Uncertainty Quantification in Multiscale Materials Modeling

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Mechanics of Advanced Materials Series
	Series editor-in-chief: Vadim V. Silberschmidt
	Series editor: Thomas Böhlke
	Series editor: David L. McDowell
	Series editor: Zhong Chen
Uncertainty Quantification in Multiscale Materials Modeling
Copyright
Contributors
About the Series editors
	Editor-in-Chief
	Series editors
Preface
1 - Uncertainty quantification in materials modeling
	1.1 Materials design and modeling
	1.2 Sources of uncertainty in multiscale materials modeling
		1.2.1 Sources of epistemic uncertainty in modeling and simulation
		1.2.2 Sources of model form and parameter uncertainties in multiscale models
			1.2.2.1 Models at different length and time scales
		1.2.3 Linking models across scales
	1.3 Uncertainty quantification methods
		1.3.1 Monte Carlo simulation
		1.3.2 Global sensitivity analysis
		1.3.3 Surrogate modeling
		1.3.4 Gaussian process regression
		1.3.5 Bayesian model calibration and validation
		1.3.6 Polynomial chaos expansion
		1.3.7 Stochastic collocation and sparse grid
		1.3.8 Local sensitivity analysis with perturbation
		1.3.9 Polynomial chaos for stochastic Galerkin
		1.3.10 Nonprobabilistic approaches
	1.4 UQ in materials modeling
		1.4.1 UQ for ab initio and DFT calculations
		1.4.2 UQ for MD simulation
		1.4.3 UQ for meso- and macroscale materials modeling
		1.4.4 UQ for multiscale modeling
		1.4.5 UQ in materials design
	1.5 Concluding remarks
	Acknowledgments
	References
2 - The uncertainty pyramid for electronic-structure methods
	2.1 Introduction
	2.2 Density-functional theory
		2.2.1 The Kohn–Sham formalism
		2.2.2 Computational recipes
	2.3 The DFT uncertainty pyramid
		2.3.1 Numerical errors
		2.3.2 Level-of-theory errors
		2.3.3 Representation errors
	2.4 DFT uncertainty quantification
		2.4.1 Regression analysis
		2.4.2 Representative error measures
	2.5 Two case studies
		2.5.1 Case 1: DFT precision for elemental equations of state
		2.5.2 Case 2: DFT precision and accuracy for the ductility of a W–Re alloy
	2.6 Discussion and conclusion
	Acknowledgment
	References
3 - Bayesian error estimation in density functional theory
	3.1 Introduction
	3.2 Construction of the functional ensemble
	3.3 Selected applications
	3.4 Conclusion
	References
4 - Uncertainty quantification of solute transport coefficients
	4.1 Introduction
	4.2 Diffusion model
	4.3 Methodology for uncertainty quantification
	4.4 Computational details
	4.5 Results and discussion
		4.5.1 Distribution of parameters
		4.5.2 Distribution of diffusivities
		4.5.3 Distribution of drag ratios
	4.6 Conclusion
	References
5 - Data-driven acceleration of first-principles saddle point and local minimum search based on scalable Gaussian processes
	5.1 Introduction
	5.2 Literature review
	5.3 Concurrent search of local minima and saddle points
		5.3.1 Concurrent searching method
		5.3.2 Curve swarm searching method
		5.3.3 Concurrent searching method assisted by GP model
		5.3.4 Benchmark on synthetic examples
	5.4 GP-DFT: a physics-based symmetry-enhanced local Gaussian process
		5.4.1 Symmetry invariance in materials systems
		5.4.2 Efficient exploit of symmetry property
		5.4.3 Dynamic clustering algorithm for GP-DFT
		5.4.4 Prediction using multiple local GP
	5.5 Application: hydrogen embrittlement in iron systems
		5.5.1 Hydrogen embrittlement in FeTiH
		5.5.2 Hydrogen embrittlement in pure bcc iron, Fe8H
		5.5.3 Hydrogen embrittlement in pure bcc iron, Fe8H, using GP-DFT
	5.6 Discussions
	5.7 Conclusion
	Acknowledgments
	References
6 - Bayesian calibration of force fields for molecular simulations
	6.1 Introduction
	6.2 Bayesian calibration
		6.2.1 The standard Bayesian scheme
			6.2.1.1 Bayes' theorem
			6.2.1.2 Validation
			6.2.1.3 Model selection
		6.2.2 Limitations of the standard scheme
			6.2.2.1 Modeling of the error sources
			6.2.2.2 Data inconsistency
			6.2.2.3 Model inadequacy/model errors
		6.2.3 Advanced Bayesian schemes
			6.2.3.1 Additive model correction
			6.2.3.2 Hierarchical models
			6.2.3.3 Stochastic Embedding models
			6.2.3.4 Approximate Bayesian Computation
	6.3 Computational aspects
		6.3.1 Sampling from the posterior PDF
		6.3.2 Metamodels
			6.3.2.1 Kriging
			6.3.2.2 Adaptive learning of kriging metamodels
			6.3.2.3 Polynomial Chaos expansions
		6.3.3 Approximation of intractable posterior PDFs
		6.3.4 High-performance computing for Bayesian inference
	6.4 Applications
		6.4.1 Introductory example: two-parameter Lennard-Jones fluids
