Mathematical modelling Books

Book SynopsisNew advanced modeling methods for simulating the electromagnetic properties of complex three-dimensional electronic systems Based on the author''s extensive research, this book sets forth tested and proven electromagnetic modeling and simulation methods for analyzing signal and power integrity as well as electromagnetic interference in large complex electronic interconnects, multilayered package structures, integrated circuits, and printed circuit boards. Readers will discover the state of the technology in electronic package integration and printed circuit board simulation and modeling. In addition to popular full-wave electromagnetic computational methods, the book presents new, more sophisticated modeling methods, offering readers the most advanced tools for analyzing and designing large complex electronic structures. Electrical Modeling and Design for 3D System Integration begins with a comprehensive review of current modeling and simulation methods fTable of ContentsForeword xi Preface xiii 1. Introduction 1 1.1 Introduction of Electronic Package Integration 1 1.2 Review of Modeling Technologies 6 1.3 Organization of the Book 10 References 11 2. Macromodeling of Complex Interconnects in 3D Integration 16 2.1 Introduction 16 2.1.1 Scope of macromodeling 18 2.1.2 Macromodeling in the picture of electrical modeling of interconnects 19 2.2 Network Parameters: Impedance Admittance and Scattering Matrices 19 2.2.1 Impedance matrix 21 2.2.2 Admittance matrix 22 2.2.3 Scattering matrix 23 2.2.4 Conversion between Z Y and S matrices 24 2.3 Rational Function Approximation with Partial Fractions 25 2.3.1 Introduction 25 2.3.2 Iterative weighted linear least-squares estimator 27 2.4 Vector Fitting (VF) Method 29 2.4.1 Two steps in vector fitting method 29 2.4.2 Fitting vectors with common poles 35 2.4.3 Selection of initial poles 37 2.4.4 Enhancement to the original vector fitting method 38 2.5 Macromodel Synthesis 41 2.5.1 Jordan canonical method for macromodel synthesis 42 2.5.2 Equivalent circuits 46 2.6 Stability Causality and Passivity of Macromodel 48 2.6.1 Stability 48 2.6.2 Causality 50 2.6.3 Passivity assessment 54 2.6.4 Passivity enforcement 58 2.6.5 Other issues 78 2.7 Macromodeling Applied to High-Speed Interconnects and Circuits 79 2.7.1 A lumped circuit with nonlinear components 79 2.7.2 Vertically natural capacitors (VNCAPs) 83 2.7.3 Stripline-to-microstrip line transition with vias 87 2.8 Conclusion 91 References 92 3. 2.5D Simulation Method for 3D Integrated Systems 97 3.1 Introduction 97 3.2 Multiple Scattering Method for Electronic Package Modeling with Open Boundary Problems 98 3.2.1 Modal expansion of fields in a parallel-plate waveguide (PPWG) 98 3.2.2 Multiple scattering coefficients among cylindrical PEC and perfect magnetic conductor (PMC) vias 101 3.2.3 Excitation source and network parameter extraction 109 3.2.4 Implementation of effective matrix-vector multiplication (MVM) in linear equations 117 3.2.5 Numerical examples for single-layer power-ground planes 121 3.3 Novel Boundary Modeling Method for Simulation of Finite-Domain Power-Ground Planes 127 3.3.1 Perfect magnetic conductor (PMC) boundary 128 3.3.2 Frequency-dependent cylinder layer (FDCL) 128 3.3.3 Validations of FDCL 131 3.4 Numerical Simulations for Finite Structures 133 3.4.1 Extended scattering matrix method (SMM) algorithm for finite structure simulation 133 3.4.2 Modeling of arbitrarily shaped boundary structures 139 Contents vii 3.5 Modeling of 3D Electronic Package Structure 142 3.5.1 Modal expansions and boundary conditions 143 3.5.2 Mode matching in PPWGs 150 3.5.3 Generalized T-matrix for two-layer problem 158 3.5.4 Formulae summary for two-layer problem 164 3.5.5 Formulae summary for 3D structure problem 169 3.5.6 Numerical simulations for multilayered power-ground planes with multiple vias 176 3.6 Conclusion 182 References 183 4. Hybrid Integral Equation Modeling Methods for 3D Integration 185 4.1 Introduction 185 4.2 2D Integral Equation Equivalent Circuit (IEEC) Method 186 4.2.1 Overview of the algorithm 186 4.2.2 Modal decoupling inside the power distribution network (PDN) 187 4.2.3 2D integral equation solution of parallel plate mode in power-ground planes (PGPs) 189 4.2.4 Combinations of