Maths for engineers Books

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 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 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 newTable 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 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 SynopsisGlyn James was most recently Emeritus Professor in Mathematics at Coventry University and was previously Dean of the School of Mathematical and Information Sciences at Coventry University. He had an active interest in mathematics education and was a past chairman of the Education Committee of the Institute of Mathematics and its Applications, and a member of the Royal Society Mathematics Education Subcommittee. In 1995 he was chairman of the Working Group that produced the report Mathematics Matters in Engineering on behalf of UK professional bodies in engineering and mathematics. He also co-edited five mathematics books and was a member of the advisory boards for three international journals. Sadly, Glyn James passed away in October 2019 during the production of this edition; his enthusiastic input was sorely missed, but this and its companion text Advanced Modern Engineering Mathematics remain a fitting legacy.     Phil Dyke

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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 SynopsisLinear algebra has become the subject to know for people in quantitative disciplines of all kinds. No longer the exclusive domain of mathematicians and engineers, it is now used everywhere there is data and everybody who works with data needs to know more. This new book from Professor Gilbert Strang, author of the acclaimed Introduction to Linear Algebra, now in its fifth edition, makes linear algebra accessible to everybody, not just those with a strong background in mathematics. It takes a more active start, beginning by finding independent columns of small matrices, leading to the key concepts of linear combinations and rank and column space. From there it passes on to the classical topics of solving linear equations, orthogonality, linear transformations and subspaces, all clearly explained with many examples and exercises. The last major topics are eigenvalues and the important singular value decomposition, illustrated with applications to differential equations and image compression. A final optional chapter explores the ideas behind deep learning.

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Book SynopsisThe aim of this book is the study of signals and deterministic systems, linear, time-invariant, finite dimensions and causal. A set of useful tools is selected for the automatic and signal processing and methods of representation of dynamic linear systems are exposed, and analysis of their behavior. Finally we discuss the estimation, identification and synthesis of control laws for the purpose of stabilization and regulation.Table of ContentsPreface ix Chapter 1 Control, Servo-mechanisms and System Regulation 1 1.1. Introduction 1 1.1.1. Generalities and definitions 1 1.1.2. Control law synthesis 5 1.1.3. Comprehension and application exercises 7 1.2. Process control 11 1.2.1. Correction in the frequency domain 11 1.2.2. Phase advance controller and PD controller 12 1.2.3. Phase delay controller and integrator compensator 14 1.2.4. Proportional, integral and derivative (PID) control 17 1.3. Some application exercises 23 1.3.1. Identification of the transfer function and control 23 1.3.2. PI control 30 1.3.3. Phase advance control 33 1.4. Some application exercises 36 1.5. Application 1: stabilization of a rigid robot with pneumatic actuator 39 1.5.1. Conventional approach 41 1.6. Application 2: temperature control of an oven 51 1.6.1. Modeling and identification study 51 Chapter 2 System Process Control 55 2.1. Introduction 55 2.2. Modeling 55 2.2.1. Introduction 55 2.3. Governability, controllability and observability 56 2.3.1. Characteristic polynomial, minimal polynomial and Cayley–Hamilton theorem 56 2.3.2. Governability or controllability 56 2.3.3. Observability 63 2.3.4. Observer 68 2.3.5. Observer for state reconstruction 69 2.3.6. Minimal state–space representation 76 2.4. State feedback, control by poles placement and stability 79 2.4.1. State feedback control 79 2.4.2. Poles placement and stabilizability 80 2.4.3. Finite-time response for a discrete system, deadbeat response 83 2.4.4. Use of observers in control: separation principle 85 2.5. Linear quadratic (LQ) control 86 2.5.1. Linear quadratic regulator 89 2.6. Optimal control (LQ) 90 2.7. Comprehension and application exercises 94 Chapter 3 Actuators: Modeling and Analysis 117 3.1. Introduction: electric, hydraulic and pneumatic actuators 117 3.1.1. Representation methods for physical systems 118 3.1.2. Modeling of a few constituents of physical systems 120 3.2. Transmission chains, actuators and sensors 126 3.2.1. Electric actuators in robotics 126 3.2.2. Motor speed torque characteristic 131 3.2.3. Dynamic behavior or transient behavior 131 3.2.4. Electric systems motor load 134 3.3. Pneumatic actuators 137 3.3.1. Pneumatic system modeling 137 3.3.2. Frictions model 145 3.4. Hydraulic actuators 149 3.4.1. System description 149 3.4.2. Mechanical model 151 3.4.3. Hydraulic actuator model 152 3.5. Application exercises 155 Chapter 4 Digital Control and Polynomial Approach 161 4.1. Introduction to digital control 161 4.1.1. Digital controller synthesis by transposition 162 4.1.2. Euler’s transposition 164 4.1.3. Choice of the sampling period (Shannon’s theorem) 170 4.2. PID controller synthesis and its equivalent digital RST 171 4.2.1. Standard controllers 171 4.2.2. Study of digital PIDs 172 4.2.3. Digital RST controller synthesis 178 4.2.4. Choice of poles and zeros to compensate 179 4.2.5 Computation of polynomials R, S and T 180 4.2.6. Additional objectives for synthesis 181 4.3. Digital control by poles placement 182 4.3.1. Choice of the sampling period 183 4.4. Diophantine, Bézout, greatest common divisor, least common multiple and division 183 4.4.1. Polynomial arithmetic 183 4.4.2. Diophantine equation ax + by = c and Bachet–Bézout theorem 184 4.4.3. Bézout’s identity 185 4.4.4. Greatest common divisor 185 4.4.5. Least common multiple 185 4.5. A few comprehension and application exercises 186 Chapter 5 NAO Robot 193 5.1. Introduction 193 5.2. Home care project 194 5.2.1. Choregraphe software 194 5.2.2. Nao Matlab SDK research 199 5.2.3. Nao and home care 206 5.2.4. The actions to be made 207 5.3. Details of the various programs 208 5.3.1. Ask for news 208 5.3.2. CallFirefighters box 212 5.3.3. CallNeighbor box 213 5.3.4. CallFamily box 215 5.3.5. Collision detection 215 5.3.6. Special actions: waking-up 216 5.3.7. Morning hygiene 220 5.3.8. Gymnastics 221 5.3.9. Nurse call 225 5.3.10. Memory game 227 5.3.11. Drugs reminder 232 5.3.12. Reading 233 5.3.13. Listening to music 235 5.3.14. Multiplication game 239 5.3.15. Nao’s dance 243 5.3.16. Memory game 245 5.3.17. Detect person on the ground 247 5.3.18. At any time 251 5.4. Conclusion 253 5.4.1. Nao’s limitations and possible improvements 253 Chapter 6 Application Problems with Solutions 255 6.1. Exercise 6.1: car suspension 255 6.1.1. Modeling 256 6.1.2. Analysis 257 6.2. Exercise 6.2: electromechanical system 259 6.2.1. Modeling 260 6.2.2. Analysis 262 6.3. Exercises: identification and state–space representation 263 6.3.1. Exercise 6.3 263 6.3.2. Exercise 6.4 265 6.3.3. Exercise 6.5 268 6.3.4. Exercise 6.6 270 6.3.5. Exercise 6.7 276 6.4. Exercises: observation and control of nonlinear systems 278 6.4.1. Exercise 6.8 278 6.4.2. Exercise 6.9 280 6.4.3. Exercise 6.10 288 6.4.4. Exercise 6.11 291 6.4.5. Exercise 6.12 293 6.4.6. Exercise 6.13 296 6.4.7. Exercise 6.14 300 6.4.8. Exercise 6.15 300 Bibliography 307 Index 313

