Uncertainty Quantification in Multiscale Materials Modeling

Front Cover......Page 1
Uncertainty Quantification in Multiscale Materials Modeling......Page 2
 Series editor: Zhong Chen......Page 3
Uncertainty Quantification in Multiscale Materials Modeling......Page 4
Copyright......Page 5
Contents......Page 6
Contributors......Page 12
 Series editors......Page 16
Preface......Page 18
 1.1 Materials design and modeling......Page 20
 1.2 Sources of uncertainty in multiscale materials modeling......Page 25
  1.2.1 Sources of epistemic uncertainty in modeling and simulation......Page 26
   1.2.2.1 Models at different length and time scales......Page 27
  1.2.3 Linking models across scales......Page 29
  1.3.1 Monte Carlo simulation......Page 31
  1.3.2 Global sensitivity analysis......Page 32
  1.3.4 Gaussian process regression......Page 34
  1.3.5 Bayesian model calibration and validation......Page 38
  1.3.6 Polynomial chaos expansion......Page 39
  1.3.7 Stochastic collocation and sparse grid......Page 40
  1.3.9 Polynomial chaos for stochastic Galerkin......Page 41
  1.3.10 Nonprobabilistic approaches......Page 42
  1.4.2 UQ for MD simulation......Page 43
  1.4.3 UQ for meso- and macroscale materials modeling......Page 44
  1.4.5 UQ in materials design......Page 46
 1.5 Concluding remarks......Page 47
 References......Page 48
 2.1 Introduction......Page 60
  2.2.1 The Kohn–Sham formalism......Page 61
  2.2.2 Computational recipes......Page 62
  2.3.1 Numerical errors......Page 64
  2.3.2 Level-of-theory errors......Page 68
  2.3.3 Representation errors......Page 69
  2.4.1 Regression analysis......Page 71
  2.4.2 Representative error measures......Page 74
  2.5.1 Case 1: DFT precision for elemental equations of state......Page 76
  2.5.2 Case 2: DFT precision and accuracy for the ductility of a W–Re alloy......Page 81
 2.6 Discussion and conclusion......Page 86
 References......Page 88
 3.1 Introduction......Page 96
 3.2 Construction of the functional ensemble......Page 97
 3.3 Selected applications......Page 102
 References......Page 107
 4.1 Introduction......Page 112
 4.2 Diffusion model......Page 113
 4.3 Methodology for uncertainty quantification......Page 118
 4.4 Computational details......Page 122
  4.5.1 Distribution of parameters......Page 123
  4.5.2 Distribution of diffusivities......Page 125
  4.5.3 Distribution of drag ratios......Page 129
 4.6 Conclusion......Page 131
 References......Page 133
 5.1 Introduction......Page 138
 5.2 Literature review......Page 142
  5.3.1 Concurrent searching method......Page 145
  5.3.2 Curve swarm searching method......Page 147
  5.3.3 Concurrent searching method assisted by GP model......Page 149
  5.3.4 Benchmark on synthetic examples......Page 153
  5.4.1 Symmetry invariance in materials systems......Page 159
  5.4.2 Efficient exploit of symmetry property......Page 160
  5.4.3 Dynamic clustering algorithm for GP-DFT......Page 161
  5.4.4 Prediction using multiple local GP......Page 162
  5.5.1 Hydrogen embrittlement in FeTiH......Page 164
  5.5.2 Hydrogen embrittlement in pure bcc iron, Fe8H......Page 169
  5.5.3 Hydrogen embrittlement in pure bcc iron, Fe8H, using GP-DFT......Page 173
 5.6 Discussions......Page 176
 References......Page 180
 6.1 Introduction......Page 188
   6.2.1.1 Bayes' theorem......Page 190
   6.2.1.3 Model selection......Page 195
   6.2.2.2 Data inconsistency......Page 196
   6.2.2.3 Model inadequacy/model errors......Page 197
   6.2.3.1 Additive model correction......Page 198
   6.2.3.2 Hierarchical models......Page 199
   6.2.3.3 Stochastic Embedding models......Page 200
   6.2.3.4 Approximate Bayesian Computation......Page 202
  6.3.1 Sampling from the posterior PDF......Page 203
  6.3.2 Metamodels......Page 205
   6.3.2.1 Kriging......Page 206
   6.3.2.2 Adaptive learning of kriging metamodels......Page 207
   6.3.2.3 Polynomial Chaos expansions......Page 209
  6.3.3 Approximation of intractable posterior PDFs......Page 211
  6.3.4 High-performance computing for Bayesian inference......Page 213
 6.4 Applications......Page 214
   6.4.1.2 Bayesian calibration......Page 215
   6.4.1.4 Uncertainty propagation through molecular simulations......Page 218