transmission and parallel plate modes 194 4.2.5 Cascade connections of equivalent networks 205 4.2.6 Simulation results 214 4.3 3D Hybrid Integral Equation Method 220 4.3.1 Overview of the algorithm 220 4.3.2 Equivalent electromagnetic currents and dyadic green’s functions 224 4.3.3 Simulation results 231 4.4 Conclusion 238 References 238 5. Systematic Microwave Network Analysis for 3D Integrated Systems 241 5.1 Intrinsic Via Circuit Model for Multiple Vias in an Irregular Plate Pair 242 5.1.1 Introduction 242 5.1.2 Segmentation of vias and a plate pair 245 5.1.3 An intrinsic 3-port via circuit model 248 5.1.4 Determination of the virtual via boundary 263 5.1.5 Complete model for multiple vias in an irregular plate pair 267 5.1.6 Validation and measurements 269 5.1.7 Conclusion 280 5.2 Parallel Plane Pair Model 281 5.2.1 Introduction 281 5.2.2 Overview of two conventional Z pp definitions 283 5.2.3 New Z pp definition using the zero-order parallel plate waves 285 5.2.4 Analytical formula for radial scattering matrix S pp in a circular plate pair 290 5.2.5 BIE method to evaluate S R pp for an irregular plate pair 292 5.2.6 Numerical examples and measurements 296 5.2.7 Conclusion 303 5.3 Cascaded Multiport Network Analysis of Multilayer Structure with Multiple Vias 305 5.3.1 Introduction 305 5.3.2 Multilayer PCB with vias and decoupling capacitors 307 5.3.3 Systematic microwave network method 308 5.3.4 Validations and discussion 316 5.3.5 Conclusion 324 Appendix: Properties of the Auxiliary Function W mn (x y) 326 References 327 6. Modeling of Through-Silicon Vias (TSV) in 3D Integration 331 6.1 Introduction 331 6.1.1 Overview of process and fabrication of TSV 332 6.1.2 Modeling of TSV 335 6.2 Equivalent Circuit Model for TSV 336 6.2.1 Overview 337 6.2.2 Problem statement: Two-TSV configuration 338 6.2.3 Wideband Pi-type equivalent-circuit model 339 6.2.4 Rigorous closed-form formulae for resistance and inductance 341 6.2.5 Scattering parameters of two-TSV system 345 6.2.6 Results and discussion 346 6.3 MOS Capacitance Effect of TSV 351 6.3.1 MOS capacitance effect 351 6.3.2 Bias voltage-dependent MOS capacitance of TSVs 351 6.3.3 Results and analysis 355 6.4 Conclusion 356 References 358 Index 361

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Book SynopsisThe decision to implement environmental protection options is a political one. These, and other political and social decisions affect the balance of the ecosystem and how the point of equilibrium desired is to be reached. This book develops a stochastic, temporal model of how political processes influence and are influenced by ecosystem processes and looks at how to find the most politically feasible plan for managing an at-risk ecosystem. Finding such a plan is accomplished by first fitting a mechanistic political and ecological model to a data set composed of observations on both political actions that impact an ecosystem and variables that describe the ecosystem. The parameters of this fitted model are perturbed just enough to cause human behaviour to change so that desired ecosystem states occur. This perturbed model gives the ecosystem management plan needed to reach desired ecosystem states. To construct such a set of interacting models, topics from political science, ecology, prTrade Review"This said, I acknowledge that the goal that stands behind this book was very challenging and that the EMT model can represent a good starting point." (Journal of Artificial Societies and Social Simulation, 2011) Table of ContentsPreface. List of Figures. List of Tables. Nomenclature. Part I Managing a Political-Ecological System. 1 Introduction. 1.1 The Problem to be Addressed. 1.2 The Book's Running Example: East African Cheetah. 1.3 The EMT's Simulator. 1.4 How to Use the EMT to Manage an Ecosystem. 1.5 Chapter Topics and Order. 1.6 The Book's Accompanying Web Resources. 2 Simulator Architecture, Operation, and Example Output. 2.1 Introduction. 2.2 Theory for Agent-Based Simulation. 2.3 Action Messages and IntIDs Model Operation. 2.4 A Plot for Displaying an Actions History. 2.5 Conclusions. 2.6 Exercises. 3 Blue Whale Population Management. 3.1 Introduction. 3.2 Current Status of Blue Whales. 3.3 Groups that Affect Blue Whale Populations. 