1 Meso-Scale VPSC Code	208
20.2.2 Macro-Scale ABAQUS Code	209
20.2.3 Coupling Between VPSC and ABAQUS	209
20.3 Methodology	209
20.4 Experimental Campaign	210
20.4.1 Uniaxial Tension and Compression Test of a Zirconium Cylinder	211
20.4.2 Four Point Bending of a Zirconium Beam	211
20.5 Model Development, Calibration and Uncertainty Quantification	211
20.6 Results and Discussion	214
20.7 Conclusion	215
References	215
21 Model Calibration and Uncertainty of A600 Wind Turbine Blades	217
21.1 Introduction	217
21.2 Blade Dynamics Characterization	218
21.3 Model Calibration	218
21.3.1 FE-Modeling	219
21.3.2 Calibration Method	220
21.3.3 Calibration Results	220
21.4 Validation	221
21.4.1 Skin Material	221
21.4.2 Core Material	221
21.4.3 Blade Mass Properties	223
21.4.4 Blade Twist Angles	224
21.5 Conclusions	225
A.1 Appendix	226
References	229
22 Validation Assessment for Joint Problem Using an Energy Dissipation Model	230
22.1 Introduction	230
22.2 Description of Experimental Setup	230
22.3 Transform of Measured Response to Quantity of Interest	233
22.4 Methodology for Validation Assessment	234
22.5 Results	235
22.6 Summary	236
References	236
23 A Bayesian Damage Prognosis Approach Applied to Bearing Failure	237
23.1 Introduction	237
23.2 Bayesian Prediction and Particle Filter Estimation	238
23.3 Implementation of Particle Filter for Damage Prognosis	239
23.4 Summary and Conclusion	241
References	242
24 Sensitivity Analysis of Beams Controlled by Shunted Piezoelectric Transducers	243
24.1 Introduction	243
24.2 The System	243
24.3 Sensitivity Analysis	244
24.3.1 Model Parameters	244
24.3.2 Design Objectives	245
24.3.3 Morris Method	245
24.4 Results	245
24.5 Conclusion and Perspectives	245
References	248
25 A Principal Component Analysis (PCA) Decomposition Based Validation Metric for Use with Full Field Measurement Situations	249
25.1 Introduction	249
25.2 Background and Relevance	250
25.3 General Methodology	252
25.3.1 Explanation of New/Unusual Techniques	253
25.4 Specific Methodology: Principal Component Analysis	253
25.4.1 Application Example: Principal Component Analysis	254
25.4.2 Issues, Concerns	258
25.4.3 Results/Significance	259
25.5 Summary and Future Work	259
A.1 Appendix	259
References	262
26 FEM Calibration with FRF Damping Equalization	264
26.1 Introduction	264
26.2 Theory	265
26.2.1 A Frequency Response Based Calibration Metric	266
26.2.2 Damping Equalization	267
26.2.3 Surrogate Modeling	269
26.2.4 Randomized Parameter Initiation, Parameter Identifiability and Minimization	271
26.3 Numerical Examples	271
26.3.1 Simple Spring-Mass System	271
26.3.2 A Generic Communications Satellite	274
26.4 Concluding Remarks	275
References	277
27 Evaluating Initial Model for Dynamic Model Updating: Criteria and Application	278
27.1 Introduction	278
27.2 Criteria for Initial Model Evaluation	279
27.3 Numerical Simulation: Model Evaluation	280
27.3.1 Elastic Boundary Beam	280
27.3.2 Evaluation of Initial Model with Erroneous Boundary Condition	280
27.3.3 Evaluation of Initial Model with Correct Boundary Condition	281
27.4 Numerical Simulation: Model Updating	282
27.4.1 Updating of Initial Model with Erroneous Boundary Condition	282
27.4.2 Updating of Initial Model with Correct Boundary Condition	283
27.5 Conclusions	284
References	284
28 Evaluating Convergence of Reduced Order Models Using Nonlinear Normal Modes	285
28.1 Introduction	285
28.2 Theoretical Development	287
28.2.1 Reduced Order Models with Local Nonlinearities	287
28.2.2 Nonlinear Normal Modes	288
