Mathematical Modeling and Simulation - Introduction for Scientist and Engineers

Mathematical Modeling and Simulation......Page 3
Contents......Page 7
Preface......Page 15
1.1 A Complex World Needs Models......Page 17
1.2 Systems, Models, Simulations......Page 19
1.2.2 Modeling and Simulation Scheme......Page 20
1.2.4 System......Page 23
1.2.5 Conceptual and Physical Models......Page 24
1.3.1 Input–Output Systems......Page 25
1.3.3 Distinguished Role of Numerical Data......Page 26
1.4 Definition of Mathematical Models......Page 27
1.5 Examples and Some More Definitions......Page 29
1.5.1 State Variables and System Parameters......Page 31
1.5.2 Using Computer Algebra Software......Page 34
1.5.3 The Problem Solving Scheme......Page 35
1.5.4 Strategies to Set up Simple Models......Page 36
1.5.4.1 Mixture Problem......Page 40
1.5.4.2 Tank Labeling Problem......Page 43
1.5.5 Linear Programming......Page 46
1.5.6 Modeling a Black Box System......Page 47
1.6.1 Phenomenological and Mechanistic Models......Page 50
1.6.3 Distributed and Lumped models......Page 54
1.7 Classification of Mathematical Models......Page 55
1.7.1 From Black to White Box Models......Page 56
1.7.2 SQM Space Classification: S Axis......Page 57
1.7.3 SQM Space Classification: Q Axis......Page 58
1.7.4 SQM Space Classification: M Axis......Page 59
1.8 Everything Looks Like a Nail?......Page 61
2 Phenomenological Models......Page 63
2.1.1 Descriptive Statistics......Page 64
2.1.1.1 Using Calc......Page 65
2.1.1.2 Using the R Commander......Page 67
2.1.2 Random Processes and Probability......Page 68
2.1.2.2 Probability......Page 69
2.1.2.3 Densities and Distributions......Page 71
2.1.2.5 The Normal Distribution......Page 73
2.1.2.6 Expected Value and Standard Deviation......Page 74
2.1.3 Inferential Statistics......Page 76
2.1.3.2 Structure of a Hypothesis Test......Page 77
2.1.3.3 The t test......Page 78
2.1.3.5 Analysis of Variance......Page 79
2.2.1 The Linear Regression Problem......Page 81
2.2.2 Solution Using Software......Page 82
2.2.3 The Coefficient of Determination......Page 84
2.2.5 Understanding LinRegEx1.r......Page 86
2.2.6 Nonlinear Linear Regression......Page 88
2.3.1 The Multiple Linear Regression Problem......Page 90
2.3.2 Solution Using Software......Page 92
2.3.3 Cross-Validation......Page 94
2.4.1 The Nonlinear Regression Problem......Page 96
2.4.2 Solution Using Software......Page 97
2.4.3 Multiple Nonlinear Regression......Page 99
2.4.4 Implicit and Vector-Valued Problems......Page 102
2.5.1 General Idea......Page 103
2.5.2 Feed-Forward Neural Networks......Page 105
2.5.3 Solution Using Software......Page 107
2.5.4 Interpretation of the Results......Page 108
2.5.5 Generalization and Overfitting......Page 111
2.5.6 Several Inputs Example......Page 113
2.6 Design of Experiments......Page 115
2.6.1 Completely Randomized Design......Page 116
2.6.2 Randomized Complete Block Design......Page 119
2.6.3 Latin Square and More Advanced Designs......Page 120
2.6.4 Factorial Designs......Page 122
2.6.5 Optimal Sample Size......Page 124
2.7.1 Soft Computing......Page 125
2.7.1.1 Fuzzy Model of a Washing Machine......Page 126
2.7.2 Discrete Event Simulation......Page 127
2.7.3 Signal Processing......Page 129
3.1 Distinguished Role of Differential Equations......Page 133
3.2.1 Archaeology Analogy......Page 134
3.2.2.1 Phenomenological Model......Page 136
3.2.2.2 Application......Page 137
3.2.3.1 Need for a Mechanistic Model......Page 138
3.2.3.2 Applying the Modeling and Simulation Scheme......Page 139