			6.4.1.1 The Lennard-Jones potential
			6.4.1.2 Bayesian calibration
			6.4.1.3 Hierarchical model
			6.4.1.4 Uncertainty propagation through molecular simulations
			6.4.1.5 Model improvement and model selection
			6.4.1.6 Summary
		6.4.2 Use of surrogate models for force field calibration
			6.4.2.1 Polynomial Chaos expansions
				6.4.2.1.1 Calibration using an uncertain PC surrogate model
			6.4.2.2 Gaussian processes and efficient global Optimization strategies
		6.4.3 Model selection and model inadequacy
	6.5 Conclusion and perspectives
	Abbreviations and symbols
	References
7 - Reliable molecular dynamics simulations for intrusive uncertainty quantification using generalized interval analysis
	7.1 Introduction
	7.2 Generalized interval arithmetic
	7.3 Reliable molecular dynamics mechanism
		7.3.1 Interval interatomic potential
			7.3.1.1 Interval potential: Lennard-Jones
			7.3.1.2 Interval potential: Morse potential
			7.3.1.3 Interval potential: embedded atomic method potential
		7.3.2 Interval-valued position, velocity, and force
		7.3.3 Uncertainty propagation schemes in R-MD
			7.3.3.1 Midpoint–radius or nominal–radius scheme
			7.3.3.2 Lower–upper bound scheme
			7.3.3.3 Total uncertainty principle scheme
			7.3.3.4 Interval statistical ensemble scheme: interval isothermal-isobaric (NPT) ensemble
	7.4 An example of R-MD: uniaxial tensile loading of an aluminum single crystal oriented in <100﹥ direction
		7.4.1 Simulation settings
		7.4.2 Interval EAM potential for aluminum based on Mishin's potential
		7.4.3 Numerical results
		7.4.4 Comparisons of numerical results for different schemes
		7.4.5 Verification and validation
		7.4.6 Finite size effect
	7.5 Discussion
	7.6 Conclusions
	Acknowledgment
	References
8 - Sensitivity analysis in kinetic Monte Carlo simulation based on random set sampling
	8.1 Introduction
	8.2 Interval probability and random set sampling
	8.3 Random set sampling in KMC
		8.3.1 Event selection
		8.3.2 Clock advancement
			8.3.2.1 When events are independent
			8.3.2.2 When events are correlated
		8.3.3 R-KMC sampling algorithm
	8.4 Demonstration
		8.4.1 Escherichia coli reaction network
		8.4.2 Methanol decomposition on Cu
		8.4.3 Microbial fuel cell
	8.5 Summary
	Acknowledgment
	References
9 - Quantifying the effects of noise on early states of spinodal decomposition: Cahn–Hilliard–Cook equation and energy-based me ...
	9.1 Introduction
	9.2 Cahn–Hilliard–Cook model
	9.3 Methodology
		9.3.1 Formulation
		9.3.2 Weak formulation
		9.3.3 Galerkin approximation
		9.3.4 Time scheme
	9.4 Morphology characterization
	9.5 Numerical implementation
		9.5.1 Spatial discretization
		9.5.2 Time discretization
		9.5.3 Parallel space–time noise generation
		9.5.4 Scalability analysis
	9.6 Results
		9.6.1 Energy-driven analysis and noise effects
			9.6.1.1 Effect of initial noise
		9.6.2 Domain size analysis
		9.6.3 Nonperiodic domains
		9.6.4 Enforcing fluctuation–dissipation
	9.7 Conclusions
	References
10 - Uncertainty quantification of mesoscale models of porous uranium dioxide
	10.1 Introduction
	10.2 Applying UQ at the mesoscale
	10.3 Grain growth
		10.3.1 Introduction
		10.3.2 Model summaries
		10.3.3 Sensitivity analysis
		10.3.4 Uncertainty quantification
	10.4 Thermal conductivity
		10.4.1 Introduction
		10.4.2 Model summaries
		10.4.3 Sensitivity analysis
		10.4.4 Uncertainty quantification
	10.5 Fracture
		10.5.1 Introduction
		10.5.2 Model summaries
		10.5.3 Sensitivity analysis
		10.5.4 Uncertainty quantification
	10.6 Conclusions
	Acknowledgments
	References
11 - Multiscale simulation of fiber composites with spatially varying uncertainties
	11.1 Background and literature review
	11.2 Our approach for multiscale UQ and UP
		11.2.1 Multiresponse Gaussian processes for uncertainty quantification
		11.2.2 Top-down sampling for uncertainty propagation
	11.3 Uncertainty quantification and propagation in cured woven fiber composites
		11.3.1 Uncertainty sources
		11.3.2 Multiscale finite element simulations
		11.3.3 Top-down sampling, coupling, and random field modeling of uncertainty sources
		11.3.4 Dimension reduction at the mesoscale via sensitivity analysis
		11.3.5 Replacing meso- and microscale simulations via metamodels
		11.3.6 Results on macroscale uncertainty
	11.4 Conclusion and future works
	Appendix
		Details on the sensitivity studies at the mesoscale
		Conditional distribution
	Acknowledgments
	References
12 - Modeling non-Gaussian random fields of material properties in multiscale mechanics of materials
	12.1 Introduction
