Maths for engineers Books

Book SynopsisThis book provides an excellent introduction to the subject designed for readers from a variety of mathematical backgrounds. It features introductory properties of metric spaces and Banach spaces, and an appendix contains a summary of the concepts of set theory.Trade Review"...deserves to be on the bookshelf of everyone who wants to know about fixed-point theory in metric and Banach spaces and experts who want to read an insightful survey of some basic ideas..." (Mathematical Reviews, 2002b) "Clear, friendly exposition." (American Mathematical Monthly, August/September 2002)Table of ContentsPreface ix I Metric Spaces 1 Introduction 3 1.1 The real numbers R 3 1.2 Continuous mappings in R 5 1.3 The triangle inequality in R 7 1.4 The triangle inequality in R" 8 1.5 Brouwer's Fixed Point Theorem 10 Exercises 11 2 Metric Spaces 13 2.1 The metric topology 15 2.2 Examples of metric spaces 19 2.3 Completeness 26 2.4 Separability and connectedness 33 2.5 Metric convexity and convexity structures 35 Exercises 38 3 Metric Contraction Principles 41 3.1 Banach's Contraction Principle 41 3.2 Further extensions of Banach's Principle 46 3.3 The Caristi-Ekeland Principle 55 3.4 Equivalents of the Caristi-Ekeland Principle 58 3.5 Set-valued contractions 61 3.6 Generalized contractions 64 Exercises 67 4 Hyperconvex Spaces 71 4.1 Introduction 71 4.2 Hyperconvexity 77 4.3 Properties of hyperconvex spaces 80 4.4 A fixed point theorem 84 4.5 Intersections of hyperconvex spaces 87 4.6 Approximate fixed points 89 4.7 Isbell's hyperconvex hull 91 Exercises 98 5 "Normal" Structures in Metric Spaces 101 5.1 A fixed point theorem 101 5.2 Structure of the fixed point set 103 5.3 Uniform normal structure 106 5.4 Uniform relative normal structure 110 5.5 Quasi-normal structure 112 5.6 Stability and normal structure 115 5.7 Ultrametric spaces 116 5.8 Fixed point set structure—separable case 120 Exercises 123 II Banach Spaces 6 Banach Spaces: Introduction 127 6.1 The definition 127 6.2 Convexity 131 6.3 £2 revisited 132 6.4 The modulus of convexity 136 6.5 Uniform convexity of the tp spaces 138 6.6 The dual space: Hahn-Banach Theorem 142 6.7 The weak and weak* topologies 144 6.8 The spaces c, CQ, t\ and ^ 146 6.9 Some more general facts 148 6.10 The Schur property and £j 150 6.11 More on Schauder bases in Banach spaces 154 6.12 Uniform convexity and reflexivity 163 6.13 Banach lattices 165 Exercises 168 7 Continuous Mappings in Banach Spaces 171 7.1 Introduction 171 7.2 Brouwer's Theorem 173 7.3 Further comments on Brouwer's Theorem 176 7.4 Schauder's Theorem 179 7.5 Stability of Schauder's Theorem 180 7.6 Banach algebras: Stone Weierstrass Theorem 182 7.7 Leray-Schauder degree 183 7.8 Condensing mappings 187 7.9 Continuous mappings in hyperconvex spaces 191 Exercises 195 8 Metric Fixed Point Theory 197 8.1 Contraction mappings 197 8.2 Basic theorems for nonexpansive mappings 199 8.3 A closer look at ßë 205 8.4 Stability results in arbitrary spaces 207 8.5 The Goebel-Karlovitz Lemma 211 8.6 Orthogonal convexity 213 8.7 Structure of the fixed point set 215 8.8 Asymptotically regular mappings 219 8.9 Set-valued mappings 222 8.10 Fixed point theory in Banach lattices 225 Exercises 238 9 Banach Space Ultrapowers 243 9.1 Finite representability 243 9.2 Convergence of ultranets 248 9.3 The Banach space ultrapower X 249 9.4 Some properties of X 252 9.5 Extending mappings to X 255 9.6 Some fixed point theorems 257 9.7 Asymptotically nonexpansive mappings 262 9.8 The demiclosedness principle 263 9.9 Uniformly non-creasy spaces 264 Exercises 270 Appendix: Set Theory 273 A.l Mappings 273 A.2 Order relations and Zermelo's Theorem 274 A.3 Zorn's Lemma and the Axiom Of Choice 275 A.4 Nets and subnets 277 A.5 Tychonoff's Theorem 278 A.6 Cardinal numbers 280 A. 7 Ordinal numbers and transfinite induction 281 A.8 Zermelo's Fixed Point Theorem 284 A.9 A remark about constructive mathematics 286 Exercises 287 Bibliography 289 Index 301

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Book SynopsisGives you what you need to know - basic essential concepts - about calculus. Suitable for those looking for a readable alternative to the usual unwieldy calculus text, this title provides in a condensed format the material covered in the standard two-year calculus course. It also covers vectors, lines, and planes in space; and line integrals.Trade Review"...expands coverage to vectors and calculus of several variables...plenty of worked out problems..." (American Mathematical Monthly, August/September 2003) "...material included is well formulated and approachable...recommended." (Choice, Vol. 41, No. 1, September 2003)Table of ContentsAUTHOR'S MESSAGE TO THE READER vii ANNOTATED TABLE OF CONTENTS ix ACKNOWLEDGMENTS xv CHAPTER 1 Lines 1 CHAPTER 2 Parabolas, Ellipses, Hyperbolas 7 CHAPTER 3 Differentiation 13 CHAPTER 4 Differentiation Formulas 19 CHAPTER 5 The Chain Rule 25 CHAPTER 6 Trigonometric Functions 31 CHAPTER 7 Exponential Functions and Logarithms 39 CHAPTER 8 Inverse Functions 45 CHAPTER 9 Derivatives and Graphs 51 CHAPTER 10 Following the Tangent Line 57 CHAPTER 11 The Indefinite Integral 63 CHAPTER 12 The Definite Integral 69 CHAPTER 13 Work, Volume, and Force 75 CHAPTER 14 Parametric Equations 81 CHAPTER 15 Change of Variable 87 CHAPTER 16 Integrating Rational Functions 91 CHAPTER 17 Integration By Parts 97 CHAPTER 18 Trigonometric Integrals 101 CHAPTER 19 Trigonometric Substitution 107 CHAPTER 20 Numerical Integration 115 CHAPTER 21 Limits At oo; Sequences 119 CHAPTER 22 Improper Integrals 127 CHAPTER 23 Series 133 CHAPTER 24 Power Series 141 CHAPTER 25 Taylor Polynomials 149 CHAPTER 26 Taylor Series 155 CHAPTER 27 Separable Differential Equations 161 CHAPTER 28 First-Order Linear Equations 167 CHAPTER 29 Homogeneous Second-Order Linear Equations 173 CHAPTER 30 Nonhomogeneous Second-Order Equations 179 CHAPTER 31 Vectors 185 CHAPTER 32 The Dot Product 195 CHAPTER 33 Lines and Planes in Space 201 CHAPTER 34 Surfaces 211 CHAPTER 35 Partial Derivatives 217 CHAPTER 36 Tangent Plane and Differential Approximation CHAPTER 37 Chain Rules 227 CHAPTER 38 Gradient and Directional Derivatives 233 CHAPTER 39 Maxima and Minima 239 CHAPTER 40 Double Integrals 245 CHAPTER 41 Line Integrals 255 CHAPTER 42 Green's Theorem 259 CHAPTER 43 Exact Differentials 267 ANSWERS 273 INDEX 299 ABOUT THE AUTHOR 303

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Book SynopsisToday''s need-to-know optimization techniques, at your fingertips The use of optimization methods is familiar territory to academicians and researchers. Yet, in today''s world of deregulated electricity markets, it''s just as important for electric power professionals to have a solid grasp of these increasingly relied upon techniques. Making those techniques readily accessible is the hallmark of Optimization Principles: Practical Applications to the Operation and Markets of the Electric Power Industry. With deregulation, market rules and economic principles dictate that commodities be priced at the marginal value of their production. As a result, it''s necessary to work with ever-more-sophisticated algorithms using optimization techniques-either for the optimal dispatch of the system itself, or for pricing commodities and the settlement of markets. Succeeding in this new environment takes a good understanding of methods that involve linear and nonTrade Review"...an important contribution to the field of power system analysis...should provide the reader with a pleasant learning experience." (IEEE Power & Energy Magazine, November/December 2005)Table of ContentsPreface. 1. Introduction. PART I: MATHEMATICAL BACKGROUND. 2. Fundamentals of Matrix Algebra. PART II: LINEAR OPTIMIZATION. 3. Solution of Equations, Inequalities, and Linear Programs. 4. Solved Linear Program Problems. PART III: NONLINEAR OPTIMIZATION. 5. Mathematical Background to Nonlinear Programs. 6. Unconstrained Nonlinear Optimization. 7. Constrained Nonlinear Optimization. 8. Solved Nonlinear Optimization Problems. Appendix A: Basic Principles of Electricity. Appendix B: Network Equations. Appendix C: Relation Between Pseudo-Inverse and Least-Square Error Fit. Bibliography. Index. About the Author.

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Book SynopsisBayesian statistics uses information from past experience to infer the results of future events. With recent advances in computing power and the development of computer intensive methods for statistical estimation, Bayesian approaches to model estimation have become more feasible and popular.Trade Review"I recommend…highly to statisticians, [and] health researchers...among others to consider keeping on their bookshelf." (Journal of Statistical Computation and Simulation, April 2005) "…a great book…fills a critical gap in existing literature. It is an excellent book for anyone interested in Bayesian modeling…" (Journal of the American Statistical Association, March 2005) "It is certainly a fine choice as a supporting reference in either a first or second Bayesian methods course…” (Technometrics, May 2004) "...has a contemporary feel, with recent developments in financial time series modelling and epidemiology included..." (Short Book Reviews, Vol 23(3), December 2003)Table of ContentsPreface. The Basis for, and Advantages of, Bayesian Model Estimation via Repeated Sampling. Hierarchical Mixture Models. Regression Models. Analysis of Multi-Level Data. Models for Time Series. Analysis of Panel Data. Models for Spatial Outcomes and Geographical Association. Structural Equation and Latent Variable Models. Survival and Event History Models. Modelling and Establishing Causal Relations: Epidemiological Methods and Models. Index.

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Book SynopsisIn modern society, we are ever more aware of the environmentalissues we face, whether these relate to global warming, depletionof rivers and oceans, despoliation of forests, pollution of land,poor air quality, environmental health issues, etc. At the mostfundamental level it is necessary to monitor what is happening inthe environment - collecting data to describe the changingscene. More importantly, it is crucial to formally describe theenvironment with sound and validated models, and to analyse andinterpret the data we obtain in order to take action. Environmental Statistics provides a broad overview of thestatistical methodology used in the study of the environment,written in an accessible style by a leading authority on thesubject. It serves as both a textbook for students of environmentalstatistics, as well as a comprehensive source of reference foranyone working in statistical investigation of environmentalissues. * Provides broad coverage of the methodology used in tTrade Review"Inspired by the Encyclopedia of Statistical Sciences, SecondEdition (ESS2e), this volume presents a concise, well-rounded focuson the statistical concepts and applications that are essential forunderstanding gathered data in the fields of engineering, qualitycontrol, and the physical sciences. The book successfully upholdsthe goals of ESS2e by combining both previously-published and newlydeveloped contributions written by over 100 leading academics,researchers, and practitioner in a comprehensive, approachableformat. The result is a succinct reference that unveils modern,cutting-edge approaches to acquiring and analyzing data acrossdiverse subject areas within these three disciplines, includingoperations research, chemistry, physics, the earth sciences,electrical engineering, and quality assurance." (Finwin, 7September 2011) "In this book, Vic Barnett, a distinguished environmentalstatistician, provides an overview of statistical methods that havebeen used on such problems in the environmental sciences."(Journal of the American Statistical Association, September2006) "...combines sound fundamentals and their applications."(European Journal of Soil Science, No.56, April 2005) "Many tables, graphs and figures illustrate the environmentalapplications of the statistical methods that are described."(Journal of the Royal Statistical Society, Series A,Vol.168, No.2, March 2005) "...well written...methods are illustrated with interestingexamples...a comprehensive reference source for anyone working onenvironmental issues..." (Short Book Reviews, Vol.24, No.3,December 2004) "Statisticians should enjoy the book. The author is an extremelyknowledgeable statistician, and he is writing about an applicationdomain that he clearly knows." (Technometrics, November2004) "An excellent book. Highly recommended." (Choice, July2004) "...this provides an excellent sketch of the current state ofdevelopment for new statistical methodologies...a valuableresource..." (Statistics in Medicine, 15th August 2005)Table of ContentsPreface. Chapter 1: Introduction. 1.1 Tomorrow is too Late! 1.2 Environmental Statistics. 1.3 Some Examples. 1.3.1 ‘Getting it all together’. 1.3.2 ‘In time and space’. 1.3.3 ‘Keep it simple’. 1.3.4 ‘How much can we take?’ 1.3.5 ‘Over the top’. 1.4 Fundamentals. 1.5 Bibliography. PART I: EXTREMAL STRESSES: EXTREMES, OUTLIERS, ROBUSTNESS. Chapter 2: Ordering and Extremes: Applications, models, inference. 2.1 Ordering the Sample. 2.1.1 Order statistics. 2.2 Order-based Inference. 2.3 Extremes and Extremal Processes. 2.3.1 Practical study and empirical models; generalized extreme-value distributions. 2.4 Peaks over Thresholds and the Generalized Pareto Distribution. Chapter 3: Outliers and Robustness. 3.1 What is an Outlier? 3.2 Outlier Aims and Objectives. 3.3 Outlier-Generating Models. 3.3.1 Discordancy and models for outlier generation. 3.3.2 Tests of discordancy for specific distributions. 3.4 Multiple Outliers: Masking and Swamping. 3.5 Accommodation: Outlier-Robust Methods. 3.6 A Possible New Approach to Outliers. 3.7 Multivariate Outliers. 3.8 Detecting Multivariate Outliers. 3.8.1 Principles. 3.8.2 Informal methods. 3.9 Tests of Discordancy. 3.10 Accommodation. 3.11 Outliers in linear models. 3.12 Robustness in General. PART II: COLLECTING ENVIRONMENTAL DATA: SAMPLING AND MONITORING. Chapter 4: Finite-Population Sampling. 4.1 A Probabilistic Sampling Scheme. 4.2 Simple Random Sampling. 4.2.1 Estimating the mean, &Xmacr;. 4.2.2 Estimating the variance, S2. 4.2.3 Choice of sample size, n. 4.2.4 Estimating the population total, XT. 4.2.5 Estimating a proportion, P. 4.3 Ratios and Ratio Estimators. 4.3.1 The estimation of a ratio. 4.3.2 Ratio estimator of a population total or mean. 4.4 Stratified (simple) Random Sampling. 4.4.1 Comparing the simple random sample mean and the stratified sample mean. 4.4.2 Choice of sample sizes. 4.4.3 Comparison of proportional allocation and optimum allocation. 4.4.4 Optimum allocation for estimating proportions. 4.5 Developments of Survey Sampling. Chapter 5: Inaccessible and Sensitive Data. 5.1 Encountered Data. 5.2 Length-Biased or Size-Biased Sampling and Weighted Distributions. 5.2.1 Weighted distribution methods. 5.3 Composite Sampling. 5.3.1 Attribute Sampling. 5.3.2 Continuous variables. 5.3.3 Estimating mean and variance. 5.4 Ranked-Set Sampling. 5.4.1 The ranked-set sample mean. 5.4.2 Optimal estimation. 5.4.3 Ranked-set sampling for normal and exponential distributions. 5.4.4 Imperfect ordering. Chapter 6: Sampling in the Wild. 6.1 Quadrat Sampling. 6.2 Recapture Sampling. 6.2.1 The Petersen and Chapman estimators. 6.2.2 Capture–recapture methods in open populations. 6.3 Transect Sampling. 6.3.1 The simplest case: strip transects. 6.3.2 Using a detectability function. 6.3.3 Estimating f (y). 6.3.4 Modifications of approach. 6.3.5 Point transects or variable circular plots. 6.4 Adaptive Sampling. 6.4.1 Simple models for adaptive sampling. Part III: EXAMINING ENVIRONMENTAL EFFECTS: STIMULUS–RESPONSE RELATIONSHIPS. Chapter 7: Relationship: regression-type models and methods. 7.1 Linear Models. 7.1.1 The linear model. 7.1.2 The extended linear model. 7.1.3 The normal linear model. 7.2 Transformations. 7.2.1 Looking at the data. 7.2.2 Simple transformations. 7.2.3 General transformations. 7.3 The Generalized Linear Model. Chapter 8: Special Relationship Models, Including Quantal Response and Repeated Measures. 8.1 Toxicology Concerns. 8.2 Quantal Response. 8.3 Bioassay. 8.4 Repeated Measures. Part IV: STANDARDS AND REGULATIONS. Chapter 9: Environmental Standards. 9.1 Introduction. 9.2 The Statistically Verifiable Ideal Standard. 9.2.1 Other sampling methods. 9.3 Guard Point Standards. 9.4 Standards Along the Cause–Effect Chain. Part V: A MANY-DIMENSIONAL ENVIRONMENT: SPATIAL AND TEMPORAL PROCESSES. Chapter 10: Time-Series Methods. 10.1 Space and Time Effects. 10.2 Time Series. 10.3 Basic Issues. 10.4 Descriptive Methods. 10.4.1 Estimating or eliminating trend. 10.4.2 Periodicities. 10.4.3 Stationary time series. 10.5 Time-Domain Models and Methods. 10.6 Frequency-Domain Models and Methods. 10.6.1 Properties of the spectral representation. 10.6.2 Outliers in time series. 10.7 Point Processes. 10.7.1 The Poisson process. 10.7.2 Other point processes. Chapter 11: Spatial Methods for Environmental Processes. 11.1 Spatial Point Process Models and Methods. 11.2 The General Spatial Process. 11.2.1 Predication, interpolation and kriging. 11.2.2 Estimation of the variogram. 11.2.3 Other forms of kriging. 11.3 More about Standards Over Space and Time. 11.4 Relationship. 11.5 More about Spatial Models. 11.5.1 Types of spatial model. 11.5.2 Harmonic analysis of spatial processes. 11.6 Spatial Sampling and Spatial Design. 11.6.1 Spatial sampling. 11.6.2 Spatial design. 11.7 Spatial-Temporal Models and Methods. References. Index.

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Book SynopsisContact mechanics is a specialist area in engineering mechanics. It deals with non standard mechanics which frequently appear in real technical applications. Examples include the simulation of car crashes, human joints, car tyres, rubber seals and metal forming processes.Table of ContentsPreface. Introduction. Introduction to Contact Mechanics. Continuum Solid Mechanics and Weak Forms. Contact Kinematics. Constitutive Equations for Contact Interfaces. Contact Boundary Value Problem and Weak Form. Discretization of the Continuum. Discretization, Small Deformation Contact. Discretization, Large Deformation Contact. Solution Algorithms. Thermo-mechanical Contact. Beam Contact. Adaptive Finite Element Methods for Contact Problems. Computation of Critical Points with Contact Constraints. Appendix A: Gauss Integration Rules. Appendix B: Convective Coordinates. Appendix C: Parameter Identification for Friction Materials. References. Index.

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Book SynopsisThe classic introduction to engineering optimization theory and practice--now expanded and updated Engineering optimization helps engineers zero in on the most effective, efficient solutions to problems. This text provides a practical, real-world understanding of engineering optimization. Rather than belaboring underlying proofs and mathematical derivations, it emphasizes optimization methodology, focusing on techniques and stratagems relevant to engineering applications in design, operations, and analysis. It surveys diverse optimization methods, ranging from those applicable to the minimization of a single-variable function to those most suitable for large-scale, nonlinear constrained problems. New material covered includes the duality theory, interior point methods for solving LP problems, the generalized Lagrange multiplier method and generalization of convex functions, and goal programming for solving multi-objective optimization problems. A practical, hands-on referTable of ContentsPreface. 1 Introduction to Optimization. 1.1 Requirements for the Application of Optimization Methods. 1.2 Applications of Optimization in Engineering. 1.3 Structure of Optimization Problems. 1.4 Scope of This Book. References. 2 Functions of a Single Variable. 2.1 Properties of Single-Variable Functions. 2.2 Optimality Criteria. 2.3 Region Elimination Methods. 2.4 Polynomial Approximation or Point Estimation Methods. 2.5 Methods Requiring Derivatives. 2.6 Comparison of Methods. 2.7 Summary. References. Problems. 3 Functions of Several Variables. 3.1 Optimality Criteria. 3.2 Direct-Search Methods. 3.3 Gradient-Based Methods. 3.4 Comparison of Methods and Numerical Results. 3.5 Summary. References. Problems. 4 Linear Programming. 4.1 Formulation of Linear Programming Models. 4.2 Graphical Solution of Linear Programs in Two Variables. 4.3 Linear Program in Standard Form. 4.5 Computer Solution of Linear Programs. 4.5.1 Computer Codes. 4.6 Sensitivity Analysis in Linear Programming. 4.7 Applications. 4.8 Additional Topics in Linear Programming. 4.9 Summary. References. Problems. 5 Constrained Optimality Criteria. 5.1 Equality-Constrained Problems. 5.2 Lagrange Multipliers. 5.3 Economic Interpretation of Lagrange Multipliers. 5.4 Kuhn-Tucker Conditions. 5.5 Kuhn-Tucker Theorems. 5.6 Saddlepoint Conditions. 5.7 Second-Order Optimality Conditions. 5.8 Generalized Lagrange Multiplier Method. 5.9 Generalization of Convex Functions. 5.10 Summary. References. Problems. 6 Transformation Methods. 6.1 Penalty Concept. 6.2 Algorithms, Codes, and Other Contributions. 6.3 Method of Multipliers. 6.4 Summary. References. Problems. 7 Constrained Direct Search. 7.1 Problem Preparation. 7.2 Adaptations of Unconstrained Search Methods. 7.3 Random-Search Methods. 7.4 Summary. References. Problems. 8 Linearization Methods for Constrained Problems. 8.1 Direct Use of Successive Linear Programs. 8.2 Separable Programming. 8.3 Summary. References. Problems. 9 Direction Generation Methods Based on Linearization. 9.1 Method of Feasible Directions. 9.2 Simplex Extensions for Linearly Constrained Problems. 9.3 Generalized Reduced Gradient Method. 9.4 Design Application. 9.5 Summary. References. Problems. 10 Quadratic Approximation Methods for Constrained Problems. 10.1 Direct Quadratic Approximation. 10.2 Quadratic Approximation of the Lagrangian Function. 10.3 Variable Metric Methods for Constrained Optimization. 10.4 Discussion. 10.5 Summary. References. Problems. 11 Structured Problems and Algorithms. 11.1 Integer Programming. 11.2 Quadratic Programming. 11.3 Complementary Pivot Problems. 11.4 Goal Programming. 11.5 Summary. References. Problems. 12 Comparison of Constrained Optimization Methods. 12.1 Software Availability. 12.2 A Comparison Philosophy. 12.3 Brief History of Classical Comparative Experiments. 12.4 Summary. References. 13 Strategies for Optimization Studies. 13.1 Model Formulation. 13.2 Problem Implementation. 13.3 Solution Evaluation. 13.4 Summary. References. Problems. 14 Engineering Case Studies. 14.1 Optimal Location of Coal-Blending Plants by Mixed-Integer Programming. 14.2 Optimization of an Ethylene Glycol-Ethylene Oxide Process. 14.3 Optimal Design of a Compressed Air Energy Storage System. 14.4 Summary. References. Appendix A Review of Linear Algebra. A.1 Set Theory. A.2 Vectors. A.3 Matrices. A.3.1 Matrix Operations. A.3.2 Determinant of a Square Matrix. A.3.3 Inverse of a Matrix. A.3.4 Condition of a Matrix. A.3.5 Sparse Matrix. A.4 Quadratic Forms. A.4.1 Principal Minor. A.4.2 Completing the Square. A.5 Convex Sets. Appendix B Convex and Concave Functions. Appendix C Gauss-Jordan Elimination Scheme. Author Index. Subject Index.