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Book SynopsisMost physical problems can be written in the form of mathematical equations (differential, integral, etc.). Mathematicians have always sought to find analytical solutions to the equations encountered in the different sciences of the engineer (mechanics, physics, biology, etc.). These equations are sometimes complicated and much effort is required to simplify them. In the middle of the 20th century, the arrival of the first computers gave birth to new methods of resolution that will be described by numerical methods. They allow solving numerically as precisely as possible the equations encountered (resulting from the modeling of course) and to approach the solution of the problems posed. The approximate solution is usually computed on a computer by means of a suitable algorithm. The objective of this book is to introduce and study the basic numerical methods and those advanced to be able to do scientific computation. The latter refers to the implementation of approaches adapted to the treatment of a scientific problem arising from physics (meteorology, pollution, etc.) or engineering (structural mechanics, fluid mechanics, signal processing, etc.) .Table of ContentsPreface xi Part 1 Introduction 1 Chapter 1 Review of Linear Algebra 3 1.1. Vector spaces 3 1.1.1. General definitions 3 1.1.2. Free families, generating families and bases 4 1.2. Linear mappings 5 1.3. Matrices 7 1.3.1. Operations on matrices 7 1.3.2. Change-of-basis matrices 8 1.3.3. Matrix notations 9 1.4. Determinants 10 1.5. Scalar product 12 1.6. Vector norm 12 1.7. Matrix eigenvectors and eigenvalues 13 1.7.1. Definitions and properties 13 1.7.2. Matrix diagonalization 15 1.7.3. Triangularization of matrices 15 1.8 Using Matlab 16 Chapter 2 Numerical Precision 21 2.1. Introduction 21 2.2. Machine representations of numbers 22 2.3. Integers 23 2.3.1. External representation 23 2.3.2. Internal representation of positive integers 24 2.4. Real numbers 25 2.4.1. External representation 25 2.4.2. Internal encoding of real numbers 25 2.5. Representation errors 26 2.5.1. Properties of computer-based arithmetic 27 2.5.2. Operation of subtraction 28 2.5.3. Stability 29 2.6. Determining the best algorithm 29 2.7 Using Matlab 30 2.7.1. Definition of variables 30 2.7.2. Manipulating numbers 30 Part 2 Approximating Functions 35 Chapter 3 Polynomial Interpolation 37 3.1. Introduction 37 3.2. Interpolation problems 37 3.2.1. Linear interpolation 38 3.3. Polynomial interpolation techniques 38 3.4. Interpolation with the Lagrange basis 39 3.4.1. Polynomial interpolation error 43 3.4.2. Neville–Aitken method 46 3.5. Interpolation with the Newton basis 46 3.6. Interpolation using spline functions 48 3.6.1. Hermite interpolation 50 3.6.2. Spline interpolation error 55 3.7 Using Matlab 58 3.7.1. Operations on polynomials 58 3.7.2. Manipulating polynomials 59 3.7.3. Evaluation of polynomials 60 3.7.4. Linear and nonlinear interpolation 60 3.7.5. Lagrange function 63 3.7.6. Newton function 64 Chapter 4 Numerical Differentiation 67 4.1. First-order numerical derivatives and the truncation error 67 4.2. Higher-order numerical derivatives 70 4.3. Numerical derivatives and interpolation 71 4.4. Studying the differentiation error 73 4.5. Richardson extrapolation 77 4.6. Application to the heat equation 78 4.7 Using Matlab 81 Chapter 5 Numerical Integration 83 5.1. Introduction 83 5.2. Rectangle method 84 5.3. Trapezoidal rule 84 5.4. Simpson’s rule 87 5.5. Hermite’s rule 90 5.6. Newton–Côtes rules 91 5.7. Gauss–Legendre method 92 5.7.1. Problem statement 92 5.7.2. Legendre polynomials 94 5.7.3 Choosing the αi and xi (i = 0, . . . , n) 99 5.8 Using Matlab 100 5.8.1. Matlab functions for numerical integration 100 5.8.2. Trapezoidal rule 101 5.8.3. Simpson’s rule 103 Part 3 Solving Linear Systems 107 Chapter 6 Matrix Norm and Conditioning 109 6.1. Introduction 109 6.2. Matrix norm 109 6.3. Condition number of a matrix 113 6.3.1 Approximation of K(A) 116 6.4. Preconditioning 116 6.5 Using Matlab 117 6.5.1. Matrices and vectors 117 6.5.2. Condition number of a matrix 119 Chapter 7 Direct Methods 123 7.1. Introduction 123 7.2. Method of determinants or Cramer’s method 123 7.2.1. Matrix inversion by Cramer’s method 124 7.3. Systems with upper triangular matrices 124 7.4. Gaussian method 125 7.4.1. Solving multiple systems in parallel 129 7.5. Gauss–Jordan method 129 7.5.1. Underlying principle 129 7.5.2. Computing the inverse of a matrix with the Gauss–Jordan algorithm 131 7.6. LU decomposition 132 7.7. Thomas algorithm 133 7.8. Cholesky decomposition 134 7.9 Using Matlab 136 7.9.1. Matrix operations 136 7.9.2. Systems of linear equations 138 Chapter 8 Iterative Methods 147 8.1. Introduction 147 8.2. Classical iterative techniques 148 8.2.1. Jacobi method 149 8.2.2. Gauss–Seidel method 151 8.2.3. Relaxation method 152 8.2.4. Block forms of the Jacobi, Gauss–Seidel and relaxation methods 154 8.3. Convergence of iterative methods 155 8.4. Conjugate gradient method 157 8.5 Using Matlab 159 8.5.1. Jacobi method 159 8.5.2. Relaxation method 160 Chapter 9 Numerical Methods for Computing Eigenvalues and Eigenvectors 163 9.1. Introduction 163 9.2. Computing det (A − λI) directly 164 9.3. Krylov methods 166 9.4. LeVerrier method 167 9.5. Jacobi method 168 9.6. Power iteration method 171 9.6.1. Deflation algorithm 172 9.7. Inverse power method 173 9.8. Givens–Householder method 174 9.8.1. Givens algorithm 175 9.9 Using Matlab 176 9.9.1. Application to a buckling beam 177 Chapter 10 Least-squares Approximation 185 10.1. Introduction 185 10.2. Analytic formulation 185 10.3. Algebraic formulation 191 10.3.1. Standard results on orthogonality 191 10.3.2. Least-squares problem 191 10.3.3. Solving by orthogonalization 192 10.4. Numerically solving linear equations by QR factorization 193 10.4.1. Householder transformations 193 10.4.2. QR factorization 193 10.4.3. Application to the least-squares problem 193 10.5. Applications 194 10.5.1. Curve fitting 194 10.5.2. Approximations of derivatives 195 10.6 Using Matlab 195 Part 4 Appendices 199 Appendix 1 Introduction to Matlab 201 Appendix 2 Introduction to Optimization 209 Bibliography 215 Index 217

