Table Of Content

Uncertainty Quantification of Stochastic Defects in Materials Emerging Materials and Technologies Series Editor: Boris I. Kharissov Biomaterials and Materials for Medicine: Innovations in Research, Devices, and Applications Jingan Li Advanced Materials and Technologies for Wastewater Treatment Sreedevi Upadhyayula and Amita Chaudhary Green Tribology: Emerging Technologies and Applications T.V.V.L.N. Rao, Salmiah Binti Kasolang, Xie Guoxin, Jitendra Kumar Katiyar, and Ahmad Majdi Abdul Rani Biotribology: Emerging Technologies and Applications T.V.V.L.N. Rao, Salmiah Binti Kasolang, Xie Guoxin, Jitendra Kumar Katiyar, and Ahmad Majdi Abdul Rani Bioengineering and Biomaterials in Ventricular Assist Devices Eduardo Guy Perpétuo Bock Semiconducting Black Phosphorus: From 2D Nanomaterial to Emerging 3D Architecture Han Zhang, Nasir Mahmood Abbasi, and Bing Wang Biomass for Bioenergy and Biomaterials Nidhi Adlakha, Rakesh Bhatnagar, and Syed Shams Yazdani Energy Storage and Conversion Devices: Supercapacitors, Batteries, and Hydroelectric Cell Anurag Gaur, A.L. Sharma, and Anil Arya Nanomaterials for Water Treatment and Remediation Srabanti Ghosh, Aziz Habibi-Yangjeh, Swati Sharma, and Ashok Kumar Nadda 2D Materials for Surface Plasmon Resonance-Based Sensors Sanjeev Kumar Raghuwanshi, Santosh Kumar, and Yadvendra Singh Functional Nanomaterials for Regenerative Tissue Medicines Mariappan Rajan Uncertainty Quantification of Stochastic Defects in Materials Liu Chu For more information about this series, please visit: https://www.routledge.com/Emerging- Materials-and-Technologies/book-series/CRCEMT Uncertainty Quantification of Stochastic Defects in Materials Liu Chu First edition published 2022 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2022 Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. 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For works that are not available on CCC please contact m pkbookspermissions@ tandf.co.uk Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. 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