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Book SynopsisThis comprehensive reference presents the latest techniques for numerical analysis of forming operations. This is the perfect tool for those who wish to investigate new analytical methods for forming.Table of ContentsThe Tensile Test and Basic Material Behavior. Tensors, Matrices, Notation. Stress. Strain. Standard Mechanical Principles. Elasticity. Plasticity. Crystal-Based Plasticity. Friction. Classical Forming Analysis. Index.

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Book SynopsisNumerical Computation of Internal and External Flows Volume 2: Computational Methods for Inviscid and Viscous Flows C. Hirsch, Vrije Universiteit Brussel, Brussels, Belgium This second volume deals with the applications of computational methods to the problems of fluid dynamics.Table of ContentsPreface xv Nomenclature xix Part V: The Numerical Computation of Potential Flows 1 Chapter 13 The Mathematical Formulations of the Potential Flow Model 4 13.1 Conservative Form of the Potential Equation 4 13.2 The Non-conservative Form of the Isentropic Potential Flow Model 6 13.2.1 Small-perturbation potential equation 7 13.3 The Mathematical Properties of the Potential Equation 9 13.3.1 Unsteady potential flow 9 13.3.2 Steady potential flow 9 13.4 Boundary Conditions 14 13.4.1 Solid wall boundary condition 14 13.4.2 Far field conditions 15 13.4.3 Cascade and channel flows 17 13.4.4 Circulation and Kutta condition 18 13.5 Integral or Weak Formulation of the Potential Model 18 13.5.1 Bateman variational principle 19 13.5.2 Analysis of some properties of the variational integral 20 Chapter 14 The Discretization of the Subsonic Potential Equation 26 14.1 Finite Difference Formulation 27 14.1.1 Numerical estimation of the density 29 14.1.2 Curvilinear mesh 31 14.1.3 Consistency of the discretization of metric coefficients 34 14.1.4 Boundary conditions—curved solid wall 36 14.2 Finite Volume Formulation 38 14.2.1 Jameson and Caughey’s finite volume method 39 14.3 Finite Element Formulation 42 14.3.1 The finite element—Galerkin method 43 14.3.2 Least squares or optimal control approach 47 14.4 Iteration Scheme for the Density 47 Chapter 15 The Computation of Stationary Transonic Potential Flows 57 15.1 The Treatment of the Supersonic Region: Artificial Viscosity—Density and Flux Upwinding 61 15.1.1 Artificial viscosity—non-conservative potential equation 62 15.1.2 Artificial viscosity—conservative potential equation 66 15.1.3 Artificial compressibility 67 15.1.4 Artificial flux or flux upwinding 70 15.2 Iteration Schemes for Potential Flow Computations 77 15.2.1 Line relaxation schemes 77 15.2.2 Guidelines for resolution of the discretized potential equation 81 15.2.3 The alternating direction implicit method—approximate factorization schemes 88 15.2.4 Other techniques—multigrid methods 98 15.3 Non-uniqueness and Non-isentropic Potential Models 104 15.3.1 Isentropic shocks 105 15.3.2 Non-uniqueness and breakdown of the transonic potential flow model 105 15.3.3 Non-isentropic potential models 112 15.4 Conclusions 117 Part VI: The Numerical Solution of the System of Euler Equations 125 Chapter 16 The Mathematical Formulation of the System of Euler Equations 132 16.1 The Conservative Formulation of the Euler Equations 132 16.1.1 Integral conservative formulation of the Euler equations 133 16.1.2 Differential conservative formulation 134 16.1.3 Cartesian system of coordinates 134 16.1.4 Discontinuities and Rankine-Hugoniot relations—entropy condition 135 16.2 The Quasi-linear Formulation of the Euler Equations 138 16.2.l The Jacobian matrices for conservative variables 138 16.2.2 The Jacobian matrices for primitive variables 145 16.2.3 Transformation matrices between conservative and non-conservative variables 147 16.3 The Characteristic Formulation of the Euler Equations—Eigenvalues and Compatibility Relations 150 16.3.1 General properties of characteristics 151 16.3.2 Diagonalization of the Jacobian matrices 153 16.3.3 Compatibility equations 154 16.4 Characteristic Variables and Eigenvalues for One-dimensional Flows 157 16.4.1 Eigenvalues and eigenvectors of Jacobian matrix 158 16.4.2 Characteristic variables 162 16.4.3 Characteristics in the xt-plane—shocks and contact discontinuities 168 16.4.4 Physical boundary conditions 171 16.4.5 Characteristics and simple wave solutions 173 16.5 Eigenvalues and Compatibility Relations in Multidimensional Flows 176 16.5.1 Jacobian eigenvalues and eigenvectors in primitive variables 177 16.5.2 Diagonalization of the conservative Jacobians 180 16.5.3 Mach cone and compatibility relations 184 16.5.4 Boundary conditions 191 16.6 Some Simple Exact Reference Solutions for One-dimensional Inviscid Flows 196 16.6.1 The linear wave equation 196 16.6.2 The inviscid Burgers equation 196 16.6.3 The shock tube problem or Riemann problem 204 16.6.4 The quasi-one-dimensional nozzle flow 211 Chapter 17 The Lax–Wendroff Family of Space-centred Schemes 224 17.1 The Space-centred Explicit Schemes of First Order 226 17.1.1 The one-dimensional Lax–Friedrichs scheme 226 17.1.2 The two-dimensional Lax–Friedrichs scheme 229 17.1.3 Corrected viscosity scheme 233 17.2 The Space-centred Explicit Schemes of Second Order 234 17.2.1 The basic one-dimensional Lax–Wendroff scheme 234 17.2.2 The two-step Lax–Wendroff schemes in one dimension 238 17.2.3 Lerat and Peyret’s family of non-linear two-step Lax–Wendroff schemes 246 17.2.4 One-step Lax–Wendroff schemes in two dimensions 251 17.2.5 Two-step Lax–Wendroff schemes in two dimensions 258 17.3 The Concept of Artificial Dissipation or Artificial Viscosity 272 17.3.1 General form of artificial dissipation terms 273 17.3.2 Von Neumann–Richtmyer artificial viscosity 274 17.3.3 Higher-order artificial viscosities 279 17.4 Lerat’s Implicit Schemes of Lax–Wendroff Type 283 17.4.1 Analysis for linear systems in one dimension 285 17.4.2 Construction of the family of schemes 288 17.4.3 Extension to non-linear systems in conservation form 292 17.4.4 Extension to multi-dimensional flows 296 17.5 Summary 296 Chapter 18 The Central Schemes with Independent Time Integration 307 18.1 The Central Second-order Implicit Schemes of Beam and Warming in One Dimension 309 18.1.1 The basic Beam and Warming schemes 310 18.1.2 Addition of artificial viscosity 315 18.2 The Multidimensional Implicit Beam and Warming Schemes 326 18.2.1 The diagonal variant of Pulliam and Chaussee 328 18.3 Jameson’s Multistage Method 334 18.3.1 Time integration 334 18.3.2 Convergence acceleration to steady state 335 Chapter 19 The Treatment of Boundary Conditions 344 19.1 One-dimensional Boundary Treatment for Euler Equations 345 19.1.1 Characteristic boundary conditions 346 19.1.2 Compatibility relations 347 19.1.3 Characteristic boundary conditions as a function of conservative and primitive variables 349 19.1.4 Extrapolation methods 353 19.1.5 Practical implementation methods for numerical boundary conditions 357 19.1.6 Nonreflecting boundary conditions 369 19.2 Multidimensional Boundary Treatment 372 19.2.1 Physical and numerical boundary conditions 372 19.2.2 Multidimensional compatibility relations 376 19.2.3 Farfield treatment for steadystate flows 377 19.2.4 Solid wall boundary 379 19.2.5 Nonreflective boundary conditions 384 19.3 The Far-field Boundary Corrections 385 19.4 The Kutta Condition 395 19.5 Summary 401 Chapter 20 Upwind Schemes for the Euler Equations 408 20.1 The Basic Principles of Upwind Schemes 409 20.2 One-dimensional Flux Vector Splitting 415 20.2.1 Steger and Warming flux vector splitting 415 20.2.2 Properties of split flux vectors 417 20.2.3 Van Leer’s flux splitting 420 20.2.4 Non-reflective boundary conditions and split fluxes 425 20.3 One-dimensional Upwind Discretizations Based on Flux Vector Splitting 426 20.3.1 First-order explicit upwind schemes 426 20.3.2 Stability conditions for first-order flux vector splitting schemes 428 20.3.3 Non-conservative firstorder upwind schemes 438 20.4 Multi-dimensional Flux Vector Splitting 438 20.4.1 Steger and Warming flux splitting 440 20.4.2 Van Leer flux splitting 440 20.4.3 Arbitrary meshes 441 20.5 The Godunov-type Schemes 443 20.5.1 The basic Godunov scheme 444 20.5.2 Osher’s approximate Riemann solver 453 20.5.3 Roe’s approximate Riemann solver 460 20.5.4 Other Godunov-type methods 469 20.5.5 Summary 472 20.6 First-order Implicit Upwind Schemes 473 20.7 Multi-dimensional First-order Upwind Schemes 475 Chapter 21 Second-order Upwind and High-resolution Schemes 493 21.1 General Formulation of Higher-order Upwind Schemes 494 21.1.1 Higher-order projection stages-variable extrapolation or MUSCL approach 495 21.1.2 Numerical flux for higher-order upwind schemes 498 21.1.3 Second-order space- and time-accurate upwind schemes based on variable extrapolation 499 21.1.4 Linearized analysis of second-order upwind schemes 502 21.1.5 Numerical flux for higher-order upwind schemes—flux extrapolation 504 21.1.6 Implicit second-order upwind schemes 512 21.1.7 Implicit second-order upwind schemes in two dimensions 514 21.1.8 Summary 516 21.2 The Definition of High-resolution Schemes 517 21.2.1 The generalized entropy condition for inviscid equations 519 21.2.2 Monotonicity condition 525 21.2.3 Total variation diminishing (TVD)schemes 528 21.3 Second-order TVD Semi-discretized Schemes with Limiters 536 21.3.1 Definition of limiters for the linear convection equation 537 21.3.2 General definition of flux limiters 550 21.3.3 Limiters for variable extrapolation—MUSCL—method 552 21.4 Timeintegration Methods for TVD Schemes 556 21.4.1 Explicit TVD schemes of first-order accuracy in time 557 21.4.2 Implicit TVD schemes 558 21.4.3 Explicit second-order TVD schemes 560 21.4.4 TVD schemes and artificial dissipation 564 21.4.5 TVD limiters and the entropy condition 568 21.5 Extension to Non-linear Systems and to Multi-dimensions 570 21.6 Conclusions to Part VI 583 Part VII: The Numerical Solution of the Navier-Stokes Equations 595 Chapter 22 The Properties of the System of Navier–Stokes Equations 597 22.1 Mathematical Formulation of the Navier–Stokes Equations 597 22.1.1 Conservative form of the Navier–Stokes equations 597 22.1.2 Integral form of the Navier–Stokes equations 599 22.1.3 Shock waves and contact layers 600 22.1.4 Mathematical properties and boundary conditions 601 22.2 Reynolds-averaged Navier–Stokes Equations 603 22.2.1 Turbulent-averaged energy equation 604 22.3 Turbulence Models 606 22.3.1 Algebraic models 608 22.3.2 One- and two-equation models—k–ε models 613 22.3.3 Algebraic Reynolds stress models 615 22.4 Some Exact One-dimensional Solutions 618 22.4.1 Solutions to the linear convection-diffusion equation 618 22.4.2 Solutions to Burgers equation 620 22.4.3 Other simple test cases 621 Chapter 23 Discretization Methods for the Navier–Stokes Equations 624 23.1 Discretization of Viscous and Heat Conduction Terms 625 23.2 Time-dependent Methods for Compressible Navier–Stokes Equations 627 23.2.1 First-order explicit central schemes 628 23.2.2 One-step Lax–Wendroff schemes 629 23.2.3 Two-step Lax–Wendroff schemes 630 23.2.4 Central schemes with separate space and time discretization 636 23.2.5 Upwind schemes 648 23.3 Discretization of the Incompressible Navier–Stokes Equations 654 23.3.1 Incompressible Navier–Stokes equations 654 23.3.2 Pseudo-compressibility method 656 23.3.3 Pressure correction methods 661 23.3.4 Selection of the space discretization 666 23.4 Conclusions to Part VII 674 Index 685

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Book SynopsisChange-point problems arise in a variety of experimental andmathematical sciences, as well as in engineering and healthsciences. This rigorously researched text provides a comprehensivereview of recent probabilistic methods for detecting various typesof possible changes in the distribution of chronologically orderedobservations. Further developing the already well-establishedtheory of weighted approximations and weak convergence, the authorsprovide a thorough survey of parametric and non-parametric methods,regression and time series models together with sequential methods.All but the most basic models are carefully developed with detailedproofs, and illustrated by using a number of data sets. Contains athorough survey of: * The Likelihood Approach * Non-Parametric Methods * Linear Models * Dependent Observations This book is undoubtedly of interest to all probabilists andstatisticians, experimental and health scientists, engineers, andessential for those wTrade Review"This book is suitable for Ph.D. students who wish to establish a solid grounding in the field, and researchers who need a reliable reference to precisely formulated results and their proofs. The book contains a very extensive list of references reading into the late 1990's." (Mathematical Reviews, 2011)Table of ContentsThe Likelihood Approach. Nonparametric Methods. Linear Models. Dependent Observations. Appendix. References. Indexes.

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Book SynopsisThe concept of a system as an entity in its own right has emergedwith increasing force in the past few decades in, for example, theareas of electrical and control engineering, economics, ecology,urban structures, automaton theory, operational research andindustry. The more definite concept of a large-scale system isimplicit in these applications, but is particularly evident infields such as the study of communication networks, computernetworks and neural networks. The Wiley-Interscience Series inSystems and Optimization has been established to serve the needs ofresearchers in these rapidly developing fields. It is intended forworks concerned with developments in quantitative systems theory,applications of such theory in areas of interest, or associatedmethodology. This is the first book-length treatment of risk-sensitive control,with many new results. The quadratic cost function of the standardLQG (linear/quadratic/Gaussian) treatment is replaced by theexponential of a quadratTable of ContentsBASICS. Deterministic Models. Stochastic Models. BEYOND. Risk-Sensitive and H infinity Criteria. Time-Integral Methods and Optimal Stationary Policies. Near-Determinism and Large Deviation Theory. Appendices. References. Index.

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Book SynopsisClearly written and free of statistical jargon, this invaluable guide concentrates on the practicalities of statistical analysis for anyone involved with agricultural research. Each section starts with the key points, giving a quick reference to the contents and plenty of examples using a reala data.Table of ContentsAcknowledgements INTRODUCTION Notation A little history Population versus samples PLANNING Formulating the idea Defining objectives Defining the population Formulating hypotheses Hypothesis testing Anticipating treatment differences DESIGN Variables Choosing the treatments Constraints Replication Blocking Randomization Covariants Confounding TRIAL STRUCTURE Considerations Single-treatment factor designs Multi-treatment factor designs Some other designs DATA ENTRY AND EXPLORATION Data entry Data Data checking Data exploration ANALYTICAL TECHNIQUES Parametric techniques Non-parametric techniques Comparison of parametric and non-parametric techniques OTHER STATISTICAL TECHNIQUES Multivariate analysis Time series analysis ASPECTS OF COMPUTING APPENDICES Glossary of Statistical Terms Analysis of Variance Formulae INDEX

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Book SynopsisMatrices are used in many fields such as statistics, econometrics, mathematics, natural sciences and engineering. They provide a concise, simple method for describing long and complicated computations. This is a comprehensive handbook and dictionary of terms for matrix theory.Table of ContentsDefinitions, Notations, Terminology. Rules for Matrix Operations. Matrix Valued Functions of a Matrix. Trace, Determinant and Rank of a Matrix. Eigenvalues and Singular Values. Matrix Decompositions and Canonical Forms. Vectorization Operators. Vector and Matrix Norms. Properties of Special Matrices. Vector and Matrix Derivatives. Polynomials, Power Series and Matrices. Appendix. References. Index.

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Book SynopsisThe finite element method and the boundary element method are two computational methods available for designing structures ranging from aircraft and ships to dams and tunnels. This text presents the mathematical basis of the joint use of both methods and their computer implementation.Table of ContentsBasic principle and domains of application. I. BOUNDARY INTEGRAL EQUATIONS FOR STATIC PROBLEMS : Integral Equations and Representations for the Poisson Equation; Numerical Solution using Boundary Elements; Integral Equations and Representations for Elastostatics; Integral Representations of Gradients and Stresses on the Boundary; Some Classical Mathematical Results II. BOUNDARY INTEGRAL EQUATIONS FOR WAVE AND EVOLUTION PROBLEMS: Waves and Elastodynamics in Time Domain; Waves and Elastodynamics in Frequency Domain; Diffusion, Fluid Flow. III. ADVANCED TOPICS : Variational Boundary Integral Formulations; Exploitation of Geometrical Symmetry; Domain Derivative and Boundary Integral Eequations. IV. ADDITIONAL TOPICS IN SOLID MECHANICS : Boundary Integral Equations for Cracked Solids; Initial Strain or Stress: Inclusions, Elastoplasticity. APPENDICES : Tangential Differential Operators and Integration by Parts; Interpolation Functions and Numerical Integration. Bibliography. Index.

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Book SynopsisThis book presents ordinary differential equations based on Lie group analysis and related invariance principles. The author provides students and teachers with a text for one-semester undergraduate and graduate courses that spans a variety of topics, from the basic theory through to applications.Trade Review"…this is the first self-contained university text on ordinary differential equations…" (Zentralblatt Math, Vol.1047, No.22, 2004)Table of ContentsIntroduction to Differential Equations. Transformation Groups. Lie Group Analysis of Ordinary Differential Equations. Brief on Lie Algebras. First Order Differential Equations. Integration of Second Order Equations. Basic Theory of Linear Equations. Nonlinear Second Order Equations. Integration of Third Order Equations. Nonlinear Superposition Principle. Index.

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Book SynopsisThis text shows how stochastic geometry can be applied to real structural problems in materials science and technology. It pays particular attention to describing spatial sizes and shapes of grains and particles, developments in stochastic geometry, and relevant computer simulation techniques.Trade Review"...provides many examples...comprehensive discussions...an introduction to the analysis of two-dimensional and three-dimensional microscopic images...references are comprehensive..." (Short Book Reviews, Vol. 21, No. 2, August 2001) "There is no book I know in our own field that deals with the subject in anything like the depth and breadth as this one does." (European Journal of Soil Science, No. 52 2001) "It can be expected that this unusually careful work will soon be acknowledged as an authoritative treatment, and certainly it will remain a major reference of applied stereology in the next two decades at least. Scientific and technical libraries should have multiple copies available." (Ceramics, Vol.45 No.3, 2001) "...an ideal textbook for a one-semester course...also an excellent reference book..." (Technometrics, February 2002)Table of ContentsDedication to Günter Bach. Preface. Series Preface. Acknowledgements. List of Notation. List of Source Codes. Introduction. Methodological Tools. Statistical Estimation of Basic Characteristics. Basic Characteristics and Digitalization. Covariance and Spectral Density. Size Distribution of Spherical Particles. Nonspherical Particles of Constant Shape. Size-Shape Distribution of Particles. Arrangement of Objects. Single-Phase Polyhedral Microstructures. Appendix A: Characteristics of Geometric Objects. Appendix B: Software Utilities. References. Index.

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Book SynopsisThis volume presents a unified framework for systems engineering and a systematic and rigorous source for a comprehensive description of the utilization of Monte Carlo methods in practical engineering problems. The author suggests that efficiency can be improved through such an integrated approach.Table of Contents1. Introduction - Probability and statistics 2. Basic concepts in system engineering 3. Basic concepts in Monte Carlo methods 4. Additional applications 5. Elements of uncertainty and uncertainty analysis 6. System transport 7. Realization of system transport Appendix

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Book SynopsisThis text examines synthetic and dynamical properties of fuzzy control systems in a quantitative manner. It includes fuzzy dynamical systems, controllability and sensitivity analysis and how these affect parameters in membership functions, fuzzification, defuzzification and inferencing.Trade Review"Design and control engineers will value the advanced control techniques, new design and analysis tools presented. Post-graduates...a useful reference." (Engineering Design, July 2000) "...a good read...it boldly tackles the stability issue of fuzzy control systems..." (Measurement and Control, October 2000) "Design and control engineers will value the advanced control techniques and new design and analysis tools presented. Postgraduates studying fuzzy control will find this book a useful reference...." (European Power Electronics & Drives Journal September 2001)Table of ContentsMODELING. Information Granularity in the Analysis and Design of Fuzzy Controllers. Fuzzy Modeling for Predictive Control. Adaptive and Learning Schemes for Fuzzy Modeling. Fuzzy System Identification with General Parameter Radial Basis Function Neural Network. ANALYSIS. Lyapunov Stability Analysis of Fuzzy Dynamic Systems. Passivity and Stability of Fuzzy Control Systems. Frequency Domain Analysis of MIMO Fuzzy Control Systems. Analytical Study of Structure of a Mamdani Fuzzy Controller with Three Input Variables. An Approach to the Analysis of Robust Stability of Fuzzy Control Systems. Fuzzy Control Systems Stability Analysis with Application to Aircraft Systems. SYNTHESIS. Observer-Based Controller Synthesis for Model-Based Fuzzy Systems via Linear Matrix Inequalities. LMI-Based Fuzzy Control: Fuzzy Regulator and Fuzzy Observer Design via LMIs. A Framework for the Synthesis of PDC-Type Takagi-Sugano Fuzzy Control Systems: An LMI Approach. On Adaptive Fuzzy Logic Control on Non-linear Systems--Synthesis and Analysis. Stabilization of Direct Adaptive Fuzzy Control Systems: Two Approaches. Gain Scheduling Based Control of a Class of TSK Systems. Output Tracking Using Fuzzy Neural Networks. Fuzzy Life-Extending Control of Mechanical Systems. Epilogue. Index.