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Book SynopsisThe purpose of this book is to introduce and study numerical methods basic and advanced ones for scientific computing. This last refers to the implementation of appropriate approaches to the treatment of a scientific problem arising from physics (meteorology, pollution, etc.) or of engineering (mechanics of structures, mechanics of fluids, treatment signal, etc.). Each chapter of this book recalls the essence of the different methods resolution and presents several applications in the field of engineering as well as programs developed under Matlab software.Table of ContentsPreface ix Part 1. Solving Equations 1 Chapter 1. Solving Nonlinear Equations 3 1.1 Introduction 3 1.2 Separating the roots 3 1.3 Approximating a separated root 4 1.3.1 Bisection method (or dichotomy method) 4 1.3.2 Fixed-point method 6 1.3.3 First convergence criterion 7 1.3.4 Iterative stopping criteria.8 1.3.5 Second convergence criterion (local criterion) 9 1.3.6 Newton’s method (or the method of tangents) 10 1.3.7 Secant method 12 1.3.8 Regula falsi method (or false position method) 17 1.4 Order of an iterative process.19 1.5 Using Matlab 19 1.5.1 Finding the roots of polynomials 19 1.5.2 Bisection method 21 1.5.3 Newton’s method 22 Chapter 2. Numerically Solving Differential Equations 25 2.1 Introduction 25 2.2 Cauchy problem and discretization 27 2.3 Euler’s method 30 2.3.1 Interpretation 30 2.3.2 Convergence 30 2.4 One-step Runge–Kutta method 31 2.4.1 Second-order Runge–Kutta method 32 2.4.2 Fourth-order Runge–Kutta method 33 2.5 Multi-step Adams methods 36 2.5.1 Open Adams methods 36 2.5.2 Closed Adams formulas 39 2.6 Predictor–Corrector method.41 2.7 Using Matlab 43 Part 2. Solving PDEs 47 Chapter 3. Finite Difference Methods 49 3.1 Introduction 49 3.2 Presentation of the finite difference method 51 3.2.1 Convergence, consistency and stability 53 3.2.2 Courant–Friedrichs–Lewy condition 56 3.2.3 Von Neumann stability analysis 57 3.3 Hyperbolic equations 58 3.3.1 Key results 59 3.3.2 Numerical schemes for solving the transport equation 63 3.3.3 Wave equation 66 3.3.4 Burgers equation 68 3.4 Elliptic equations 72 3.4.1 Poisson equation 72 3.5 Parabolic equations 74 3.5.1 Heat equation 74 3.6 Using Matlab 76 Chapter 4. Finite Element Method 83 4.1 Introduction 83 4.2 One-dimensional finite element methods 83 4.3 Two-dimensional finite element methods 88 4.4 General procedure of the method 93 4.5 Finite element method for computing elastic structures 93 4.5.1 Linear elasticity 93 4.5.2 Variational formulation of the linear elasticity problem 97 4.5.3 Planar linear elasticity problems 99 4.5.4 Applying the finite element method to planar problems 101 4.5.5 Axisymmetric problems.105 4.5.6 Three-dimensional problems 107 4.6 Using Matlab 107 4.6.1 Solving Poisson’s equation 108 4.6.2 Solving the heat equation.111 4.6.3 Computing structures 112 Chapter 5. Finite Volume Methods 117 5.1 Introduction 117 5.2 Finite volume method (FVM) 118 5.2.1 Conservation properties of the method 118 5.2.2 The stages of the method.119 5.2.3 Convergence 120 5.2.4 Consistency 120 5.2.5 Stability 120 5.3 Advection schemes 121 5.3.1 Two-dimensional FVM. 126 5.3.2 Convection-diffusion equation 129 5.3.3 Central differencing scheme 131 5.3.4 Upwind (decentered) scheme 133 5.3.5 Hybrid scheme 136 5.3.6 Power-law scheme 136 5.3.7 QUICK scheme 137 5.3.8 Higher-order schemes 139 5.3.9 Unsteady one-dimensional convection-diffusion Equation 140 5.3.10 Explicit scheme 142 5.3.11 Crank–Nicolson scheme.142 5.3.12 Implicit scheme 143 5.4 Using Matlab 144 Chapter 6. Meshless Methods. 147 6.1 Introduction 147 6.2 Limitations of the FEM and motivation of meshless methods 148 6.3 Examples of meshless methods148 6.3.1 Advantages of meshless methods 149 6.3.2 Disadvantages of meshless methods150 6.3.3 Comparison of the finite element method and meshless methods 151 6.4 Basis of meshless methods 151 6.4.1 Approximations 151 6.4.2 Kernel (weight) functions.152 6.4.3 Completeness 152 6.4.4 Partition of unity 152 6.5 Meshless method (EFG) 153 6.5.1 Theory 153 6.5.2 Moving Least-Squares Approximation 153 6.6 Application of the meshless method to elasticity 163 6.6.1 Formulation of static linear elasticity 163 6.6.2 Imposing essential boundary conditions 165 6.7 Numerical examples 170 6.7.1 Fixed-free beam 170 6.7.2 Compressed block 171 6.8 Using Matlab 173 Part 3. Appendices 179 Appendix 1181 Appendix 2189 Bibliography 195 Index 199

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Book SynopsisAdvanced Graph Theory focuses on some of the main notions arising in graph theory with an emphasis from the very start of the book on the possible applications of the theory and the fruitful links existing with linear algebra. The second part of the book covers basic material related to linear recurrence relations with application to counting and the asymptotic estimate of the rate of growth of a sequence satisfying a recurrence relation.Table of ContentsForeword ix Introduction xi Chapter 1. A First Encounter with Graphs 1 1.1. A few definitions 1 1.1.1. Directed graphs 1 1.1.2. Unoriented graphs 9 1.2. Paths and connected components 14 1.2.1. Connected components 16 1.2.2. Stronger notions of connectivity 18 1.3. Eulerian graphs 23 1.4. Defining Hamiltonian graphs 25 1.5. Distance and shortest path 27 1.6. A few applications 30 1.7. Comments 35 1.8. Exercises 37 Chapter 2. A Glimpse at Complexity Theory 43 2.1. Some complexity classes 43 2.2. Polynomial reductions 46 2.3. More hard problems in graph theory 49 Chapter 3. Hamiltonian Graphs 53 3.1. A necessary condition 53 3.2. A theorem of Dirac 55 3.3. A theorem of Ore and the closure of a graph 56 3.4. Chvátal’s condition on degrees 59 3.5. Partition of Kn into Hamiltonian circuits 62 3.6. De Bruijn graphs and magic tricks 65 3.7. Exercises 68 Chapter 4. Topological Sort and Graph Traversals 69 4.1. Trees 69 4.2. Acyclic graphs 79 4.3. Exercises 82 Chapter 5. Building New Graphs from Old Ones 85 5.1. Some natural transformations 85 5.2. Products 90 5.3. Quotients 92 5.4. Counting spanning trees 93 5.5. Unraveling 94 5.6. Exercises 96 Chapter 6. Planar Graphs 99 6.1. Formal definitions 99 6.2. Euler’s formula 104 6.3. Steinitz’ theorem 109 6.4. About the four-color theorem 113 6.5. The five-color theorem 115 6.6. From Kuratowski’s theorem to minors 120 6.7. Exercises 123 Chapter 7. Colorings 127 7.1. Homomorphisms of graphs 127 7.2. A digression: isomorphisms and labeled vertices 131 7.3. Link with colorings 134 7.4. Chromatic number and chromatic polynomial 136 7.5. Ramsey numbers 140 7.6. Exercises 147 Chapter 8. Algebraic Graph Theory 151 8.1. Prerequisites 151 8.2. Adjacency matrix 154 8.3. Playing with linear recurrences 160 8.4. Interpretation of the coefficients 168 8.5. A theorem of Hoffman 169 8.6. Counting directed spanning trees 172 8.7. Comments 177 8.8. Exercises 178 Chapter 9. Perron–Frobenius Theory 183 9.1. Primitive graphs and Perron’s theorem 183 9.2. Irreducible graphs 188 9.3. Applications 190 9.4. Asymptotic properties 195 9.4.1. Canonical form 196 9.4.2. Graphs with primitive components 197 9.4.3. Structure of connected graphs 206 9.4.4. Period and the Perron–Frobenius theorem 214 9.4.5. Concluding examples 218 9.5. The case of polynomial growth 224 9.6. Exercises 231 Chapter 10. Google’s Page Rank 233 10.1. Defining the Google matrix 238 10.2. Harvesting the primitivity of the Google matrix 241 10.3. Computation 246 10.4. Probabilistic interpretation 246 10.5. Dependence on the parameter α 247 10.6. Comments 248 Bibliography 249 Index 263

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Book SynopsisThis volume contains contributions from speakers at the 4th International Symposium on Finite Volumes for Complex Applications, held in Marrakech, Morocco, in July 2005. The subject of these papers ranges from theoretical and numerical results to physical applications. Topics covered include: Theoretical and numerical results • theoretical foundation • convergence • new finite volume schemes • adaptivity • higher order discretization and parallelization Physical applications • multiphase flow and flows through porous media • turbulent flows • shallow water problems • stiff source terms • cryogenic applications • medical and biological applications • image processing Papers on Industrial codes, as well as interdisciplinary approaches are also included in these proceedings.Table of ContentsTheoretical and numerical results; theoretical foundation; convergence; new finite volume schemes; adaptivity; higher order discretization and parallelization; Physical applications; multiphase flow and flows through porous media; turbulent flows; shallow water problems; stiff source terms; cryogenic applications; medical and biological applications; image processing; Papers on Industrial codes, as well as interdisciplinary approaches are also included in these proceedings.

