Mathematical Modeling and Simulation

Kai Velten is a professor of mathematics at the University of Applied Sciences, Wiesbaden, Germany, and a modeling and simulation consultant. Having studied mathematics, physics and economics at the Universities of Gottingen and Bonn, he worked at Braunschweig Technical University (Institute of Geoecology, 1990-93) and at Erlangen University (Institute of Applied Mathematics, 1994-95). From 1996-2000, he held a post as project manager and group leader at the Fraunhofer-ITWM in Kaiserslautern (consultant projects for the industry). His research emphasizes differential equation models and is documented in 34 scientific publications and one patent.

• Preface xiii1 Principles of Mathematical Modeling 11.1 A Complex World Needs Models 11.2 Systems, Models, Simulations 31.2.1 Teleological Nature of Modeling and Simulation 41.2.2 Modeling and Simulation Scheme 41.2.3 Simulation 71.2.4 System 71.2.5 Conceptual and Physical Models 81.3 Mathematics as a Natural Modeling Language 91.3.1 Input–Output Systems 91.3.2 General Form of Experimental Data 101.3.3 Distinguished Role of Numerical Data 101.4 Definition of Mathematical Models 111.5 Examples and Some More Definitions 131.5.1 State Variables and System Parameters 151.5.2 Using Computer Algebra Software 181.5.3 The Problem Solving Scheme 191.5.4 Strategies to Set up Simple Models 201.5.4.1 Mixture Problem 241.5.4.2 Tank Labeling Problem 271.5.5 Linear Programming 301.5.6 Modeling a Black Box System 311.6 Even More Definitions 341.6.1 Phenomenological and Mechanistic Models 341.6.2 Stationary and Instationary models 381.6.3 Distributed and Lumped models 381.7 Classification of Mathematical Models 391.7.1 From Black to White Box Models 401.7.2 SQM Space Classification: S Axis 411.7.3 SQM Space Classification: Q Axis 421.7.4 SQM Space Classification: M Axis 431.8 Everything Looks Like a Nail? 452 Phenomenological Models 472.1 Elementary Statistics 482.1.1 Descriptive Statistics 482.1.1.1 Using Calc 492.1.1.2 Using the R Commander 512.1.2 Random Processes and Probability 522.1.2.1 Random Variables 532.1.2.2 Probability 532.1.2.3 Densities and Distributions 552.1.2.4 The Uniform Distribution 572.1.2.5 The Normal Distribution 572.1.2.6 Expected Value and Standard Deviation 582.1.2.7 More on Distributions 602.1.3 Inferential Statistics 602.1.3.1 Is Crop A’s Yield Really Higher? 612.1.3.2 Structure of a Hypothesis Test 612.1.3.3 The t test 622.1.3.4 Testing Regression Parameters 632.1.3.5 Analysis of Variance 632.2 Linear Regression 652.2.1 The Linear Regression Problem 652.2.2 Solution Using Software 662.2.3 The Coefficient of Determination 682.2.4 Interpretation of the Regression Coefficients 702.2.5 Understanding LinRegEx1.r 702.2.6 Nonlinear Linear Regression 722.3 Multiple Linear Regression 742.3.1 The Multiple Linear Regression Problem 742.3.2 Solution Using Software 762.3.3 Cross-Validation 782.4 Nonlinear Regression 802.4.1 The Nonlinear Regression Problem 802.4.2 Solution Using Software 812.4.3 Multiple Nonlinear Regression 832.4.4 Implicit and Vector-Valued Problems 862.5 Neural Networks 872.5.1 General Idea 872.5.2 Feed-Forward Neural Networks 892.5.3 Solution Using Software 912.5.4 Interpretation of the Results 922.5.5 Generalization and Overfitting 952.5.6 Several Inputs Example 972.6 Design of Experiments 992.6.1 Completely Randomized Design 1002.6.2 Randomized Complete Block Design 1032.6.3 Latin Square and More Advanced Designs 1042.6.4 Factorial Designs 1062.6.5 Optimal Sample