3.4 Blue Whale Ecosystem ID. 3.5 Interactions Between IDs. 3.6 Data Sets for the Blue Whale EMT. 3.7 Main Points of this Chapter's Example. 3.8 Exercises. 4 Finding the Most Practical Ecosystem Management Plan. 4.1 Introduction. 4.2 Some Methods for Developing Ecosystem Management Plans. 4.3 Consistency Analysis Parameter Estimator Overview. 4.4 The MPEMP: Definition and Construction. 4.5 The MPEMP for East African Cheetah. 4.6 Conclusions. 4.7 Exercises. 5 An Open, Web-Based Ecosystem Management Tool. 5.1 Introduction. 5.2 Components of a Politically Realistic EMT. 5.3 Id Language and Software System. 5.4 How the EMT Website Would be Used. Part II Model Formulation, Estimation, and Reliability. 6 Influence Diagrams of Political Decision Making. 6.1 Introduction. 6.2 Theories of Political Decision Making. 6.3 Architecture of a Group Decision Making ID. 6.4 Related Modeling Efforts. 6.5 Conclusions. 6.6 Exercises. 7 Group IDs for the East African Cheetah EMT. 7.1 Introduction. 7.2 Country Backgrounds. 7.3 Selection of Groups to Model. 7.4 President IDs. 7.5 EPA IDs. 7.6 Rural Residents IDs. 7.7 Pastoralists IDs. 7.8 Conservation NGOs ID. 7.9 Conclusions. 7.10 Exercises. 8 Modeling Wildlife Population Dynamics with an Influence Diagram. 8.1 Introduction. 8.2 Model of Cheetah and Prey Population Dynamics. 8.3 Solving SDEs within an ID. 8.4 Example of Ecosystem ID Output. 8.5 Conclusions. 8.6 Exercises. 9 Political Action Taxonomies, Collection Protocols, and an Actions History Example. 9.1 Introduction. 9.2 Political Action Taxonomies. 9.3 Adapting the BCOW Taxonomy to Ecosystem Management Actions. 9.4 EMAT Coding Protocol. 9.5 Actions History Data for the East African Cheetah EMT. 9.6 Conclusions. 10 Ecosystem Data. 10.1 Introduction. 10.2 Wildlife Monitoring. 10.3 Wildlife Abundance Estimation Methods. 10.4 East African Cheetah Prey Abundance Data. 10.5 Data on Cheetah Habitat Suitability Nodes. 10.6 Conclusions. 10.7 Exercises. 11 Statistical Fitting of the Political-Ecological System Simulator. 11.1 Introduction. 11.2 Consistency Analysis Applied to an Actions History. 11.3 Consistency Analysis of the East African Cheetah EMT Simulator. 11.4 Conclusions and Another Collection Initialization Algorithm. 11.5 Exercises. 12 Assessing the Simulator's Reliability and Improving Its Construct Validity. 12.1 Introduction. 12.2 Steps for Assessing Simulator Reliability. 12.3 Sensitivity Analysis. 12.4 One-Step-Ahead Prediction Error Rates. 12.5 MC Hypothesis Tests. 12.6 Sensitivity to Hidden Bias Analysis. 12.7 Conclusions. 12.8 Exercises. Part III Assessment. 13 Current Capabilities and Limitations of the Politically Realistic EMT. 13.1 Introduction. 13.2 Current Capabilities of the EMT. 13.3 Current Limitations of the EMT. 13.4 Supporting the EMT in the Real World. 13.5 Consequences of Using a Politically Realistic EMT. Appendices. Appendix A Heuristics Used to Assign Hypothesis Values to Parameters. Appendix B Cluster Computing Version of Hooke and Jeeves Search. References. Index.

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Book SynopsisSince mathematical models express our understanding of how nature behaves, we use them to validate our understanding of the fundamentals about systems (which could be processes, equipment, procedures, devices, or products). Also, when validated, the model is useful for engineering applications related to diagnosis, design, and optimization.Table of ContentsSeries Preface xiii Preface xv Acknowledgments xxiii Nomenclature xxv Symbols xxxvii Part I INTRODUCTION 1 Introductory Concepts 3 1.1 Illustrative Example – Traditional Linear Least-Squares Regression 3 1.2 How Models Are Used 7 1.3 Nonlinear Regression 7 1.4 Variable Types 8 1.5 Simulation 12 1.6 Issues 13 1.7 Takeaway 15 Exercises 15 2 Model Types 16 2.1 Model Terminology 16 2.2 A Classification of Mathematical Model Types 17 2.3 Steady-State and Dynamic Models 21 2.3.1 Steady-State Models 22 2.3.2 Dynamic Models (Time-Dependent, Transient) 24 2.4 Pseudo-First Principles – Appropriated First Principles 26 2.5 Pseudo-First Principles – Pseudo-Components 28 2.6 Empirical Models with Theoretical Grounding 28 2.6.1 Empirical Steady State 28 2.6.2 Empirical Time-Dependent 30 2.7 Empirical Models with No Theoretical Grounding 31 2.8 Partitioned Models 31 2.9 Empirical or Phenomenological? 