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Book SynopsisDC programming and DCA: 40 Years of breakthroughs driving Intelligence and Technology.- A DC approach for the chance-constrained portfolio optimization problem.- Consolidation of Biomimicry and Swarm-based Metaheuristic Algorithms with Supervised Learning Neural Network for Function Approximation.

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Book SynopsisMathematics course with 60 Java-based interactive mathematic simulations by the author Comprehensive and systematically organized collection of 2,000 Java-based physics simulations All simulations are runnable, and can be accessed both on- and offline Visualization of mathematic relationships Facilitates an experiment-based understanding of problems, including suggestions for your own mathematical experiments Calculation procedures can be adjusted in a variety of ways Introduction to simulation techniques with the EJS (Easy Java Simulation) tool Visual interface for simple and transparent modeling and programming Building block library for programming one's own simulations Quick access to simulations from links embedded in the digital text Mathematics is the language of physics and technology. Yet in the age of computers, mathematic skill is not based on mastery of arithmetic. Rather, it depends on understanding relationships in time and space, and expressing them with precise and clear formulas. In this regard, one cannot rely on the rote memorization of rules and formulas - insight and intuitive understanding are crucial. But how can this understanding be achieved in higher mathematics, which depends on abstract concepts such as complex numbers, real and complex infinite series, infinitesimal calculus, 2, 3, and 4 dimensional functions, conformal maps, vectors, and linear and nonlinear ordinary and partial differential equations? The author takes a highly practical approach to facilitating the insight essential for true learning in mathematics. Students can work directly with the simulation programs, can visualize relationships, and creatively interact with the calculation procedures. Proceeding in textbook fashion, the work makes use of a broad palette of multimedia tools, and features numerous interactive calculation programs for mathematical experimentation. Students merely have to select one of the many predefined examples and set the relevant parameters - and in a flash the results are graphically displayed in 2 or 3 dimensions. In addition, the specific functions used can be changed or even newly formulated according to user preferences. For example, a procedure developed for a fourth degree power function for the numerical calculation of zero points can be adapted for use with another function. Each simulation is accompanied by a detailed description, instructions for use, and numerous suggestions for experimentation. The mathematical simulations are based on the Easy Java Simulation (EJS) programming tool. All of the files developed with EJS are completely open and transparent. The user can even draw on the examples as building blocks for the development his or her own calculation procedures. The appendix contains a short introduction to EJS. The work is enriched by a comprehensive collection of cosmological simulations as well as models from the Open Source Physics project, organized by subject area. Intended as a systematic collection of methods and materials for upper-secondary school teachers and as a course for students of physics and mathematics, the work facilitates hands-on and experiment-driven learning in higher mathematics. The print version contains the electronic text and simulations for offline use. For questions concerning download or online access to the simulations, please contact service@degruyter.com.

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Book SynopsisDas Standardwerk HÜTTE erscheint in der 35., aktualisierten Auflage in der Reihe Springer Reference Technik. Diese Reihe bietet allen Ingenieurinnen und Ingenieuren – ob im Studium oder in allen Formen des Berufslebens – zielführendes Fachwissen in aktueller, kompakter und verständlicher Form. Die HÜTTE enthält in drei Bänden die Grundlagen des Ingenieurwissens sowie ökonomisch-gesellschaftliche Kapitel, die das Ingenieurwissen ergänzen. Band 1 umfasst Mathematisch-naturwissenschaftliche und allgemeine Grundlagen: Mathematik Physik Chemie Recht Patente Technik- und Risikokommunikation Band 2 Grundlagen des Maschinenbaus und ergänzende FächerBand 3 Elektro- und informationstechnische GrundlagenTable of ContentsMathematik.-Physik.-Chemie.-Recht.-Patente.-Technikkommunikation, Risikobewertung und Risikokommunikation.

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Book SynopsisDas Standardwerk HÜTTE erscheint in der 35., aktualisierten Auflage in der Reihe Springer Reference Technik. Diese Reihe bietet allen Ingenieurinnen und Ingenieuren – ob im Studium oder in allen Formen des Berufslebens – zielführendes Fachwissen in aktueller, kompakter und verständlicher Form. Die HÜTTE enthält in drei Bänden die Grundlagen des Ingenieurwissens sowie ökonomisch-gesellschaftliche Kapitel, die das Ingenieurwissen ergänzen. Band 2 umfasst Grundlagen des Maschinenbaus und ergänzende Fächer: Werkstoffe Technische Mechanik Technische Thermodynamik Entwicklung Konstruktion Produktion Logistik Betriebswirtschaft Management Normung Band 1 Mathematisch-naturwissenschaftliche und allgemeine GrundlagenBand 3 Elektro- und informationstechnische GrundlagenTable of ContentsMaterialwissenschaften.- Technische Mechanik.- Strömungsmechanik.- Thermodynamik.- Entwicklung und Konstruktion.- Produktion.- Logistik.- BWL.- Management.- Normung.

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Book SynopsisDas Standardwerk HÜTTE erscheint in der 35., aktualisierten Auflage in der Reihe Springer Reference Technik. Diese Reihe bietet allen Ingenieurinnen und Ingenieuren – ob im Studium oder in allen Formen des Berufslebens – zielführendes Fachwissen in aktueller, kompakter und verständlicher Form. Die HÜTTE enthält in drei Bänden die Grundlagen des Ingenieurwissens sowie ökonomisch-gesellschaftliche Kapitel, die das Ingenieurwissen ergänzen. Band 3 umfasst Elektro- und informationstechnische Grundlagen: Elektrotechnik Messtechnik Regelungs- und Steuerungstechnik Informatik Informationsmanagement Band 1 Mathematisch-naturwissenschaftliche und allgemeine GrundlagenBand 2 Grundlagen des Maschinenbaus und ergänzende FächerTable of ContentsElektrotechnik.- Messtechnik.- Regelungs- und Steuerungstechnik.- Technische Informatik.

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Book SynopsisThis book is a new edition of Tensors and Manifolds: With Applications to Mechanics and Relativity which was published in 1992. It is based on courses taken by advanced undergraduate and beginning graduate students in mathematics and physics, giving an introduction to the expanse of modern mathematics and its application in modern physics. It aims to fill the gap between the basic courses and the highly technical and specialised courses which both mathematics and physics students require in their advanced training, while simultaneously trying to promote, at an early stage, a better appreciation and understanding of each other''s discipline. The book sets forth the basic principles of tensors and manifolds, describing how the mathematics underlies elegant geometrical models of classical mechanics, relativity and elementary particle physics. The existing material from the first edition has been reworked and extended in some sections to provide extra clarity, as well as additional problemTrade ReviewReview from previous edition Clearly written and self-contained and, in particular, the author has succeeded in combining mathematical rigor with a certain degree of informality in a satisfactory way. As such, this work will certainly be appreciated by a wide audience. * Mathematical Reviews, August 1993 *Table of Contents1. Vector spaces ; 2. Multilinear mappings and dual spaces ; 3. Tensor product spaces ; 4. Tensors ; 5. Symmetric and skew-symmetric tensors ; 6. Exterior (Grassmann) algebra ; 7. The tangent map of real cartesian spaces ; 8. Topological spaces ; 9. Differentiable manifolds ; 10. Submanifolds ; 11. Vector fields, 1-forms and other tensor fields ; 12. Differentiation and integration of differential forms ; 13. The flow and the Lie derivative of a vector field ; 14. Integrability conditions for distributions and for pfaffian systems ; 15. Pseudo-Riemannian manifolds ; 16. Connection 1-forms ; 17. Connection on manifolds ; 18. Mechanics ; 19. Additional topics in mechanics ; 20. A spacetime ; 21. Some physics on Minkowski spacetime ; 22. Einstein spacetimes ; 23. Spacetimes near an isolated star ; 24. Nonempty spacetimes ; 25. Lie groups ; 26. Fiber bundles ; 27. Connections on fiber bundles ; 28. Gauge theory