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Book SynopsisThis work is a guide to the principles behind sensitivity analysis. It suggests suitable methods for particular types of problem, which allows a greater understanding of the entire causal assessment chain. This makes the impact of source uncertainties and framing assumptions more transparent.Trade Review"The book has a fair price...I think this is a book that everyone who does modeling should buy. It can readily be read piecemeal...so it is ideal for leisurely self-study..." (Technometrics Vol. 42, No. 4 May 2001) "...this book will prove helpful in the solution of many modeling problems." (La Doc Sti, September 2000) "...presents many different sensitivity analysis methodologies and demonstrates their usefulness in scientific research." (Zentralblatt MATH, Vol. 961, 2001/11)Table of ContentsWhat is Sensitivity Analysis. Hitchhiker's Guide to Sensitivity Analysis. METHODS. Designs of Experiments. Screening Methods. Local Methods. Sampling-Based Methods. Reliability Algorithms: FORM and SORM Methods. Variance-Based Methods. Managing the Tyranny of Parameters in Mathematical Modelling of Physical Systems. Bayesian Sensitivity Analysis. Graphical Methods. APPLICATIONS. Practical Experience in Applying Sensitivity and Uncertainty Analysis. Scenario and Parametric Sensitivity and Uncertainty Analysis in Nuclear Waste Disposal Risk Assessment: The Case of GESAMAC. Sensitivity Analysis for Signal Extraction in Economic Time Series. A Dataless Precalibration Analysis in Solid State Physics. Appplication of First-Order (FORM) and Second-Order (SORM) Reliability Methods: Analysis and Interpretation of Sensitivity Measures Related to Groundwater Pressure Decreases and Resulting Ground Subsidence. One-at-a-Time and Mini-Global Analyses for Characterizing Model Sensitivity in the Nonlinear Ozone Predictions from the US EPA Regional Acid Deposition Model (RADM). Comparing Different Sensitivity Analysis Methods on a Chemical Reactions Model. An Application of Sensitivity Analysis to Fish Population Dynamics. Global Sensitivity Analysis: A Quality Assurance Tool in Environmental Policy Modelling. CONCLUSIONS. Assuring the Quality of Models Designed for Predictive Tasks. Fortune and Future of Sensitivity Analysis. References. Appendix. Index.

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Book SynopsisAt present, most books on ecological modelling rely on very complex mathematics, resulting in students and researchers shying away from investigating the potential uses of ecological models and their methods of construction. This new book aims to open up this exciting area to a much wider audience.Trade Review"Teachers of courses on ecological modelling will find [this book] a useful source-book at a competitive price."Table of ContentsIntroduction: Themes Of Ecological Modelling. Probability Of Population Extinction. Looking For Cycles: The Dynamics Of Predators And Their Prey. Population Dynamics Of Species With Complex Life-Histories. Dynamics Of Ecological Communities. Spatial Models And Thresholds. Disease And Biological Control. Answers To Questions. Glossary Of Symbols And Terms. References. Index

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Book SynopsisIntroduces the distributed control of robotic networks. This book presents a set of tools for understanding coordination algorithms, determining their correctness, and assessing their complexity. It analyzes various cooperative strategies for tasks such as consensus, rendezvous, connectivity maintenance, deployment, and boundary estimation.Trade Review"This book covers its subject very thoroughly. The framework the authors have established is very elegant and, if it catches on, this book could be the primary reference for this approach. I don't know of any other book that covers this set of topics."—Richard M. Murray, California Institute of Technology"The authors do an excellent job of clearly describing the problems and presenting rigorous, provably correct algorithms with complexity bounds for each problem. The authors also do a fantastic job of providing the mathematical insight necessary for such complex problems."—Ali Jadbabaie, University of Pennsylvania"The order of presentation makes much sense, and the book thoroughly covers what it sets out to cover. The algorithms and results are presented using a clear mathematical and computer science formalism, which allows a uniform presentation. The formalism used and the way of presenting the algorithms may be helpful for structuring the presentation of new algorithms in the future."—Vincent Blondel, Université catholique de LouvainTable of ContentsPreface ix Chapter 1. An introduction to distributed algorithms 1 1.1 Elementary concepts and notation 1 1.2 Matrix theory 6 1.3 Dynamical systems and stability theory 12 1.4 Graph theory 20 1.5 Distributed algorithms on synchronous networks 37 1.6 Linear distributed algorithms 52 1.7 Notes 66 1.8 Proofs 69 1.9 Exercises 85 Chapter 2. Geometric models and optimization 95 2.1 Basic geometric notions 95 2.2 Proximity graphs 104 2.3 Geometric optimization problems and multicenter functions 111 2.4 Notes 124 2.5 Proofs 125 2.6 Exercises 133 Chapter 3. Robotic network models and complexity notions 139 3.1 A model for synchronous robotic networks 139 3.2 Robotic networks with relative sensing 151 3.3 Coordination tasks and complexity notions 158 3.4 Complexity of direction agreement and equidistance 165 3.5 Notes 166 3.6 Proofs 169 3.7 Exercises 176 Chapter 4. Connectivity maintenance and rendezvous 179 4.1 Problem statement 180 4.2 Connectivity maintenance algorithms 182 4.3 Rendezvous algorithms 191 4.4 Simulation results 200 4.5 Notes 201 4.6 Proofs 204 4.7 Exercises 215 Chapter 5. Deployment 219 5.1 Problem statement 220 5.2 Deployment algorithms 222 5.3 Simulation results 233 5.4 Notes 237 5.5 Proofs 239 5.6 Exercises 245 Chapter 6. Boundary estimation and tracking 247 6.1 Event-driven asynchronous robotic networks 248 6.2 Problem statement 252 6.3 Estimate update and cyclic balancing law 256 6.4 Simulation results 266 6.5 Notes 268 6.6 Proofs 270 6.7 Exercises 275 Bibliography 279 Algorithm Index 305 Subject Index 307 Symbol Index 313

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Book SynopsisCo-published with Oxford University Press. This highly technical and thought-provoking book stresses the development of mathematical foundations for the application of the electromagnetic model to problems of research and technology.Table of ContentsPreface. Linear Analysis. The Green's Function Method. The Spectral Representation Method. Electromagnetic Sources. Electromagnetic Boundary Value Problems. Index.

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Book SynopsisSPATIAL ERROR ANALYSIS is an all-in-one sourcebook on error measurements in one-, two-, and three-dimensional spaces. This book features exhaustive, systematic coverage of error measurement relationships, techniques, and solutions used to solve general, correlated cases. It is packed with 62 figures and 24 tables. MATLAB-based M-files* for practical applications created especially for this volume are available on the Web at ftp://ftp.mathworks.com/pub/books/hsu. Solutions to two- and three-dimensional problems are presented without relying on equal standard deviations from each channel. They also make no assumption that the random variables of interest are independent or uncorrelated. * MATLAB (developed by MathWorks, Inc.) must be purchased separately. Sponsored by: IEEE Aerospace and Electronic Systems Society.Table of ContentsPreface. List of Figures. List of Tables. Introduction. Prameter Estimation from Samples. One-Dimensional Error Analysis. Two-Dimensional Error Analysis. Three-Dimensional Error Analysis. Maximum Likelihood Estimation of Radial Error PDF. Position Location Problems. Risk Analysis. Appendix A: Probability Density Functions. Appendix B: Method of Confidence Intervals. Appendix C: Function of N Random Variables. Appendix D: GPS Dilution of Precisions. Appendix E: Listing of Author-Generated M-files. Bibliography. Index. About the Author.

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Book SynopsisThis book is an adventure into the computer analysis of three dimensional composite structures using the finite element method (FEM). Once the basic philosophy of the method is understood, the reader may expand its application and modify the computer programs to suit particular needs.Trade Review`The book is highly recommended as a reference text for advanced undergraduate students, as a graduate course on the FE analysis of composites, and as a reference work for both researchers in laboratories and practising engineers in industry.' Zentralblatt MATH, 906 Table of ContentsPreface. 1. Some Results from Continuum Mechanics. 2. A Brief History of FEM. 3. Natural Modes for Finite Elements. 4. Composites. 5. Composite Beam Element. 6. Composite Plate and Shell Element. 7. Computational Statistics. 8. Nonlinear Analysis of Anisotropic Shells. 9. Programming Aspects. Appendices: A. Geometry of the Bema Element in Space. B. Contents of the Floppy Disk. Bibliography. Index.

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a huge range and FREE tracked UK delivery on ALL orders.

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Book SynopsisThis text is intended for the undergraduate course in math methods, with an audience of physics and engineering majors. As a required course in most departments, the text relies heavily on explained examples, real-world applications and student engagement.Trade Review"[Mathematical Methods in Engineering and Physics] is my book of choice for teaching undergraduates...I honestly never thought that I could be so enchanted by the heat equation before seeing how Felder and Felder effectively have students derive it as part of honing their intuition for how to think about partial differential equations." - Christine Aidala, PhD, Associate Professor of Physics at University of Michigan for the American Journal of Physics Table of ContentsPreface xi 1 Introduction to Ordinary Differential Equations 1 1.1 Motivating Exercise: The Simple Harmonic Oscillator 2 1.2 Overview of Differential Equations 3 1.3 Arbitrary Constants 15 1.4 Slope Fields and Equilibrium 25 1.5 Separation of Variables 34 1.6 Guess and Check, and Linear Superposition 39 1.7 Coupled Equations (see felderbooks.com) 1.8 Differential Equations on a Computer (see felderbooks.com) 1.9 Additional Problems (see felderbooks.com) 2 Taylor Series and Series Convergence 50 2.1 Motivating Exercise: Vibrations in a Crystal 51 2.2 Linear Approximations 52 2.3 Maclaurin Series 60 2.4 Taylor Series 70 2.5 Finding One Taylor Series from Another 76 2.6 Sequences and Series 80 2.7 Tests for Series Convergence 92 2.8 Asymptotic Expansions (see felderbooks.com) 2.9 Additional Problems (see felderbooks.com) 3 Complex Numbers 104 3.1 Motivating Exercise: The Underdamped Harmonic Oscillator 104 3.2 Complex Numbers 105 3.3 The Complex Plane 113 3.4 Euler’s Formula I—The Complex Exponential Function 117 3.5 Euler’s Formula II—Modeling Oscillations 126 3.6 Special Application: Electric Circuits (see felderbooks.com) 3.7 Additional Problems (see felderbooks.com) 4 Partial Derivatives 136 4.1 Motivating Exercise: The Wave Equation 136 4.2 Partial Derivatives 137 4.3 The Chain Rule 145 4.4 Implicit Differentiation 153 4.5 Directional Derivatives 158 4.6 The Gradient 163 4.7 Tangent Plane Approximations and Power Series (see felderbooks.com) 4.8 Optimization and the Gradient 172 4.9 Lagrange Multipliers 181 4.10 Special Application: Thermodynamics (see felderbooks.com) 4.11 Additional Problems (see felderbooks.com) 5 Integrals in Two or More Dimensions 188 5.1 Motivating Exercise: Newton’s Problem (or) The Gravitational Field of a Sphere 188 5.2 Setting Up Integrals 189 5.3 Cartesian Double Integrals over a Rectangular Region 204 5.4 Cartesian Double Integrals over a Non-Rectangular Region 211 5.5 Triple Integrals in Cartesian Coordinates 216 5.6 Double Integrals in Polar Coordinates 221 5.7 Cylindrical and Spherical Coordinates 229 5.8 Line Integrals 240 5.9 Parametrically Expressed Surfaces 249 5.10 Surface Integrals 253 5.11 Special Application: Gravitational Forces (see felderbooks.com) 5.12 Additional Problems (see felderbooks.com) 6 Linear Algebra I 266 6.1 The Motivating Example on which We’re Going to Base the Whole Chapter: The Three-Spring Problem 266 6.2 Matrices: The Easy Stuff 276 6.3 Matrix Times Column 280 6.4 Basis Vectors 286 6.5 Matrix Times Matrix 294 6.6 The Identity and Inverse Matrices 303 6.7 Linear Dependence and the Determinant 312 6.8 Eigenvectors and Eigenvalues 325 6.9 Putting It Together: Revisiting the Three-Spring Problem 336 6.10 Additional Problems (see felderbooks.com) 7 Linear Algebra II 346 7.1 Geometric Transformations 347 7.2 Tensors 358 7.3 Vector Spaces and Complex Vectors 369 7.4 Row Reduction (see felderbooks.com) 7.5 Linear Programming and the Simplex Method (see felderbooks.com) 7.6 Additional Problems (see felderbooks.com) 8 Vector Calculus 378 8.1 Motivating Exercise: Flowing Fluids 378 8.2 Scalar and Vector Fields 379 8.3 Potential in One Dimension 387 8.4 From Potential to Gradient 396 8.5 From Gradient to Potential: The Gradient Theorem 402 8.6 Divergence, Curl, and Laplacian 407 8.7 Divergence and Curl II—The Math Behind the Pictures 416 8.8 Vectors in Curvilinear Coordinates 419 8.9 The Divergence Theorem 426 8.10 Stokes’ Theorem 432 8.11 Conservative Vector Fields 437 8.12 Additional Problems (see felderbooks.com) 9 Fourier Series and Transforms 445 9.1 Motivating Exercise: Discovering Extrasolar Planets 445 9.2 Introduction to Fourier Series 447 9.3 Deriving the Formula for a Fourier Series 457 9.4 Different Periods and Finite Domains 459 9.5 Fourier Series with Complex Exponentials 467 9.6 Fourier Transforms 472 9.7 Discrete Fourier Transforms (see felderbooks.com) 9.8 Multivariate Fourier Series (see felderbooks.com) 9.9 Additional Problems (see felderbooks.com) 10 Methods of Solving Ordinary Differential Equations 484 10.1 Motivating Exercise: A Damped, Driven Oscillator 485 10.2 Guess and Check 485 10.3 Phase Portraits (see felderbooks.com) 10.4 Linear First-Order Differential Equations (see felderbooks.com) 10.5 Exact Differential Equations (see felderbooks.com) 10.6 Linearly Independent Solutions and the Wronskian (see felderbooks.com) 10.7 Variable Substitution 494 10.8 Three Special Cases of Variable Substitution 505 10.9 Reduction of Order and Variation of Parameters (see felderbooks.com) 10.10 Heaviside, Dirac, and Laplace 512 10.11 Using Laplace Transforms to Solve Differential Equations 522 10.12 Green’s Functions 531 10.13 Additional Problems (see felderbooks.com) 11 Partial Differential Equations 541 11.1 Motivating Exercise: The Heat Equation 542 11.2 Overview of Partial Differential Equations 544 11.3 Normal Modes 555 11.4 Separation of Variables—The Basic Method 567 11.5 Separation of Variables—More than Two Variables 580 11.6 Separation of Variables—Polar Coordinates and Bessel Functions 589 11.7 Separation of Variables—Spherical Coordinates and Legendre Polynomials 607 11.8 Inhomogeneous Boundary Conditions 616 11.9 The Method of Eigenfunction Expansion 623 11.10 The Method of Fourier Transforms 636 11.11 The Method of Laplace Transforms 646 11.12 Additional Problems (see felderbooks.com) 12 Special Functions and ODE Series Solutions 652 12.1 Motivating Exercise: The Circular Drum 652 12.2 Some Handy Summation Tricks 654 12.3 A Few Special Functions 658 12.4 Solving Differential Equations with Power Series 666 12.5 Legendre Polynomials 673 12.6 The Method of Frobenius 682 12.7 Bessel Functions 688 12.8 Sturm-Liouville Theory and Series Expansions 697 12.9 Proof of the Orthgonality of Sturm-Liouville Eigenfunctions (see felderbooks.com) 12.10 Special Application: The Quantum Harmonic Oscillator and Ladder Operators (see felderbooks.com) 12.11 Additional Problems (see felderbooks.com) 13 Calculus with Complex Numbers 708 13.1 Motivating Exercise: Laplace’s Equation 709 13.2 Functions of Complex Numbers 710 13.3 Derivatives, Analytic Functions, and Laplace’s Equation 716 13.4 Contour Integration 726 13.5 Some Uses of Contour Integration 733 13.6 Integrating Along Branch Cuts and Through Poles (see felderbooks.com) 13.7 Complex Power Series 742 13.8 Mapping Curves and Regions 747 13.9 Conformal Mapping and Laplace’s Equation 754 13.10 Special Application: Fluid Flow (see felderbooks.com) 13.11 Additional Problems (see felderbooks.com) Appendix A Different Types of Differential Equations 765 Appendix B Taylor Series 768 Appendix C Summary of Tests for Series Convergence 770 Appendix D Curvilinear Coordinates 772 Appendix E Matrices 774 Appendix F Vector Calculus 777 Appendix G Fourier Series and Transforms 779 Appendix H Laplace Transforms 782 Appendix I Summary: Which PDE Technique Do I Use? 787 Appendix J Some Common Differential Equations and Their Solutions 790 Appendix K Special Functions 798 Appendix L Answers to “Check Yourself” in Exercises 801 Appendix M Answers to Odd-Numbered Problems (see felderbooks.com) Index 805

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Book SynopsisAlgebraic Identification and Estimation Methods in Feedback Control Systems presents a model-based algebraic approach to online parameter and state estimation in uncertain dynamic feedback control systems. This approach evades the mathematical intricacies of the traditional stochastic approach, proposing a direct model-based scheme with several easy-to-implement computational advantages. The approach can be used with continuous and discrete, linear and nonlinear, mono-variable and multi-variable systems. The estimators based on this approach are not of asymptotic nature, and do not require any statistical knowledge of the corrupting noises to achieve good performance in a noisy environment. These estimators are fast, robust to structured perturbations, and easy to combine with classical or sophisticated control laws. This book uses module theory, differential algebra, and operational calculus in an easy-to-understand manner and also details how to apply these in the coTable of ContentsSeries Preface xiii Preface xv 1 Introduction 1 1.1 Feedback Control of Dynamic Systems 2 1.1.1 Feedback 2 1.1.2 Why Do We Need Feedback? 3 1.2 The Parameter Identification Problem 3 1.2.1 Identifying a System 4 1.3 A Brief Survey on Parameter Identification 4 1.4 The State Estimation Problem 5 1.4.1 Observers 6 1.4.2 Reconstructing the State via Time Derivative Estimation 7 1.5 Algebraic Methods in Control Theory: Differences from Existing Methodologies 8 1.6 Outline of the Book 9 References 12 2 Algebraic Parameter Identification in Linear Systems 15 2.1 Introduction 15 2.1.1 The Parameter-Estimation Problem in Linear Systems 16 2.2 Introductory Examples 17 2.2.1 Dragging an Unknown Mass in Open Loop 17 2.2.2 A Perturbed First-Order System 24 2.2.3 The Visual Servoing Problem 30 2.2.4 Balancing of the Plane Rotor 35 2.2.5 On the Control of the Linear Motor 38 2.2.6 Double-Bridge Buck Converter 42 2.2.7 Closed-Loop Behavior 43 2.2.8 Control of an unknown variable gain motor 47 2.2.9 Identifying Classical Controller Parameters 50 2.3 A Case Study Introducing a “Sentinel” Criterion 53 2.3.1 A Suspension System Model 54 2.4 Remarks 67 References 68 3 Algebraic Parameter Identification in Nonlinear Systems 71 3.1 Introduction 71 3.2 Algebraic Parameter Identification for Nonlinear Systems 72 3.2.1 Controlling an Uncertain Pendulum 74 3.2.2 A Block-Driving Problem 80 3.2.3 The Fully Actuated Rigid Body 84 3.2.4 Parameter Identification Under Sliding Motions 90 3.2.5 Control of an Uncertain Inverted Pendulum Driven by a DC Motor 92 3.2.6 Identification and Control of a Convey Crane 96 3.2.7 Identification of a Magnetic Levitation System 103 3.3 An Alternative Construction of the System of Linear Equations 105 3.3.1 Genesio–Tesi Chaotic System 107 3.3.2 The Ueda Oscillator 108 3.3.3 Identification and Control of an Uncertain Brushless DC Motor 112 3.3.4 Parameter Identification and Self-tuned Control for the Inertia Wheel Pendulum 119 3.3.5 Algebraic Parameter Identification for Induction Motors 128 3.3.6 A Criterion to Determine the Estimator Convergence: The Error Index 136 3.4 Remarks 141 References 141 4 Algebraic Parameter Identification in Discrete-Time Systems 145 4.1 Introduction 145 4.2 Algebraic Parameter Identification in Discrete-Time Systems 145 4.2.1 Main Purpose of the Chapter 146 4.2.2 Problem Formulation and Assumptions 147 4.2.3 An Introductory Example 148 4.2.4 Samuelson’s Model of the National Economy 150 4.2.5 Heating of a Slab from Two Boundary Points 155 4.2.6 An Exact Backward Shift Reconstructor 157 4.3 A Nonlinear Filtering Scheme 160 4.3.1 Hénon System 161 4.3.2 A Hard Disk Drive 164 4.3.3 The Visual Servo Tracking Problem 166 4.3.4 A Shape Control Problem in a Rolling Mill 170 4.3.5 Algebraic Frequency Identification of a Sinusoidal Signal by Means of Exact Discretization 175 4.4 Algebraic Identification in Fast-Sampled Linear Systems 178 4.4.1 The Delta-Operator Approach: A Theoretical Framework 179 4.4.2 Delta-Transform Properties 181 4.4.3 A DC Motor Example 181 4.5 Remarks 188 References 188 5 State and Parameter Estimation in Linear Systems 191 5.1 Introduction 191 5.1.1 Signal Time Derivation Through the “Algebraic Derivative Method” 192 5.1.2 Observability of Nonlinear Systems 192 5.2 Fast State Estimation 193 5.2.1 An Elementary Second-Order Example 193 5.2.2 An Elementary Third-Order Example 194 5.2.3 A Control System Example 198 5.2.4 Control of a Perturbed Third-Order System 201 5.2.5 A Sinusoid Estimation Problem 203 5.2.6 Identification of Gravitational Wave Parameters 205 5.2.7 A Power Electronics Example 210 5.2.8 A Hydraulic Press 213 5.2.9 Identification and Control of a Plotter 218 5.3 Recovering Chaotically Encrypted Signals 222 5.3.1 State Estimation for a Lorenz System 227 5.3.2 State Estimation for Chen’s System 229 5.3.3 State Estimation for Chua’s Circuit 231 5.3.4 State Estimation for Rossler’s System 232 5.3.5 State Estimation for the Hysteretic Circuit 234 5.3.6 Simultaneous Chaotic Encoding–Decoding with Singularity Avoidance 239 5.3.7 Discussion 240 5.4 Remarks 241 References 242 6 Control of Nonlinear Systems via Output Feedback 245 6.1 Introduction 245 6.2 Time-Derivative Calculations 246 6.2.1 An Introductory Example 247 6.2.2 Identifying a Switching Input 253 6.3 The Nonlinear Systems Case 255 6.3.1 Control of a Synchronous Generator 256 6.3.2 Control of a Multi-variable Nonlinear System 261 6.3.3 Experimental Results on a Mechanical System 267 6.4 Remarks 278 References 279 7 Miscellaneous Applications 281 7.1 Introduction 281 7.1.1 The Separately Excited DC Motor 282 7.1.2 Justification of the ETEDPOF Controller 285 7.1.3 A Sensorless Scheme Based on Fast Adaptive Observation 287 7.1.4 Control of the Boost Converter 292 7.2 Alternative Elimination of Initial Conditions 298 7.2.1 A Bounded Exponential Function 299 7.2.2 Correspondence in the Frequency Domain 300 7.2.3 A System of Second Order 301 7.3 Other Functions of Time for Parameter Estimation 304 7.3.1 A Mechanical System Example 304 7.3.2 A Derivative Approach to Demodulation 310 7.3.3 Time Derivatives via Parameter Identification 312 7.3.4 Example 314 7.4 An Algebraic Denoising Scheme 318 7.4.1 Example 321 7.4.2 Numerical Results 322 7.5 Remarks 325 References 326 Appendix A Parameter Identification in Linear Continuous Systems: A Module Approach 329 A.1 Generalities on Linear Systems Identification 329 A.1.1 Example 330 A.1.2 Some Definitions and Results 330 A.1.3 Linear Identifiability 331 A.1.4 Structured Perturbations 333 A.1.5 The Frequency Domain Alternative 337 References 338 Appendix B Parameter Identification in Linear Discrete Systems: A Module Approach 339 B.1 A Short Review of Module Theory over Principal Ideal Rings 339 B.1.1 Systems 340 B.1.2 Perturbations 340 B.1.3 Dynamics and Input–Output Systems 341 B.1.4 Transfer Matrices 341 B.1.5 Identifiability 342 B.1.6 An Algebraic Setting for Identifiability 342 B.1.7 Linear identifiability of transfer functions 344 B.1.8 Linear Identification of Perturbed Systems 345 B.1.9 Persistent Trajectories 347 References 348 Appendix C Simultaneous State and Parameter Estimation: An Algebraic Approach 349 C.1 Rings, Fields and Extensions 349 C.2 Nonlinear Systems 350 C.2.1 Differential Flatness 351 C.2.2 Observability and Identifiability 352 C.2.3 Observability 352 C.2.4 Identifiable Parameters 352 C.2.5 Determinable Variables 352 C.3 Numerical Differentiation 353 C.3.1 Polynomial Time Signals 353 C.3.2 Analytic Time Signals 353 C.3.3 Noisy Signals 354 References 354 Appendix D Generalized Proportional Integral Control 357 D.1 Generalities on GPI Control 357 D.2 Generalization to MIMO Linear Systems 365 References 368 Index 369