32 2.10 Ensemble Models 32 2.11 Simulators 33 2.12 Stochastic and Probabilistic Models 33 2.13 Linearity 34 2.14 Discrete or Continuous 36 2.15 Constraints 36 2.16 Model Design (Architecture, Functionality, Structure) 37 2.17 Takeaway 37 Exercises 37 Part II PREPARATION FOR UNDERLYING SKILLS 3 Propagation of Uncertainty 43 3.1 Introduction 43 3.2 Sources of Error and Uncertainty 44 3.2.1 Estimation 45 3.2.2 Discrimination 45 3.2.3 Calibration Drift 45 3.2.4 Accuracy 45 3.2.5 Technique 46 3.2.6 Constants and Data 46 3.2.7 Noise 46 3.2.8 Model and Equations 46 3.2.9 Humans 47 3.3 Significant Digits 47 3.4 Rounding Off 48 3.5 Estimating Uncertainty on Values 49 3.5.1 Caution 50 3.6 Propagation of Uncertainty – Overview – Two Types, Two Ways Each 51 3.6.1 Maximum Uncertainty 51 3.6.2 Probable Uncertainty 56 3.6.3 Generality 58 3.7 Which to Report? Maximum or Probable Uncertainty 59 3.8 Bootstrapping 59 3.9 Bias and Precision 61 3.10 Takeaway 65 Exercises 66 4 Essential Probability and Statistics 67 4.1 Variation and Its Role in Topics 67 4.2 Histogram and Its PDF and CDF Views 67 4.3 Constructing a Data-Based View of PDF and CDF 70 4.4 Parameters that Characterize the Distribution 71 4.5 Some Representative Distributions 72 4.5.1 Gaussian Distribution 72 4.5.2 Log-Normal Distribution 72 4.5.3 Logistic Distribution 74 4.5.4 Exponential Distribution 74 4.5.5 Binomial Distribution 75 4.6 Confidence Interval 76 4.7 Central Limit Theorem 77 4.8 Hypothesis and Testing 78 4.9 Type I and Type II Errors, Alpha and Beta 80 4.10 Essential Statistics for This Text 82 4.10.1 t-Test for Bias 83 4.10.2 Wilcoxon Signed Rank Test for Bias 83 4.10.3 r-lag-1 Autocorrelation Test 84 4.10.4 Runs Test 87 4.10.5 Test for Steady State in a Noisy Signal 87 4.10.6 Chi-Square Contingency Test 89 4.10.7 Kolmogorov–Smirnov Distribution Test 89 4.10.8 Test for Proportion 90 4.10.9 F-Test for Equal Variance 90 4.11 Takeaway 91 Exercises 91 5 Simulation 93 5.1 Introduction 93 5.2 Three Sources of Deviation: Measurement, Inputs, Coefficients 93 5.3 Two Types of Perturbations: Noise (Independent) and Drifts (Persistence) 95 5.4 Two Types of Influence: Additive and Scaled with Level 98 5.5 Using the Inverse CDF to Generate n and u from UID(0, 1) 99 5.6 Takeaway 100 Exercises 100 6 Steady and Transient State Detection 101 6.1 Introduction 101 6.1.1 General Applications 101 6.1.2 Concepts and Issues in Detecting Steady State 104 6.1.3 Approaches and Issues to SSID and TSID 104 6.2 Method 106 6.2.1 Conceptual Model 106 6.2.2 Equations 107 6.2.3 Coefficient, Threshold, and Sample Frequency Values 108 6.2.4 Noiseless Data 111 6.3 Applications 112 6.3.1 Applications of the R-Statistic Approach for Process Monitoring 112 6.3.2 Applications of the R-Statistic Approach for Determining Regression Convergence 112 6.4 Takeaway 114 Exercises 114 Part III REGRESSION, VALIDATION, DESIGN 7 Regression Target – Objective Function 119 7.1 Introduction 119 7.2 Experimental and Measurement Uncertainty – Static and Continuous Valued 119 7.3 Likelihood 122 7.4 Maximum Likelihood 124 7.5 Estimating σx and σy Values 127 7.6 Vertical SSD – A Limiting Consideration of Variability Only in the Response Measurement 127 7.7 r-Square as a Measure of Fit 128 7.8 Normal, Total, or Perpendicular SSD 130 7.9 Akaho’s Method 132 7.10 Using a Model Inverse for Regression 134 7.11 Choosing the Dependent Variable 135 7.12 Model Prediction with Dynamic Models 136 7.13 Model Prediction with Classification Models 137 7.14 Model Prediction with Rank Models 138 7.15 Probabilistic Models 139 7.16 Stochastic Models 139 7.17 Takeaway 139 Exercises 140 8 Constraints 141 8.1 Introduction 141 8.2 Constraint Types 141 8.3 Expressing Hard Constraints in the Optimization Statement 142 8.4 Expressing Soft Constraints in the Optimization Statement 143 8.5 Equality Constraints 147 8.6 Takeaway 148 Exercises 148 9 The Distortion of Linearizing Transforms 149 9.1 Linearizing Coefficient Expression in Nonlinear Functions 149 9.2 The Associated Distortion 151 9.3 Sequential Coefficient Evaluation 154 9.4 Takeaway 155 Exercises 155 10 Optimization Algorithms 157 10.1 Introduction 157 10.2 Optimization Concepts 157 10.3 Gradient-Based Optimization 159 10.3.1 Numerical Derivative Evaluation 159 10.3.2 Steepest Descent – The Gradient 161 10.3.3 Cauchy’s Method 162 10.3.4 Incremental Steepest Descent (ISD) 163 10.3.5 Newton–Raphson (NR) 163 10.3.6 Levenberg–Marquardt (LM) 165 10.3.7 Modified LM 166 10.3.8 Generalized Reduced Gradient (GRG) 167 10.3.9 Work Assessment 167 10.3.10 Successive Quadratic (SQ) 167 10.3.11 Perspective 168 10.4 Direct Search Optimizers 168 10.4.1 Cyclic Heuristic Direct Search 169 10.4.2 Multiplayer Direct Search Algorithms 170 10.4.3 Leapfrogging 171 10.5 Takeaway 173 11 Multiple Optima 176 11.1 Introduction 176 11.2 Quantifying the Probability of Finding the Global Best 178 11.3 Approaches to Find the Global Optimum 179 11.4 Best-of-N Rule for Regression Starts 180 11.5 Interpreting the CDF 182 11.6 Takeaway 184 12 Regression Convergence Criteria 185 12.1 Introduction 185 12.2 Convergence versus Stopping 185 12.3 Traditional Criteria for Claiming Convergence 186 12.4 Combining DV Influence on OF 188 12.5 Use Relative Impact as Convergence Criterion 189 12.6 Steady-State Convergence Criterion 190 12.7 Neural Network Validation 197 12.8 Takeaway 198 Exercises 198 13 Model Design – Desired and Undesired Model Characteristics and Effects 199 13.1 Introduction 199 13.2 Redundant Coefficients 199 13.3 Coefficient Correlation 201 13.4 Asymptotic and Uncertainty Effects When Model is Inverted 203 13.5 Irrelevant Coefficients 205 13.6 Poles and Sign Flips w.r.t. the DV 206 13.7 Too Many Adjustable Coefficients or Too Many Regressors 206 13.8 Irrelevant Model Coefficients 215 13.8.1 Standard Error of the Estimate 216 13.8.2 Backward Elimination 216 13.8.3 Logical Tests 216 13.8.4 Propagation of Uncertainty 216 13.8.5 Bootstrapping 217 13.9 Scale-Up or Scale-Down Transition to New Phenomena 217 13.10 Takeaway 218 Exercises 218 14 Data Pre- and Post-processing 220 14.1 Introduction 220 14.2 Pre-processing Techniques 221 14.2.1 Steady- and Transient-State Selection 221 14.2.2 Internal Consistency 221 14.2.3 Truncation 222 14.2.4 Averaging and Voting 222 14.2.5 Data Reconciliation 223 14.2.6 Real-Time Noise Filtering for Noise Reduction (MA, FoF, STF) 224 14.2.7 Real-Time Noise filtering for Outlier Removal (Median Filter) 227 14.2.8 Real-Time Noise Filtering, Statistical Process Control 228 14.2.9 Imputation of Input Data 230 14.3 Post-processing 231 14.3.1 Outliers and Rejection Criterion 231 14.3.2 Bimodal Residual Distributions 233 14.3.3 Imputation of Response Data 235 14.4 Takeaway 235 Exercises 235 15 Incremental Model Adjustment 237 15.1 Introduction 237 15.2 Choosing the Adjustable Coefficient in Phenomenological Models 238 15.3 Simple Approach 238 15.4 An Alternate Approach 240 15.5 Other Approaches 241 15.6 Takeaway 241 Exercises 241 16 Model and Experimental Validation 242 16.1 Introduction 242 16.1.1 Concepts 242 16.1.2 Deterministic Models 244 16.1.3 Stochastic Models 246 16.1.4 Reality! 249 16.2 Logic-Based Validation Criteria 250 16.3 Data-Based Validation Criteria and Statistical Tests 251 16.3.1 Continuous-Valued, Deterministic, Steady State, or End-of-Batch 251 16.3.2 Continuous-Valued, Deterministic, Transient 263 16.3.3 Class/Discrete/Rank-Valued, Deterministic, Batch, or Steady State 264 16.3.4 Continuous-Valued, Stochastic, Batch, or Steady State 265 16.3.5 Test for Normally Distributed Residuals 266 16.3.6 Experimental Procedure Validation 266 16.4 Model Discrimination 267 16.4.1 Mechanistic Models 267 16.4.2 Purely Empirical Models 268 16.5 Procedure Summary 268 16.6 Alternate Validation Approaches 269 16.7 Takeaway 270 Exercises 270 17 Model Prediction Uncertainty 272 17.1 Introduction 272 17.2 Bootstrapping 273 17.3 Takeaway 276 18 Design of Experiments for Model Development and Validation 277 18.1 Concept – Plan and Data 277 18.2 Sufficiently Small Experimental Uncertainty – Methodology 277 18.3 Screening Designs – A Good Plan for an Alternate Purpose 281 18.4 Experimental Design – A Plan for Validation and Discrimination 282 18.4.1 Continually Redesign 282 18.4.2 Experimental Plan 283 18.5 EHS&LP 286 18.6 Visual Examples of Undesired Designs 287 18.7 Example for an Experimental Plan 289 18.8 Takeaway 291 Exercises 292 19 Utility versus Perfection 293 19.1 Competing and Conflicting Measures of Excellence 293 19.2 Attributes for Model Utility Evaluation 294 19.3 Takeaway 295 Exercises 296 20 Troubleshooting 