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Book SynopsisThis monograph provides a treatment of the theory of algebraic Riccati equations, an area of increasing interest in the mathematics and engineering communities. A range of applications are covered, demonstrating the use of these equations for providing solutions to complex problems.Table of Contents1. Preliminaries from the theory of matrices ; 2. Indefinite scalar products ; 3. Skew-symmetric scalar products ; 4. Matrix theory and control ; 5. Linear matrix equations ; 6. Rational matrix functions ; 7. Geometric theory: the complex case ; 8. Geometric theory: the real case ; 9. Constructive existence and comparison theorems ; 10. Hermitian solutions and factorizations of rational matrix functions ; 11. Perturbation theory ; 12. Geometric theory for the discrete algebraic Riccati equation ; 13. Constructive existence and comparison theorems ; 14. Perturbation theory for discrete algebraic Riccati equations ; 15. Discrete algebraic Riccati equations and matrix pencils ; 16. Linear-quadratic regulator problems ; 17. The discrete Kalman filter ; 18. The total least squares technique ; 19. Canonical factorization ; 20. Hoo control problems ; 21. Contractive rational matrix functions ; 22. The matrix sign function ; 23. Structured stability radius ; Bibliography ; List of notations ; Index

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Book SynopsisThis book is aimed at those readers who already have some knowledge of mathematical methods and have also been introduced to the basic ideas of quantum optics. It should be attractive to students who have already explored one of the more introductory texts such as Loudon''s The quantum theory of light (2/e, 1983, OUP) and are seeking to acquire the mathematical skills used in real problems. This book is not primarily about the physics of quantum optics but rather presents the mathematical methods widely used by workers in this field. There is no comparable book which covers either the range or the depth of mathematical techniques.Trade Review... the authors are well-known for their work on topics, such as the quantum-phase operator and quasi-probability distribution so theory PhD students will be able to learn these subjects direct from the horse's mouth. The authors have [] included, for pedagogic purposes, extra detail of the mathematical workings that a PhD student would not be able find in the research literature. * New Scientist, 6 June 1998 *The reader will find here a very clear presentation of material not readily found elsewhere. Postgraduate students of quantum optics will find this work to be of the greatest utility... Care has been taken to present quite difficult topics in the simplest and most straightforward way; and yet the treatment is concise and focused... Experienced researchers will find that this text is a most convenient handbook of techniques, and will want it close to their elbow. * Contemporary Physics, 1998, vol. 39, no. 4 *Table of Contents1. Foundations ; 2. Coherent interactions ; 3. Operators and states ; 4. Quantum statistics of fields ; 5. Dissipative processes ; 6. Dressed states ; Appendices ; Selected bibliography ; Index

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Book SynopsisThis monograph explores the equivalence of signal functions with their sets of values taken at discrete points. Beginning with an introduction to the main ideas, and background material on Fourier analysis and Hilbert spaces and their bases, it covers a wide variety of topics.Trade Review...the text is written by use of LATEX and its beautiful graphics reveal the power and the advantages of this system. * Zentralblatt fuer Mathematik 827/97 *Table of Contents1. An introduction to sampling theory ; 1.1 General introduction ; 1.2 Introduction - continued ; 1.3 The seventeenth to the mid twentieth century - a brief review ; 1.4 Interpolation and sampling from the seventeenth century to the mid twentieth century - a brief review ; 1.5 Introduction - concluding remarks ; 2. Background in Fourier analysis ; 2.1 The Fourier Series ; 2.2 The Fourier transform ; 2.3 Poisson's summation formula ; 2.4 Tempered distributions - some basic facts ; 3. Hilbert spaces, bases and frames ; 3.1 Bases for Banach and Hilbert spaces ; 3.2 Riesz bases and unconditional bases ; 3.3 Frames ; 3.4 Reproducing kernel Hilbert spaces ; 3.5 Direct sums of Hilbert spaces ; 3.6 Sampling and reproducing kernels ; 4. Finite sampling ; 4.1 A general setting for finite sampling ; 4.2 Sampling on the sphere ; 5. From finite to infinite sampling series ; 5.1 The change to infinite sampling series ; 5.2 The Theorem of Hinsen and Kloosters ; 6. Bernstein and Paley-Weiner spaces ; 6.1 Convolution and the cardinal series ; 6.2 Sampling and entire functions of polynomial growth ; 6.3 Paley-Weiner spaces ; 6.4 The cardinal series for Paley-Weiner spaces ; 6.5 The space ReH1 ; 6.6 The ordinary Paley-Weiner space and its reproducing kernel ; 6.7 A convergence principle for general Paley-Weiner spaces ; 7. More about Paley-Weiner spaces ; 7.1 Paley-Weiner theorems - a review ; 7.2 Bases for Paley-Weiner spaces ; 7.3 Operators on the Paley-Weiner space ; 7.4 Oscillatory properties of Paley-Weiner functions ; 8. Kramer's lemma ; 8.1 Kramer's Lemma ; 8.2 The Walsh sampling therem ; 9. Contour integral methods ; 9.1 The Paley-Weiner theorem ; 9.2 Some formulae of analysis and their equivalence ; 9.3 A general sampling theorem ; 10. Ireggular sampling ; 10.1 Sets of stable sampling, of interpolation and of uniqueness ; 10.2 Irregular sampling at minimal rate ; 10.3 Frames and over-sampling ; 11. Errors and aliasing ; 11.1 Errors ; 11.2 The time jitter error ; 11.3 The aliasing error ; 12. Multi-channel sampling ; 12.1 Single channel sampling ; 12.3 Two channels ; 13. Multi-band sampling ; 13.1 Regular sampling ; 13.3 An algorithm for the optimal regular sampling rate ; 13.4 Selectively tiled band regions ; 13.5 Harmonic signals ; 13.6 Band-ass sampling ; 14. Multi-dimensional sampling ; 14.1 Remarks on multi-dimensional Fourier analysis ; 14.2 The rectangular case ; 14.3 Regular multi-dimensional sampling ; 15. Sampling and eigenvalue problems ; 15.1 Preliminary facts ; 15.2 Direct and inverse Sturm-Liouville problems ; 15.3 Further types of eigenvalue problem - some examples ; 16. Campbell's generalised sampling theorem ; 16.1 L.L. Campbell's generalisation of the sampling theorem ; 16.2 Band-limited functions ; 16.3 Non band-limited functions - an example ; 17. Modelling, uncertainty and stable sampling ; 17.1 Remarks on signal modelling ; 17.2 Energy concentration ; 17.3 Prolate Spheroidal Wave functions ; 17.4 The uncertainty principle of signal theory ; 17.5 The Nyquist-Landau minimal sampling rate

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Book SynopsisThis book gives a detailed mathematical and statistical analysis of the cointegrated vector autoregresive model. This model had gained popularity because it can at the same time capture the short-run dynamic properties as well as the long-run equilibrium behaviour of many non-stationary time series. It also allows relevant economic questions to be formulated in a consistent statistical framework.Part I of the book is planned so that it can be used by those who want to apply the methods without going into too much detail about the probability theory. The main emphasis is on the derivation of estimators and test statistics through a consistent use of the Guassian likelihood function. It is shown that many different models can be formulated within the framework of the autoregressive model and the interpretation of these models is discussed in detail. In particular, models involving restrictions on the cointegration vectors and the adjustment coefficients are discussed, as well as the role