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Book SynopsisTheory and Application of Multiphase Lattice Boltzmann Methods presents a comprehensive review of all popular multiphase Lattice Boltzmann Methods developed thus far and is aimed at researchers and practitioners within relevant Earth Science disciplines as well as Petroleum, Chemical, Mechanical and Geological Engineering. Clearly structured throughout, this book will be an invaluable reference on the current state of all popular multiphase Lattice Boltzmann Methods (LBMs). The advantages and disadvantages of each model are presented in an accessible manner to enable the reader to choose the model most suitable for the problems they are interested in. The book is targeted at graduate students and researchers who plan to investigate multiphase flows using LBMs. Throughout the text most of the popular multiphase LBMs are analyzed both theoretically and through numerical simulation. The authors present many of the mathematical derivations of the models in greater detail tTable of ContentsPreface xi About the companion website xiii 1 Introduction 1 1.1 History of the Lattice Boltzmann method 2 1.2 The Lattice Boltzmann method 3 1.3 Multiphase LBM 6 1.3.1 Color-gradient model 7 1.3.2 Shan–Chen model 7 1.3.3 Free-energy model 8 1.3.4 Interface tracking model 9 1.4 Comparison of models 9 1.5 Units in this book and parameter conversion 11 1.6 Appendix: Einstein summation convention 14 1.6.1 Kronecker δ function 15 1.6.2 Lattice tensors 15 1.7 Use of the Fortran code in the book 16 2 Single-component multiphase Shan–Chen-type model 18 2.1 Introduction 18 2.1.1 "Equilibrium" velocity in the SC model 20 2.1.2 Inter-particle forces in the SC SCMP LBM 20 2.2 Typical equations of state 21 2.2.1 Parameters in EOS 27 2.3 Thermodynamic consistency 28 2.3.1 The SCMP LBM EOS 29 2.3.2 Incorporating other EOS into the SC model 31 2.4 Analytical surface tension 32 2.4.1 Inter-particle Force Model A 32 2.4.2 Inter-particle Force Model B 33 2.5 Contact angle 34 2.6 Capillary rise 36 2.7 Parallel flow and relative permeabilities 39 2.8 Forcing term in the SC model 40 2.8.1 Schemes to incorporate the body force 42 2.8.2 Scheme overview 44 2.8.3 Theoretical analysis 44 2.8.4 Numerical results and discussion 46 2.9 Multirange pseudopotential (Inter-particle Force Model B) 55 2.10 Conclusions 58 2.11 Appendix A: Analytical solution for layered multiphase flow in a channel 58 2.12 Appendix B: FORTRAN code to simulate single component multiphase droplet contacting a wall as shown in Figure 2.7(c) 60 3 Shan and Chen-type multi-component multiphase models 71 3.1 Multi-component multiphase SC LBM 71 3.1.1 Fluid–fluid cohesion and fluid–solid adhesion 73 3.2 Derivation of the pressure 73 3.2.1 Pressure in popular papers (2D) 74 3.2.2 Pressure in popular papers (3D) 75 3.3 Determining Gc and the surface tension 76 3.4 Contact angle 78 3.4.1 Application of Young's equation to MCMP LBM 79 3.4.2 Contact angle measurement 79 3.4.3 Verification of proposed equation 80 3.5 Flow through capillary tubes 83 3.6 Layered two-phase flow in a 2D channel 85 3.7 Pressure or velocity boundary conditions 87 3.7.1 Boundary conditions for 2D simulations 87 3.7.2 Boundary conditions for 3D simulations 89 3.8 Displacement in a 3D porous medium 91 4 Rothman–Keller multiphase Lattice Boltzmann model 94 4.1 Introduction 94 4.2 RK color-gradient model 96 4.3 Theoretical analysis (Chapman–Enskog expansion) 99 4.3.1 Discussion of above formulae 103 4.4 Layered two-phase flow in a 2D channel 103 4.4.1 Cases of two fluids with identical densities 104 4.4.2 Cases of two fluids with different densities 106 4.5 Interfacial tension and isotropy of the RK model 110 4.5.1 Interfacial tension 110 4.5.2 Isotropy 110 4.6 Drainage and capillary filling 111 4.7 MRT RK model 113 4.8 Contact angle 114 4.8.1 Spurious currents 115 4.9 Tests of inlet/outlet boundary conditions 117 4.10 Immiscible displacements in porous media 118 4.11 Appendix A 121 4.12 Appendix B 122 5 Free-energy-based multiphase Lattice Boltzmann model 136 5.1 Swift free-energy based single-component multiphase LBM 136 5.1.1 Derivation of the coefficients in the equilibrium distribution function 138 5.2 Chapman–Enskog expansion 143 5.3 Issue of Galilean invariance 146 5.4 Phase separation 149 5.5 Contact angle 154 5.5.1 How to specify a desired contact angle 154 5.5.2 Numerical verification 155 5.6 Swift free-energy-based multi-component multiphase LBM 158 5.7 Appendix 158 6 Inamuro's multiphase Lattice Boltzmann model 167 6.1 Introduction 167 6.1.1 Inamuro's method 167 6.1.2 Comment on the presentation 169 6.1.3 Chapman–Enskog expansion analysis 170 6.1.4 Cahn–Hilliard equation (equation for order parameter) 173 6.1.5 Poisson equation 174 6.2 Droplet collision 175 6.3 Appendix 178 7 He–Chen–Zhang multiphase Lattice Boltzmann model 196 7.1 Introduction 196 7.2 HCZ model 196 7.3 Chapman–Enskog analysis 199 7.3.1 N–S equations 199 7.3.2 CH equation 202 7.4 Surface tension and phase separation 202 7.5 Layered two-phase flow in a channel 204 7.6 Rayleigh–Taylor instability 205 7.7 Contact angle 210 7.8 Capillary rise 213 7.9 Geometric scheme to specify the contact angle and its hysteresis 215 7.9.1 Examples of droplet slipping in shear flows 218 7.10 Oscillation of an initially ellipsoidal droplet 219 7.11 Appendix A 222 7.12 Appendix B: 2D code 223 7.13 Appendix C: 3D code 238 8 Axisymmetric multiphase HCZ model 253 8.1 Introduction 253 8.2 Methods 253 8.2.1 Macroscopic governing equations 253 8.2.2 Axisymmetric HCZ LBM (Premnath and Abraham 2005a) 255 8.2.3 MRT version of the axisymmetric LBM (McCracken and Abraham 2005) 256 8.2.4 Axisymmetric boundary conditions 258 8.3 The Laplace law 258 8.4 Oscillation of an initially ellipsoidal droplet 259 8.5 Cylindrical liquid column break 263 8.6 Droplet collision 265 8.6.1 Effect of gradient and Laplacian calculation 267 8.6.2 Effect of BGK and MRT 274 8.7 A revised axisymmetric HCZ model (Huang et al. 2014) 276 8.7.1 MRT collision 276 8.7.2 Calculation of the surface tension 277 8.7.3 Mass correction 278 8.8 Bubble rise 279 8.8.1 Numerical validation 281 8.8.2 Surface-tension calculation effect 283 8.8.3 Terminal bubble shape 284 8.8.4 Wake behind the bubble 284 8.9 Conclusion 286 8.10 Appendix A: Chapman–Enskog analysis 288 8.10.1 Preparation for derivation 288 8.10.2 Mass conservation 289 8.10.3 Momentum conservation 289 8.10.4 CH equation 291 9 Extensions of the HCZ model for high-density ratio two-phase flows 292 9.1 Introduction 292 9.2 Model I (Lee and Lin 2005) 293 9.2.1 Stress and potential form of intermolecular forcing terms 293 9.2.2 Model description 294 9.2.3 Implementation 297 9.2.4 Directional derivative 298 9.2.5 Droplet splashing on a thin liquid film 299 9.3 Model II (Amaya-Bower and Lee 2010) 301 9.3.1 Implementation 302 9.4 Model III (Lee and Liu 2010) 304 9.5 Model IV 305 9.6 Numerical tests for different models 306 9.6.1 A drop inside a box with periodic boundary conditions 306 9.6.2 Layered two-phase flows in a channel 311 9.6.3 Galilean invariance 313 9.7 Conclusions 316 9.8 Appendix A: Analytical solutions for layered two-phase flow in a channel 317 9.9 Appendix B: 2D code based on Amaya-Bower and Lee (2010) 319 10 Axisymmetric high-density ratio two-phase LBMs (extension of the HCZ model) 334 10.1 Introduction 334 10.2 The model based on Lee and Lin (2005) 334 10.2.1 The equilibrium distribution functions I 336 10.2.2 The equilibrium distribution functions II 336 10.2.3 Source terms 337 10.2.4 Stress and potential form of intermolecular forcing terms 337 10.2.5 Chapman–Enskog analysis 338 10.2.6 Implementation 340 10.2.7 Droplet splashing on a thin liquid film 342 10.2.8 Head-on droplet collision 342 10.3 Axisymmetric model based on Lee and Liu (2010) 345 10.3.1 Implementation 347 10.3.2 Head-on droplet collision 348 10.3.3 Bubble rise 353 Index 371

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Book SynopsisAn introduction to mathematical techniques used in engineering with an emphasis on applications in linear circuits and systems This book provides an integrated approach to learning the necessary mathematical tools specifically used for linear circuits and systems.Table of ContentsPreface xiii Notation and Bibliography xvii About the Companion Website xix 1 Overview and Background 1 1.1 Introduction 1 1.2 Mathematical Models 3 1.3 Frequency Content 12 1.4 Functions and Properties 16 1.5 Derivatives and Integrals 22 1.6 Sine, Cosine, and 𝜋 33 1.7 Napier’s Constant e and Logarithms 38 PART I CIRCUITS, MATRICES, AND COMPLEX NUMBERS 51 2 Circuits and Mechanical Systems 53 2.1 Introduction 53 2.2 Voltage, Current, and Power 54 2.3 Circuit Elements 60 2.4 Basic Circuit Laws 67 2.4.1 Mesh-Current and Node-Voltage Analysis 69 2.4.2 Equivalent Resistive Circuits 71 2.4.3 RC and RL Circuits 75 2.4.4 Series RLC Circuit 78 2.4.5 Diode Circuits 82 2.5 Mechanical Systems 85 2.5.1 Simple Pendulum 86 2.5.2 Mass on a Spring 92 2.5.3 Electrical and Mechanical Analogs 95 3 Linear Equations and Matrices 105 3.1 Introduction 105 3.2 Vector Spaces 106 3.3 System of Linear Equations 108 3.4 Matrix Properties and Special Matrices 113 3.5 Determinant 122 3.6 Matrix Subspaces 128 3.7 Gaussian Elimination 135 3.7.1 LU and LDU Decompositions 146 3.7.2 Basis Vectors 148 3.7.3 General Solution of 𝐀𝐲 = 𝐱 151 3.8 Eigendecomposition 152 3.9 MATLAB Functions 156 4 Complex Numbers and Functions 163 4.1 Introduction 163 4.2 Imaginary Numbers 165 4.3 Complex Numbers 167 4.4 Two Coordinates 169 4.5 Polar Coordinates 171 4.6 Euler’s Formula 175 4.7 Matrix Representation 182 4.8 Complex Exponential Rotation 183 4.9 Constant Angular Velocity 189 4.10 Quaternions 192 PART II SIGNALS, SYSTEMS, AND TRANSFORMS 203 5 Signals, Generalized Functions, and Fourier Series 205 5.1 Introduction 205 5.2 Energy and Power Signals 206 5.3 Step and Ramp Functions 208 5.4 Rectangle and Triangle Functions 211 5.5 Exponential Function 214 5.6 Sinusoidal Functions 217 5.7 Dirac Delta Function 220 5.8 Generalized Functions 223 5.9 Unit Doublet 233 5.10 Complex Functions and Singularities 240 5.11 Cauchy Principal Value 242 5.12 Even and Odd Functions 245 5.13 Correlation Functions 248 5.14 Fourier Series 251 5.15 Phasor Representation 261 5.16 Phasors and Linear Circuits 265 6 Differential Equation Models for Linear Systems 275 6.1 Introduction 275 6.2 Differential Equations 276 6.3 General Forms of The Solution 278 6.4 First-Order Linear ODE 280 6.4.1 Homogeneous Solution 283 6.4.2 Nonhomogeneous Solution 285 6.4.3 Step Response 287 6.4.4 Exponential Input 287 6.4.5 Sinusoidal Input 289 6.4.6 Impulse Response 290 6.5 Second-Order Linear ODE 294 6.5.1 Homogeneous Solution 296 6.5.2 Damping Ratio 304 6.5.3 Initial Conditions 306 6.5.4 Nonhomogeneous Solution 307 6.6 Second-Order ODE Responses 311 6.6.1 Step Response 311 6.6.2 Step Response (Alternative Method) 313 6.6.3 Impulse Response 319 6.7 Convolution 319 6.8 System of ODEs 323 7 Laplace Transforms and Linear Systems 335 7.1 Introduction 335 7.2 Solving ODEs Using Phasors 336 7.3 Eigenfunctions 339 7.4 Laplace Transform 340 7.5 Laplace Transforms and Generalized Functions 347 7.6 Laplace Transform Properties 352 7.7 Initial and Final Value Theorems 364 7.8 Poles and Zeros 367 7.9 Laplace Transform Pairs 372 7.9.1 Constant Function 372 7.9.2 Rectangle Function 373 7.9.3 Triangle Function 374 7.9.4 Ramped Exponential Function 376 7.9.5 Sinusoidal Functions 376 7.10 Transforms and Polynomials 377 7.11 Solving Linear ODEs 380 7.12 Impulse Response and Transfer Function 382 7.13 Partial Fraction Expansion 387 7.13.1 Distinct Real Poles 388 7.13.2 Distinct Complex Poles 391 7.13.3 Repeated Real Poles 396 7.13.4 Repeated Complex Poles 402 7.14 Laplace Transforms and Linear Circuits 409 8 Fourier Transforms and Frequency Responses 423 8.1 Introduction 423 8.2 Fourier Transform 425 8.3 Magnitude and Phase 435 8.4 Fourier Transforms and Generalized Functions 437 8.5 Fourier Transform Properties 442 8.6 Amplitude Modulation 449 8.7 Frequency Response 453 8.7.1 First-Order Low-Pass Filter 455 8.7.2 First-Order High-Pass Filter 459 8.7.3 Second-Order Band-Pass Filter 460 8.7.4 Second-Order Band-Reject Filter 463 8.8 Frequency Response of Second-Order Filters 466 8.9 Frequency Response of Series RLC Circuit 475 8.10 Butterworth Filters 478 8.10.1 Low-Pass Filter 481 8.10.2 High-Pass Filter 484 8.10.3 Band-Pass Filter 487 8.10.4 Band-Reject Filter 490 APPENDICES 499 Introduction to Appendices 500 A Extended Summaries of Functions and Transforms 501 A.1 Functions and Notation 501 A.2 Laplace Transform 502 A.3 Fourier Transform 504 A.4 Magnitude and Phase 506 A.5 Impulsive Functions 511 A.5.1 Dirac Delta Function (Shifted) 511 A.5.2 Unit Doublet (Shifted) 514 A.6 Piecewise Linear Functions 514 A.6.1 Unit Step Function 514 A.6.2 Signum Function 517 A.6.3 Constant Function (Two-Sided) 517 A.6.4 Ramp Function 521 A.6.5 Absolute Value Function (Two-Sided Ramp) 523 A.6.6 Rectangle Function 524 A.6.7 Triangle Function 528 A.7 Exponential Functions 529 A.7.1 Exponential Function (Right-Sided) 529 A.7.2 Exponential Function (Ramped) 531 A.7.3 Exponential Function (Two-Sided) 533 A.7.4 Gaussian Function 537 A.8 Sinusoidal Functions 539 A.8.1 Cosine Function (Two-Sided) 539 A.8.2 Cosine Function (Right-Sided) 541 A.8.3 Cosine Function (ExponentiallyWeighted) 544 A.8.4 Cosine Function (ExponentiallyWeighted and Ramped) 544 A.8.5 Sine Function (Two-Sided) 549 A.8.6 Sine Function (Right-Sided) 550 A.8.7 Sine Function (ExponentiallyWeighted) 553 A.8.8 Sine Function (ExponentiallyWeighted and Ramped) 554 B Inverse Laplace Transforms 559 B.1 Improper Rational Function 559 B.2 Unbounded System 562 B.3 Double Integrator and Feedback 563 C Identities, Derivatives, and Integrals 565 C.1 Trigonometric Identities 565 C.2 Summations 566 C.3 Miscellaneous 567 C.4 Completing the Square 567 C.5 Quadratic and Cubic Formulas 568 C.6 Derivatives 571 C.7 Indefinite Integrals 573 C.8 Definite Integrals 574 D Set Theory 577 D.1 Sets and Subsets 577 D.2 Set Operations 579 E Series Expansions 583 E.1 Taylor Series 583 E.2 Maclaurin Series 585 E.3 Laurent Series 588 F Lambert W-Function 593 F.1 LambertW-Function 593 F.2 Nonlinear Diode Circuit 597 F.3 System of Nonlinear Equations 598 Glossary 601 Bibliography 609 Index 615