297 20.1 Introduction 297 20.2 Bimodal and Multimodal Residuals 297 20.3 Trends in the Residuals 298 20.4 Parameter Correlation 298 20.5 Convergence Criterion – Too Tight, Too Loose 299 20.6 Overfitting (Memorization) 300 20.7 Solution Procedure Encounters Execution Errors 300 20.8 Not a Sharp CDF (OF) 300 20.9 Outliers 301 20.10 Average Residual Not Zero 302 20.11 Irrelevant Model Coefficients 302 20.12 Data Work-Up after the Trials 302 20.13 Too Many rs! 303 20.14 Propagation of Uncertainty Does Not Match Residuals 303 20.15 Multiple Optima 304 20.16 Very Slow Progress 304 20.17 All Residuals are Zero 304 20.18 Takeaway 305 Exercises 305 Part IV CASE STUDIES AND DATA 21 Case Studies 309 21.1 Valve Characterization 309 21.2 CO2 Orifice Calibration 311 21.3 Enrollment Trend 312 21.4 Algae Response to Sunlight Intensity 314 21.5 Batch Reaction Kinetics 316 Appendix A: VBA Primer: Brief on VBA Programming – Excel in Office 2013 319 Appendix B: Leapfrogging Optimizer Code for Steady-State Models 328 Appendix C: Bootstrapping with Static Model 341 References and Further Reading 350 Index 355

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Book SynopsisAn Introduction to Models and Modeling in the Earth and Environmental Sciences offers students and professionals the opportunity to learn about groundwater modeling, starting from the basics. Using clear, physically-intuitive examples, the author systematically takes us on a tour that begins with the simplest representations of fluid flow and builds through the most important equations of groundwater hydrology. Along the way, we learn how to develop a conceptual understanding of a system, how to choose boundary and initial conditions, and how to exploit model symmetry. Other important topics covered include non-dimensionalization, sensitivity, and finite differences. Written in an eclectic and readable style that will win over even math-phobic students, this text lays the foundation for a successful career in modeling and is accessible to anyone that has completed two semesters of Calculus. Although the popular imTable of ContentsAbout the companion website, xi Introduction, 1 1 Modeling basics, 4 1.1 Learning to model, 4 1.2 Three cardinal rules of modeling, 5 1.3 How can I evaluate my model?, 7 1.4 Conclusions, 8 2 A model of exponential decay, 9 2.1 Exponential decay, 9 2.2 The Bandurraga Basin, Idaho, 10 2.3 Getting organized, 10 2.4 Nondimensionalization, 17 2.5 Solving for θ, 19 2.6 Calibrating the model to the data, 21 2.7 Extending the model, 23 2.8 A numerical solution for exponential decay, 26 2.9 Conclusions, 28 2.10 Problems, 29 3 A model of water quality, 31 3.1 Oases in the desert, 31 3.2 Understanding the problem, 32 3.3 Model development, 32 3.4 Evaluating the model, 37 3.5 Applying the model, 38 3.6 Conclusions, 39 3.7 Problems, 40 4 The Laplace equation, 42 4.1 Laplace’s equation, 42 4.2 The Elysian Fields, 43 4.3 Model development, 44 4.4 Quantifying the conceptual model, 47 4.5 Nondimensionalization, 48 4.6 Solving the governing equation, 49 4.7 What does it mean?, 50 4.8 Numerical approximation of the second derivative, 54 4.9 Conclusions, 57 4.10 Problems, 58 5 The Poisson equation, 62 5.1 Poisson’s equation, 62 5.2 Alcatraz island, 63 5.3 Understanding the problem, 65 5.4 Quantifying the conceptual model, 74 5.5 Nondimensionalization, 76 5.6 Seeking a solution, 79 5.7 An alternative nondimensionalization, 82 5.8 Conclusions, 84 5.9 Problems, 85 6 The transient diffusion equation, 87 6.1 The diffusion equation, 87 6.2 The Twelve Labors of Hercules, 88 6.3 The Augean Stables, 90 6.4 Carrying out the plan, 92 6.5 An analytical solution, 100 6.6 Evaluating the solution, 109 6.7 Transient finite differences, 114 6.8 Conclusions, 118 6.9 Problems, 119 7 The Theis equation, 122 7.1 The Knight of the Sorrowful Figure, 122 7.2 Statement of the problem, 124 7.3 The governing equation, 125 7.4 Boundary conditions, 127 7.5 Nondimensionalization, 128 7.6 Solving the governing equation, 132 7.7 Theis and the “well function”, 134 7.8 Back to the beginning, 135 7.9 Violating the model assumptions, 138 7.10 Conclusions, 139 7.11 Problems, 140 8 The transport equation, 141 8.1 The advection–dispersion equation, 141 8.2 The problem child, 143 8.3 The Augean Stables, revisited, 