Mathematical Modeling and Simulation: Introduction for Scientists and Engineers

✍ Scribed by Kai Velten, Dominik M. Schmidt, Katrin Kahlen

• Publisher: Wiley-VCH
• Year: 2024
• Tongue: English
• Leaves: 499
• Edition: 2
• Category: Library

✦ Synopsis

Learn to use modeling and simulation methods to attack real-world problems, from physics to engineering, from life sciences to process engineering

Reviews of the first edition (2009):

"Perfectly fits introductory modeling courses [...] and is an enjoyable reading in the first place. Highly recommended [...]"
Zentralblatt MATH, European Mathematical Society, 2009

"This book differs from almost all other available modeling books in that [the authors address] both mechanistic and statistical models as well as 'hybrid' models. [...] The modeling range is enormous."
SIAM Society of Industrial and Applied Mathematics, USA, 2011

This completely revised and substantially extended second edition answers the most important questions in the field of modeling: What is a mathematical model? What types of models do exist? Which model is appropriate for a particular problem? What are simulation, parameter estimation, and validation? What kind of mathematical problems appear and how can these be efficiently solved using professional free of charge open source software?

The book addresses undergraduates and practitioners alike. Although only basic knowledge of calculus and linear algebra is required, the most important mathematical structures are discussed in sufficient detail, ranging from statistical models to partial differential equations and accompanied by examples from biology, ecology, economics, medicine, agricultural, chemical, electrical, mechanical, and process engineering.

About 200 pages of additional material include a unique chapter on virtualization, Crash Courses on the data analysis and programming languages R and Python and on the computer algebra language Maxima, many new methods and examples scattered throughout the book, an update of all software-related procedures, and a comprehensive book software providing templates for typical modeling tasks in thousands of code lines. The book software includes GmLinux, an operating system specifically designed for this book providing preconfigured and ready-to-use installations of OpenFOAM, Salome, FreeCAD/CfdOF workbench, ParaView, R, Maxima/wxMaxima, Python, Rstudio, Quarto/Markdown and other free of charge open source software used in the book.

