Table Of ContentUncertainty Quantification
of Stochastic Defects
in Materials
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Uncertainty Quantification of Stochastic Defects in Materials
Liu Chu
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Uncertainty Quantification
of Stochastic Defects
in Materials
Liu Chu
First edition published 2022
by CRC Press
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Library of Congress Cataloging‑in‑Publication Data
Names: Chu, Liu (Materials scientist), author.
Title: Uncertainty quantification of stochastic defects in materials / Liu Chu.
Description: Boca Raton, FL : CRC Press, 2022. | Series: Emerging materials
and technologies | Includes bibliographical references and index. |
Summary: “This book investigates uncertainty quantification methods for
stochastic defects in material microstructure. Pursuing a comprehensive
numerical analytical system, the book establishes a fundamental
framework for this topic, while emphasizing the importance of stochastic
and uncertainty quantification analysis and the significant influence of
microstructure defects in the material macro properties. The book is
intended for researchers, engineers, and advanced students interested in
material defect quantification methods, material reliability assessment,
artificial material microstructure optimization, and defect testing”—
Provided by publisher.
Identifiers: LCCN 2021035516 (print) | LCCN 2021035517 (ebook) |
ISBN 9781032128733 (hbk) | ISBN 9781032128757 (pbk) |
ISBN 9781003226628 (ebk)
Subjects: LCSH: Materials—Mathematical models | Quantitative research. |
Stochastic processes.
Classification: LCC TA404.23.C478 2022 (print) | LCC TA404.23 (ebook) |
DDC 620.1/1—dc23
LC record available at https://lccn.loc.gov/2021035516
LC ebook record available at https://lccn.loc.gov/2021035517
ISBN: 978-1-032-12873-3 (hbk)
ISBN: 978-1-032-12875-7 (pbk)
ISBN: 978-1-003-22662-8 (ebk)
DOI: 10.1201/9781003226628
Typeset in Times
by codeMantra
This book is a gift for my sons, Hanlin Shi and Chudi Shi.
Contents
Preface.......................................................................................................................xi
Author ....................................................................................................................xiii
Chapter 1 Overview ..............................................................................................1
References ............................................................................................3
Chapter 2 Uncertainty Quantification ...................................................................5
2.1 S tochastic Defects......................................................................5
2.2 Uncertainty Quantification ........................................................6
2.2.1 Probability Theory .......................................................6
2.2.2 Evidence Theory ..........................................................7
2.2.3 Possibility Theory ........................................................9
2.2.4 I nterval Analysis ..........................................................9
2.2.5 Convex Modeling .........................................................9
2.3 Correlation Analysis ................................................................10
2.3.1 Pearson Correlation Coefficient .................................10
2.3.2 Spearman Correlation Coefficient ..............................11
2.3.3 Kendall Correlation Coefficient .................................12
2.4 Sensitivity Analysis .................................................................13
References ..........................................................................................15
Section i Methods and theories
Chapter 3 Monte Carlo Methods .........................................................................21
3.1 Introduction .............................................................................21
3.1.1 Mathematical Formulation of MC Integration ...........21
3.1.2 Plain (Crude) MC Algorithm .....................................22
3.1.3 Geometric MC Algorithm ..........................................23
3.2 Advanced MC Methods ...........................................................24
3.2.1 Importance Sampling Algorithm ...............................24
3.2.2 Weight Function Approach.........................................26
3.2.3 Latin Hypercube Sampling Approach ........................26
3.3 Random Interpolation Quadrature ..........................................27
3.4 Iterative MC Methods for Linear Equations ...........................28
3.4.1 Iterative MC Algorithms ............................................29
3.4.2 Convergence and Mapping .........................................31
3.5 Markov Chain MC Methods for the Eigenvalue Problem .......32
3.5.1 Formulation of the Eigenvalue Problem .....................32
vii
viii Contents
3.5.2 Method for Choosing the Number of Iterations k ......34
3.5.3 Method for Choosing the Number of Chains .............35
References ..........................................................................................36
Chapter 4 Polynomial Chaos Expansion .............................................................37
4.1 Fundamental Description of PCE............................................37
4.2 S tochastic Approximation .......................................................38
4.3 Hermite Polynomials and Gram–Charlier Series ...................39
4.4 Karhunen–Loeve Transform ...................................................42
4.5 Karhunen–Loeve Expansion ...................................................44