   6.4.1.5 Model improvement and model selection......Page 219
   6.4.2.1 Polynomial Chaos expansions......Page 223
    6.4.2.1.1 Calibration using an uncertain PC surrogate model......Page 228
   6.4.2.2 Gaussian processes and efficient global Optimization strategies......Page 231
  6.4.3 Model selection and model inadequacy......Page 234
 6.5 Conclusion and perspectives......Page 236
 Abbreviations and symbols......Page 238
 References......Page 239
 7.1 Introduction......Page 248
 7.2 Generalized interval arithmetic......Page 251
 7.3 Reliable molecular dynamics mechanism......Page 253
   7.3.1.1 Interval potential: Lennard-Jones......Page 254
   7.3.1.3 Interval potential: embedded atomic method potential......Page 255
  7.3.2 Interval-valued position, velocity, and force......Page 258
   7.3.3.1 Midpoint–radius or nominal–radius scheme......Page 260
   7.3.3.3 Total uncertainty principle scheme......Page 262
   7.3.3.4 Interval statistical ensemble scheme: interval isothermal-isobaric (NPT) ensemble......Page 263
  7.4.1 Simulation settings......Page 265
  7.4.3 Numerical results......Page 267
  7.4.4 Comparisons of numerical results for different schemes......Page 269
  7.4.5 Verification and validation......Page 277
  7.4.6 Finite size effect......Page 280
 7.5 Discussion......Page 282
 7.6 Conclusions......Page 286
 References......Page 287
 8.1 Introduction......Page 292
 8.2 Interval probability and random set sampling......Page 295
 8.3 Random set sampling in KMC......Page 299
  8.3.1 Event selection......Page 300
  8.3.2 Clock advancement......Page 302
   8.3.2.1 When events are independent......Page 303
   8.3.2.2 When events are correlated......Page 304
  8.3.3 R-KMC sampling algorithm......Page 305
  8.4.1 Escherichia coli reaction network......Page 308
  8.4.2 Methanol decomposition on Cu......Page 309
  8.4.3 Microbial fuel cell......Page 313
 8.5 Summary......Page 315
 References......Page 316
 9.1 Introduction......Page 320
 9.2 Cahn–Hilliard–Cook model......Page 324
 9.3 Methodology......Page 325
  9.3.3 Galerkin approximation......Page 326
  9.3.4 Time scheme......Page 327
 9.4 Morphology characterization......Page 328
  9.5.1 Spatial discretization......Page 329
  9.5.3 Parallel space–time noise generation......Page 330
  9.5.4 Scalability analysis......Page 332
  9.6.1 Energy-driven analysis and noise effects......Page 333
  9.6.2 Domain size analysis......Page 335
  9.6.4 Enforcing fluctuation–dissipation......Page 338
 9.7 Conclusions......Page 343
 References......Page 344
 10.1 Introduction......Page 348
 10.2 Applying UQ at the mesoscale......Page 349
  10.3.1 Introduction......Page 351
  10.3.2 Model summaries......Page 352
  10.3.3 Sensitivity analysis......Page 355
  10.3.4 Uncertainty quantification......Page 356
  10.4.2 Model summaries......Page 358
  10.4.3 Sensitivity analysis......Page 360
  10.4.4 Uncertainty quantification......Page 362
  10.5.2 Model summaries......Page 365
  10.5.3 Sensitivity analysis......Page 366
  10.5.4 Uncertainty quantification......Page 367
 References......Page 369
 11.1 Background and literature review......Page 374
 11.2 Our approach for multiscale UQ and UP......Page 378
  11.2.1 Multiresponse Gaussian processes for uncertainty quantification......Page 380
  11.2.2 Top-down sampling for uncertainty propagation......Page 381
  11.3.1 Uncertainty sources......Page 382
  11.3.2 Multiscale finite element simulations......Page 384
  11.3.3 Top-down sampling, coupling, and random field modeling of uncertainty sources......Page 386
  11.3.4 Dimension reduction at the mesoscale via sensitivity analysis......Page 389
  11.3.5 Replacing meso- and microscale simulations via metamodels......Page 391
  11.3.6 Results on macroscale uncertainty......Page 393
 11.4 Conclusion and future works......Page 396
  Details on the sensitivity studies at the mesoscale......Page 397
 References......Page 399
 12.1 Introduction......Page 404
  12.2.1 Definition of scales......Page 405
  12.2.2 On the representation of random fields......Page 407
  12.2.3 Information-theoretic description of random fields......Page 410