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Book SynopsisA comprehensive guide to using energy principles and variational methods for solving problems in solid mechanics This book provides a systematic, highly practical introduction to the use of energy principles, traditional variational methods, and the finite element method for the solution of engineering problems involving bars, beams, torsion, plane elasticity, trusses, and plates. It begins with a review of the basic equations of mechanics, the concepts of work and energy, and key topics from variational calculus. It presents virtual work and energy principles, energy methods of solid and structural mechanics, Hamilton's principle for dynamical systems, and classical variational methods of approximation. And it takes a more unified approach than that found in most solid mechanics books, to introduce the finite element method. Featuring more than 200 illustrations and tables, this Third Edition has been extensively reorganized and contains much new Table of ContentsAbout the Author xvii About the Companion Website xix Preface to the Third Edition xxi Preface to the Second Edition xxiii Preface to the First Edition xxv 1. Introduction and Mathematical Preliminaries 1 1.1 Introduction 1 1.1.1 Preliminary Comments 1 1.1.2 The Role of Energy Methods and Variational Principles 1 1.1.3 A Brief Review of Historical Developments 2 1.1.4 Preview 4 1.2 Vectors 5 1.2.1 Introduction 5 1.2.2 Definition of a Vector 6 1.2.3 Scalar and Vector Products 8 1.2.4 Components of a Vector 12 1.2.5 Summation Convention 13 1.2.6 Vector Calculus 17 1.2.7 Gradient, Divergence, and Curl Theorems 22 1.3 Tensors 26 1.3.1 Second-Order Tensors 26 1.3.2 General Properties of a Dyadic 29 1.3.3 Nonion Form and Matrix Representation of a Dyad 30 1.3.4 Eigenvectors Associated with Dyads 34 1.4 Summary 39 Problems 40 2. Review of Equations of Solid Mechanics 47 2.1 Introduction 47 2.1.1 Classification of Equations 47 2.1.2 Descriptions of Motion 48 2.2 Balance of Linear and Angular Momenta 50 2.2.1 Equations of Motion 50 2.2.2 Symmetry of Stress Tensors 54 2.3 Kinematics of Deformation 56 2.3.1 Green-Lagrange Strain Tensor 56 2.3.2 Strain Compatibility Equations 62 2.4 Constitutive Equations 65 2.4.1 Introduction 65 2.4.2 Generalized Hooke's Law 66 2.4.3 Plane Stress-Reduced Constitutive Relations 68 2.4.4 Thermoelastic Constitutive Relations 70 2.5 Theories of Straight Beams 71 2.5.1 Introduction 71 2.5.2 The Bernoulli-Euler Beam Theory 73 2.5.3 The Timoshenko Beam Theory 76 2.5.4 The von Ka’rma’n Theory of Beams 81 2.5.4.1 Preliminary Discussion 81 2.5.4.2 The Bernoulli-Euler Beam Theory 82 2.5.4.3 The Timoshenko Beam Theory 84 2.6 Summary 85 Problems 88 3. Work, Energy, and Variational Calculus 97 3.1 Concepts of Work and Energy 97 3.1.1 Preliminary Comments 97 3.1.2 External and Internal Work Done 98 3.2 Strain Energy and Complementary Strain Energy 102 3.2.1 General Development 102 3.2.2 Expressions for Strain Energy and Complementary Strain Energy Densities of Isotropic Linear Elastic Solids 107 3.2.2.1 Stain energy density 107 3.2.2.2 Complementary stain energy density 108 3.2.3 Strain Energy and Complementary Strain Energy for Trusses 109 3.2.4 Strain Energy and Complementary Strain Energy for Torsional Members 114 3.2.5 Strain Energy and Complementary Strain Energy for Beams 117 3.2.5.1 The Bernoulli-Euler Beam Theory 117 3.2.5.2 The Timoshenko Beam Theory 119 3.3 Total Potential Energy and Total Complementary Energy 123 3.3.1 Introduction 123 3.3.2 Total Potential Energy of Beams 124 3.3.3 Total Complementary Energy of Beams 125 3.4 Virtual Work 126 3.4.1 Virtual Displacements 126 3.4.2 Virtual Forces 131 3.5 Calculus of Variations 135 3.5.1 The Variational Operator 135 3.5.2 Functionals 138 3.5.3 The First Variation of a Functional 139 3.5.4 Fundamental Lemma of Variational Calculus 140 3.5.5 Extremum of a Functional 141 3.5.6 The Euler Equations 143 3.5.7 Natural and Essential Boundary Conditions 146 3.5.8 Minimization of Functionals with Equality Constraints 151 3.5.8.1 The Lagrange Multiplier Method 151 3.5.8.2 The Penalty Function Method 153 3.6 Summary 156 Problems 159 4. Virtual Work and Energy Principles of Mechanics 167 4.1 Introduction 167 4.2 The Principle of Virtual Displacements 167 4.2.1 Rigid Bodies 167 4.2.2 Deformable Solids 168 4.2.3 Unit Dummy-Displacement Method 172 4.3 The Principle of Minimum Total Potential Energy and Castigliano's Theorem I 179 4.3.1 The Principle of Minimum Total Potential Energy179 4.3.2 Castigliano's Theorem I 188 4.4 The Principle of Virtual Forces 196 4.4.1 Deformable Solids 196 4.4.2 Unit Dummy-Load Method 198 4.5 Principle of Minimum Total Complementary Potential Energy and Castigliano's Theorem II 204 4.5.1 The Principle of the Minimum total Complementary Potential Energy 204 4.5.2 Castigliano's Theorem II 206 4.6 Clapeyron's, Betti's, and Maxwell's Theorems 217 4.6.1 Principle of Superposition for Linear Problems 217 4.6.2 Clapeyron's Theorem 220 4.6.3 Types of Elasticity Problems and Uniqueness of Solutions 224 4.6.4 Betti's Reciprocity Theorem 226 4.6.5 Maxwell's Reciprocity Theorem 230 4.7 Summary 232 Problems 235 5. Dynamical Systems: Hamilton's Principle 243 5.1 Introduction 243 5.2 Hamilton's Principle for Discrete Systems 243 5.3 Hamilton's Principle for a Continuum 249 5.4 Hamilton's Principle for Constrained Systems 255 5.5 Rayleigh's Method 260 5.6 Summary 262 Problems 263 6. Direct Variational Methods 269 6.1 Introduction 269 6.2 Concepts from Functional Analysis 270 6.2.1 General Introduction 270 6.2.2 Linear Vector Spaces 271 6.2.3 Normed and Inner Product Spaces 276 6.2.3.1 Norm 276 6.2.3.2 Inner product 279 6.2.3.3 Orthogonality 280 6.2.4 Transformations, and Linear and Bilinear Forms 281 6.2.5 Minimum of a Quadratic Functional 282 6.3 The Ritz Method 287 6.3.1 Introduction 287 6.3.2 Description of the Method 288 6.3.3 Properties of Approximation Functions 293 6.3.3.1 Preliminary Comments 293 6.3.3.2 Boundary Conditions 293 6.3.3.3 Convergence 294 6.3.3.4 Completeness 294 6.3.3.5 Requirements on ɸ0 and ɸi 295 6.3.4 General Features of the Ritz Method 299 6.3.5 Examples 300 6.3.6 The Ritz Method for General Boundary-Value Problems 323 6.3.6.1 Preliminary Comments 323 6.3.6.2 Weak Forms 323 6.3.6.3 Model Equation 1 324 6.3.6.4 Model Equation 2 328 6.3.6.5 Model Equation 3 330 6.3.6.6 Ritz Approximations 332 6.4 Weighted-Residual Methods 337 6.4.1 Introduction 337 6.4.2 The General Method of Weighted Residuals 339 6.4.3 The Galerkin Method 44 6.4.4 The Least-Squares Method 349 6.4.5 The Collocation Method 356 6.4.6 The Subdomain Method 359 6.4.7 Eigenvalue and Time-Dependent Problems 361 6.4.7.1 Eigenvalue Problems 361 6.4.7.2 Time-Dependent Problems 362 6.5 Summary 381 Problems 383 7. Theory and Analysis of Plates 391 7.1 Introduction 391 7.1.1 General Comments 391 7.1.2 An Overview of Plate Theories 393 7.1.2.1 The Classical Plate Theory 394 7.1.2.2 The First-Order Plate Theory 395 7.1.2.3 The Third-Order Plate Theory 396 7.1.2.4 Stress-Based Theories 397 7.2 The Classical Plate Theory 398 7.2.1 Governing Equations of Circular Plates 398 7.2.2 Analysis of Circular Plates 405 7.2.2.1 Analytical Solutions For Bending 405 7.2.2.2 Analytical Solutions For Buckling 411 7.2.2.3 Variational Solutions 414 7.2.3 Governing Equations in Rectangular Coordinates 427 7.2.4 Navier Solutions of Rectangular Plates 435 7.2.4.1 Bending 438 7.2.4.2 Natural Vibration 443 7.2.4.3 Buckling Analysis 445 7.2.4.4 Transient Analysis 447 7.2.5 Lévy Solutions of Rectangular Plates 449 7.2.6 Variational Solutions: Bending 454 7.2.7 Variational Solutions: Natural Vibration 470 7.2.8 Variational Solutions: Buckling 475 7.2.8.1 Rectangular Plates Simply Supported along Two Opposite Sides and Compressed in the Direction Perpendicular to Those Sides 475 7.2.8.2 Formulation for Rectangular Plates with Arbitrary Boundary Conditions 478 7.3 The First-Order Shear Deformation Plate Theory 486 7.3.1 Equations of Circular Plates 486 7.3.2 Exact Solutions of Axisymmetric Circular Plates 488 7.3.3 Equations of Plates in Rectangular Coordinates 492 7.3.4 Exact Solutions of Rectangular Plates 496 7.3.4.1 Bending Analysis 498 7.3.4.2 Natural Vibration 501 7.3.4.3 Buckling Analysis 502 7.3.5 Variational Solutions of Circular and Rectangular Plates 503 7.3.5.1 Axisymmetric Circular Plates 503 7.3.5.2 Rectangular Plates 505 7.4 Relationships Between Bending Solutions of Classical and Shear Deformation Theories 507 7.4.1 Beams 507 7.4.1.1 Governing Equations 508 7.4.1.2 Relationships Between BET and TBT 508 7.4.2 Circular Plates 512 7.4.3 Rectangular Plates 516 7.5 Summary 521 Problems 521 8. The Finite Element Method 527 8.1 Introduction 527 8.2 Finite Element Analysis of Straight Bars 529 8.2.1 Governing Equation 529 8.2.2 Representation of the Domain by Finite Elements 530 8.2.3 Weak Form over an Element 531 8.2.4 Approximation over an Element 532 8.2.5 Finite Element Equations 537 8.2.5.1 Linear Element 538 8.2.5.2 Quadratic Element 539 8.2.6 Assembly (Connectivity) of Elements 539 8.2.7 Imposition of Boundary Conditions 542 8.2.8 Postprocessing 543 8.3 Finite Element Analysis of the Bernoulli-Euler Beam Theory 549 8.3.1 Governing Equation 549 8.3.2 Weak Form over an Element 549 8.3.3 Derivation of the Approximation Functions 550 8.3.4 Finite Element Model 552 8.3.5 Assembly of Element Equations 553 8.3.6 Imposition of Boundary Conditions 555 8.4 Finite Element Analysis of the Timoshenko Beam Theory 558 8.4.1 Governing Equations 558 8.4.2 Weak Forms 558 8.4.3 Finite Element Models 559 8.4.4 Reduced Integration Element (RIE) 559 8.4.5 Consistent Interpolation Element (CIE) 561 8.4.6 Superconvergent Element (SCE) 562 8.5 Finite Element Analysis of the Classical Plate Theory 565 8.5.1 Introduction 565 8.5.2 General Formulation 566 8.5.3 Conforming and Nonconforming Plate Elements 568 8.5.4 Fully Discretized Finite Element Models 569 8.5.4.1 Static Bending 569 8.5.4.2 Buckling 569 8.5.4.3 Natural Vibration 570 8.5.4.4 Transient Response 570 8.6 Finite Element Analysis of the First-Order Shear Deformation Plate Theory 574 8.6.1 Governing Equations and Weak Forms 574 8.6.2 Finite Element Approximations 576 8.6.3 Finite Element Model 577 8.6.4 Numerical Integration 579 8.6.5 Numerical Examples 582 8.6.5.1 Isotropic Plates 582 8.6.5.2 Laminated Plates 584 8.7 Summary 587 Problems 588 9. Mixed Variational and Finite Element Formulations 595 9.1 Introduction 595 9.1.1 General Comments 595 9.1.2 Mixed Variational Principles 595 9.1.3 Extremum and Stationary Behavior of Functionals 597 9.2 Stationary Variational Principles 599 9.2.1 Minimum Total Potential Energy 599 9.2.2 The Hellinger-Reissner Variational Principle 601 9.2.3 The Reissner Variational Principle 605 9.3 Variational Solutions Based on Mixed Formulations 606 9.4 Mixed Finite Element Models of Beams 610 9.4.1 The Bernoulli-Euler Beam Theory 610 9.4.1.1 Governing Equations And Weak Forms 610 9.4.1.2 Weak-Form Mixed Finite Element Model 610 9.4.1.3 Weighted-Residual Finite Element Models 613 9.4.2 The Timoshenko Beam Theory 615 9.4.2.1 Governing Equations 615 9.4.2.2 General Finite Element Model 615 9.4.2.3 ASD-LLCC Element 617 9.4.2.4 ASD-QLCC Element 617 9.4.2.5 ASD-HQLC Element 618 9.5 Mixed Finite Element Analysis of the Classical Plate Theory 620 9.5.1 Preliminary Comments 620 9.5.2 Mixed Model I 620 9.5.2.1 Governing Equations 620 9.5.2.2 Weak Forms 621 9.5.2.3 Finite Element Model 622 9.5.3 Mixed Model II 625 9.5.3.1 Governing Equations 625 9.5.3.2 Weak Forms 625 9.5.3.3 Finite Element Model 626 9.6 Summary 630 Problems 631 10. Analysis of Functionally Graded Beams and Plates 635 10.1 Introduction 635 10.2 Functionally Graded Beams 638 10.2.1 The Bernoulli-Euler Beam Theory 638 10.2.1.1 Displacement and strain fields 638 10.2.1.2 Equations of motion and boundary conditions 638 10.2.2 The Timoshenko Beam Theory 639 10.2.2.1 Displacement and strain fields 639 10.2.2.2 Equations of motion and boundary conditions 640 10.2.3 Equations of Motion in terms of Generalized Displacements 641 10.2.3.1 Constitutive Equations 641 10.2.3.2 Stress Resultants of BET 641 10.2.3.3 Stress Resultants of TBT 642 10.2.3.4 Equations of Motion of the BET 642 10.2.3.5 Equations of Motion of the TBT 642 10.2.4 Stiffiness Coefficients643 10.3 Functionally Graded Circular Plates 645 10.3.1 Introduction 645 10.3.2 Classical Plate Theory 646 10.3.2.1 Displacement and Strain Fields 646 10.3.2.2 Equations of Motion 646 10.3.3 First-Order Shear Deformation Theory 647 10.3.3.1 Displacement and Strain Fields 647 10.3.3.2 Equations of Motion 648 10.3.4 Plate Constitutive Relations 649 10.3.4.1 Classical Plate Theory 649 10.3.4.2 First-Order Plate Theory 649 10.4 A General Third-Order Plate Theory 650 10.4.1 Introduction 650 10.4.2 Displacements and Strains 651 10.4.3 Equations of Motion 653 10.4.4 Constitutive Relations 657 10.4.5 Specialization to Other Theories 658 10.4.5.1 A General Third-Order Plate Theory with Traction-Free Top and Bottom Surfaces 658 10.4.5.2 The Reddy Third-Order Plate Theory 661 10.4.5.3 The First-Order Plate Theory 663 10.4.5.4 The Classical Plate Theory 664 10.5 Navier's Solutions 664 10.5.1 Preliminary Comments 664 10.5.2 Analysis of Beams 665 10.5.2.1 Bernoulli-Euler Beams 665 10.5.2.2 Timoshenko Beams 667 10.5.2.3 Numerical Results 669 10.5.3 Analysis of Plates 671 10.5.3.1 Boundary Conditions 672 10.5.3.2 Expansions of Generalized Displacements 672 10.5.3.3 Bending Analysis 673 10.5.3.4 Free Vibration Analysis 676 10.5.3.5 Buckling Analysis 677 10.5.3.6 Numerical Results 679 10.6 Finite Element Models 681 10.6.1 Bending of Beams 681 10.6.1.1 Bernoulli-Euler Beam Theory 681 10.6.1.2 Timoshenko Beam Theory 683 10.6.2 Axisymmetric Bending of Circular Plates 684 10.6.2.1 Classical Plate Theory 681 10.6.2.2 First-Order Shear Deformation Plate Theory 686 10.6.3 Solution of Nonlinear Equations 688 10.6.3.1 Times approximation 688 10.6.3.2 Newton's Iteration Approach 688 10.6.3.3 Tangent Stiffiness Coefficients for the BET 690 10.6.3.4 Tangent Stiffiness Coefficients for the TBT 692 10.6.3.5 Tangent Stiffiness Coefficients for the CPT 693 10.6.3.6 Tangent Stiffiness Coefficients for the FSDT 693 10.6.4 Numerical Results for Beams and Circular Plates 694 10.6.4.1 Beams 694 10.6.4.2 Circular Plates 697 10.7 Summary 699 Problems 700 References 701 Answers to Most Problems 711 Index 723

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Book SynopsisCovers multivariable calculus, starting from the basics and leading up to the three theorems of Green, Gauss, and Stokes, but always with an eye on practical applications. Written for a wide spectrum of undergraduate students by an experienced author, this book provides a very practical approach to advanced calculusstarting from the basics and leading up to the theorems of Green, Gauss, and Stokes. It explains, clearly and concisely, partial differentiation, multiple integration, vectors and vector calculus, and provides end-of-chapter exercises along with their solutions to aid the readers' understanding. Written in an approachable style and filled with numerous illustrative examples throughout, Two and Three Dimensional Calculus: with Applications in Science and Engineering assumes no prior knowledge of partial differentiation or vectors and explains difficult concepts with easy to follow examples. Rather than concentrating on mathematical structures, tTable of ContentsPreface xi 1 Revision of One-Dimensional Calculus 1 1.1 Limits and Convergence 1 1.2 Differentiation 3 1.2.1 Rules for Differentiation 5 1.2.2 Mean Value Theorem 7 1.2.3 Taylor’s Series 8 1.2.4 Maxima and Minima 12 1.2.5 Numerical Differentiation 13 1.3 Integration 16 Exercises 22 2 Partial Differentiation 25 2.1 Introduction 25 2.2 Differentials 29 2.2.1 Small Errors 30 2.3 Total Derivative 33 2.4 Chain Rule 36 2.4.1 Leibniz Rule 39 2.4.2 Chain Rule in n Dimensions 41 2.4.3 Implicit Functions 42 2.5 Jacobian 43 2.6 Higher Derivatives 46 2.6.1 Higher Differentials 49 2.7 Taylor’sTheorem 50 2.8 Conjugate Functions 52 2.9 Case Study:Thermodynamics 54 Exercises 58 3 Maxima and Minima 61 3.1 Introduction 61 3.2 Maxima, Minima and Saddle Points 63 3.3 Lagrange Multipliers 74 3.3.1 Generalisations 77 3.4 Optimisation 81 3.4.1 Hill Climbing Techniques 81 Exercises 85 4 Vector Algebra 89 4.1 Introduction 89 4.2 Vector Addition 90 4.3 Components 92 4.4 Scalar Product 94 4.5 Vector Product 97 4.5.1 Scalar Triple Product 102 4.5.2 Vector Triple Product 105 Exercises 106 5 Vector Differentiation 109 5.1 Introduction 109 5.2 Differential Geometry 111 5.2.1 Space Curves 112 5.2.2 Surfaces 120 5.3 Mechanics 129 Exercises 135 6 Gradient, Divergence, and Curl 139 6.1 Introduction 139 6.2 Gradient 139 6.3 Divergence 143 6.4 Curl 145 6.5 Vector Identities 146 6.6 Conjugate Functions 151 Exercises 154 7 Curvilinear Co-ordinates 157 7.1 Introduction 157 7.2 Curved Axes and Scale Factors 157 7.3 Curvilinear Gradient, Divergence, and Curl 161 7.3.1 Gradient 161 7.3.2 Divergence 163 7.3.3 Curl 165 7.4 Further Results and Tensors 166 7.4.1 Tensor Notation 166 7.4.2 Covariance and Contravariance 168 Exercises 171 8 PathIntegrals 173 8.1 Introduction 173 8.2 Integration Along a Curve 173 8.3 Practical Applications 181 Exercises 186 9 Multiple Integrals 191 9.1 Introduction 191 9.2 The Double Integral 191 9.2.1 Rotation and Translation 199 9.2.2 Change of Order of Integration 201 9.2.3 Plane Polar Co-ordinates 203 9.2.4 Applications of Double Integration 208 9.3 Triple Integration 213 9.3.1 Cylindrical and Spherical Polar Co-ordinates 219 9.3.2 Applications of Triple Integration 227 Exercises 233 10 Surface Integrals 241 10.1 Introduction 241 10.2 Green’s Theorem in the Plane 242 10.3 Integration over a Curved Surface 246 10.4 Applications of Surface Integration 253 Exercises 256 11 Integral Theorems 259 11.1 Introduction 259 11.2 Stokes’ Theorem 260 11.3 Gauss’ DivergenceTheorem 268 11.3.1 Green’s Second Identity 275 11.4 Co-ordinate-Free Definitions 277 11.5 Applications of Integral Theorems 279 11.5.1 Electromagnetic Theory 279 11.5.1.1 Maxwell’s Equations 279 11.5.2 Fluid Mechanics 283 11.5.3 ElasticityTheory 287 11.5.4 Heat Transfer 297 Exercises 298 12 Solutions and Answers to Exercises 301 References 375 Index 377

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Book SynopsisExplore the diverse electrical engineering application of polymer composite materials with this in-depth collection edited by leaders in the field Polymer Composites for Electrical Engineering delivers a comprehensive exploration of the fundamental principles, state-of-the-art research, and future challenges of polymer composites. Written from the perspective of electrical engineering applications, like electrical and thermal energy storage, high temperature applications, fire retardance, power cables, electric stress control, and others, the book covers all major application branches of these widely used materials. Rather than focus on polymer composite materials themselves, the distinguished editors have chosen to collect contributions from industry leaders in the area of real and practical electrical engineering applications of polymer composites. The book?s relevance will only increase as advanced polymer composites receive more attention and interest iTable of ContentsAbout the Author xi Preface to the Fourth Edition xiii 1 Introduction 1 The Case for Imprecision 2 A Historical Perspective 4 The Utility of Fuzzy Systems 7 Limitations of Fuzzy Systems 9 The Illusion: Ignoring Uncertainty and Accuracy 11 Uncertainty and Information 13 Fuzzy Sets and Membership 14 Chance versus Fuzziness 17 Intuition of Uncertainty: Fuzzy versus Probability 19 Sets as Points in Hypercubes 21 Summary 23 References 23 Problems 24 2 Classical Sets and Fuzzy Sets 27 Classical Sets 28 Fuzzy Sets 36 Summary 45 References 46 Problems 46 3 Classical Relations and Fuzzy Relations 51 Cartesian Product 52 Crisp Relations 53 Fuzzy Relations 58 Tolerance and Equivalence Relations 67 Fuzzy Tolerance and Equivalence Relations 70 Value Assignments 72 Other Forms of the Composition Operation 76 Summary 77 References 77 Problems 77 4 Properties of Membership Functions, Fuzzification, and Defuzzification 84 Features of the Membership Function 85 Various Forms 87 Fuzzification 88 Defuzzification to Crisp Sets 90 λ-Cuts for Fuzzy Relations 92 Defuzzification to Scalars 93 Summary 102 References 103 Problems 104 5 Logic and Fuzzy Systems 107 Part I: Logic 107 Classical Logic 108 Fuzzy Logic 122 Part II: Fuzzy Systems 132 Summary 151 References 153 Problems 154 6 Historical Methods of Developing Membership Functions 163 Membership Value Assignments 164 Intuition 164 Inference 165 Rank Ordering 167 Neural Networks 168 Genetic Algorithms 179 Inductive Reasoning 188 Summary 195 References 196 Problems 197 7 Automated Methods for Fuzzy Systems 201 Definitions 202 Batch Least Squares Algorithm 205 Recursive Least Squares Algorithm 210 Gradient Method 213 Clustering Method 218 Learning from Examples 221 Modified Learning from Examples 224 Summary 233 References 235 Problems 235 8 Fuzzy Systems Simulation 237 Fuzzy Relational Equations 242 Nonlinear Simulation Using Fuzzy Systems 243 Fuzzy Associative Memories (FAMs) 246 Summary 257 References 258 Problems 259 9 Decision Making with Fuzzy Information 265 Fuzzy Synthetic Evaluation 267 Fuzzy Ordering 269 Nontransitive Ranking 272 Preference and Consensus 275 Multiobjective Decision Making 279 Fuzzy Bayesian Decision Method 285 Decision Making under Fuzzy States and Fuzzy Actions 295 Summary 309 References 310 Problems 311 10 Fuzzy Classification and Pattern Recognition 323 Fuzzy Classification 324 Classification by Equivalence Relations 324 Cluster Analysis 332 Cluster Validity 332 c-Means Clustering 333 Hard c-Means (HCM) 333 Fuzzy c-Means (FCM) 343 Classification Metric 351 Hardening the Fuzzy c-Partition 354 Similarity Relations from Clustering 356 Fuzzy Pattern Recognition 357 Single-Sample Identification 357 Multifeature Pattern Recognition 365 Summary 378 References 379 Problems 380 11 Fuzzy Control Systems 388 Control System Design Problem 390 Examples of Fuzzy Control System Design 393 Fuzzy Engineering Process Control 404 Fuzzy Statistical Process Control 417 Industrial Applications 431 Summary 434 References 437 Problems 438 12 Applications of Fuzzy Systems Using Miscellaneous Models 455 Fuzzy Optimization 455 Fuzzy Cognitive Mapping 462 Agent-Based Models 477 Fuzzy Arithmetic and the Extension Principle 481 Fuzzy Algebra 487 Data Fusion 491 Summary 498 References 498 Problems 500 13 Monotone Measures: Belief, Plausibility, Probability, and Possibility 505 Monotone Measures 506 Belief and Plausibility 507 Evidence Theory 512 Probability Measures 515 Possibility and Necessity Measures 517 Possibility Distributions as Fuzzy Sets 525 Possibility Distributions Derived from Empirical Intervals 528 Summary 548 References 549 Problems 550 Index 554