Size 1082.7 Other Phenomenological Modeling Approaches 1092.7.1 Soft Computing 1092.7.1.1 Fuzzy Model of a Washing Machine 1102.7.2 Discrete Event Simulation 1112.7.3 Signal Processing 1133 Mechanistic Models I: ODEs 1173.1 Distinguished Role of Differential Equations 1173.2 Introductory Examples 1183.2.1 Archaeology Analogy 1183.2.2 Body Temperature 1203.2.2.1 Phenomenological Model 1203.2.2.2 Application 1213.2.3 Alarm Clock 1223.2.3.1 Need for a Mechanistic Model 1223.2.3.2 Applying the Modeling and Simulation Scheme 1233.2.3.3 Setting Up the Equations 1253.2.3.4 Comparing Model and Data 1263.2.3.5 Validation Fails – What Now? 1273.2.3.6 A Different Way to Explain the Temperature Memory 1283.2.3.7 Limitations of the Model 1293.3 General Idea of ODE’s 1303.3.1 Intrinsic Meaning of π 1303.3.2 E X Solves An Ode 1303.3.3 Infinitely Many Degrees of Freedom 1313.3.4 Intrinsic Meaning of the Exponential Function 1323.3.5 ODEs as a Function Generator 1343.4 Setting Up ODE Models 1353.4.1 Body Temperature Example 1353.4.1.1 Formulation of an ODE Model 1353.4.1.2 ODE Reveals the Mechanism 1363.4.1.3 ODE’s Connect Data and Theory 1373.4.1.4 Three Ways to Set up ODEs 1383.4.2 Alarm Clock Example 1393.4.2.1 A System of Two ODEs 1393.4.2.2 Parameter Values Based on A priori Information 1403.4.2.3 Result of a Hand-fit 1413.4.2.4 A Look into the Black Box 1423.5 Some Theory You Should Know 1433.5.1 Basic Concepts 1433.5.2 First-order ODEs 1453.5.3 Autonomous, Implicit, and Explicit ODEs 1463.5.4 The Initial Value Problem 1463.5.5 Boundary Value Problems 1473.5.6 Example of Nonuniqueness 1493.5.7 ODE Systems 1503.5.8 Linear versus Nonlinear 1523.6 Solution of ODE’s: Overview 1533.6.1 Toward the Limits of Your Patience 1533.6.2 Closed Form versus Numerical Solutions 1543.7 Closed Form Solutions 1563.7.1 Right-hand Side Independent of the Independent Variable 1563.7.1.1 General and Particular Solutions 1563.7.1.2 Solution by Integration 1573.7.1.3 Using Computer Algebra Software 1583.7.1.4 Imposing Initial Conditions 1603.7.2 Separation of Variables 1613.7.2.1 Application to the Body Temperature Model 1643.7.2.2 Solution Using Maxima and Mathematica 1653.7.3 Variation of Constants 1663.7.3.1 Application to the Body Temperature Model 1673.7.3.2 Using Computer Algebra Software 1693.7.3.3 Application to the Alarm Clock Model 1703.7.3.4 Interpretation of the Result 1713.7.4 Dust Particles in the ODE Universe 1733.8 Numerical Solutions 1743.8.1 Algorithms 1753.8.1.1 The Euler Method 1753.8.1.2 Example Application 1763.8.1.3 Order of Convergence 1783.8.1.4 Stiffness 1793.8.2 Solving ODE’s Using Maxima 1803.8.2.1 Heuristic Error Control 1813.8.2.2 ODE Systems 1823.8.3 Solving ODEs Using R 1843.8.3.1 Defining the ODE 1843.8.3.2 Defining Model and Program Control Parameters 1863.8.3.3 Local Error Control in lsoda 1863.8.3.4 Effect of the Local Error Tolerances 1873.8.3.5 A Rule of Thumb to Set the Tolerances 1883.8.3.6 The Call of lsoda 1893.8.3.7 Example Applications 1903.9 Fitting ODE’s to Data 1943.9.1 Parameter Estimation in the Alarm Clock Model 1943.9.1.1 Coupling lsoda with nls 1953.9.1.2 Estimating One Parameter 1973.9.1.3 Estimating Two Parameters 1983.9.1.4 Estimating Initial Values 1993.9.1.5 Sensitivity of the Parameter Estimates 2003.9.2 The General Parameter Estimation Problem 2013.9.2.1 One State Variable Characterized by Data 2023.9.2.2 Several State Variables Characterized by Data 2033.9.3 Indirect Measurements Using Parameter Estimation 2043.10 More Examples 2053.10.1 