144 8.4 Defining the problem, 144 8.5 The governing equation, 146 8.6 Nondimensionalization, 148 8.7 Analytical solutions, 152 8.8 Cauchy conditions, 165 8.9 Retardation and dispersion, 167 8.10 Numerical solution of the ADE, 169 8.11 Conclusions, 173 8.12 Problems, 174 9 Heterogeneity and anisotropy, 177 9.1 Understanding the problem, 177 9.2 Heterogeneity and the representative elemental volume, 179 9.3 Heterogeneity and effective properties, 180 9.4 Anisotropy in porous media, 187 9.5 Layered media, 188 9.6 Numerical simulation, 189 9.7 Some additional considerations, 191 9.8 Conclusions, 192 9.9 Problems, 192 10 Approximation, error, and sensitivity, 195 10.1 Things we almost know, 195 10.2 Approximation using derivatives, 196 10.3 Improving our estimates, 197 10.4 Bounding errors, 199 10.5 Model sensitivity, 201 10.6 Conclusions, 206 10.7 Problems, 207 11 A case study, 210 11.1 The Borax Lake Hot Springs, 210 11.2 Study motivation and conceptual model, 212 11.3 Defining the conceptual model, 213 11.4 Model development, 215 11.5 Evaluating the solution, 224 11.6 Conclusions, 229 11.7 Problems, 230 12 Closing remarks, 233 12.1 Some final thoughts, 233 Appendix A A heuristic approach to nondimensionalization, 236 Appendix B Evaluating implicit equations, 238 B.1 Trial and error, 239 B.2 The graphical method, 239 B.3 Iteration, 240 B.4 Newton’s method, 241 Appendix C Matrix solution for implicit algorithms, 243 C.1 Solution of 1D equations, 243 C.2 Solution for higher dimensional problems, 244 C.3 The tridiagonal matrix routine TDMA, 244 Index, 247

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Book SynopsisTriangulations, and more precisely meshes, are at the heart of many problems relating to a wide variety of scientific disciplines, and in particular numerical simulations of all kinds of physical phenomena. In numerical simulations, the functional spaces of approximation used to search for solutions are defined from meshes, and in this sense these meshes play a fundamental role. This strong link between meshes and functional spaces leads us to consider advanced simulation methods in which the meshes are adapted to the behaviors of the underlying physical phenomena. This book presents the basic elements of this vision of meshing. These mesh adaptations are generally governed by a posteriori error estimators representing an increase of the error with respect to a size or metric. Independently of this metric of calculation, compliance with a geometry can also be calculated using a so-called geometric metric. The notion of mesh thus finds its meaning in the metric of its elements. Table of ContentsForeword ix Introduction xi Chapter 1. Metrics, Definitions and Properties 1 1.1. Definitions and properties 2 1.2. Metric interpolation and intersection 6 1.2.1. Metric interpolation 7 1.2.2. Metric intersection 13 1.3. Geometric metrics 14 1.3.1. Geometric metric for a curve 16 1.3.2. Geometric metric for a surface 17 1.3.3. Turning any metric into a geometric metric 23 1.4. Meshing metrics 23 1.5. Metrics gradation 24 1.6. Element metric 31 1.6.1. Metric of a simplicial element 31 1.6.2. Metric of a non-simplicial element 37 1.6.3. Metric of an element of arbitrary degree 38 1.7. Element shape and metric quality 38 1.8. Practical computations in the presence of a metric 46 1.8.1. Calculation of the length 46 1.8.2. The calculation of an angle, area or volume 49 Chapter 2. Interpolation Errors and Metrics 53 2.1. Some properties 54 2.2. Interpolation error of a quadratic function 55 2.3. Bézier formulation and interpolation error 62 2.3.1. For a quadratic function 63 2.3.2. For a cubic function 66 2.3.3. For a polynomial function of arbitrary degree 80 2.3.4. Error threshold or mesh density 85 2.4. Computations of discrete derivatives 86 2.4.1. The L2 double projection method 86 2.4.2. Green formula 88 2.4.3. Least square and Taylor 89 Chapter 3. Curve Meshing 93 3.1. Parametric curve meshing 95 3.1.1. Curve in R3 95 3.1.2. About metrics used and computations of lengths 99 3.1.3. Curve plotted on a patch 103 3.2. Discrete curve meshing 104 3.3. Remeshing a meshed curve 104 Chapter 4. Simplicial Meshing 107 4.1. Definitions 108 4.2. Variety (surface) meshing 109 4.2.1. Patch-based meshing 110 4.2.2. Discrete surface remeshing 119 4.2.3. Meshing using a volume mesher 120 4.3. The meshing of a plane or of a volume domain 122 4.3.1. Tree-based method 123 4.3.2. Front-based method 126 4.3.3. Delaunay-based method 129 4.3.4. Remeshing of a meshed domain 134 4.4. Other generation methods? 