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Book SynopsisThis is a thoroughly updated and expanded 4th edition of the classic text Nonlinear Ordinary Differential Equations by Dominic Jordan and Peter Smith. Including numerous worked examples and diagrams, further exercises have been incorporated into the text and answers are provided at the back of the book. Topics include phase plane analysis, nonlinear damping, small parameter expansions and singular perturbations, stability, Liapunov methods, Poincare sequences, homoclinic bifurcation and Liapunov exponents.Over 500 end-of-chapter problems are also included and as an additional resource fully-worked solutions to these are provided in the accompanying text Nonlinear Ordinary Differential Equations: Problems and Solutions, (OUP, 2007). Both texts cover a wide variety of applications whilst keeping mathematical prequisites to a minimum making these an ideal resource for students and lecturers in engineering, mathematics and the sciences.Trade ReviewReview from previous edition "...classic book...The book succeeds as an exceptionally well written test for its intended audience...No doubt one of its strongest features is over 500 problems...throughout the entire book only important physical processes are described... The new edition is greatly enhanced...I strongly recommend that you take a look. The presentation is exquisitely straightforward with numerous physically interesting examples, and it is carefully and well written" * SIAM *Table of ContentsAPPENDICES

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Book SynopsisThis book provides an introduction to MIP modeling and to planning systems, a unique collection of reformulation results, and an easy to use problem-solving library. This approach is demonstrated through a series of real life case studies, exercises and detailed illustrations.Trade ReviewFrom the reviews: "The book provides a complete overview of different models existing in the literature as well as in practice. … The authors also analyze MIP (mixed integer programming) based algorithms … . Practitioners who are interested in using MIP … can use the book to identify the most efficient way to formulate the problems and to choose the most efficient solution method. … it also can serve as a good reference for students and researchers. Overall, this is an excellent book." (Panos M. Pardalos, Mathematical Reviews, Issue 2006 k) "Recently published Production Planning by Mixed Integer Programming by Yves Pochet and Laurence Wolsey has raised considerable expectations. Firstly, problems of production planning are among the most interesting in Operations Research. … Secondly, both authors are renowned experts in the field. … There is no doubt that this volume offers the present best introduction to integer programming formulations of lot-sizing problems, encountered in production planning." (Jakub Marecek, The Computer Journal, September, 2007)Table of ContentsProduction Planning and MIP.- The Modeling and Optimization Approach.- Production Planning Models and Systems.- Mixed Integer Programming Algorithms.- Classification and Reformulation.- Reformulations in Practice.- Basic Polyhedral Combinatorics for Production Planning and MIP.- Mixed Integer Programming Algorithms and Decomposition Approaches.- Single-Item Uncapacitated Lot-Sizing.- Basic MIP and Fixed Cost Flow Models.- Single-Item Lot-Sizing.- Lot-Sizing with Capacities.- Backlogging and Start-Ups.- Single-Item Variants.- Multi-Item Lot-Sizing.- Multi-Item Single-Level Problems.- Multi-Level Lot-Sizing Problems.- Problem Solving.- Test Problems.

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Book SynopsisRigid Body Dynamics Algorithms presents the subject of computational rigid-body dynamics through the medium of spatial 6D vector notation. The use of spatial vector notation facilitates the implementation of dynamics algorithms on a computer: shorter, simpler code that is easier to write, understand and debug, with no loss of efficiency.Trade ReviewFrom the reviews: "This book deals with a numerical treatment of different problems in the dynamics of rigid-body systems which arise mainly in robotics ... . is centered on mechanical models made up of many rigid bodies connected by joints. ... The book is written in a clear way. Each chapter begins by stating the objectives to be achieved. The algorithms presented are well documented and worked examples are also given. ... the bibliography close this useful book on the computational approach to the dynamics of rigid-body systems." (A. San Miguel, Mathematical Reviews, Issue 2011 h)Table of ContentsIntroduction.- Spatial Vector Algebra.- Dynamics of Rigid Body Systems.- Modelling Rigid Body Systems.- Inverse Dynamics.- Forward Dynamics - Inertia Matrix Methods.- Forward Dynamics - Propagation Methods.- Closed Loop Systems.- Hybrid Dynamics and Other Topics.- Accuracy and Efficiency.- Contact and Impact.

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Book SynopsisExamples.- A Model for Longitudinal Data.- Exploratory Data Analysis.- Estimation of the Marginal Model.- Inference for the Marginal Model.- Inference for the Random Effects.- Fitting Linear Mixed Models with SAS.- General Guidelines for Model Building.- Exploring Serial Correlation.- Local Influence for the Linear Mixed Model.- The Heterogeneity Model.- Conditional Linear Mixed Models.- Exploring Incomplete Data.- Joint Modeling of Measurements and Missingness.- Simple Missing Data Methods.- Selection Models.- Pattern-Mixture Models.- Sensitivity Analysis for Selection Models.- Sensitivity Analysis for Pattern-Mixture Models.- How Ignorable Is Missing At Random ?.- The Expectation-Maximization Algorithm.- Design Considerations.- Case Studies.Trade ReviewFrom the reviews: MATHEMATICAL REVIEWS "This book emphasizes practice rather than mathematical rigor and the majority of the chapters are explanatory rather than research oriented. In this respect, guidance and advice on practical issues are the main focus of the text. Hence it will be of interest to applied statisticians and biomedical researchers in industry, particularly in the pharmaceutical industry, medical public health organizations, contract research organizations, and academia." "This book provides a comprehensive treatment of linear mixed models for continuous longitudinal data. Over 125 illustrations are included in the book. … I do believe that the book may serve as a useful reference to a broader audience. Since practical examples are provided as well as discussion of the leading software utilization, it may also be appropriate as a textbook in an advanced undergraduate-level or a graduate-level course in an applied statistics program." (Ana Ivelisse Avil és, Technometrics, Vol. 43 (3), 2001) "A practical book with a great many examples, including worked computer code and access to the datasets. … The authors state that the book covers ‘linear mixed models for continuous outcomes’ … . The book has four main strengths: its practical bent, its emphasis on exploratory analysis, its description of tools for model checking, and its treatment of dropout and missingness … . my impression of the book was … positive. Its strong practical nature and emphasis on dropout modelling are particularly welcome … ." (Harry Southworth, ISCB Newsletter, June, 2002) "This book is devoted to linear mixed-effects models with strong emphasis on the SAS procedure. Guidance and advice on practical issues are the main focus of the text. … It is of value to applied statisticians and biomedical researchers. … I recommend this book as a reference to applied statisticians and biomedical researchers, particularly in the pharmaceutical industry, medical and public organizations." (Wang Songgui, Zentralblatt MATH, Vol. 956, 2001)Table of ContentsIntroduction * Examples * A model for Longitudinal Data * Exploratory Data Analysis * Estimation of the Marginal Model * Inference for the Marginal Model * Inference for the Random Effects * Fitting Linear Mixed Models with SAS * General Guidelines for Model Building * Exploring Serial Correlation * Local Influence for the Linear Mixed Model * The Heterogeneity Model * Conditional Linear Mixed Models * Exploring Incomplete Data * Joint Modeling of Measurements and Missingness * Simple Missing Data Methods * Selection Models * Pattern-Mixture Models * Sensitivity Analysis for Selection Models * Sensitivity Analysis for Models * How Ignorable is Missing at Random? * The Expectation-Maximization Algorithm * Design Considerations * Case Studies

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Book SynopsisIntended as a companion for textbooks in mathematical methods for science and engineering, this book presents a large number of numerical topics and exercises together with discussions of methods for solving such problems using Mathematica(R).Trade ReviewFrom the reviews:"From a stylistic perspective the book strikes a comfortable balance between explanation and example which makes it easy to dip into and attractive to work through. For the eager reader there is always the promise of an interesting result after half an hour of labour. … The target audience of this book is likely to be a Physics undergraduate finishing his or her first year of study." (Dr. E. J. Grace, Contemporary Physics, Vol. 45 (2), 2004)"Initially this book has been designed as a companion to the undergraduate textbook ‘Mathematical methods’ … and later on developed into a self-contained introduction to the use of computer algebra system (CAS) Mathematica tailored specifically for undergraduate students in physics and related fields. … The book is written in a transparent manner and does not require any prior knowledge of physics for mastering computational techniques. … thanks to a massive array of carefully selected and nicely explained examples from undergraduate physics." (Yuri V. Rogovchenko, Zentralblatt MATH, Vol. 1028, 2004)"This book is intended to be a companion for textbooks in mathematical methods for undergraduate science and engineering students. It presents a number of numerical topics and exercises together with discussions of methods needed for solving problems with Mathematica. … In conclusion, this very well produced and illustrated book is heartily recommended … ." (André Hautot, Gary J. Long, Physicalia, Vol. 26 (1), 2004)Table of ContentsMathematica in a Nutshell / Vectors and Matrices in Mathematica / Integration / Infinite Series and Finite Sums / Numerical Solutions of ODE's: Theory / Numerical Solutions of ODE's: Examples Using Mathematica