  12.2.4 Getting started with a toy problem......Page 412
  12.3.1 Preliminaries......Page 415
  12.3.2 Setting up the MaxEnt formulation......Page 416
  12.3.3 Defining the non-Gaussian random field......Page 419
   12.3.4.1 Formulation......Page 421
   12.3.4.2 Two-dimensional numerical illustration......Page 424
  12.4.1 Background......Page 426
  12.4.2 Setting up the MaxEnt formulation......Page 427
  12.4.3 Defining random field models for strain energy functions......Page 431
 12.5 Conclusion......Page 434
 References......Page 435
 13.1 Introduction......Page 440
  13.2.1 Finite element model of composite plate......Page 444
  13.2.2 Matrix crack modeling......Page 445
  13.2.3 Fractal dimension......Page 448
  13.2.4 Spatial uncertainty in material property......Page 450
  13.3.1 Localized damage detection based on fractal dimension–based approach......Page 451
  13.3.2 Spatial uncertainty......Page 458
 13.4 Conclusions......Page 460
 References......Page 463
 14.1 Introduction......Page 468
 14.2 Multiresponse, multiscale TDBU HMM calibration......Page 469
  14.2.2 Formulation......Page 470
 14.3 Usage: TDBU calibration of CP of bcc Fe......Page 474
  14.3.2 Crystal plasticity model......Page 475
  14.3.3 Parameter estimates and data......Page 476
  14.3.4 Implementation of the method......Page 477
 14.4 Between the models: connection testing......Page 479
  14.4.1 Background......Page 480
  14.4.2 Formulation......Page 481
  14.5.1 Background......Page 483
  14.5.2 Implementation......Page 484
 14.6 Discussion and extensions to validation......Page 485
 References......Page 488
 15.1 Introduction......Page 492
 15.2 Hierarchical reliability approach......Page 496
  15.3.1 Construction of a stochastic reduced–order model......Page 499
  15.3.2 SROM-based surrogate model and Monte Carlo simulation......Page 504
 15.4 Concurrent coupling......Page 506
 15.5 Applications examples......Page 509
   15.5.1.2 Model definition......Page 510
   15.5.1.4 Results......Page 512
   15.5.2.1 Objective......Page 516
   15.5.2.3 Uncertainty......Page 517
   15.5.2.4 Results......Page 520
   15.5.3.2 Model definition......Page 523
   15.5.3.3 Uncertainty......Page 528
   15.5.3.4 Results......Page 529
  15.5.4 Summary and discussion......Page 531
 15.6 Conclusions......Page 532
 Nomenclature......Page 533
 References......Page 534
 16.1 Introduction......Page 538
  16.2.1 Surrogate model with uncertainties......Page 541
  16.2.2 Utility functions......Page 543
 16.3 Design of new shape-memory alloys......Page 546
  16.3.1 Searching for NiTi-based shape-memory alloys with high transformation temperature......Page 547
  16.3.2 Search for very low thermal hysteresis NiTi-based shape-memory alloys......Page 549
 References......Page 554
 17.1 Introduction......Page 558
 17.2 A strategy for predicting the mechanical properties of additively manufactured metallic lattice structures via strut-level  .........Page 562
 17.3 Experimental investigation of the mechanical properties of DMLS octet lattice structures......Page 565
  17.3.2 Dimensional accuracy and relative density analysis of octet truss lattice structures......Page 566
  17.3.3 Tension testing of standard tensile bars......Page 567
  17.3.4 Compression testing of octet truss lattice structures......Page 568
 17.4 Finite element analysis of the DMLS octet lattice structures based on bulk material properties......Page 571
 17.5 Experimental investigation of the mechanical properties of DMLS lattice struts......Page 574
 17.6 Finite element analysis of the DMLS octet lattice structures based on strut-level properties......Page 577
 17.7 Opportunities for expanding the experimental study to better inform the finite element modeling......Page 578
 17.8 Discussion......Page 579
 Appendix......Page 580
 References......Page 581
 B......Page 586
 C......Page 587
 D......Page 589
 E......Page 590
 G......Page 591
 I......Page 593
 L......Page 594
 M......Page 595
 N......Page 597
 P......Page 598
 Q......Page 599
 R......Page 600
 S......Page 601
 T......Page 602
 U......Page 603
 V......Page 604
 Z......Page 605
Back Cover......Page 606