28.3 Numerical Results	289
28.3.1 Nonlinear Normal Mode Convergence	290
28.3.2 Impulse Loading Verification	294
28.3.3 Random Loading Verification	296
28.4 Conclusion	297
References	297
29 Approximate Bayesian Computation for Finite Element Model Updating	299
29.1 Introduction	299
29.2 Parametric Uncertainty	300
29.3 Approximate Bayesian Computation	300
29.4 Bayesian Model Updating	301
29.4.1 Finite Element Model	301
29.4.2 Prior Distribution of θ1 and θ2	301
29.5 Numerical Experiments	302
29.6 Remarks and Future Work	304
References	304
30 An Efficient Method for the Quantification of the Frequency Domain Statistical Properties of Short Response Time Series of Dynamic Systems	305
30.1 Introduction	305
30.2 Uncertainty Quantification of Discrete Response Fourier Transforms	306
30.2.1 General Concept and Problem Description	306
30.2.2 Approach 1: Analytical Uncertainty Propagation Based on a Time Domain Operator	307
30.2.3 Approach 2: Analytical Uncertainty Propagation Based on Frequency Domain Estimator	309
30.2.4 Approach 3: Sample-Based Uncertainty Propagation	310
30.3 Benchmark Study: Three-Degree-of-Freedom System	310
30.3.1 System Description	310
30.3.2 General Example	312
30.3.3 Influence of Time Frame Length	313
30.4 Conclusions	314
References	314
31 Quantifying Uncertainty in Modal Parameters Estimated Using Higher Order Time Domain Algorithms	315
31.1 Introduction	316
31.2 Theoretical Background	316
31.2.1 Modal Parameter Estimation Procedure	316
31.2.1.1 Covariance Matrix of Noise or Residuals ( Σ)	318
31.2.1.2 Covariance Matrix of Polynomial Coefficient Matrices ( Σ)	319
31.2.2 Estimation of Confidence Intervals	319
31.3 Results	320
31.4 Conclusions	322
References	322
32 Testing and Model Correlation of a Plexiplate with a WaterBoundary Condition	324
32.1 Introduction	324
32.2 Experimental Modal Analysis	324
32.2.1 Model Development and Solution	326
32.2.2 Comparison of Predicted and Experimental Modes	327
32.3 Conclusions	328
References	331
33 Detection of Stress-Stiffening Effect on Automotive Components	332
33.1 Introduction	332
33.2 Experimental Setup	333
33.3 From Component to Assembly	334
33.4 Modal Meta-Modelling Through Stochastic Expansion	336
33.5 Conclusions	339
References	340
34 Approach to Evaluate Uncertainty in Passive and Active Vibration Reduction	341
34.1 Introduction	341
34.2 Simple Example for Mathematical Evaluation of Uncertainty in Passive and Active Vibration Reduction Design	342
34.2.1 Basic Mathematical Dynamic Model of a Simple Mass-Damper-Spring System	342
34.2.2 Mathematical Simulation of Vibration Reduction Under Parameter Uncertainty	343
34.2.3 Case Studies for Passive and Active Vibration Reduction Under Uncertainty	345
34.3 Conclusion	348
References	348
35 Project-Oriented Validation on a Cantilever Beam Under Vibration Active Control	349
35.1 Basic Theory	350
35.1.1 FE Modeling of the Controlled Cantilever Beam	350
35.1.2 Identification of Mean Values of Supporting Parameters via RSM Based Model Updating	351
35.1.3 Estimation of the Standard Deviations of the Response Features and Supporting Parameters	352
35.2 Experimental Case Study	352
35.2.1 The Detailed Modeling of the Cantilever Beam Without and with Electromagnetic Actuator	352
35.2.2 EMA and Modal Frequency Error Estimation of the Cantilever Beam with One Fixed End	353
35.2.2.1 Experimental Modal Analysis	353
35.2.2.2 Modal Frequency Errors Estimation of the Cantilever Beam with One Fixed End	353