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Book SynopsisThis volume deals with controllability and observability properties of nonlinear systems, as well as various ways to obtain input-output representations. The emphasis is on fundamental notions as (controlled) invariant distributions and submanifolds, together with algorithms to compute the required feedbacks. Table of Contents1 Introduction.- 2 Manifolds, Vectorfields, Lie Brackets, Distributions.- 3 Controllability and Observability, Local Decompositions.- 4 Input-Output Representations.- 5 State Space Transformation and Feedback.- 6 Feedback Linearization of Nonlinear Systems.- 7 Controlled Invariant Distribution and the Disturbance Decoupling Problem.- 8 The Input-Output Decoupling Problem.- 9 The Input-Output Decoupling Problem.- 10 Local Stability and Stabilization of Nonlinear Systems.- 11 Controlled Invariant Submanifolds and Nonlinear Zero Dynamics.- 12 Mechanical Nonlinear Control Systems.- 13 Controlled Invariance and Decoupling for General Nonlinear Systems.- 14 Discrete-Time Nonlinear Control Systems.

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Book SynopsisThis book is the result of our teaching over the years an undergraduate course on Linear Optimal Systems to applied mathematicians and a first-year graduate course on Linear Systems to engineers.Table of Contents1 Introduction.- 1.1 Science and Engineering.- 1.2 Physical Systems, Models, and Representations.- 1.3 Robustness.- 2 The System RepresentationR(•) = [A(•),B(•),C(•),D(•)].- 2.1 Fundamental Properties ofR(•).- 2.1.1 Definitions.- 2.1.2 Structure ofR(•).- 2.1.3 State Transition Matrix.- 2.1.4 State Transition Map and Response Map.- 2.1.5 Impulse Response Matrix.- 2.1.6 Adjoint Equations.- 2.1.7 Linear-Quadratic Optimization.- 2.2 Applications.- 2.2.1 Variational Equation.- 2.2.2 Control Correction Example.- 2.2.3 Optimization Example.- 2.2.4 Periodically Varying Differential Equations.- 2d The Discrete-Time System RepresentationRd(•) = [A(•),B(•),C(•),D(•)].- 2d.1 Fundamental Properties ofRd(•).- 2d.2 Application: Periodically Varying Recursion Equations.- 3 The System RepresentationR= [A,B,C,D], Part I.- 3.1 Preliminaries.- 3.2 General Properties ofR= [A,B,C,D].- 3.2.1 Definition.- 3.2.2 State Transition Matrix.- 3.2.3 The State Transition and Response Map of R.- 3.3 Properties of R when A has a Basis of Eigenvectors.- 3d The Discrete-Time System Representation Rd = [A,B,C,D].- 3d.1 Preliminaries.- 3d.2 General Properties of Rd.- 3d.3 Properties of Rd when A has a Basis of Eigenvectors.- 4 The System Representation R = [A,B,C,D], Part II.- 4.1 Preliminaries.- 4.2 Minimal Polynomial.- 4.3 Decomposition Theorem.- 4.4 The Decomposition of a Linear Map.- 4.5 Jordan Form.- 4.6 Function of a Matrix.- 4.7 Spectral Mapping Theorem.- 4.8 The Linear Map X ? AX+XB.- 5 General System Concepts.- 5.1 Dynamical Systems.- 5.2 Time-Invariant Dynamical Systems.- 5.3 Linear Dynamical Systems.- 5.4 Equivalence.- 6 Sampled Data Systems.- 6.1 Relation BetweenL- and z-Transforms.- 6.2 D/A Converter.- 6.3 A/D Converter.- 6.4 Sampled-Data System.- 6.5 Example.- 7 Stability.- 7.1 I/O Stability.- 7.2 State Related Stability Concepts and Applications.- 7.2.1 Stability of x = A(t)x.- 7.2.2 Bounded Trajectories and Regulation.- 7.2.3 Response to T-Periodic Inputs.- 7.2.4 Periodically Varying System with Periodic Input.- 7.2.5 Slightly Nonlinear Systems.- 7d Stability: The Discrete-Time Case.- 7d.1 I/O Stability.- 7d.2 State Related Stability Concepts.- 7d.2.1 Stability of x(k+1) = A(k)x(k).- 7d.2.2 Bounded Trajectories and Regulation.- 7d.2.3 Response to q-Periodic Inputs.- 8 Controllability and Observability.- 8.1 Controllability and Observability of Dynamical Systems.- 8.2 Controllability of the Pair (A(•),B(•)).- 8.2.1 Controllability of the Pair (A(•),B(•)).- 8.2.2 The Cost of Control.- 8.2.3 Stabilization by Linear State Feedback.- 8.3 Observability of the Pair (C(•),A(•)).- 8.4 Duality.- 8.5 Linear Time-Invariant Systems.- 8.5.1 Observability Properties of the Pair (C,A).- 8.5.2 Controllability of the Pair (A,B).- 8.6 Kalman Decomposition Theorem.- 8.7 Hidden Modes, Stabilizability, and Detectability.- 8.8 Balanced Representations.- 8.9 Robustness of Controllability.- 8d Controllability and Observability: The Discrete-Time Case.- 8d.1 Controllability and Observability of Dynamical Systems.- 8d.2 Reachability and Controllability of the Pair (A(•),B(•)).- 8d.2.1 Controllability of the Pair (A(•),B(•)).- 8d.2.2 The Cost of Control.- 8d.3 Observability of the Pair (C(•),A(•)).- 8d.4 Duality.- 8d.5 Linear Time-Invariant Systems.- 8d.5.1 Observability of the Pair (C,A).- 8d.5.2 Reachability and Controllability of the Pair(A,B).- 8d.6 Kalman Decomposition Theorem.- 8d.7 Stabilizability and Detectability.- 9 Realization Theory.- 9.1 Minimal Realizations.- 9.2 Controllable Canonical Form.- 10 Linear State Feedback and Estimation.- 10.1 Linear State Feedback.- 10.2 Linear Output Injection and State Estimation.- 10.3 State Feedback of the Estimated State.- 10.4 Infinite Horizon Linear Quadratic Optimization.- 10d.4 Infinite Horizon Linear Quadratic Optimization. The Discrete-Time Case.- 11 Unity Feedback Systems.- 11.1 The Feedback System ?c.- 11.1.1 State Space Analysis.- 11.1.2 Special Case:R1andR2have no Unstable Hidden Modes.- 11.1.3 The Discrete-Time Case.- 11.2 Nyquist Criterion.- 11.2.1 The Nyquist Criterion.- 11.2.2 Remarks on the Nyquist Criterion.- 11.2.3 Proof of Nyquist Criterion.- 11.2.4 The Discrete-Time Case.- 11.3 Robustness.- 11.3.1 Robustness With Respect to Plant Perturbations.- 11.3.2 Robustness With Respect to Exogenous Disturbances.- 11.3.3 Robust Regulation.- 11.3.4 Bandwidth-Robustness Tradeoff.- 11.3.5 The Discrete-Time Case.- 11.4 Kharitonov’s Theorem.- 11.4.1 Hurwitz Polynomials.- 11.4.2 Kharitonov’s Theorem.- 11.5 Robust Stability Under Structured Perturbations.- 11.5.1 General Robustness Theorem.- 11.5.2 Special Case: Affine Maps and Convexity.- 11.5.3 The Discrete Time Case.- 11.6 Stability Under Arbitrary Additive Plant Perturbations.- 11.7 Transmission Zeros.- 11.7.1 Single-Input Single-Output Case.- 11.7.2 Multi-Input Multi-Output Case: Assumptions and Definitions.- 11.7.3 Characterization of the Zeros.- 11.7.4 Application to Unity Feedback Systems.- Appendix A Linear Maps and Matrix Analysis.- A.1 Preliminary Notions.- A.2 Rings and Fields.- A.3 Linear Spaces.- A4. Linear Maps.- AS. Matrix Representation.- A.5.1 The Concept of Matrix Representation.- A.5.2 Matrix Representation and Change of Basis.- A.5.3 Range and Null Space: Rank and Nullity.- A.5.4 Echelon Forms of a Matrix.- A.6 Notmed Linear Spaces.- A.6.1 Norms.- A.6.2 Convergence.- A.6.3 Equivalent Norms.- A.6.4 The Lebesgue Spaces 1P and LP [Tay.1].- A.6.5 Continuous Linear Transformations.- A.7 The Adjoint of a Linear Map.- A.7.1 Inner Products.- A.7.2 Adjoints of Continuous Linear Maps.- A.7.3 Properties of the Adjoint.- A.7.4 The Finite Rank Operator Fundamental Lemma.- A.7.5 Singular Value Decomposition (SVD).- Appendix B Differential Equations.- BA Existence and Uniqueness of Solutions.- B.1.1 Assumptions.- B.1.2 Fundamental Theorem.- B.1.3 Construction of a Solution by Iteration.- B.1.4 The Bellman-Gronwall Inequality.- B.1.5 Uniqueness.- B.2 Initial Conditions and Parameter Perturbations.- B.3 Geometric Interpretation and Numerical Calculations.- Appendix C Laplace Transforms.- C.1 Definition of the Laplace Transform.- C.2 Properties of Laplace Transforms.- Appendix D the z-Transform.- D.1 Definition of the z-Transform.- D.2 Properties of the z-Transform.- References.- Abbreviations.- Mathematical Symbols.