136 Chapter 5. Non-simplicial Meshing 141 5.1. Definitions 142 5.2. Variety meshing 143 5.3. Construction methods for meshing a planar or volume domain 145 5.3.1. Cylindrical geometry and extrusion method 147 5.3.2. Algebraic methods and block-based methods 148 5.3.3. Tree-based method 172 5.3.4. Pairing method 174 5.3.5. Polygonal or polyhedral cell meshing 176 5.3.6. Construction of boundary layers 177 5.4. Other generation methods 182 5.4.1. “Q-morphism” or “H-morphism” meshing 182 5.4.2. Meshing using a reference frame field 183 5.5. Topological invariants (quadrilaterals and hexahedra) 185 Chapter 6. High-order Mesh Construction 195 6.1. Straight meshes 196 6.1.1. Local node numbering 196 6.1.2. Overall node numeration 201 6.1.3. Node positions 204 6.1.4. On filling up matrices according to element degrees 207 6.2. Construction of curved meshes 208 6.2.1. First-degree mesh 209 6.2.2. Node creation 209 6.2.3. Deformation and validation 210 6.2.4. General scheme 211 6.3. Curved meshes on a variety, curve or surface 215 Chapter 7. Mesh Optimization 225 7.1. Toward a definition of quality 226 7.2. Optimization process 233 7.2.1. Global methods 233 7.2.1.1. Optimization of a cost function 233 7.2.1.2. Iterative relaxation of the position of vertices by duality (simplices) 234 7.2.1.3. Global optimization of the position of vertices (quadrilaterals and hexahedra) 235 7.2.2. Local operators and local methods 236 7.2.2.1. Vertex moves by barycentering 236 7.2.2.2. Vertex moves and Laplacian operator 237 7.2.2.3. Moving or removing vertices and flips by insertion or reinsertion 241 7.2.2.4. Edge flips 241 7.2.2.5. Cluster of edge flips 243 7.2.2.6. Edge or face flip by reinsertion 244 7.2.2.7. Edge slicing 244 7.2.2.8. Removal of an edge by merging 245 7.2.2.9. Metric field update 246 7.2.2.10. Topological and metric criteria 246 7.2.2.11. Strategies 246 7.3. Planar mesh 248 7.4. Surface mesh 250 7.5. Volume meshing 251 7.6. High-degree meshing 254 Chapter 8. Mesh Adaptation 265 8.1. Generic framework for adaptive computation, the continuous mesh 266 8.1.1. Duality between discrete and continuous geometric entities 267 8.1.2. Duality between discrete and continuous interpolation error 269 8.1.3. Discrete–continuous duality in one diagram 272 8.2. Optimal control of the interpolation error in Lp-norm 272 8.3. Generic scheme of stationary adaptation 279 8.3.1. Error estimators 282 8.3.2. Interpolation of solution fields 287 8.4. Unsteady adaptation 289 8.4.1. Space–time error estimators based on the characteristics of the solution 290 8.4.2. Extension of the error analysis for the fixed-point algorithm for unsteady mesh adaptation 291 8.4.3. Mesh adaptation for unsteady problems 292 8.4.4. Unsteady mesh adaptation targeted at a function of interest 294 8.4.5. Conservative interpolation of solution fields 295 8.5. Mobile geometry with or without deformation 297 8.5.1. General context of the adaptation for mobile and/or deformable geometries 297 8.5.2. ALE continuous optimal mesh minimizing the interpolation error in Lp-norm 298 8.5.3. Space–time error estimator for moving geometry problems 300 Chapter 9. Meshing and Parallelism 303 9.1. Renumbering via a filling curve 304 9.2. Parallelism: two memory paradigms and different strategies 307 9.3. Algorithm parallelization for mesh construction 312 9.4. Parallelization of a mesh construction process, partition then meshing 324 9.5. Mesh parallelization, meshing then partition 326 Chapter 10. Applications 331 10.1. Surface meshing 332 10.2. In computational fluid dynamics 334 10.3. Computational solid mechanics 341 10.4. Computational electromagnetism 345 10.5. Renumbering and parallelism 346 10.6. Other more exotic applications 349 Chapter 11. Some Algorithms and Formulas 353 11.1. Local numbering of nodes of high-order elements 354 11.2. Length computations etc., in the presence of a metric field 364 11.3. Quality 369 Conclusions and Perspectives 373 Bibliography 375 Index 387

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