✦ Table of Contents

Cover
Title Page
Copyright
Contents
Preface
Chapter 1 Principles of Mathematical Modeling
1.1 A Complex World Needs Models
1.2 Systems, Models, Simulations
1.2.1 Teleological Nature of Modeling and Simulation
1.2.2 Modeling and Simulation Scheme
1.2.3 Simulation
1.2.4 System
1.2.5 Conceptual and Physical Models
1.3 Mathematics as a Natural Modeling Language
1.3.1 Input–Output Systems
1.3.2 General Form of Experimental Data
1.3.3 Distinguished Role of Numerical Data
1.4 Definition of Mathematical Models
1.5 Examples and Some More Definitions
1.5.1 State Variables and System Parameters
1.5.2 Using Computer Algebra Software
1.5.3 The Problem‐Solving Scheme
1.5.4 Strategies to Set Up Simple Models
1.5.4.1 Mixture Problem
1.5.4.2 Tank Labeling Problem
1.5.4.3 Financial Mathematics
1.5.5 Linear Programming
1.5.6 Modeling a Black Box System
1.6 Even More Definitions
1.6.1 Phenomenological and Mechanistic Models
1.6.2 Stationary and Instationary Models
1.6.3 Distributed and Lumped Models
1.7 Classification of Mathematical Models
1.7.1 From Black to White Box Models
1.7.2 SQM Space Classification: S Axis
1.7.3 SQM Space Classification: Q Axis
1.7.4 SQM Space Classification: M Axis
1.8 Everything Looks Like a Nail?
Chapter 2 Phenomenological Models
2.1 Elementary Statistics
2.1.1 Descriptive Statistics
2.1.1.1 Using Calc or Excel
2.1.1.2 Using R in RStudio
2.1.1.3 Roadmap for a First Analysis
2.1.2 Random Processes and Probability
2.1.2.1 Random Variables
2.1.2.2 Probability
2.1.2.3 Densities and Distributions
2.1.2.4 The Uniform Distribution
2.1.2.5 The Normal Distribution
2.1.2.6 Expected Value and Standard Deviation
2.1.2.7 More on Distributions
2.1.2.8 Quantiles and Confidence Intervals
2.1.3 Inferential Statistics
2.1.3.1 Is Crop A's Yield Really Higher?
2.1.3.2 Structure of a Hypothesis Test
2.1.3.3 The t‐test
2.1.3.4 Testing Normality
2.1.3.5 Type I/II Errors, Power, and Effect Size
2.1.3.6 Testing Regression Parameters
2.1.3.7 Analysis of Variance
2.2 Linear Regression
2.2.1 The Linear Regression Problem
2.2.2 Solution Using Software
2.2.3 The Coefficient of Determination
2.2.4 Interpretation of the Regression Coefficients
2.2.5 Checking Assumptions
2.2.6 Nonlinear Linear Regression
2.3 Multiple Linear Regression
2.3.1 The Multiple Linear Regression Problem
2.3.2 Solution Using Software
2.3.3 Cross‐Validation
2.4 Nonlinear Regression
2.4.1 The Nonlinear Regression Problem
2.4.2 Solution Using Software
2.4.3 Multiple Nonlinear Regression
2.4.4 Implicit and Vector‐Valued Problems
2.5 Smoothing Splines
2.6 Neural Networks
2.6.1 General Idea
2.6.2 Feed‐Forward Neural Networks
2.6.3 Solution Using Software
2.6.4 Interpretation of the Results
2.6.5 Generalization and Overfitting
2.6.6 Several Inputs Example
2.7 Big Data Analysis
2.7.1 From Data to Knowledge
2.7.2 Artificial Data
2.7.3 Influencing Factors and Interactions for z1
2.7.4 Influencing Factors and Interactions for z2 and z3
2.7.5 Dimensional Reduction and Classification
2.7.5.1 Principal Component Analysis, Factor Analysis, and Correspondence Analysis
2.7.5.2 Classification
2.7.6 Conclusions
2.8 Signal Processing
2.8.1 Example, Idea, and Useful R Packages
2.8.2 Time‐Series Classification Using tsfresh, Python and R
2.9 Design of Experiments
2.9.1 Completely Randomized Design
2.9.2 Randomized Complete Block Design
2.9.3 Latin Square Design
2.9.4 Factorial Designs
2.9.5 Optimal Sample Size
2.9.6 DOE Workflow
2.9.7 Optimal Designs
2.10 Other Phenomenological Modeling Approaches
2.10.1 Soft Computing
2.10.1.1 Fuzzy Model of a Washing Machine
2.10.2 Discrete Event Simulation
Chapter 3 Mechanistic Models I: ODEs
3.1 Distinguished Role of Differential Equations
3.2 Introductory Examples
3.2.1 Archaeology Analogy
3.2.2 Body Temperature
3.2.2.1 Phenomenological Model
3.2.2.2 Application
3.2.3 Alarm Clock
3.2.3.1 Need for a Mechanistic Model
3.2.3.2 Applying the Modeling and Simulation Scheme
3.2.3.3 Setting Up the Equations
3.2.3.4 Comparing Model and Data
3.2.3.5 Validation Fails – What Now?
3.2.3.6 A Different Way to Explain the Temperature Memory
3.2.3.7 Limitations of the Model
3.3 General Idea of ODE's
3.3.1 Intrinsic Meaning of pi
3.3.2 ex Solves an ODE
3.3.3 Infinitely Many Degrees of Freedom
3.3.4 Intrinsic Meaning of the Exponential Function
