Mathematical Modeling and Simulation

Transcription

1 Kai Velten Mathematical Modeling and Simulation Introduction for Scientists ar d Engineers WILLYNCH Verlag (irribli & Co. l<gaa

2 I v Contents Preface XIII 1 Principles of Mathematical Modeling A Complex World Needs Models Systems, Models, Simulafions Teleological Nature of Modeling and Simulation Modeling and Simulation Scheme Simulation System Conceptual and Physical Models Mathematics as a Natural Modeling Language Input Output Systems General Form of Experimental Data Distinguished Role of Numerical Data Definition of Mathematical Models Examples and Some More Definitions State Variables and System Parameters Using Computer Algebra Software The Problem Solving Scheme Strategies to Set up Simple Models Mixture Problem Tank Labeling Problem Linear Programming Modeling a Black Box System Even More Definitions Phenomenological and Mechanistic Models Stationary and Instationary models Distributed and Lumped models Classification of Mathematical Models From Black to White Box Models SQM Space Classification: S Axis SQM Space Classification: Q Axis 42 Mathematical Modeling and Simulation: Introduction für Scientists and Engineers. Kai Velten Copyright 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN:

3 VII Contents SQM Space Classification: M Axis Everything Looks Like a Nail? 45 2 Phenomenological Models Elementary Statistics Descriptive Statistics Using Calc Using the R Commander Random Processes and Probability Random Variables Probability Densities and Distributions The Uniform Distribution The Normal Distribution Expected Value and Standard Deviation More an Distributions Inferential Statistics Is Crop A's Yield Really Higher? Structure of a Hypothesis Test The t test Testing Regression Parameters Analysis of Variance Linear Regression The Linear Regression Problem Solution Using Software The Coefficient of Determination Interpretation of the Regression Coefficients Understanding Lin Reg E x 1. r Nonlinear Linear Regression Multiple Linear Regression The Multiple Linear Regression Problem Solution Using Software Cross-Validation Nonlinear Regression The Nonlinear Regression Problem Solution Using Software Multiple Nonlinear Regression Implicit and Vector-Valued Problems Neural Networks General Idea Feed-Porward Neural Networks Solution Using Software Interpretation of the Results Generalization and Overfitting Several Inputs Example 97

4 Contents IVII 2.6 Design of Experiments Completely Randomized Design Randomized Complete Block Design Latin Square and More Advanced Designs Factorial Designs Optimal Sample Size Other Phenomenological Modeling Approaches Soft Computing Fuzzy Model of a Washing Machine Discrete Event Simulation Signal Processing Mechanistic Models I: ODEs Distinguished Role of Differential Equations Introductory Examples Archaeology Analogy Body Temperature Phenomenological Model Application Alarm Clock Need for a Mechanistic Model Applying the Modeling and Simulation Scheme Setting Up the Equations Comparing Model and Data Validation Fails What Now? A Different Way to Explain the Temperature Memory Limitations of the Model General Idea of ODE's Intrinsic Meaning of g ex Solves an ODE Infinitely Many Degrees of Freedom Intrinsic Meaning of the Exponential Function ODEs as a Function Generator Setting Up ODE Models Body Temperature Example Formulation of an ODE Model ODE Reveals the Mechanism ODE's Connect Data and Theory Three Ways to Set up ODEs Alarm Clock Example A System of Two ODEs Parameter Values Based an A priori Information Result of a Hand-fit A Look into the Black Box Some Theory You Should Know 143