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Book SynopsisA problem-solving approach to statistical signal processing for practicing engineers, technicians, and graduate students This book takes a pragmatic approach in solving a set of common problems engineers and technicians encounter when processing signals.Table of ContentsList of Figures xvii List of Tables xxiii Preface xxv List of Abbreviations xxix How to Use the Book xxxi About the Companion Website xxxiii Prerequisites xxxv Why are there so many matrixes in this book? xxxvii 1 Manipulations on Matrixes 1 1.1 Matrix Properties 1 1.1.1 Elementary Operations 2 1.2 Eigen-Decomposition 6 1.3 Eigenvectors in Everyday Life 9 1.3.1 Conversations in a Noisy Restaurant 9 1.3.2 Power Control in a Cellular System 12 1.3.3 Price Equilibrium in the Economy 14 1.4 Derivative Rules 15 1.4.1 Derivative with respect to x 16 1.4.2 Derivative with respect to x 17 1.4.3 Derivative with respect to the Matrix X 18 1.5 Quadratic Forms 19 1.6 Diagonalization of a Quadratic Form 20 1.7 Rayleigh Quotient 21 1.8 Basics of Optimization 22 1.8.1 Quadratic Function with Simple Linear Constraint (M=1) 23 1.8.2 Quadratic Function with Multiple Linear Constraints 23 Appendix A: Arithmetic vs. Geometric Mean 24 2 Linear Algebraic Systems 27 2.1 Problem Definition and Vector Spaces 27 2.1.1 Vector Spaces in Tomographic Radiometric Inversion 29 2.2 Rotations 31 2.3 Projection Matrixes and Data-Filtering 33 2.3.1 Projections and Commercial FM Radio 34 2.4 Singular Value Decomposition (SVD) and Subspaces 34 2.4.1 How to Choose the Rank of Afor Gaussian Model? 35 2.5 QR and Cholesky Factorization 36 2.6 Power Method for Leading Eigenvectors 38 2.7 Least Squares Solution of Overdetermined Linear Equations 39 2.8 Efficient Implementation of the LS Solution 41 2.9 Iterative Methods 42 3 Random Variables in Brief 45 3.1 Probability Density Function (pdf), Moments, and Other Useful Properties 45 3.2 Convexity and Jensen Inequality 49 3.3 Uncorrelatedness and Statistical Independence 49 3.4 Real-Valued Gaussian Random Variables 51 3.5 Conditional pdf for Real-Valued Gaussian Random Variables 54 3.6 Conditional pdf in Additive Noise Model 56 3.7 Complex Gaussian Random Variables 56 3.7.1 Single Complex Gaussian Random Variable 56 3.7.2 Circular Complex Gaussian Random Variable 57 3.7.3 Multivariate Complex Gaussian Random Variables 58 3.8 Sum of Square of Gaussians: Chi-Square 59 3.9 Order Statistics for N rvs 60 4 Random Processes and Linear Systems 63 4.1 Moment Characterizations and Stationarity 64 4.2 Random Processes and Linear Systems 66 4.3 Complex-Valued Random Processes 68 4.4 Pole-Zero and Rational Spectra (Discrete-Time) 69 4.4.1 Stability of LTI Systems 70 4.4.2 Rational PSD 71 4.4.3 Paley–Wiener Theorem 72 4.5 Gaussian Random Process (Discrete-Time) 73 4.6 Measuring Moments in Stochastic Processes 75 Appendix A: Transforms for Continuous-Time Signals 76 Appendix B: Transforms for Discrete-Time Signals 79 5 Models and Applications 83 5.1 Linear Regression Model 84 5.2 Linear Filtering Model 86 5.2.1 Block-Wise Circular Convolution 88 5.2.2 Discrete Fourier Transform and Circular Convolution Matrixes 89 5.2.3 Identification and Deconvolution 90 5.3 MIMO systems and Interference Models 91 5.3.1 DSL System 92 5.3.2 MIMO in Wireless Communication 92 5.4 Sinusoidal Signal 97 5.5 Irregular Sampling and Interpolation 97 5.5.1 Sampling With Jitter 100 5.6 Wavefield Sensing System 101 6 Estimation Theory 105 6.1 Historical Notes 105 6.2 Non-Bayesian vs. Bayesian 106 6.3 Performance Metrics and Bounds 107 6.3.1 Bias 107 6.3.2 Mean Square Error (MSE) 108 6.3.3 Performance Bounds 109 6.4 Statistics and Sufficient Statistics 110 6.5 MVU and BLU Estimators 111 6.6 BLUE for Linear Models 112 6.7 Example: BLUE of the Mean Value of Gaussian rvs 114 7 Parameter Estimation 117 7.1 Maximum Likelihood Estimation (MLE) 117 7.2 MLE for Gaussian Model 119 7.2.1 Additive Noise Model with 119 7.2.2 Additive Noise Model with 120 7.2.3 Additive Noise Model with Multiple Observations with Known 121 7.2.3.1 Linear Model 121 7.2.3.2 Model 122 7.2.3.3 Model 123 7.2.4 Model 123 7.2.5 Additive Noise Model with Multiple Observations with Unknown 124 7.3 Other Noise Models 125 7.4 MLE and Nuisance Parameters 126 7.5 MLE for Continuous-Time Signals 128 7.5.1 Example: Amplitude Estimation 129 7.5.2 MLE for Correlated Noise 130 7.6 MLE for Circular Complex Gaussian 131 7.7 Estimation in Phase/Frequency Modulations 131 7.7.1 MLE Phase Estimation 132 7.7.2 Phase Locked Loops 133 7.8 Least Square (LS) Estimation 135 7.8.1 Weighted LS with 136 7.8.2 LS Estimation and Linear Models 137 7.8.3 Under or Over-Parameterizing? 138 7.8.4 Constrained LS Estimation 139 7.9 Robust Estimation 140 8 Cramér–Rao Bound 143 8.1 Cramér–Rao Bound and Fisher Information Matrix 143 8.1.1 CRB for Scalar Problem (P=1) 143 8.1.2 CRB and Local Curvature of Log-Likelihood 144 8.1.3 CRB for Multiple Parameters (p 1) 144 8.2 Interpretation of CRB and Remarks 146 8.2.1 Variance of Each Parameter 146 8.2.2 Compactness of the Estimates 146 8.2.3 FIM for Known Parameters 147 8.2.4 Approximation of the Inverse of FIM 148 8.2.5 Estimation Decoupled From FIM 148 8.2.6 CRB and Nuisance Parameters 149 8.2.7 CRB for Non-Gaussian rv and Gaussian Bound 149 8.3 CRB and Variable Transformations 150 8.4 FIM for Gaussian Parametric Model 151 8.4.1 FIM for with 151 8.4.2 FIM for Continuous-Time Signals in Additive White Gaussian Noise 152 8.4.3 FIM for Circular Complex Model 152 Appendix A: Proof of CRB 154 Appendix B: FIM for Gaussian Model 156 Appendix C: Some Derivatives for MLE and CRB Computations 157 9 MLE and CRB for Some Selected Cases 159 9.1 Linear Regressions 159 9.2 Frequency Estimation 162 9.3 Estimation of Complex Sinusoid 164 9.3.1 Proper, Improper, and Non-Circular Signals 165 9.4 Time of Delay Estimation 166 9.5 Estimation of Max for Uniform pdf 170 9.6 Estimation of Occurrence Probability for Binary pdf 172 9.7 How to Optimize Histograms? 173 9.8 Logistic Regression 176 10 Numerical Analysis and Montecarlo Simulations 179 10.1 System Identification and Channel Estimation 181 10.1.1 Matlab Code and Results 184 10.2 Frequency Estimation 184 10.2.1 Variable (Coarse/Fine) Sampling 187 10.2.2 Local Parabolic Regression 189 10.2.3 Matlab Code and Results 190 10.3 Time of Delay Estimation 192 10.3.1 Granularity of Sampling in ToD Estimation 193 10.3.2 Matlab Code and Results 194 10.4 Doppler-Radar System by Frequency Estimation 196 10.4.1 EM Method 197 10.4.2 Matlab Code and Results 199 11 Bayesian Estimation 201 11.1 Additive Linear Model with Gaussian Noise 203 11.1.1 Gaussian A-priori: 204 11.1.2 Non-Gaussian A-Priori 206 11.1.3 Binary Signals: MMSE vs. MAP Estimators 207 11.1.4 Example: Impulse Noise Mitigation 210 11.2 Bayesian Estimation in Gaussian Settings 212 11.2.1 MMSE Estimator 213 11.2.2 MMSE Estimator for Linear Models 213 11.3 LMMSE Estimation and Orthogonality 215 11.4 Bayesian CRB 218 11.5 Mixing Bayesian and Non-Bayesian 220 11.5.1 Linear Model with Mixed Random/Deterministic Parameters 220 11.5.2 Hybrid CRB 222 11.6 Expectation-Maximization (EM) 223 11.6.1 EM of the Sum of Signals in Gaussian Noise 224 11.6.2 EM Method for the Time of Delay Estimation of Multiple Waveforms 227 11.6.3 Remarks 228 Appendix A: Gaussian Mixture pdf 229 12 Optimal Filtering 231 12.1 Wiener Filter 231 12.2 MMSE Deconvolution (or Equalization) 233 12.3 Linear Prediction 234 12.3.1 Yule–Walker Equations 235 12.4 LS Linear Prediction 237 12.5 Linear Prediction and AR Processes 239 12.6 Levinson Recursion and Lattice Predictors 241 13 Bayesian Tracking and Kalman Filter 245 13.1 Bayesian Tracking of State in Dynamic Systems 246 13.1.1 Evolution of the A-posteriori pdf 247 13.2 Kalman Filter (KF) 249 13.2.1 KF Equations 251 13.2.2 Remarks 253 13.3 Identification of Time-Varying Filters in Wireless Communication 255 13.4 Extended Kalman Filter (EKF) for Non-Linear Dynamic Systems 257 13.5 Position Tracking by Multi-Lateration 258 13.5.1 Positioning and Noise 260 13.5.2 Example of Position Tracking 263 13.6 Non-Gaussian Pdf and Particle Filters264 14 Spectral Analysis 267 14.1 Periodogram 268 14.1.1 Bias of the Periodogram 268 14.1.2 Variance of the Periodogram 271 14.1.3 Filterbank Interpretation 273 14.1.4 Pdf of the Periodogram (White Gaussian Process) 274 14.1.5 Bias and Resolution 275 14.1.6 Variance Reduction and WOSA 278 14.1.7 Numerical Example: Bandlimited Process and (Small) Sinusoid 280 14.2 Parametric Spectral Analysis 282 14.2.1 MLE and CRB 284 14.2.2 General Model for AR, MA, ARMA Spectral Analysis 285 14.3 AR Spectral Analysis 286 14.3.1 MLE and CRB 286 14.3.2 A Good Reason to Avoid Over-Parametrization in AR 289 14.3.3 Cramér–Rao Bound of Poles in AR Spectral Analysis 291 14.3.4 Example: Frequency Estimation by AR Spectral Analysis 293 14.4 MA Spectral Analysis 296 14.5 ARMA Spectral Analysis 298 14.5.1 Cramér–Rao Bound for ARMA Spectral Analysis 300 Appendix A: Which Sample Estimate of the Autocorrelation to Use? 302 Appendix B: Eigenvectors and Eigenvalues of Correlation Matrix 303 Appendix C: Property of Monic Polynomial 306 Appendix D: Variance of Pole in AR(1) 307 15 Adaptive Filtering 309 15.1 Adaptive Interference Cancellation 311 15.2 Adaptive Equalization in Communication Systems 313 15.2.1 Wireless Communication Systems in Brief 313 15.2.2 Adaptive Equalization 315 15.3 Steepest Descent MSE Minimization 317 15.3.1 Convergence Analysis and Step-Size 318 15.3.2 An Intuitive View of Convergence Conditions 320 15.4 From Iterative to Adaptive Filters 323 15.5 LMS Algorithm and Stochastic Gradient 324 15.6 Convergence Analysis of LMS Algorithm 325 15.6.1 Convergence in the Mean 326 15.6.2 Convergence in the Mean Square 326 15.6.3 Excess MSE 329 15.7 Learning Curve of LMS 331 15.7.1 Optimization of the Step-Size 332 15.8 NLMS Updating and Non-Stationarity 333 15.9 Numerical Example: Adaptive Identification 334 15.10 RLS Algorithm 338 15.10.1 Convergence Analysis 339 15.10.2 Learning Curve of RLS 341 15.11 Exponentially-Weighted RLS 342 15.12 LMS vs. RLS 344 Appendix A: Convergence in Mean Square 344 16 Line Spectrum Analysis 347 16.1 Model Definition 349 16.1.1 Deterministic Signals 350 16.1.2 Random Signals 350 16.1.3 Properties of Structured Covariance 351 16.2 Maximum Likelihood and Cramér–Rao Bounds 352 16.2.1 Conditional ML 353 16.2.2 Cramér–Rao Bound for Conditional Model 354 16.2.3 Unconditional ML 356 16.2.4 Cramér–Rao Bound for Unconditional Model 356 16.2.5 Conditional vs. Unconditional Model & Bounds 357 16.3 High-Resolution Methods 357 16.3.1 Iterative Quadratic ML (IQML) 358 16.3.2 Prony Method 360 16.3.3 MUSIC 360 16.3.4 ESPRIT 363 16.3.5 Model Order 365 17 Equalization in Communication Engineering 367 17.1 Linear Equalization 369 17.1.1 Zero Forcing (ZF) Equalizer 370 17.1.2 Minimum Mean Square Error (MMSE) Equalizer 371 17.1.3 Finite-Length/Finite-Block Equalizer 371 17.2 Non-Linear Equalization 372 17.2.1 ZF-DFE 373 17.2.2 MMSE–DFE 374 17.2.3 Finite-Length MMSE–DFE 375 17.2.4 Asymptotic Performance for Infinite-Length Equalizers 376 17.3 MIMO Linear Equalization 377 17.3.1 ZF MIMO Equalization 377 17.3.2 MMSE MIMO Equalization 379 17.4 MIMO–DFE Equalization 379 17.4.1 Cholesky Factorization and Min/Max Phase Decomposition 379 17.4.2 MIMO–DFE 380 18 2D Signals and Physical Filters 383 18.1 2D Sinusoids 384 18.1.1 Moiré Pattern 386 18.2 2D Filtering 388 18.2.1 2D Random Fields 390 18.2.2 Wiener Filtering 391 18.2.3 Image Acquisition and Restoration 392 18.3 Diffusion Filtering 394 18.3.1 Evolution vs. Time: Fourier Method 394 18.3.2 Extrapolation of the Density 395 18.3.3 Effect of Phase-Shift 396 18.4 Laplace Equation and Exponential Filtering 398 18.5 Wavefield Propagation 400 18.5.1 Propagation/Backpropagation 400 18.5.2 Wavefield Extrapolation and Focusing 402 18.5.3 Exploding Reflector Model 402 18.5.4 Wavefield Extrapolation 404 18.5.5 Wavefield Focusing (or Migration) 406 Appendix A: Properties of 2D Signals 406 Appendix B: Properties of 2D Fourier Transform 410 Appendix C: Finite Difference Method for PDE-Diffusion 412 19 Array Processing 415 19.1 Narrowband Model 415 19.1.1 Multiple DoAs and Multiple Sources 419 19.1.2 Sensor Spacing Design 420 19.1.3 Spatial Resolution and Array Aperture 421 19.2 Beamforming and Signal Estimation 422 19.2.1 Conventional Beamforming 425 19.2.2 Capon Beamforming (MVDR) 426 19.2.3 Multiple-Constraint Beamforming 429 19.2.4 Max-SNR Beamforming 431 19.3 DoA Estimation 432 19.3.1 ML Estimation and CRB 433 19.3.2 Beamforming and Root-MVDR 434 20 Multichannel Time of Delay Estimation 435 20.1 Model Definition for ToD 440 20.2 High Resolution Method for ToD (L=1) 441 20.2.1 ToD in the Fourier Transformed Domain 441 20.2.2 CRB and Resolution 444 20.3 Difference of ToD (DToD) Estimation 445 20.3.1 Correlation Method for DToD 445 20.3.2 Generalized Correlation Method 448 20.4 Numerical Performance Analysis of DToD 452 20.5 Wavefront Estimation: Non-Parametric Method (L=1) 454 20.5.1 Wavefront Estimation in Remote Sensing and Geophysics 456 20.5.2 Narrowband Waveforms and 2D Phase Unwrapping 457 20.5.3 2D Phase Unwrapping in Regular Grid Spacing 458 20.6 Parametric ToD Estimation and Wideband Beamforming 460 20.6.1 Delay and Sum Beamforming 462 20.6.2 Wideband Beamforming After Fourier Transform 464 20.7 Appendix A: Properties of the Sample Correlations 465 20.8 Appendix B: How to Delay a Discrete-Time Signal? 466 20.9 Appendix C: Wavefront Estimation for 2D Arrays 467 21 Tomography 467 21.1 X-ray Tomography 471 21.1.1 Discrete Model 471 21.1.2 Maximum Likelihood 473 21.1.3 Emission Tomography 473 21.2 Algebraic Reconstruction Tomography (ART) 475 21.3 Reconstruction From Projections: Fourier Method 475 21.3.1 Backprojection Algorithm 476 21.3.2 How Many Projections to Use? 479 21.4 Traveltime Tomography 480 21.5 Internet (Network) Tomography 483 21.5.1 Latency Tomography 484 21.5.2 Packet-Loss Tomography 484 22 Cooperative Estimation 487 22.1 Consensus and Cooperation 490 22.1.1 Vox Populi: The Wisdom of Crowds 490 22.1.2 Cooperative Estimation as Simple Information Consensus 490 22.1.3 Weighted Cooperative Estimation ( ) 493 22.1.4 Distributed MLE ( ) 495 22.2 Distributed Estimation for Arbitrary Linear Models (p>1) 496 22.2.1 Centralized MLE 497 22.2.2 Distributed Weighted LS 498 22.2.3 Distributed MLE 500 22.2.4 Distributed Estimation for Under-Determined Systems 501 22.2.5 Stochastic Regressor Model 503 22.2.6 Cooperative Estimation in the Internet of Things (IoT) 503 22.2.7 Example: Iterative Distributed Estimation 505 22.3 Distributed Synchronization 506 22.3.1 Synchrony-States for Analog and Discrete-Time Clocks 507 22.3.2 Coupled Clocks 510 22.3.3 Internet Synchronization and the Network Time Protocol (NTP) 512 Appendix A: Basics of Undirected Graphs 515 23 Classification and Clustering 521 23.1 Historical Notes 522 23.2 Classification 523 23.2.1 Binary Detection Theory 523 23.2.2 Binary Classification of Gaussian Distributions 528 23.3 Classification of Signals in Additive Gaussian Noise 529 23.3.1 Detection of Known Signal 531 23.3.2 Classification of Multiple Signals 532 23.3.3 Generalized Likelihood Ratio Test (GLRT) 533 23.3.4 Detection of Random Signals 535 23.4 Bayesian Classification 536 23.4.1 To Classify or Not to Classify? 537 23.4.2 Bayes Risk 537 23.5 Pattern Recognition and Machine Learning 538 23.5.1 Linear Discriminant 539 23.5.2 Least Squares Classification 540 23.5.3 Support Vectors Principle 541 23.6 Clustering 543 23.6.1 K-Means Clustering 544 23.6.2 EM Clustering 545 References 549 Index 557