3.3.5 ODEs as a Function Generator
3.4 Setting Up ODE Models
3.4.1 Body Temperature Example
3.4.1.1 Formulation of an ODE Model
3.4.1.2 ODE Reveals the Mechanism
3.4.1.3 ODE's Connect Data and Theory
3.4.1.4 Three Ways to Set Up ODEs
3.4.2 Alarm Clock Example
3.4.2.1 A System of Two ODEs
3.4.2.2 Parameter Values Based on A Priori Information
3.4.2.3 Result of a Hand‐Fit
3.4.2.4 A Look into the Black Box
3.5 Some Theory You Should Know
3.5.1 Basic Concepts
3.5.2 First‐Order ODEs
3.5.3 Autonomous, Implicit, and Explicit ODEs
3.5.4 The Initial Value Problem
3.5.5 Boundary Value Problems
3.5.6 Example of Nonuniqueness
3.5.7 ODE Systems
3.5.8 Linear Versus Nonlinear
3.6 Solution of ODE's: Overview
3.6.1 Toward the Limits of Your Patience
3.6.2 Closed Form Versus Numerical Solutions
3.7 Closed Form Solutions
3.7.1 Right‐Hand Side Independent of the Independent Variable
3.7.1.1 General and Particular Solutions
3.7.1.2 Solution by Integration
3.7.1.3 Using Computer Algebra Software
3.7.1.4 Imposing Initial Conditions
3.7.2 Separation of Variables
3.7.2.1 Application to the Body Temperature Model
3.7.2.2 Solution Using Maxima and Mathematica
3.7.3 Variation of Constants
3.7.3.1 Application to the Body Temperature Model
3.7.3.2 Using Computer Algebra Software
3.7.3.3 Application to the Alarm Clock Model
3.7.3.4 Interpretation of the Result
3.7.4 Dust Particles in the ODE Universe
3.8 Numerical Solutions
3.8.1 Algorithms
3.8.1.1 The Euler Method
3.8.1.2 Example Application
3.8.1.3 Order of Convergence
3.8.1.4 Stiffness
3.8.2 Solving ODE's Using Maxima
3.8.2.1 Heuristic Error Control
3.8.2.2 ODE Systems
3.8.3 Solving ODEs Using R and lsoda
3.8.3.1 Local Error Control in lsoda
3.8.3.2 Effect of the Local Error Tolerances
3.8.3.3 A Rule of Thumb to Set the Tolerances
3.8.3.4 Example Applications
3.9 Fitting ODE's to Data
3.9.1 Parameter Estimation in the Alarm Clock Model
3.9.1.1 Estimating Two Parameters
3.9.1.2 Estimating Initial Values
3.9.1.3 Sensitivity of the Parameter Estimates
3.9.2 The General Parameter Estimation Problem
3.9.2.1 One State Variable Characterized by Data
3.9.2.2 Several State Variables Characterized by Data
3.9.3 Indirect Measurements Using Parameter Estimation
3.10 More Examples
3.10.1 Predator–Prey Interaction
3.10.1.1 Lotka–Volterra Model
3.10.1.2 General Dynamical Behavior
3.10.1.3 Nondimensionalization
3.10.1.4 Phase Plane Plots
3.10.2 Wine Fermentation
3.10.2.1 Setting Up a Mathematical Model
3.10.2.2 Yeast
3.10.2.3 Ethanol and Sugar
3.10.2.4 Nitrogen
3.10.2.5 Using a Hand‐Fit to Estimate N0
3.10.2.6 Parameter Estimation
3.10.2.7 Problems with Nonautonomous Models
3.10.2.8 Converting Data into a Function
3.10.2.9 Using Weighting Factors
3.10.3 Pharmacokinetics
3.10.4 Plant Growth
Chapter 4 Mechanistic Models II: PDEs
4.1 Introduction
4.1.1 Limitations of ODE Models
4.1.2 Overview: Strange Animals, Sounds, and Smells
4.1.3 Two Problems You Should Be Able to Solve
4.2 The Heat Equation
4.2.1 Fourier's Law
4.2.2 Conservation of Energy
4.2.3 Heat Equation = Fourier's Law + Energy Conservation
4.2.4 Heat Equation in Multidimensions
4.2.5 Anisotropic Case
4.2.6 Understanding Off‐diagonal Conductivities
4.3 Some Theory You Should Know
4.3.1 Partial Differential Equations
4.3.1.1 First‐Order PDEs
4.3.1.2 Second‐Order PDEs
4.3.1.3 Linear Versus Nonlinear
4.3.1.4 Elliptic, Parabolic, and Hyperbolic Equations
4.3.2 Initial and Boundary Conditions
4.3.2.1 Well Posedness
4.3.2.2 A Rule of Thumb
4.3.2.3 Dirichlet and Neumann Conditions
4.3.3 Symmetry and Dimensionality
4.3.3.1 1D Example
4.3.3.2 2D Example
4.3.3.3 3D Example
4.3.3.4 Rotational Symmetry
4.3.3.5 Mirror Symmetry
4.3.3.6 Symmetry and Periodic Boundary Conditions
4.4 Closed‐Form Solutions
4.4.1 Problem 1
4.4.2 Separation of Variables
4.4.3 A Particular Solution for Validation
4.5 Numerical Solution of PDEs
4.6 The Finite Difference Method
4.6.1 Replacing Derivatives with Finite Differences
4.6.2 Formulating an Algorithm
4.6.3 Implementation in R
4.6.4 Error and Stability Issues
4.6.5 Explicit and Implicit Schemes
4.6.6 Computing Electrostatic Potentials
4.6.7 Iterative Methods for the Linear Equations
4.6.8 Billions of Unknowns
4.7 The Finite Element Method
4.7.1 Weak Formulation of PDEs