Predator–Prey Interaction 2053.10.1.1 Lotka–Volterra Model 2053.10.1.2 General Dynamical Behavior 2073.10.1.3 Nondimensionalization 2083.10.1.4 Phase Plane Plots 2093.10.2 Wine Fermentation 2113.10.2.1 Setting Up a Mathematical Model 2123.10.2.2 Yeast 2133.10.2.3 Ethanol and Sugar 2153.10.2.4 Nitrogen 2163.10.2.5 Using a Hand-fit to Estimate N 0 2173.10.2.6 Parameter Estimation 2193.10.2.7 Problems with Nonautonomous Models 2203.10.2.8 Converting Data into a Function 2223.10.2.9 Using Weighting Factors 2223.10.3 Pharmacokinetics 2233.10.4 Plant Growth 2264 Mechanistic Models II: PDEs 2294.1 Introduction 2294.1.1 Limitations of ODE Models 2294.1.2 Overview: Strange Animals, Sounds, and Smells 2304.1.3 Two Problems You Should Be Able to Solve 2314.2 The Heat Equation 2334.2.1 Fourier’s Law 2344.2.2 Conservation of Energy 2354.2.3 Heat Equation = Fourier’s Law + Energy Conservation 2364.2.4 Heat Equation in Multidimensions 2384.2.5 Anisotropic Case 2384.2.6 Understanding Off-diagonal Conductivities 2394.3 Some Theory You Should Know 2414.3.1 Partial Differential Equations 2414.3.1.1 First-order PDEs 2424.3.1.2 Second-order PDEs 2434.3.1.3 Linear versus Nonlinear 2434.3.1.4 Elliptic, Parabolic, and Hyperbolic Equations 2444.3.2 Initial and Boundary Conditions 2454.3.2.1 Well Posedness 2464.3.2.2 A Rule of Thumb 2464.3.2.3 Dirichlet and Neumann Conditions 2474.3.3 Symmetry and Dimensionality 2484.3.3.1 1D Example 2494.3.3.2 2D Example 2514.3.3.3 3D Example 2524.3.3.4 Rotational Symmetry 2524.3.3.5 Mirror Symmetry 2534.3.3.6 Symmetry and Periodic Boundary Conditions 2534.4 Closed Form Solutions 2544.4.1 Problem 1 2554.4.2 Separation of Variables 2554.4.3 A Particular Solution for Validation 2574.5 Numerical Solution of PDE’s 2574.6 The Finite Difference Method 2584.6.1 Replacing Derivatives with Finite Differences 2584.6.2 Formulating an Algorithm 2594.6.3 Implementation in R 2604.6.4 Error and Stability Issues 2624.6.5 Explicit and Implicit Schemes 2634.6.6 Computing Electrostatic Potentials 2644.6.7 Iterative Methods for the Linear Equations 2644.6.8 Billions of Unknowns 2654.7 The Finite-Element Method 2664.7.1 Weak Formulation of PDEs 2674.7.2 Approximation of the Weak Formulation 2694.7.3 Appropriate Choice of the Basis Functions 2704.7.4 Generalization to Multidimensions 2714.7.5 Summary of the Main Steps 2724.8 Finite-element Software 2744.9 A Sample Session Using Salome-Meca 2764.9.1 Geometry Definition Step 2774.9.1.1 Organization of the GUI 2774.9.1.2 Constructing the Geometrical Primitives 2784.9.1.3 Excising the Sphere 2794.9.1.4 Defining the Boundaries 2814.9.2 Mesh Generation Step 2814.9.3 Problem Definition and Solution Step 2834.9.4 Postprocessing Step 2854.10 A Look Beyond the Heat Equation 2864.10.1 Diffusion and Convection 2884.10.2 Flow in Porous Media 2904.10.2.1 Impregnation Processes 2914.10.2.2 Two-phase Flow 2934.10.2.3 Water Retention and Relative Permeability 2934.10.2.4 Asparagus Drip Irrigation 2954.10.2.5 Multiphase Flow and Poroelasticity 2964.10.3 Computational Fluid Dynamics (CFD) 2964.10.3.1 Navier–Stokes Equations 2964.10.3.2 Backward Facing Step Problem 2984.10.3.3 Solution Using Code-Saturne 2994.10.3.4 Postprocessing Using Salome-Meca 3014.10.3.5 Coupled Problems 3024.10.4 Structural Mechanics 3034.10.4.1 Linear Static Elasticity 3034.10.4.2 Example: Eye Tonometry 3064.11 Other Mechanistic Modeling Approaches 3094.11.1 Difference Equations 3094.11.2 Cellular Automata 3104.11.3 Optimal Control Problems 3124.11.4 Differential-algebraic Problems 3144.11.5 Inverse Problems 314A CAELinux and the Book Software 317B R (Programming Language and Software Environment) 321B.1 Using R in a Konsole Window 321B.1.1 Batch Mode 321B.1.2 Command Mode 322B.2 R Commander 322C Maxima 323C. 1 Using Maxima in a Konsole Window 323C.1. 1 Batch Mode 323C.1. 2 Command Mode 323C. 2 wxMaxima 324References 325Index 335