	12.2 Methodology and elementary example
		12.2.1 Definition of scales
		12.2.2 On the representation of random fields
		12.2.3 Information-theoretic description of random fields
		12.2.4 Getting started with a toy problem
	12.3 Application to matrix-valued non-Gaussian random fields in linear elasticity
		12.3.1 Preliminaries
		12.3.2 Setting up the MaxEnt formulation
		12.3.3 Defining the non-Gaussian random field
		12.3.4 Application to transversely isotropic materials
			12.3.4.1 Formulation
			12.3.4.2 Two-dimensional numerical illustration
	12.4 Application to vector-valued non-Gaussian random fields in nonlinear elasticity
		12.4.1 Background
		12.4.2 Setting up the MaxEnt formulation
		12.4.3 Defining random field models for strain energy functions
	12.5 Conclusion
	Acknowledgments
	References
13 - Fractal dimension indicator for damage detection in uncertain composites
	13.1 Introduction
	13.2 Formulation
		13.2.1 Finite element model of composite plate
		13.2.2 Matrix crack modeling
		13.2.3 Fractal dimension
		13.2.4 Spatial uncertainty in material property
	13.3 Numerical results
		13.3.1 Localized damage detection based on fractal dimension–based approach
		13.3.2 Spatial uncertainty
	13.4 Conclusions
	References
14 - Hierarchical multiscale model calibration and validation for materials applications
	14.1 Introduction
	14.2 Multiresponse, multiscale TDBU HMM calibration
		14.2.1 Background
		14.2.2 Formulation
	14.3 Usage: TDBU calibration of CP of bcc Fe
		14.3.1 Background
		14.3.2 Crystal plasticity model
		14.3.3 Parameter estimates and data
		14.3.4 Implementation of the method
	14.4 Between the models: connection testing
		14.4.1 Background
		14.4.2 Formulation
	14.5 Usage: test of TDBU connection in CP of bcc Fe
		14.5.1 Background
		14.5.2 Implementation
		14.5.3 Sensitivity study of σp2
	14.6 Discussion and extensions to validation
	Acknowledgments
	References
15 - Efficient uncertainty propagation across continuum length scales for reliability estimates
	15.1 Introduction
	15.2 Hierarchical reliability approach
	15.3 Stochastic reduced–order models
		15.3.1 Construction of a stochastic reduced–order model
		15.3.2 SROM-based surrogate model and Monte Carlo simulation
	15.4 Concurrent coupling
	15.5 Applications examples
		15.5.1 Failure of a housing assembly
			15.5.1.1 Objective
			15.5.1.2 Model definition
			15.5.1.3 Uncertainty
			15.5.1.4 Results
		15.5.2 Apparent modulus of a cubic-elastic, micron-size plate in plane strain
			15.5.2.1 Objective
			15.5.2.2 Model definition
			15.5.2.3 Uncertainty
			15.5.2.4 Results
		15.5.3 Multiscale uncertainty propagation
			15.5.3.1 Objective
			15.5.3.2 Model definition
			15.5.3.3 Uncertainty
			15.5.3.4 Results
		15.5.4 Summary and discussion
	15.6 Conclusions
	Nomenclature
	Acknowledgments
	References
16 - Bayesian Global Optimization applied to the design of shape-memory alloys
	16.1 Introduction
	16.2 Bayesian Global Optimization
		16.2.1 Surrogate model with uncertainties
		16.2.2 Utility functions
	16.3 Design of new shape-memory alloys
		16.3.1 Searching for NiTi-based shape-memory alloys with high transformation temperature
		16.3.2 Search for very low thermal hysteresis NiTi-based shape-memory alloys
	16.4 Summary
	References
17 - An experimental approach for enhancing the predictability of mechanical properties of additively manufactured architected m ...
	17.1 Introduction
	17.2 A strategy for predicting the mechanical properties of additively manufactured metallic lattice structures via strut-level  ...
	17.3 Experimental investigation of the mechanical properties of DMLS octet lattice structures
		17.3.1 Fabrication of octet truss lattice structures and tensile bars
		17.3.2 Dimensional accuracy and relative density analysis of octet truss lattice structures
		17.3.3 Tension testing of standard tensile bars
		17.3.4 Compression testing of octet truss lattice structures
	17.4 Finite element analysis of the DMLS octet lattice structures based on bulk material properties
	17.5 Experimental investigation of the mechanical properties of DMLS lattice struts
	17.6 Finite element analysis of the DMLS octet lattice structures based on strut-level properties
	17.7 Opportunities for expanding the experimental study to better inform the finite element modeling
	17.8 Discussion
	Appendix
	Acknowledgments
	References
Index
	A
	B
	C
	D
	E
	F
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	H
	I
	J
	K
	L
	M
	N
	O
	P
	Q
	R
	S
	T
	U
	V
	W
	X
	Y
	Z
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