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Book SynopsisIntroduces basic concepts in probability and statistics to data science students, as well as engineers and scientists Aimed at undergraduate/graduate-level engineering and natural science students, this timely, fully updated edition of a popular book on statistics and probability shows how real-world problems can be solved using statistical concepts. It removes Excel exhibits and replaces them with R software throughout, and updates both MINITAB and JMP software instructions and content. A new chapter discussing data miningincluding big data, classification, machine learning, and visualizationis featured. Another new chapter covers cluster analysis methodologies in hierarchical, nonhierarchical, and model based clustering. The book also offers a chapter on Response Surfaces that previously appeared on the book's companion website. Statistics and Probability with Applications for Engineers and Scientists using MINITAB, R and JMP, Second Edition is broken iTable of ContentsPreface xvii Acknowledgments xxi About The Companion Site xxiii 1 Introduction 1 1.1 Designed Experiment 2 1.1.1 Motivation for the Study 2 1.1.2 Investigation 3 1.1.3 Changing Criteria 3 1.1.4 A Summary of the Various Phases of the Investigation 5 1.2 A Survey 6 1.3 An Observational Study 6 1.4 A Set of Historical Data 7 1.5 A Brief Description of What is Covered in this Book 7 Part I Fundamentals of Probability and Statistics 2 Describing Data Graphically and Numerically 13 2.1 Getting Started with Statistics 14 2.1.1 What is Statistics? 14 2.1.2 Population and Sample in a Statistical Study 14 2.2 Classification of Various Types of Data 18 2.2.1 Nominal Data 18 2.2.2 Ordinal Data 19 2.2.3 Interval Data 19 2.2.4 Ratio Data 19 2.3 Frequency Distribution Tables for Qualitative and Quantitative Data 20 2.3.1 Qualitative Data 21 2.3.2 Quantitative Data 24 2.4 Graphical Description of Qualitative and Quantitative Data 30 2.4.1 Dot Plot 30 2.4.2 Pie Chart 31 2.4.3 Bar Chart 33 2.4.4 Histograms 37 2.4.5 Line Graph 44 2.4.6 Stem-and-Leaf Plot 45 2.5 Numerical Measures of Quantitative Data 50 2.5.1 Measures of Centrality 51 2.5.2 Measures of Dispersion 56 2.6 Numerical Measures of Grouped Data 67 2.6.1 Mean of a Grouped Data 67 2.6.2 Median of a Grouped Data 68 2.6.3 Mode of a Grouped Data 69 2.6.4 Variance of a Grouped Data 69 2.7 Measures of Relative Position 70 2.7.1 Percentiles 71 2.7.2 Quartiles 72 2.7.3 Interquartile Range (IQR) 72 2.7.4 Coefficient of Variation 73 2.8 Box-Whisker Plot 75 2.8.1 Construction of a Box Plot 75 2.8.2 How to Use the Box Plot 76 2.9 Measures of Association 80 2.10 Case Studies 84 2.10.1 About St. Luke’s Hospital 85 2.11 Using JMP 86 Review Practice Problems 87 3 Elements of Probability 97 3.1 Introduction 97 3.2 Random Experiments, Sample Spaces, and Events 98 3.2.1 Random Experiments and Sample Spaces 98 3.2.2 Events 99 3.3 Concepts of Probability 103 3.4 Techniques of Counting Sample Points 108 3.4.1 Tree Diagram 108 3.4.2 Permutations 110 3.4.3 Combinations 110 3.4.4 Arrangements of n Objects Involving Several Kinds of Objects 111 3.5 Conditional Probability 113 3.6 Bayes’s Theorem 116 3.7 Introducing Random Variables 120 Review Practice Problems 122 4 Discrete Random Variables and Some Important Discrete Probability Distributions 128 4.1 Graphical Descriptions of Discrete Distributions 129 4.2 Mean and Variance of a Discrete Random Variable 130 4.2.1 Expected Value of Discrete Random Variables and Their Functions 130 4.2.2 The Moment-Generating Function-Expected Value of a Special Function of X 133 4.3 The Discrete Uniform Distribution 136 4.4 The Hypergeometric Distribution 137 4.5 The Bernoulli Distribution 141 4.6 The Binomial Distribution 142 4.7 The Multinomial Distribution 146 4.8 The Poisson Distribution 147 4.8.1 Definition and Properties of the Poisson Distribution 147 4.8.2 Poisson Process 148 4.8.3 Poisson Distribution as a Limiting Form of the Binomial 148 4.9 The Negative Binomial Distribution 153 4.10 Some Derivations and Proofs (Optional) 156 4.11 A Case Study 156 4.12 Using JMP 157 Review Practice Problems 157 5 Continuous Random Variables and Some Important Continuous Probability Distributions 164 5.1 Continuous Random Variables 165 5.2 Mean and Variance of Continuous Random Variables 168 5.2.1 Expected Value of Continuous Random Variables and Their Functions 168 5.2.2 The Moment-Generating Function and Expected Value of a Special Function of X 171 5.3 Chebyshev’s Inequality 173 5.4 The Uniform Distribution 175 5.4.1 Definition and Properties 175 5.4.2 Mean and Standard Deviation of the Uniform Distribution 178 5.5 The Normal Distribution 180 5.5.1 Definition and Properties 180 5.5.2 The Standard Normal Distribution 182 5.5.3 The Moment-Generating Function of the Normal Distribution 187 5.6 Distribution of Linear Combination of Independent Normal Variables 189 5.7 Approximation of the Binomial and Poisson Distributions by the Normal Distribution 193 5.7.1 Approximation of the Binomial Distribution by the Normal Distribution 193 5.7.2 Approximation of the Poisson Distribution by the Normal Distribution 196 5.8 A Test of Normality 196 5.9 Probability Models Commonly used in Reliability Theory 201 5.9.1 The Lognormal Distribution 202 5.9.2 The Exponential Distribution 206 5.9.3 The Gamma Distribution 211 5.9.4 The Weibull Distribution 214 5.10 A Case Study 218 5.11 Using JMP 219 Review Practice Problems 220 6 Distribution of Functions Of Random Variables 228 6.1 Introduction 229 6.2 Distribution Functions of Two Random Variables 229 6.2.1 Case of Two Discrete Random Variables 229 6.2.2 Case of Two Continuous Random Variables 232 6.2.3 The Mean Value and Variance of Functions of Two Random Variables 233 6.2.4 Conditional Distributions 235 6.2.5 Correlation between Two Random Variables 238 6.2.6 Bivariate Normal Distribution 241 6.3 Extension to Several Random Variables 244 6.4 The Moment-Generating Function Revisited 245 Review Practice Problems 249 7 Sampling Distributions 253 7.1 Random Sampling 253 7.1.1 Random Sampling from an Infinite Population 254 7.1.2 Random Sampling from a Finite Population 256 7.2 The Sampling Distribution of the Sample Mean 258 7.2.1 Normal Sampled Population 258 7.2.2 Nonnormal Sampled Population 258 7.2.3 The Central Limit Theorem 259 7.3 Sampling from a Normal Population 264 7.3.1 The Chi-Square Distribution 264 7.3.2 The Student t-Distribution 271 7.3.3 Snedecor’s F-Distribution 276 7.4 Order Statistics 279 7.4.1 Distribution of the Largest Element in a Sample 280 7.4.2 Distribution of the Smallest Element in a Sample 281 7.4.3 Distribution of the Median of a Sample and of the kth Order Statistic 282 7.4.4 Other Uses of Order Statistics 284 7.5 Using JMP 286 Review Practice Problems 286 8 Estimation of Population Parameters 289 8.1 Introduction 290 8.2 Point Estimators for the Population Mean and Variance 290 8.2.1 Properties of Point Estimators 292 8.2.2 Methods of Finding Point Estimators 295 8.3 Interval Estimators for the Mean μ of a Normal Population 301 8.3.1 σ2 Known 301 8.3.2 σ2 Unknown 304 8.3.3 Sample Size is Large 306 8.4 Interval Estimators for The Difference of Means of Two Normal Populations 313 8.4.1 Variances are Known 313 8.4.2 Variances are Unknown 314 8.5 Interval Estimators for the Variance of a Normal Population 322 8.6 Interval Estimator for the Ratio of Variances of Two Normal Populations 327 8.7 Point and Interval Estimators for the Parameters of Binomial Populations 331 8.7.1 One Binomial Population 331 8.7.2 Two Binomial Populations 334 8.8 Determination of Sample Size 338 8.8.1 One Population Mean 339 8.8.2 Difference of Two Population Means 339 8.8.3 One Population Proportion 340 8.8.4 Difference of Two Population Proportions 341 8.9 Some Supplemental Information 343 8.10 A Case Study 343 8.11 Using JMP 343 Review Practice Problems 344 9 Hypothesis Testing 352 9.1 Introduction 353 9.2 Basic Concepts of Testing a Statistical Hypothesis 353 9.2.1 Hypothesis Formulation 353 9.2.2 Risk Assessment 355 9.3 Tests Concerning the Mean of a Normal Population Having Known Variance 358 9.3.1 Case of a One-Tail (Left-Sided) Test 358 9.3.2 Case of a One-Tail (Right-Sided) Test 362 9.3.3 Case of a Two-Tail Test 363 9.4 Tests Concerning the Mean of a Normal Population Having Unknown Variance 372 9.4.1 Case of a Left-Tail Test 372 9.4.2 Case of a Right-Tail Test 373 9.4.3 The Two-Tail Case 374 9.5 Large Sample Theory 378 9.6 Tests Concerning the Difference of Means of Two Populations Having Distributions with Known Variances 380 9.6.1 The Left-Tail Test 380 9.6.2 The Right-Tail Test 381 9.6.3 The Two-Tail Test 383 9.7 Tests Concerning the Difference of Means of Two Populations Having Normal Distributions with Unknown Variances 388 9.7.1 Two Population Variances are Equal 388 9.7.2 Two Population Variances are Unequal 392 9.7.3 The Paired t-Test 395 9.8 Testing Population Proportions 401 9.8.1 Test Concerning One Population Proportion 401 9.8.2 Test Concerning the Difference Between Two Population Proportions 405 9.9 Tests Concerning the Variance of a Normal Population 410 9.10 Tests Concerning the Ratio of Variances of Two Normal Populations 414 9.11 Testing of Statistical Hypotheses using Confidence Intervals 418 9.12 Sequential Tests of Hypotheses 422 9.12.1 A One-Tail Sequential Testing Procedure 422 9.12.2 A Two-Tail Sequential Testing Procedure 427 9.13 Case Studies 430 9.14 Using JMP 431 Review Practice Problems 431 Part II Statistics in Actions 10 Elements of Reliability Theory 445 10.1 The Reliability Function 446 10.1.1 The Hazard Rate Function 446 10.1.2 Employing the Hazard Function 455 10.2 Estimation: Exponential Distribution 457 10.3 Hypothesis Testing: Exponential Distribution 465 10.4 Estimation: Weibull Distribution 467 10.5 Case Studies 472 10.6 Using JMP 474 Review Practice Problems 474 11 On Data Mining 476 11.1 Introduction 476 11.2 What is Data Mining? 477 11.2.1 Big Data 477 11.3 Data Reduction 478 11.4 Data Visualization 481 11.5 Data Preparation 490 11.5.1 Missing Data 490 11.5.2 Outlier Detection and Remedial Measures 491 11.6 Classification 492 11.6.1 Evaluating a Classification Model 493 11.7 Decision Trees 499 11.7.1 Classification and Regression Trees (CART) 500 11.7.2 Further Reading 511 11.8 Case Studies 511 11.9 Using JMP 512 Review Practice Problems 512 12 Cluster Analysis 518 12.1 Introduction 518 12.2 Similarity Measures 519 12.2.1 Common Similarity Coefficients 524 12.3 Hierarchical Clustering Methods 525 12.3.1 Single Linkage 526 12.3.2 Complete Linkage 531 12.3.3 Average Linkage 534 12.3.4 Ward’s Hierarchical Clustering 536 12.4 Nonhierarchical Clustering Methods 538 12.4.1 K-Means Method 538 12.5 Density-Based Clustering 544 12.6 Model-Based Clustering 547 12.7 A Case Study 552 12.8 Using JMP 553 Review Practice Problems 553 13 Analysis of Categorical Data 558 13.1 Introduction 558 13.2 The Chi-Square Goodness-of-Fit Test 559 13.3 Contingency Tables 568 13.3.1 The 2 × 2 Case with Known Parameters 568 13.3.2 The 2 × 2 Case with Unknown Parameters 570 13.3.3 The r × s Contingency Table 572 13.4 Chi-Square Test for Homogeneity 577 13.5 Comments on the Distribution of the Lack-of-Fit Statistics 581 13.6 Case Studies 583 13.7 Using JMP 584 Review Practice Problems 585 14 Nonparametric Tests 591 14.1 Introduction 591 14.2 The Sign Test 592 14.2.1 One-Sample Test 592 14.2.2 The Wilcoxon Signed-Rank Test 595 14.2.3 Two-Sample Test 598 14.3 Mann–Whitney (Wilcoxon) W Test for Two Samples 604 14.4 Runs Test 608 14.4.1 Runs above and below the Median 608 14.4.2 The Wald–Wolfowitz Run Test 611 14.5 Spearman Rank Correlation 614 14.6 Using JMP 618 Review Practice Problems 618 15 Simple Linear Regression Analysis 622 15.1 Introduction 623 15.2 Fitting the Simple Linear Regression Model 624 15.2.1 Simple Linear Regression Model 624 15.2.2 Fitting a Straight Line by Least Squares 627 15.2.3 Sampling Distribution of the Estimators of Regression Coefficients 631 15.3 Unbiased Estimator of σ2 637 15.4 Further Inferences Concerning Regression Coefficients (β0, β1), E(Y ), and Y 639 15.4.1 Confidence Interval for β1 with Confidence Coefficient (1 − α) 639 15.4.2 Confidence Interval for β0 with Confidence Coefficient (1 − α) 640 15.4.3 Confidence Interval for E(Y |X) with Confidence Coefficient (1 − α) 642 15.4.4 Prediction Interval for a Future Observation Y with Confidence Coefficient (1 − α) 645 15.5 Tests of Hypotheses for β0 and β1 652 15.5.1 Test of Hypotheses for β1 652 15.5.2 Test of Hypotheses for β0 652 15.6 Analysis of Variance Approach to Simple Linear Regression Analysis 659 15.7 Residual Analysis 665 15.8 Transformations 674 15.9 Inference About ρ 681 15.10A Case Study 683 15.11 Using JMP 684 Review Practice Problems 684 16 Multiple Linear Regression Analysis 693 16.1 Introduction 694 16.2 Multiple Linear Regression Models 694 16.3 Estimation of Regression Coefficients 699 16.3.1 Estimation of Regression Coefficients Using Matrix Notation 701 16.3.2 Properties of the Least-Squares Estimators 703 16.3.3 The Analysis of Variance Table 704 16.3.4 More Inferences about Regression Coefficients 706 16.4 Multiple Linear Regression Model Using Quantitative and Qualitative Predictor Variables 714 16.4.1 Single Qualitative Variable with Two Categories 714 16.4.2 Single Qualitative Variable with Three or More Categories 716 16.5 Standardized Regression Coefficients 726 16.5.1 Multicollinearity 728 16.5.2 Consequences of Multicollinearity 729 16.6 Building Regression Type Prediction Models 730 16.6.1 First Variable to Enter into the Model 730 16.7 Residual Analysis and Certain Criteria for Model Selection 734 16.7.1 Residual Analysis 734 16.7.2 Certain Criteria for Model Selection 735 16.8 Logistic Regression 740 16.9 Case Studies 745 16.10 Using JMP 748 Review Practice Problems 748 17 Analysis of Variance 757 17.1 Introduction 758 17.2 The Design Models 758 17.2.1 Estimable Parameters 758 17.2.2 Estimable Functions 760 17.3 One-Way Experimental Layouts 761 17.3.1 The Model and Its Analysis 761 17.3.2 Confidence Intervals for Treatment Means 767 17.3.3 Multiple Comparisons 773 17.3.4 Determination of Sample Size 780 17.3.5 The Kruskal–Wallis Test for One-Way Layouts (Nonparametric Method) 781 17.4 Randomized Complete Block (RCB) Designs 785 17.4.1 The Friedman Fr-Test for Randomized Complete Block Design (Nonparametric Method) 792 17.4.2 Experiments with One Missing Observation in an RCB-Design Experiment 794 17.4.3 Experiments with Several Missing Observations in an RCB-Design Experiment 795 17.5 Two-Way Experimental Layouts 798 17.5.1 Two-Way Experimental Layouts with One Observation per Cell 800 17.5.2 Two-Way Experimental Layouts with r > 1 Observations per Cell 801 17.5.3 Blocking in Two-Way Experimental Layouts 810 17.5.4 Extending Two-Way Experimental Designs to n-Way Experimental Layouts 811 17.6 Latin Square Designs 813 17.7 Random-Effects and Mixed-Effects Models 820 17.7.1 Random-Effects Model 820 17.7.2 Mixed-Effects Model 822 17.7.3 Nested (Hierarchical) Designs 824 17.8 A Case Study 831 17.9 Using JMP 832 Review Practice Problems 832 18 The 2k Factorial Designs 847 18.1 Introduction 848 18.2 The Factorial Designs 848 18.3 The 2k Factorial Designs 850 18.4 Unreplicated 2k Factorial Designs 859 18.5 Blocking in the 2k Factorial Design 867 18.5.1 Confounding in the 2k Factorial Design 867 18.5.2 Yates’s Algorithm for the 2k Factorial Designs 875 18.6 The 2k Fractional Factorial Designs 877 18.6.1 One-half Replicate of a 2k Factorial Design 877 18.6.2 One-quarter Replicate of a 2k Factorial Design 882 18.7 Case Studies 887 18.8 Using JMP 889 Review Practice Problems 889 19 Response Surfaces 897 19.1 Introduction 897 19.1.1 Basic Concepts of Response Surface Methodology 898 19.2 First-Order Designs 903 19.3 Second-Order Designs 917 19.3.1 Central Composite Designs (CCDs) 918 19.3.2 Some Other First-Order and Second-Order Designs 928 19.4 Determination of Optimum or Near-Optimum Point 936 19.4.1 The Method of Steepest Ascent 937 19.4.2 Analysis of a Fitted Second-Order Response Surface 941 19.5 Anova Table for a Second-Order Model 946 19.6 Case Studies 948 19.7 Using JMP 950 Review Practice Problems 950 20 Statistical Quality Control—Phase I Control Charts 958 21 Statistical Quality Control—Phase II Control Charts 960 Appendices 961 Appendix A Statistical Tables 962 Appendix B Answers to Selected Problems 969 Appendix C Bibliography 992 Index 1003

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Book SynopsisThis book describes the basics and developments of the new XFEM approach to fracture analysis of structures and materials, providing state of the art techniques and algorithms for fracture analysis of structures.Table of ContentsPreface xiii Nomenclature xvii 1 Introduction 1 1.1 Composite Structures 1 1.2 Failures of Composites 2 1.2.1 Matrix Cracking 2 1.2.2 Delamination 2 1.2.3 Fibre/Matrix Debonding 2 1.2.4 Fibre Breakage 3 1.2.5 Macro Models of Cracking in Composites 3 1.3 Crack Analysis 3 1.3.1 Local and Non-Local Formulations 3 1.3.2 Theoretical Methods for Failure Analysis 5 1.4 Analytical Solutions for Composites 6 1.4.1 Continuum Models 6 1.4.2 Fracture Mechanics of Composites 6 1.5 Numerical Techniques 8 1.5.1 Boundary Element Method 8 1.5.2 Finite Element Method 8 1.5.3 Adaptive Finite/Discrete Element Method 10 1.5.4 Meshless Methods 10 1.5.5 Extended Finite Element Method 11 1.5.6 Extended Isogeometric Analysis 12 1.5.7 Multiscale Analysis 13 1.6 Scope of the Book 13 2 Fracture Mechanics, A Review 17 2.1 Introduction 17 2.2 Basics of Elasticity 20 2.2.1 Stress–Strain Relations 20 2.2.2 Airy Stress Function 22 2.2.3 Complex Stress Functions 22 2.3 Basics of LEFM 23 2.3.1 Fracture Mechanics 23 2.3.2 Infinite Tensile Plate with a Circular Hole 24 2.3.3 Infinite Tensile Plate with an Elliptical Hole 26 2.3.4 Westergaard Analysis of a Line Crack 28 2.3.5 Williams Solution of a Wedge Corner 29 2.4 Stress Intensity Factor, K 30 2.4.1 Definition of the Stress Intensity Factor 30 2.4.2 Examples of Stress Intensity Factors for LEFM 33 2.4.3 Griffith Energy Theories 35 2.4.4 Mixed Mode Crack Propagation 38 2.5 Classical Solution Procedures for K and G 41 2.5.1 Displacement Extrapolation/Correlation Method 41 2.5.2 Mode I Energy Release Rate 41 2.5.3 Mode I Stiffness Derivative/Virtual Crack Model 42 2.5.4 Two Virtual Crack Extensions for Mixed Mode Cases 42 2.5.5 Single Virtual Crack Extension Based on Displacement Decomposition 43 2.6 Quarter Point Singular Elements 44 2.7 J Integral 47 2.7.1 Generalization of J 48 2.7.2 Effect of Crack Surface Traction 48 2.7.3 Effect of Body Force 49 2.7.4 Equivalent Domain Integral (EDI) Method 49 2.7.5 Interaction Integral Method 49 2.8 Elastoplastic Fracture Mechanics (EPFM) 51 2.8.1 Plastic Zone 51 2.8.2 Crack-Tip Opening Displacements (CTOD) 53 2.8.3 J Integral for EPFM 55 3 Extended Finite Element Method 57 3.1 Introduction 57 3.2 Historic Development of XFEM 58 3.2.1 A Review of XFEM Development 58 3.2.2 A Review of XFEM Composite Analysis 62 3.3 Enriched Approximations 62 3.3.1 Partition of Unity 62 3.3.2 Intrinsic and Extrinsic Enrichments 63 3.3.3 Partition of Unity Finite Element Method 66 3.3.4 MLS Enrichment 66 3.3.5 Generalized Finite Element Method 67 3.3.6 Extended Finite Element Method 67 3.3.7 Generalized PU Enrichment 67 3.4 XFEM Formulation 67 3.4.1 Basic XFEM Approximation 68 3.4.2 Signed Distance Function 69 3.4.3 Modelling the Crack 70 3.4.4 Governing Equation 71 3.4.5 XFEM Discretization 72 3.4.6 Evaluation of Derivatives of Enrichment Functions 73 3.4.7 Selection of Nodes for Discontinuity Enrichment 75 3.4.8 Numerical Integration 77 3.5 XFEM Strong Discontinuity Enrichments 79 3.5.1 A Modified FE Shape Function 79 3.5.2 The Heaviside Function 81 3.5.3 The Sign Function 84 3.5.4 Strong Tangential Discontinuity 85 3.5.5 Crack Intersection 85 3.6 XFEM Weak Discontinuity Enrichments 86 3.7 XFEM Crack-Tip Enrichments 87 3.7.1 Isotropic Enrichment 87 3.7.2 Orthotropic Enrichment Functions 88 3.7.3 Bimaterial Enrichments 88 3.7.4 Orthotropic Bimaterial Enrichments 89 3.7.5 Dynamic Enrichment 89 3.7.6 Orthotropic Dynamic Enrichments for Moving Cracks 90 3.7.7 Bending Plates 91 3.7.8 Crack-Tip Enrichments in Shells 91 3.7.9 Electro-Mechanical Enrichment 92 3.7.10 Dislocation Enrichment 93 3.7.11 Hydraulic Fracture Enrichment 94 3.7.12 Plastic Enrichment 94 3.7.13 Viscoelastic Enrichment 95 3.7.14 Contact Corner Enrichment 96 3.7.15 Modification for Large Deformation Problems 97 3.7.16 Automatic Enrichment 99 3.8 Transition from Standard to Enriched Approximation 99 3.8.1 Linear Blending 100 3.8.2 Hierarchical Transition Domain 100 3.9 Tracking Moving Boundaries 103 3.9.1 Level Set Method 103 3.9.2 Alternative Methods 106 3.10 Numerical Simulations 107 3.10.1 A Central Crack in an Infinite Tensile Plate 107 3.10.2 An Edge Crack in a Finite Plate 109 3.10.3 Tensile Plate with a Central Inclined Crack 110 3.10.4 A Bending Plate in Fracture Mode III 111 3.10.5 Crack Propagation in a Shell 112 3.10.6 Shear Band Simulation 115 3.10.7 Fault Simulation 116 3.10.8 Sliding Contact Stress Singularity by PUFEM 119 3.10.9 Hydraulic Fracture 122 3.10.10 Dislocation Dynamics 126 4 Static Fracture Analysis of Composites 131 4.1 Introduction 131 4.2 Anisotropic Elasticity 134 4.2.1 Elasticity Solution 134 4.2.2 Anisotropic Stress Functions 136 4.3 Analytical Solutions for Near Crack Tip 137 4.3.1 The General Solution 137 4.3.2 Special Solutions for Different Types of Composites 140 4.4 Orthotropic Mixed Mode Fracture 142 4.4.1 Energy Release Rate for Anisotropic Materials 142 4.4.2 Anisotropic Singular Elements 142 4.4.3 SIF Calculation by Interaction Integral 143 4.4.4 Orthotropic Crack Propagation Criteria 147 4.5 Anisotropic XFEM 149 4.5.1 Governing Equation 149 4.5.2 XFEM Discretization 150 4.5.3 Orthotropic Enrichment Functions 151 4.6 Numerical Simulations 152 4.6.1 Plate with a Crack Parallel to the Material Axis of Orthotropy 152 4.6.2 Edge Crack with Several Orientations of the Axes of Orthotropy 155 4.6.3 Inclined Edge Notched Tensile Specimen 156 4.6.4 Central Slanted Crack 160 4.6.5 An Inclined Centre Crack in a Disk Subjected to Point Loads 164 4.6.6 Crack Propagation in an Orthotropic Beam 166 5 Dynamic Fracture Analysis of Composites 169 5.1 Introduction 169 5.1.1 Dynamic Fracture Mechanics 169 5.1.2 Dynamic Fracture Mechanics of Composites 170 5.1.3 Dynamic Fracture by XFEM 172 5.2 Analytical Solutions for Near Crack Tips in Dynamic States 173 5.2.1 Analytical Solution for a Propagating Crack in Isotropic Material 174 5.2.2 Asymptotic Solution for a Stationary Crack in Orthotropic Media 175 5.2.3 Analytical Solution for Near Crack Tip of a Propagating Crack in Orthotropic Material 176 5.3 Dynamic Stress Intensity Factors 178 5.3.1 Stationary and Moving Crack Dynamic Stress Intensity Factors 178 5.3.2 Dynamic Fracture Criteria 179 5.3.3 J Integral for Dynamic Problems 180 5.3.4 Domain Integral for Orthotropic Media 181 5.3.5 Interaction Integral 182 5.3.6 Crack-Axis Component of the Dynamic J Integral 183 5.3.7 Field Decomposition Technique 185 5.4 Dynamic XFEM 185 5.4.1 Dynamic Equations of Motion 185 5.4.2 XFEM Discretization 185 5.4.3 XFEM Enrichment Functions 187 5.4.4 Time Integration Schemes 191 5.5 Numerical Simulations 195 5.5.1 Plate with a Stationary Central Crack 195 5.5.2 Mode I Plate with an Edge Crack 196 5.5.3 Mixed Mode Edge Crack in Composite Plates 199 5.5.4 A Composite Plate with Double Edge Cracks under Impulsive Loading 210 5.5.5 Pre-Cracked Three Point Bending Beam under Impact Loading 213 5.5.6 Propagating Central Inclined Crack in a Circular Orthotropic Plate 217 6 Fracture Analysis of Functionally Graded Materials (FGMs) 225 6.1 Introduction 225 6.2 Analytical Solution for Near a Crack Tip 227 6.2.1 Average Material Properties 227 6.2.2 Mode I Near Tip Fields in FGM Composites 228 6.2.3 Stress and Displacement Field (Similar to Homogeneous Orthotropic Composites) 233 6.3 Stress Intensity Factor 235 6.3.1 J Integral 235 6.3.2 Interaction Integral 236 6.3.3 FGM Auxillary Fields 236 6.3.4 Isoparametric FGM 240 6.4 Crack Propagation in FGM Composites 240 6.5 Inhomogeneous XFEM 241 6.5.1 Governing Equation 241 6.5.2 XFEM Approximation 241 6.5.3 XFEM Discretization 243 6.6 Numerical Examples 244 6.6.1 Plate with a Centre Crack Parallel to the Material Gradient 244 6.6.2 Proportional FGM Plate with an Inclined Central Crack 247 6.6.3 Non-Proportional FGM Plate with a Fixed Inclined Central Crack 250 6.6.4 Rectangular Plate with an Inclined Crack (Non-Proportional Distribution) 251 6.6.5 Crack Propagation in a Four-Point FGM Beam 253 7 Delamination/Interlaminar Crack Analysis 261 7.1 Introduction 261 7.2 Fracture Mechanics for Bimaterial Interface Cracks 264 7.2.1 Isotropic Bimaterial Interfaces 265 7.2.2 Orthotropic Bimaterial Interface Cracks 266 7.2.3 Stress Contours for a Crack between Two Dissimilar Orthotropic Materials 270 7.3 Stress Intensity Factors for Interlaminar Cracks 271 7.4 Delamination Propagation 273 7.4.1 Fracture Energy-Based Criteria 273 7.4.2 Stress-Based Criteria 273 7.4.3 Contact-Based Criteria 274 7.5 Bimaterial XFEM 275 7.5.1 Governing Equation 275 7.5.2 XFEM Discretization 276 7.5.3 XFEM Enrichment Functions for Bimaterial Problems 278 7.5.4 Discretization and Integration 280 7.6 Numerical Examples 280 7.6.1 Central Crack in an Infinite Bimaterial Plate 280 7.6.2 Isotropic-Orthotropic Bimaterial Crack 289 7.6.3 Orthotropic Double Cantilever Beam 291 7.6.4 Concrete Beams Strengthened with Fully Bonded GFRP 294 7.6.5 FRP Reinforced Concrete Cantilever Beam Subjected to Edge Loadings 295 7.6.6 Delamination of Metallic I Beams Strengthened by FRP Strips 298 7.6.7 Variable Section Beam Reinforced by FRP 300 8 New Orthotropic Frontiers 303 8.1 Introduction 303 8.2 Orthotropic XIGA 303 8.2.1 NURBS Basis Function 304 8.2.2 Extended Isogeometric Analysis 305 8.2.3 XIGA Simulations 313 8.3 Orthotropic Dislocation Dynamics 321 8.3.1 Straight Dislocations in Anisotropic Materials 321 8.3.2 Edge Dislocations in Anisotropic Materials 322 8.3.3 Curve Dislocations in Anisotropic Materials 324 8.3.4 Anisotropic Dislocation XFEM 324 8.3.5 Plane Strain Anisotropic Solution 329 8.3.6 Individual Sliding Systems s1 and s2 in an Infinite Domain 330 8.3.7 Simultaneous Sliding Systems in an Infinite Domain 330 8.4 Other Anisotropic Applications 333 8.4.1 Biomechanics 333 8.4.2 Piezoelectric 335 References 339 Index 363