4.7.2 Approximation of the Weak Formulation
4.7.3 Appropriate Choice of the Basis Functions
4.7.4 Generalization to Multidimensions
4.7.5 Summary of the Main Steps
4.8 The Finite Volume Method
4.8.1 Weak Formulations of Conservation Equations
4.8.2 Discretization
4.8.3 Evaluation of the Fluxes
4.8.4 A Simple 1D Finite Volume Method
4.8.5 Error and Stability Issues
4.8.6 Notes on the Finite Volume Method
4.9 Software Packages to Solve PDEs
4.10 A Sample Session on the Numerical Solution of Thermal Conduction
4.10.1 Geometry Definition Step
4.10.1.1 Organization of the GUI
4.10.1.2 Constructing the Geometrical Primitives
4.10.1.3 Excising the Sphere
4.10.1.4 Defining the Boundaries
4.10.2 OpenFOAM Terminal
4.10.3 Mesh Generation Step
4.10.3.1 Blockmesh
4.10.3.2 Surface Features
4.10.3.3 SnappyHexMesh
4.10.4 Case Definition Step
4.10.4.1 Transport Properties
4.10.4.2 Boundary and Initial Conditions
4.10.4.3 Solution Settings
4.10.5 Solving the Problem
4.10.6 Postprocessing
4.10.7 Excursus: Mesh Visualization
4.10.8 Excursus: Steady‐State Time Discretization with the SIMPLE Algorithm
4.10.9 List of Terminal Commands
4.11 A Look Beyond the Heat Equation
4.11.1 Diffusion and Convection
4.11.2 Flow in Porous Media
4.11.2.1 Impregnation Processes
4.11.2.2 Two‐Phase Flow in Porous Media
4.11.2.3 Water Retention and Relative Permeability
4.11.2.4 Asparagus Drip Irrigation
4.11.2.5 Multiphase Flow and Poroelasticity
4.11.3 Structural Mechanics
4.11.3.1 Linear Static Elasticity
4.11.3.2 Example: Eye Tonometry
4.12 Computational Fluid Dynamics (CFD)
4.12.1 Navier–Stokes Equations
4.12.2 Coupled Problems
4.13 Numerical Solutions of Example Flow Problems
4.13.1 Single‐Phase Flow
4.13.1.1 Backward‐Facing Step Problem
4.13.1.2 Creating the Geometry in FreeCAD
4.13.1.3 CfdOF Workbench and Mesh Generation
4.13.1.4 Physics Model
4.13.1.5 Solution Process
4.13.1.6 Postprocessing
4.13.2 Two‐Phase Flow
4.13.2.1 Problem Description
4.13.2.2 Creating and Exporting the Geometry in Salome
4.13.2.3 Geometry Import and Boundary Definition
4.13.2.4 Physics Model and Phase Definitions
4.13.2.5 Mesh Generation
4.13.2.6 Solution Process
4.13.2.7 Postprocessing
4.14 Other Mechanistic Modeling Approaches
4.14.1 Difference Equations
4.14.2 Cellular Automata
4.14.3 Optimal Control Problems
4.14.4 Differential‐Algebraic Problems
4.14.5 Inverse Problems
Chapter 5 Virtualization
5.1 Introduction
5.2 Virtual Plants
5.2.1 Static Models Using 3D‐Digitization
5.2.1.1 Deriving Geometric Characteristics
5.2.1.2 Shape Models for Plant Organs
5.2.1.3 Allometric, Growth, and Response Functions
5.2.2 Dynamic Models Using Lindenmayer Systems
5.2.3 Functional–Structural Plant Models
5.3 Examples of Advanced Applications
5.3.1 Greenhouse Cucumber
5.3.1.1 Light Distribution Within Cucumber Canopies
5.3.1.2 Photosynthesis and its Limitations
5.3.1.3 Salt Effects
5.3.1.4 How to Predict Responses of Plant Architecture to Changing Light Environments
5.3.1.5 How to Predict Fruit Growth, While Accounting for Changing Plant Architectures
5.3.2 Virtual Riesling Vineyard
5.3.2.1 Introduction to the Model Virtual Riesling and Simulation Studies on Vineyard Level
5.3.2.2 Learning from Historical Temperature Data
5.3.2.3 How to Predict Effects of Leaf Removal on Light Absorption in a Vineyard
5.3.2.4 Toward Predicting Sunburn of Grapevine Berries
5.3.2.5 Toward Better Understanding How CO2 Enrichment Impacts the Growth of Grapevine
Chapter 6 Crashcourses and Book Software
6.1 Crashcourse R
6.1.1 Preliminaries
6.1.1.1 Installation
6.1.1.2 Book Software and Course Data
6.1.2 Basic Workflow
6.1.2.1 Working Directory
6.1.2.2 Console Mode
6.1.2.3 Programming Mode
6.1.2.4 Documentation Mode
6.1.2.5 Preprocessing and Analysis of Spreadsheet Data
6.1.2.6 Tidyverse and Recommended Default Packages
6.1.2.7 Help and Documentation
6.1.3 Knowledge Database
6.1.3.1 Working Environments
6.1.3.2 Data
6.1.3.3 Plots
6.1.3.4 Programming
6.2 Crashcourse Maxima
6.2.1 Code Examples in the Book Software
6.2.2 Software, Documentation, and Interfaces
6.2.3 Basics
6.2.4 Equations
6.2.5 Functions, Derivatives, Integrals
6.2.6 Plots
6.2.7 Differential Equations
6.2.8 Complex Programs and Mixed Topics
6.3 CrashCrashcourse Python (and all the rest)
6.4 Book Software and GmLinux
References
Index
EULA