5 VIII I Contents Basic Concepts First-order ODEs Autonomous, Implicit, and Explicit ODEs The Initial Value Problem Boundary Value Problems Example of Nonuniqueness ODE Systems Linear versus Nonlinear Solution of ODE's: Overview Toward the Limits of Your Patience Closed Form versus Numerical Solutions Closed Form Solutions Right-hand Side Independent of the Independent Variable General and Particular Solutions Solution by Integration Using Computer Algebra Software Imposing Initial Conditions Separation of Variables Application to the Body Temperature Model Solution Using Maxima and Mathematica Variation of Constants Application to the Body Temperature Model Using Computer Algebra Software Application to the Alarm Clock Model Interpretation of the Result Dust Partides in the ODE Universe Numerical Solutions Algorithms The Euler Method Example Application Order of Convergence Stiffness Solving ODE's Using Maxima Heuristic Error Control ODE Systems Solving ODEs Using R Defining the ODE Defining Model and Program Control Parameters Local Error Control in lsoda Effect of the Local Error Tolerances A Rule of Thumb to Set the Tolerances The Call of lsoda Example Applications Fitting ODE's to Data Parameter Estimation in the Alarm Clock Model 194

6 Contents IIX Coupling lsoda with nls Estimating One Parameter Estimating Two Parameters Estimating Initial Values Sensitivity of the Parameter Estimates The General Parameter Estimation Problem One State Variable Characterized by Data Several State Variables Characterized by Data Indirect Measurements Using Parameter Estimation More Examples Predator Prey Interaction Lotka Volterra Model General Dynamical Behavior Nondimensionalization Phase Plane Plots Wine Fermentation Setting Up a Mathematical Model Yeast Ethanol and Sugar Nitrogen Using a Hand-fit to Estimate No Parameter Estimation Problems with Nonautonomous Models Converting Data into a Function Using Weighting Factors Pharmacokinetics Plant Growth Mechanistic Models PDEs Introduction Limitations of ODE Models Overview: Strange Animals, Sounds, and Smells Two Problems You Should Be Able to Solve The Heat Equation Fourier's Law Conservation of Energy Heat Equation = Fourier's Law Energy Conservation Heat Equation in Multidimensions Anisotropic Case Understanding Off-diagonal Conductivities Some Theory You Should Know Partial Differential Equations First-order PDEs Second-order PDEs Linear versus Nonlinear 243

7 Contents Elliptic, Parabolic, and Hyperbolic Equations Initial and Boundary Conditions Well Posedness A Rule of Thumb Dirichlet and Neumann Conditions Symmetry and Dimensionality D Example D Example D Example Rotational Symmetry Mirror Symmetry Symmetry and Periodic Boundary Conditions Closed Form Solutions Problem Separation of Variables A Particular Solution for Validation Numerical Solution of PDE's The Finite Difference Method Replacing Derivatives with Finite Differences Formulating an Algorithm Implementation in R Error and Stability Issues Explicit and Implicit Schemes Computing Electrostatic Potentials Iterative Methods for the Linear Equations Billions of Unknowns The Finite-Element Method Weak Formulation of PDEs Approximation of the Weak Formulation Appropriate Choice of the Basis Functions Generalization to Multidimensions Summary of the Main Steps Finite-element Software A Sample Session Using Salome-Meca Geometry Definition Step Organization of the GUI Constructing the Geometrical Primitives Excising the Sphere Defining the Boundaries Mesh Generation Step Problem Definition and Solution Step Postprocessing Step A Look Beyond the Heat Equation Diffusion and Convection Flow in Porous Media 290

8 Contents 'XI Impregnation Processes Two-phase Flow Water Retention and Relative Permeability Asparagus Drip Irrigation Multiphase Flow and Poroelasticity Computational Fluid Dynamics (CFD) Navier Stokes Equations Backward Facing Step Problem Solution Using Code-Saturne Postprocessing Using Salome-Meca Coupled Problems Structural Mechanics Linear Static Elasticity Example: Eye Tonometry Other Mechanistic Modeling Approaches Difference Equations Cellular Automata Optimal Control Problems Differential-algebraic Problems Inverse Problems 314 A CAELinux and the Book Software 317 B R (Programming Language and Software Environment) 321 B.1 Using R in a Konsole Window 321 B.1.1 Batch Mode 321 B.1.2 Command Mode 322 B.2 R Commander 322 C Maxima 323 C.1 Using Maxima in a Konsole Window 323 C.1.1 Batch Mode 323 C.1.2 Command Mode 323 C.2 =Maxima 324 References 325 Index 335