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Book SynopsisLikely to be the first textbook to be published on XFEM Concise, without completeness being compromised Emphasis on practical applications Comprehensive numerical examples in each chapter.Table of ContentsDedication. Preface . Nomenclature . Chapter 1 Introduction. 1.1 ANALYSIS OF STRUCTURES. 1.2 ANALYSIS OF DISCONTINUITIES. 1.3 FRACTURE MECHANICS. 1.4 CRACK MODELLING. 1.4.1 Local and non-local models. 1.4.2 Smeared crack model. 1.4.3 Discrete inter-element crack. 1.4.4 Discrete cracked element. 1.4.5 Singular elements. 1.4.6 Enriched elements. 1.5 ALTERNATIVE TECHNIQUES. 1.6 A REVIEW OF XFEM APPLICATIONS. 1.6.1 General aspects of XFEM. 1.6.2 Localisation and fracture. 1.6.3 Composites. 1.6.4 Contact. 1.6.5 Dynamics. 1.6.6 Large deformation/shells. 1.6.7 Multiscale. 1.6.8 Multiphase/solidification. 1.7 SCOPE OF THE BOOK. Chapter 2 Fracture Mechanics, a Review. 2.1 INTRODUCTION. 2.2 BASICS OF ELASTICITY. 2.2.1 Stress–strain relations. 2.2.2 Airy stress function. 2.2.3 Complex stress functions. 2.3 BASICS OF LEFM. 2.3.1 Fracture mechanics. 2.3.2 Circular hole. 2.3.3 Elliptical hole. 2.3.4 Westergaard analysis of a sharp crack. 2.4 STRESS INTENSITY FACTOR, K . 2.4.1 Definition of the stress intensity factor. 2.4.2 Examples of stress intensity factors for LEFM. 2.4.3 Griffith theories of strength and energy. 2.4.4 Brittle material. 2.4.5 Quasi-brittle material. 2.4.6 Crack stability. 2.4.7 Fixed grip versus fixed load. 2.4.8 Mixed mode crack propagation. 2.5 SOLUTION PROCEDURES FOR K AND G . 2.5.1 Displacement extrapolation/correlation method. 2.5.2 Mode I energy release rate. 2.5.3 Mode I stiffness derivative/virtual crack model. 2.5.4 Two virtual crack extensions for mixed mode cases. 2.5.5 Single virtual crack extension based on displacement decomposition. 2.5.6 Quarter point singular elements. 2.6 ELASTOPLASTIC FRACTURE MECHANICS (EPFM). 2.6.1 Plastic zone. 2.6.2 Crack tip opening displacements (CTOD). 2.6.3 J integral. 2.6.4 Plastic crack tip fields. 2.6.5 Generalisation of J . 2.7 NUMERICAL METHODS BASED ON THE J INTEGRAL. 2.7.1 Nodal solution. 2.7.2 General finite element solution. 2.7.3 Equivalent domain integral (EDI) method. 2.7.4 Interaction integral method. Chapter 3 Extended Finite Element Method for Isotropic Problems. 3.1 INTRODUCTION. 3.2 A REVIEW OF XFEM DEVELOPMENT. 3.3 BASICS OF FEM. 3.3.1 Isoparametric finite elements, a short review. 3.3.2 Finite element solutions for fracture mechanics. 3.4 PARTITION OF UNITY. 3.5 ENRICHMENT. 3.5.1 Intrinsic enrichment. 3.5.2 Extrinsic enrichment. 3.5.3 Partition of unity finite element method. 3.5.4 Generalised finite element method. 3.5.5 Extended finite element method. 3.5.6 Hp-clouds enrichment. 3.5.7 Generalisation of the PU enrichment. 3.5.8 Transition from standard to enriched approximation. 3.6 ISOTROPIC XFEM. 3.6.1 Basic XFEM approximation. 3.6.2 Signed distance function. 3.6.3 Modelling strong discontinuous fields. 3.6.4 Modelling weak discontinuous fields. 3.6.5 Plastic enrichment. 3.6.6 Selection of nodes for discontinuity enrichment. 3.6.7 Modelling the crack. 3.7 DISCRETIZATION AND INTEGRATION. 3.7.1 Governing equation. 3.7.2 XFEM discretization. 3.7.3 Element partitioning and numerical integration. 3.7.4 Crack intersection. 3.8 TRACKING MOVING BOUNDARIES. 3.8.1 Level set method. 3.8.2 Fast marching method. 3.8.3 Ordered upwind method. 3.9 NUMERICAL SIMULATIONS. 3.9.1 A tensile plate with a central crack. 3.9.2 Double edge cracks. 3.9.3 Double internal collinear cracks. 3.9.4 A central crack in an infinite plate. 3.9.5 An edge crack in a finite plate. Chapter 4 XFEM for Orthotropic Problems. 4.1 INTRODUCTION. 4.2 ANISOTROPIC ELASTICITY. 4.2.1 Elasticity solution. 4.2.2 Anisotropic stress functions. 4.2.3 Orthotropic mixed mode problems. 4.2.4 Energy release rate and stress intensity factor for anisotropic. materials. 4.2.5 Anisotropic singular elements. 4.3 ANALYTICAL SOLUTIONS FOR NEAR CRACK TIP. 4.3.1 Near crack tip displacement field (class I). 4.3.2 Near crack tip displacement field (class II). 4.3.3 Unified near crack tip displacement field (both classes). 4.4 ANISOTROPIC XFEM. 4.4.1 Governing equation. 4.4.2 XFEM discretization. 4.4.3 SIF calculations. 4.5 NUMERICAL SIMULATIONS. 4.5.1 Plate with a crack parallel to material axis of orthotropy. 4.5.2 Edge crack with several orientations of the axes of orthotropy. 4.5.3 Single edge notched tensile specimen with crack inclination. 4.5.4 Central slanted crack. 4.5.5 An inclined centre crack in a disk subjected to point loads. 4.5.6 A crack between orthotropic and isotropic materials subjected to. tensile tractions. Chapter 5 XFEM for Cohesive Cracks. 5.1 INTRODUCTION. 5.2 COHESIVE CRACKS. 5.2.1 Cohesive crack models. 5.2.2 Numerical models for cohesive cracks. 5.2.3 Crack propagation criteria. 5.2.4 Snap-back behaviour. 5.2.5 Griffith criterion for cohesive crack. 5.2.6 Cohesive crack model. 5.3 XFEM FOR COHESIVE CRACKS. 5.3.1 Enrichment functions. 5.3.2 Governing equations. 5.3.3 XFEM discretization. 5.4 NUMERICAL SIMULATIONS. 5.4.1 Mixed mode bending beam. 5.4.2 Four point bending beam. 5.4.3 Double cantilever beam. Chapter 6 New Frontiers. 6.1 INTRODUCTION. 6.2 INTERFACE CRACKS. 6.2.1 Elasticity solution for isotropic bimaterial interface. 6.2.2 Stability of interface cracks. 6.2.3 XFEM approximation for interface cracks. 6.3 CONTACT. 6.3.1 Numerical models for a contact problem. 6.3.2 XFEM modelling of a contact problem. 6.4 DYNAMIC FRACTURE. 6.4.1 Dynamic crack propagation by XFEM. 6.4.2 Dynamic LEFM. 6.4.3 Dynamic orthotropic LEFM. 6.4.4 Basic formulation of dynamic XFEM. 6.4.5 XFEM discretization. 6.4.6 Time integration. 6.4.7 Time finite element method. 6.4.8 Time extended finite element method. 6.5 MULTISCALE XFEM. 6.5.1 Basic formulation. 6.5.2 The zoom technique. 6.5.3 Homogenisation based techniques. 6.5.4 XFEM discretization. 6.6 MULTIPHASE XFEM. 6.6.1 Basic formulation. 6.6.2 XFEM approximation. 6.6.3 Two-phase fluid flow. 6.6.4 XFEM approximation. Chapter 7 XFEM Flow. 7.1 INTRODUCTION. 7.2 AVAILABLE OPEN-SOURCE XFEM. 7.3. FINITE ELEMENT ANALYSIS. 7.3.1 Defining the model. 7.3.2 Creating the finite element mesh. 7.3.3 Linear elastic analysis. 7.3.4 Large deformation. 7.3.5 Nonlinear (elastoplastic) analysis. 7.3.6 Material constitutive matrix. 7.4 XFEM. 7.4.1 Front tracking. 7.4.2 Enrichment detection. 7.4.3 Enrichment functions. 7.4.4 Ramp (transition) functions. 7.4.5 Evaluation of the B matrix. 7.5 NUMERICAL INTEGRATION. 7.5.1 Sub-quads. 7.5.2 Sub-triangles. 7.6 SOLVER. 7.6.1 XFEM degrees of freedom. 7.6.2 Time integration. 7.6.3 Simultaneous equations solver. 7.6.4 Crack length control. 7.7 POST-PROCESSING. 7.7.1 Stress intensity factor. 7.7.2 Crack growth. 7.7.3 Other applications. 7.8 CONFIGURATION UPDATE. References . Index

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Book SynopsisCivil and environmental engineers need an understanding of mathematical statistics and probability theory to deal with the variability that affects engineers' structures, soil pressures, river flows and the like. Students, too, need to get to grips with these rather difficult concepts. This book, written by engineers for engineers, tackles the subject in a clear, up-to-date manner using a process-orientated approach. It introduces the subjects of mathematical statistics and probability theory, and then addresses model estimation and testing, regression and multivariate methods, analysis of extreme events, simulation techniques, risk and reliability, and economic decision making. 325 examples and case studies from European and American practice are included and each chapter features realistic problems to be solved. For the second edition new sections have been added on Monte Carlo Markov chaiTable of ContentsPreface. Introduction. Preliminary data analysis. Basic probability concepts. Random variables and their properties. Probability distributions. Model estimation and testing. Methods of regression and multivariate analysis. Frequency analysis of extreme events. Simulation techniques for design. Risk and reliability analysis. Bayesian decision methods and parameter uncertainty. Appendixes Further mathematics Glossary of symbols Tables of selected distributions Brief answers to selected problems . Data lists. Index

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Book SynopsisThis is the first textbook to address quantified risk assessment (QRA) as specifically applied to offshore installations and operations. These minimalistic installations with no helideck and very limited safety systems will require a new approach to risk assessment and emergency planning, especially during manned periods involving W2W vessels.Trade Review“The book, which offers complete and up-to-date information about some environmental aspects and impacts, is useful for academics and students, as well as for professionals in the sector and regulatory authorities.” (Emilia Di Lorenzo, zbMATH 1427.91004, 2020)Table of ContentsPart III.- 14.Methodology for Quantified Risk Assessment.- 15.Analysis Techniques.- 16.Presentation of Risk Results from QRA Studies.- 17.Evaluation of Personnel Risk Levels.- 18.Environmental Risk Analysis.- 19.Approach to Risk Based Design.- 20.Risk based Emergency Response Planning.- Part IV.- 21.Use of Risk Analysis during the Operations Phase.- 22.Use of Risk Indicators for Major Hazard Risk.- 23.Barrier Management for Major Hazard Risk.- Appendix A.Overview of Software.- Appendix B.Overview of Fatalities in Norwegian Sector.- Appendix C.Network Resources.

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Book SynopsisOptimization is of critical importance in engineering. Engineers constantly strive for the best possible solutions, the most economical use of limited resources, and the greatest efficiency. As system complexity increases, these goals mandate the use of state-of-the-art optimization techniques.In recent years the theory and methodology of optimization have seen revolutionary improvements. Moreover, the exponential growth in computational power, along with the availability of multicore computing with virtually unlimited memory and storage capacity, has fundamentally changed what engineers can do to optimize their designs. This is a two-way process: engineers benefit from developments in optimization methodology, and challenging new classes of optimization problems arise from novel engineering applications.Advances and Trends in Optimization with Engineering Applications reviews 10 major areas of optimization and related engineering applications in a distinct part, providing a broad summary of state-of-the-art optimization techniques most important to engineering practice. Each part provides a clear overview of a specific area, followed by chapters detailing applications to a wide range of real-world problems.The book provides a solid foundation for engineers and mathematical optimizers alike who want to understand not only the importance of optimization methods to engineering but also the capabilities of current methods.

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Book SynopsisHighly oscillatory phenomena range across numerous areas in science and engineering and their computation represents a difficult challenge. A case in point is integrals of rapidly oscillating functions in one or more variables. The quadrature of such integrals has been historically considered very demanding. Research in the past 15 years (in which the authors played a major role) resulted in a range of very effective and affordable algorithms for highly oscillatory quadrature. This is the only monograph bringing together the new body of ideas in this area in its entirety.The starting point is that approximations need to be analyzed using asymptotic methods rather than by more standard polynomial expansions. As often happens in computational mathematics, once a phenomenon is understood from a mathematical standpoint, effective algorithms follow. As reviewed in this monograph, we now have at our disposal a number of very effective quadrature methods for highly oscillatory integrals—Filon-type and Levin-type methods, methods based on steepest descent, and complex-valued Gaussian quadrature. Their understanding calls for a fairly varied mathematical toolbox—from classical numerical analysis, approximation theory, and theory of orthogonal polynomials all the way to asymptotic analysis—yet this understanding is the cornerstone of efficient algorithms.

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Book SynopsisThe Portable, Extensible Toolkit for Scientific Computation (PETSc) is an open-source library of advanced data structures and methods for solving linear and nonlinear equations and for managing discretizations. This book uses these modern numerical tools to demonstrate how to solve nonlinear partial differential equations (PDEs) in parallel. It starts from key mathematical concepts, such as Krylov space methods, preconditioning, multigrid, and Newton’s method. In PETSc these components are composed at run time into fast solvers.Discretizations are introduced from the beginning, with an emphasis on finite difference and finite element methodologies. The example C programs of the first 12 chapters, listed on the inside front cover, solve (mostly) elliptic and parabolic PDE problems. Discretization leads to large, sparse, and generally nonlinear systems of algebraic equations. For such problems, mathematical solver concepts are explained and illustrated through the examples, with sufficient context to speed further development. PETSc for Partial Differential Equations addresses both discretizations and fast solvers for PDEs, emphasizing practice more than theory. Well-structured examples lead to run-time choices that result in high solver performance and parallel scalability. The last two chapters build on the reader’s understanding of fast solver concepts when applying the Firedrake Python finite element solver library. This textbook, the first to cover PETSc programming for nonlinear PDEs, provides an on-ramp for graduate students and researchers to a major area of high-performance computing for science and engineering. It is suitable as a supplement for courses in scientific computing or numerical methods for differential equations.

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Book SynopsisThis book presents the first survey of the Localized Orthogonal Decomposition (LOD) method, a pioneering approach for the numerical homogenization of partial differential equations with multiscale data beyond periodicity and scale separation. The authors provide a careful error analysis, including previously unpublished results, and a complete implementation of the method in MATLAB. They also reveal how the LOD method relates to classical homogenization and domain decomposition. Illustrated with numerical experiments that demonstrate the significance of the method, the book is enhanced by a survey of applications including eigenvalue problems and evolution problems.Numerical Homogenization by Localized Orthogonal Decomposition is appropriate for graduate students in applied mathematics, numerical analysis, and scientific computing. Researchers in the field of computational partial differential equations will find this self-contained book of interest, as will applied scientists and engineers interested in multiscale simulation.

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Book SynopsisFit for students just starting to build a background in mathematics, this textbook provides an introduction to numerical methods for linear algebra problems.Introduction to Numerical Linear Algebra is ideal for a flipped classroom, as it provides detailed explanations that allow students to read on their own and instructors to go beyond lecturing, assumes that the reader has taken a course on linear algebra, but reviews background as needed, and covers several topics not commonly addressed in related introductory books, including diffusion, a toy model of computed tomography, global positioning systems, the use of eigenvalues in analyzing stability of equilibria, a detailed derivation and careful motivation of the QR method for eigenvalues starting from power iteration, a discussion of the use of the SVD for assigning grades, and multigrid methods. This textbook is appropriate for undergraduate and beginning graduate students in mathematics and related fields. It can be used in the following courses: Advanced Numerical Analysis, Special Topics on Numerical Analysis, Topics on Data Science, Topics on Numerical Optimization, and Topics on Approximation Theory

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Book SynopsisCollecting a set of classical and emerging methods that otherwise would not be available in a single treatment, Foundations of Computational Imaging: A Model-Based Approach is the first book to define a common foundation for the mathematical and statistical methods used in computational imaging. The book is designed to bring together an eclectic group of researchers with a wide variety of applications and disciplines including applied math, physics, chemistry, optics, and signal processing, to address a collection of problems that can benefit from a common set of methods. Inside, readers will find: Basic techniques of model-based image processing. A comprehensive treatment of Bayesian and regularized image reconstruction methods. An integrated treatment of advanced reconstruction techniques such as majorization, constrained optimization, ADMM, and Plug-and-Play methods for model integration. Foundations of Computational Imaging can be used in courses on Model-Based or Computational Imaging, Advanced Numerical Analysis, Special Topics on Numerical Analysis, Topics on Data Science, Topics on Numerical Optimization, and Topics on Approximation Theory. It is also for researchers or practitioners in medical imaging, scientific imaging, commercial imaging, or industrial imaging.

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Book SynopsisReduced order modeling is an important, growing field in computational science and engineering, and this is the first book to address the subject in relation to computational fluid dynamics. It focuses on complex parametrization of shapes for their optimization and includes recent developments in advanced topics such as turbulence, stability of flows, inverse problems, optimization, and flow control, as well as applications.This book will be of interest to researchers and graduate students in the field of reduced order modeling.

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