3.2.3.3 Setting Up the Equations......Page 141
3.2.3.4 Comparing Model and Data......Page 142
3.2.3.5 Validation Fails – What Now?......Page 143
3.2.3.6 A Different Way to Explain the Temperature Memory......Page 144
3.2.3.7 Limitations of the Model......Page 145
3.3.2 e(x) Solves an ODE......Page 146
3.3.3 Infinitely Many Degrees of Freedom......Page 147
3.3.4 Intrinsic Meaning of the Exponential Function......Page 148
3.3.5 ODEs as a Function Generator......Page 150
3.4.1.1 Formulation of an ODE Model......Page 151
3.4.1.2 ODE Reveals the Mechanism......Page 152
3.4.1.3 ODE’s Connect Data and Theory......Page 153
3.4.1.4 Three Ways to Set up ODEs......Page 154
3.4.2.1 A System of Two ODEs......Page 155
3.4.2.2 Parameter Values Based on A priori Information......Page 156
3.4.2.3 Result of a Hand-fit......Page 157
3.4.2.4 A Look into the Black Box......Page 158
3.5.1 Basic Concepts......Page 159
3.5.2 First-order ODEs......Page 161
3.5.4 The Initial Value Problem......Page 162
3.5.5 Boundary Value Problems......Page 163
3.5.6 Example of Nonuniqueness......Page 165
3.5.7 ODE Systems......Page 166
3.5.8 Linear versus Nonlinear......Page 168
3.6.1 Toward the Limits of Your Patience......Page 169
3.6.2 Closed Form versus Numerical Solutions......Page 170
3.7.1.1 General and Particular Solutions......Page 172
3.7.1.2 Solution by Integration......Page 173
3.7.1.3 Using Computer Algebra Software......Page 174
3.7.1.4 Imposing Initial Conditions......Page 176
3.7.2 Separation of Variables......Page 177
3.7.2.1 Application to the Body Temperature Model......Page 180
3.7.2.2 Solution Using Maxima and Mathematica......Page 181
3.7.3 Variation of Constants......Page 182
3.7.3.1 Application to the Body Temperature Model......Page 183
3.7.3.2 Using Computer Algebra Software......Page 185
3.7.3.3 Application to the Alarm Clock Model......Page 186
3.7.3.4 Interpretation of the Result......Page 187
3.7.4 Dust Particles in the ODE Universe......Page 189
3.8 Numerical Solutions......Page 190
3.8.1.1 The Euler Method......Page 191
3.8.1.2 Example Application......Page 192
3.8.1.3 Order of Convergence......Page 194
3.8.1.4 Stiffness......Page 195
3.8.2 Solving ODE’s Using Maxima......Page 196
3.8.2.1 Heuristic Error Control......Page 197
3.8.2.2 ODE Systems......Page 198
3.8.3.1 Defining the ODE......Page 200
3.8.3.3 Local Error Control in lsoda......Page 202
3.8.3.4 Effect of the Local Error Tolerances......Page 203
3.8.3.5 A Rule of Thumb to Set the Tolerances......Page 204
3.8.3.6 The Call of lsoda......Page 205
3.8.3.7 Example Applications......Page 206
3.9.1 Parameter Estimation in the Alarm Clock Model......Page 210
3.9.1.1 Coupling lsoda with nls......Page 211
3.9.1.2 Estimating One Parameter......Page 213
3.9.1.3 Estimating Two Parameters......Page 214
3.9.1.4 Estimating Initial Values......Page 215
3.9.1.5 Sensitivity of the Parameter Estimates......Page 216
3.9.2 The General Parameter Estimation Problem......Page 217
3.9.2.1 One State Variable Characterized by Data......Page 218
3.9.2.2 Several State Variables Characterized by Data......Page 219
3.9.3 Indirect Measurements Using Parameter Estimation......Page 220
3.10.1.1 Lotka–Volterra Model......Page 221
3.10.1.2 General Dynamical Behavior......Page 223
3.10.1.3 Nondimensionalization......Page 224
3.10.1.4 Phase Plane Plots......Page 225
3.10.2 Wine Fermentation......Page 227
3.10.2.1 Setting Up a Mathematical Model......Page 228
3.10.2.2 Yeast......Page 229
3.10.2.3 Ethanol and Sugar......Page 231