4.6 Comparison and Discussion ....................................................45
References ..........................................................................................49
Chapter 5 Stochastic Finite Element Method .....................................................51
5.1 Methods for Discretization of Random Fields ........................51
5.1.1 Point Discretization Methods .....................................51
5.1.1.1 The Midpoint Method .................................51
5.1.1.2 The Shape Function Method .......................52
5.1.2 Average Discretization Methods ................................52
5.1.2.1 Spatial Average ...........................................52
5.1.2.2 The Weighted Integral Method ...................53
5.1.3 Series Expansion Methods .........................................54
5.2 Perturbation Method ................................................................56
5.3 S pectral SFEM ........................................................................59
5.4 Neumann SFEM ......................................................................64
5.5 Finite Element Reliability Analysis .........................................66
References ..........................................................................................69
Chapter 6 Machine Learning Methods ...............................................................71
6.1 Artificial Neural Networks ......................................................71
6.1.1 Components of ANN ..................................................71
6.1.2 L earning Paradigms ...................................................74
6.1.3 Theoretical Properties ................................................74
6.1.3.1 Computational Power ..................................74
6.1.3.2 Capacity ......................................................75
6.1.3.3 Convergence................................................75
6.1.3.4 Generalization and Statistics ......................75
6.2 Radial Basis Network ..............................................................76
6.3 Backpropagation Neural Network ...........................................76
6.4 Restricted Boltzmann Machine ...............................................77
6.5 Hopfield Neural Network.........................................................80
6.6 Convolutional Neural Network ................................................81
References ..........................................................................................83
Contents ix
Section ii examples
Chapter 7 Numerical Examples ..........................................................................87
7.1 Importance Sampling ..............................................................87
7.2 Orthogonal Polynomial ...........................................................90
7.3 G ram–Charlier Series..............................................................92
7.4 Kriging Surrogate Model ........................................................94
7.4.1 Numerical Issues of the Kriging Model .....................96
7.4.1.1 Issue 1: Requirement of Sufficient
Effective Samples .......................................96
7.4.1.2 Issue 2: Effects of the Correlation Matrix ...98
7.4.2 Subset Simulation .......................................................98
7.4.3 Kriging Surrogate Model with Subset Simulation ...100
7.4.3.1 The Six-H ump Camel-B ack Function.......100
7.4.3.2 Vibration Analysis of a Wing Structure ...102
References ........................................................................................106
Chapter 8 Monte Carlo-B ased Finite Element Method ....................................107
8.1 Introduction ...........................................................................107
8.2 Graphene Material Description .............................................108
8.3 Monte Carlo-B ased Finite Element Method ..........................109
8.4 Graphene with Stochastic Defects .........................................112
8.5 Buckling Results and Discussion ..........................................112
8.5.1 P robability Analysis .................................................112
8.5.2 Comparison and Discussion .....................................114
8.5.3 Displacement Results of Graphene Sheets ...............118
8.6 Conclusion .............................................................................120
References ........................................................................................120
Chapter 9 Impacts of Vacancy Defects in Resonant Vibration .........................123
9.1 Introduction ...........................................................................123
9.2 Materials and Methods ..........................................................124
9.3 Validation of the Model .........................................................128
9.4 Results and Discussion ..........................................................130
9.4.1 Amount of Vacancy Defects .....................................130
9.4.2 Geometrical Parameters ...........................................133
9.4.3 Material Parameters .................................................137
9.4.4 Graphene Sheets with Vacancy Defects ...................137
9.5 Conclusions ............................................................................142
References ........................................................................................142
Chapter 10 Uncertainty Quantification in Nanomaterials ..................................145
10.1 Introduction ...........................................................................145