35.2.3 Verification the Identified Supporting Stiffness of the Cantilever Beam with One Fixed End and Rubber Springs	355
35.2.4 Identified Mean Supporting Stiffness of the Cantilever Beam with One Fixed End and with an Electromagnetic Actuator	355
35.2.5 Estimation of the Standard Deviations of the Response Features for the Cantilever Beam with an Electromagnetic Actuator	356
35.3 Conclusions and Discussions	357
References	357
36 Inferring Structural Variability Using Modal Analysis in a Bayesian Framework	358
36.1 Introduction	358
36.2 Bayesian Model Updating	359
36.3 Experimental Structure	359
36.3.1 Finite Element Numerical Model	359
36.3.2 Finite Element Validation	359
36.3.3 Prior Distribution of θ1 and θ2	361
36.4 Metamodelling	362
36.4.1 Radial Basis Neural Network	362
36.4.2 Metamodel Benchmarking	363
36.5 Numerical Experiments	365
36.5.1 Transitional Markov Chain Monte Carlo	366
36.6 General Remarks	368
References	368
37 Including SN-Curve Uncertainty in Fatigue Reliability Analyses of Wind Turbines	369
37.1 Introduction	369
37.2 Theory	370
37.2.1 Formulation of the Damage Function	370
37.2.2 Inclusion of Material Uncertainty	371
37.3 Example	372
37.3.1 Results	373
37.4 Conclusions	374
References	374
38 Robust Design of Notching Profiles Under Epistemic Model Uncertainties	376
38.1 Introduction	376
38.2 Notching in Shaker Tests	377
38.2.1 System of Interest	377
38.2.2 Primary Notching Principle	377
38.3 Robust Design Under Lack of Knowledge	379
38.3.1 Classical Reliability Based Robust Design	379
38.3.2 Info-Gap Uncertainty Model	379
38.3.3 Reliability-Based Robust Design with Info-Gap Uncertainty	380
38.3.3.1 Worst Case Design	380
38.3.3.2 Reliability-Based Design 1	381
38.3.3.3 Reliability-Based Design 2	381
38.4 Discussion	381
38.5 Conclusion	382
References	383
39 Optimal Selection of Calibration and Validation Test Samples Under Uncertainty	384
39.1 Introduction	384
39.2 Integration of Validation and Calibration for Prediction	385
39.2.1 Calibration	386
39.2.2 Validation	386
39.2.2.1 Validation Uncertainty for Sparse Observation Data	387
39.2.2.2 Stochastic Assessment of Model Reliability	388
39.2.3 Integration for Prediction	389
39.3 Test Selection Optimization Methodology	389
39.3.1 Calibration Objective Formulation	390
39.3.2 Validation Objective Formulation	390
39.3.3 Solution Approach for the Combined Optimization Problem	390
39.4 Numerical Example	392
39.5 Conclusion	393
References	394
40 Uncertainty Quantification in Experimental Structural Dynamics Identification of Composite Material Structures	395
40.1 Introduction	395
40.2 Experimental Structural Dynamics Identification	396
40.2.1 Experimental Modal Analysis of Three GFRP Plates: Coupon Level	396
40.2.2 Experimental Modal Analysis of CFRP Panel: Component Level	398
40.2.3 Experimental Modal Analysis of Three GFRP Helicopter Blades: Fully Assembled Structure Level	399
40.3 Summary	400
References	400
41 Analysis of Numerical Errors in Strongly Coupled Numerical Models	401
41.1 Introduction	401
41.2 Quantifying Discretization Errors	403
41.3 Quantifying Discretization in the Coupled Models	403
41.4 Case Study Problem	404
41.5 Conclusions	409
References	410
42 Robust Expansion of Experimental Mode Shapes Under Epistemic Uncertainties	411
42.1 Introduction	411
42.2 Robust ECRE-Based Expansion	413
42.2.1 ECRE-Based Expansion: Formulation	413
42.2.2 Robust Expansion Process: Approach	414
42.3 Numerical Applications	414
42.4 Conclusions	417
References	419

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