3.10.2.4 Nitrogen......Page 232
3.10.2.5 Using a Hand-fit to Estimate N(0)......Page 233
3.10.2.6 Parameter Estimation......Page 235
3.10.2.7 Problems with Nonautonomous Models......Page 236
3.10.2.9 Using Weighting Factors......Page 238
3.10.3 Pharmacokinetics......Page 239
3.10.4 Plant Growth......Page 242
4.1.1 Limitations of ODE Models......Page 245
4.1.2 Overview: Strange Animals, Sounds, and Smells......Page 246
4.1.3 Two Problems You Should Be Able to Solve......Page 247
4.2 The Heat Equation......Page 249
4.2.1 Fourier’s Law......Page 250
4.2.2 Conservation of Energy......Page 251
4.2.3 Heat Equation = Fourier’s Law + Energy Conservation......Page 252
4.2.5 Anisotropic Case......Page 254
4.2.6 Understanding Off-diagonal Conductivities......Page 255
4.3.1 Partial Differential Equations......Page 257
4.3.1.1 First-order PDEs......Page 258
4.3.1.3 Linear versus Nonlinear......Page 259
4.3.1.4 Elliptic, Parabolic, and Hyperbolic Equations......Page 260
4.3.2 Initial and Boundary Conditions......Page 261
4.3.2.2 A Rule of Thumb......Page 262
4.3.2.3 Dirichlet and Neumann Conditions......Page 263
4.3.3 Symmetry and Dimensionality......Page 264
4.3.3.1 1D Example......Page 265
4.3.3.2 2D Example......Page 267
4.3.3.4 Rotational Symmetry......Page 268
4.3.3.6 Symmetry and Periodic Boundary Conditions......Page 269
4.4 Closed Form Solutions......Page 270
4.4.2 Separation of Variables......Page 271
4.5 Numerical Solution of PDE’s......Page 273
4.6.1 Replacing Derivatives with Finite Differences......Page 274
4.6.2 Formulating an Algorithm......Page 275
4.6.3 Implementation in R......Page 276
4.6.4 Error and Stability Issues......Page 278
4.6.5 Explicit and Implicit Schemes......Page 279
4.6.7 Iterative Methods for the Linear Equations......Page 280
4.6.8 Billions of Unknowns......Page 281
4.7 The Finite-Element Method......Page 282
4.7.1 Weak Formulation of PDEs......Page 283
4.7.2 Approximation of the Weak Formulation......Page 285
4.7.3 Appropriate Choice of the Basis Functions......Page 286
4.7.4 Generalization to Multidimensions......Page 287
4.7.5 Summary of the Main Steps......Page 288
4.8 Finite-element Software......Page 290
4.9 A Sample Session Using Salome-Meca......Page 292
4.9.1.1 Organization of the GUI......Page 293
4.9.1.2 Constructing the Geometrical Primitives......Page 294
4.9.1.3 Excising the Sphere......Page 295
4.9.2 Mesh Generation Step......Page 297
4.9.3 Problem Definition and Solution Step......Page 299
4.9.4 Postprocessing Step......Page 301
4.10 A Look Beyond the Heat Equation......Page 302
4.10.1 Diffusion and Convection......Page 304
4.10.2 Flow in Porous Media......Page 306
4.10.2.1 Impregnation Processes......Page 307
4.10.2.3 Water Retention and Relative Permeability......Page 309
4.10.2.4 Asparagus Drip Irrigation......Page 311
4.10.3.1 Navier–Stokes Equations......Page 312
4.10.3.2 Backward Facing Step Problem......Page 314
4.10.3.3 Solution Using Code-Saturne......Page 315
4.10.3.4 Postprocessing Using Salome-Meca......Page 317
4.10.3.5 Coupled Problems......Page 318
4.10.4.1 Linear Static Elasticity......Page 319
4.10.4.2 Example: Eye Tonometry......Page 322
4.11.1 Difference Equations......Page 325
4.11.2 Cellular Automata......Page 326
4.11.3 Optimal Control Problems......Page 328
4.11.5 Inverse Problems......Page 330
A CAELinux and the Book Software......Page 333
B.1.1 Batch Mode......Page 337
B.2 R Commander......Page 338
C.1.2 Command Mode......Page 339
C.2 wxMaxima......Page 340
References......Page 341
Index......Page 351