"Very solid introductory text at the undergraduate level aimed at wide audience. Perfectly fits introductory modeling courses at colleges and universities that prefer to use open-source software rather than commercial one, and is an enjoyable reading in the first place. Highly recommended both as a main text and a supplementary one. (...) This delightful book has two unbeatable features that should absolutely win the audience (...) First of all, it illuminates many important conceptual ideas of mathematical modelling (...) Second, (...) this book enthusiastically promotes open-source software that works on most computers and operating systems and is freely available on the web. (...) Professor Velten suggests an elegant approach to mathematical modeling, carefully going through all important steps from identification of a problem, definition of the associated system under study and analysis of the system's properties to design of a mathematical model for the system, its numerical simulation and validation."(Yuri V. Rogovchenko, Zentralblatt MATH, European Mathematical Society) "The book is certainly a reference for those, beginners or professional, who search for a complete and easy to follow step-by-step guide in the amazing world of modeling and simulation (...) it is shown that mathematical models and simulation, if adequately used, help to reduce experimental costs by a better exploration of the information content of experimental data (...) it is explained how to analyze a real problem arising from science or engineering and how to best describe it through a mathematical model. A number of examples help the reader to follow step by step the basics of modelling."(Marcello Vasta, Meccanica: International Journal of Theoretical and Applied Mechanics, Vol. 44(3), 2009) "The broad subject area covered in this book reflects the background of the author, an experienced mathematical consultant and academic (...) This book differs from almost all other available modeling books in that the author addresses both mechanistic and statistical models as well as "hybrid" models. Since many problems coming out of industrial and medical applications in recent years require hybrid models, this text is timely. The modeling range is enormous (...) In this single chapter ("Phenomenological Models") he manages to cover almost all the material one would expect to find in an undergraduate statistics program. (...) Parameter sensitivity and overfitting problems are discussed in a very simple context - very nice! (...) The author points out that, by translating a real-world problem into a mathematical form, one brings to bear on that problem the vast knowledge and powerful and free software tools available within the "mathematical universe", and his aim is to enable the reader to source this information. (...) I believe the author has succeeded in providing access to the available tools and an understanding of how to go about using these tools to solve real-world problems."Neville Fowkes (University of Western Australia) in: SIAM Rev. 53(2), 2011, pp. 387-388 (Society of Industrial and Applied Mathematics, Philadelphia, USA)