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Book Synopsis(iv) The chapter on int~gral theorems, now Chapter 5, has been expanded to include an altemative proof of Gauss's theorem, a treatmeot of Green's theorem and a more extended discussioo of the classification of vector fields.Table of Contents1 Definitions. Addition of Vectors.- 1. Scalar and Vector Quantities.- 2. Graphical Representation of Vectors.- 3. Addition and Subtraction of Vectors.- 4. Components of a Vector.- 5. Geometrical Applications.- 6. Scalar and Vector Fields.- Miscellaneous Exercises I.- 2 Products of Vectors.- 1. General.- 2. The Scalar Product.- 3. The Vector Product.- 4. Vector Area.- 5. Application to Vector Products.- 6. Products of Three Vectors.- 7. Line and Surface Integrals as Scalar Products.- Miscellaneous Exercises II.- 3 The Differentiation of Vectors.- 1. Scalar Differentiation.- 2. Differentiation of Sums and Products.- 3. Partial Differentiation.- Miscellaneous Exercises III.- 4 The Operator ? and Its Uses.- 1. The Operator ?.- 2. The Gradient of a Scalar Field.- 3. The Divergence of a Vector Field.- 4. The Operator div grad..- 5. The Operator ?2 with Vector Operand.- 6. The Curl of a Vector Field.- 7. Simple Examples of the Curl of a Vector Field.- 8. Divergence of a Vector Product.- 9. Divergence and Curl of SA.- 10. The Operator curl grad..- 11. The Operator grad div..- 12. The Operator div curl..- 13. The Operator curl curl..- 14. The Vector Field grad (k/r).- 15. Vector Operators in Terms of Polar Co-ordinates.- Miscellaneous Exercises IV.- 5 Integral Theorems.- 1. The Divergence Theorem of Gauss.- 2. Gauss’s Theorem and the Inverse Square Law.- 3. Green’s Theorem.- 4. Stokes’s Theorem.- 5. Alternative Definitions of Divergence and Curl.- 6. Classification of Vector Fields.- Miscellaneous Exercises V.- 6 The Scalar Potential Field.- 1. General Properties.- 2. The Inverse Square Law. Point Sources.- 3. Volume Distributions.- 4. Multi-valued Potentials.- 7 The Vector Potential Field.- 1. The Magnetic Field of a Steady Current.- 2. The Vector Potential.- 3. Linear Currents.- 4. Simple Examples of Vector Potential.- 8 The Electromagnetic Field Equations of Maxwell.- 1. General.- 2. Maxwell’s Equations.- 3. Energy Considerations.- Miscellaneous Exercises VIII.- Answers to Exercises.

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Book SynopsisA good grasp of the theory of structures is fundamental to structural engineers and architects. This lucid volume provides a broad introduction to the mathematics behind a range of structural processes - to help structural engineers and practising architects gain a better intuitive understanding of the structural analysis and design of buildings.Trade Review"Elements of the Theory of Structures has become one of this reviewer's prize possessions. Anyone with a love of structures, an appreciation of the practical, and adequate mathematical preparation should get this book." A.F. Martin, Applied Mechanics ReviewTable of ContentsPreface; 1. The theory of structures; 2. Virtual work; 3. Betti, Maxwell, Müller-Breslau, Melchers; 4. Jettied construction; 5. Clebsch, Macaulay, Wittrick, Lowe; 6. The elastica; 7. Mechanisms of collapse; 8. The absolute minimum-weight design of frames; 9. Inverse design of grillages; 10. The relation between incremental and static plastic collapse; 11. The bending of a beam of trapezoidal cross-section; 12. The simple plastic bending of beams; 13. Leaning walls; domes and fan vaults; the error function; Bibliography; Name index; Subject index.

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Book SynopsisThe basic theory presented in a way which emphasizes intuition, problem-solving and the connections with other fields.Trade Review'… this is unquestionably a valuable book, written at an accessible level for graduate students while providing a nice summary of the last century's - and notably the last two decades' - developments of these methods. I do not doubt that this will be a popular book; it fills a hole in the textbook/monograph literature that's needed filling for at least ten years. Moreover, the author's style is relaxed and crystal clear while maintaining mathematical precision and power.' Charles Doering, University of Michigan'… to practitioners in the field of first-passage problems and to students entering the field … I can recommend it strongly … It is clearly written and the organisation and presentation of the material are excellent. It serves as a useful repository of standard and not-so-standard techniques which anyone working in the area of stochastic processes in general, and first-passage problems in particular, will want to have on their shelves.' Alan J. Bray, The Journal of Statistical Physics'I found the author's style accessible and interesting.' Contemporary PhysicsTable of ContentsPreface; Errata; 1. First-passage fundamentals; 2. First passage in an interval; 3. Semi-infinite system; 4. Illustrations of first passage in simple geometries; 5. Fractal and nonfractal networks; 6. Systems with spherical symmetry; 7. Wedge domains; 8. Applications to simple reactions; References; Index.

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Book SynopsisIn this volume, the author covers the mathematical methods appropriate to both linear-systems theory and signal processing. The text deals with a number of topics usually found in introductory linear-systems courses, such as complex numbers and Laplace transforms, and addresses additional topics such as complex variable theory and Fourier series and transforms. Although the discussion is mathematically self-contained, it assumes that the reader has a firm background in calculus and differential equations. Each chapter contains a number of worked examples plus exercises designed to allow the student to put concepts into practice. The author writes in a mathematically elegant yet relaxed and readable style, and provides interesting historical notes along the way. Undergraduate students of electrical engineering, applied mathematics, and related disciplines - and their teachers - will welcome this book.Trade Review"...well-written and easy to read and should be useful to beginning undergraduates in engineering, particularly electronic engineering." Times Higher Education SupplementTable of ContentsPreface; 1. Complex numbers; 2. Laplace transforms; 3. Linear systems; 4. Fourier series; 5. Functions of a complex variable; 6. Fourier transforms; 7. Laplace transform revisited; Answers to selected exercises; Index.

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