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- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
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.
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Veröffentlichungsjahr: 2024
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Preface
1 Principles of Mathematical Modeling
1.1 A Complex World Needs Models
1.2 Systems, Models, Simulations
1.3 Mathematics as a Natural Modeling Language
1.4 Definition of Mathematical Models
1.5 Examples and Some More Definitions
1.6 Even More Definitions
1.7 Classification of Mathematical Models
1.8 Everything Looks Like a Nail?
2 Phenomenological Models
2.1 Elementary Statistics
2.2 Linear Regression
2.3 Multiple Linear Regression
2.4 Nonlinear Regression
2.5 Smoothing Splines
2.6 Neural Networks
2.7 Big Data Analysis
2.8 Signal Processing
2.9 Design of Experiments
2.10 Other Phenomenological Modeling Approaches
Note
3 Mechanistic Models I: ODEs
3.1 Distinguished Role of Differential Equations
3.2 Introductory Examples
3.3 General Idea of ODE's
3.4 Setting Up ODE Models
3.5 Some Theory You Should Know
3.6 Solution of ODE's: Overview
3.7 Closed Form Solutions
3.8 Numerical Solutions
3.9 Fitting ODE's to Data
3.10 More Examples
4 Mechanistic Models II: PDEs
4.1 Introduction
4.2 The Heat Equation
4.3 Some Theory You Should Know
4.4 Closed‐Form Solutions
4.5 Numerical Solution of PDEs
4.6 The Finite Difference Method
4.7 The Finite Element Method
4.8 The Finite Volume Method
4.9 Software Packages to Solve PDEs
4.10 A Sample Session on the Numerical Solution of Thermal Conduction
4.11 A Look Beyond the Heat Equation
4.12 Computational Fluid Dynamics (CFD)
4.13 Numerical Solutions of Example Flow Problems
4.14 Other Mechanistic Modeling Approaches
Notes
5 Virtualization
5.1 Introduction
5.2 Virtual Plants
5.3 Examples of Advanced Applications
6 Crashcourses and Book Software
6.1 Crashcourse R
6.2 Crashcourse Maxima
6.3 CrashCrashcourse Python (and all the rest)
6.4 Book Software and GmLinux
References
Index
End User License Agreement
List of Tables
Chapter 2
Table 2.1 Spring data (see
spring.ods
in the book software).
Table 2.2 Naively computed rates of change in the dataset
spring.csv
.
Table 2.3 Example of a factorial design.
Chapter 3
Table 3.1 Result of line 6 of Listing 3.16, , compared with the values of ...
Table 3.2 Parameter estimates obtained with Listing 3.24 for , and and c...
Table 3.3 State variables of the wine fermentation model.
Table 3.4 Parameters of the wine fermentation model.
Chapter 4
Table 4.1 Examples of finite element and finite volume software packages.
Chapter 6
Table 6.1 R packages used in the book software.
List of Illustrations
Chapter 1
Figure 1.1 Car as a real system and as a model.
Figure 1.2 (a) Communication of a system with the outside world. (b) General...
Figure 1.3 Tank problem.
Figure 1.4 (a) Potted plant. (b) The same potted plant written as a reduced ...
Figure 1.5 Result of Listing 1.1 in
wxMaxima
.
Figure 1.6 Problem‐solving scheme.
Figure 1.7 (a) Tank front side with volume labels. (b) Unknowns and auxiliar...
Figure 1.8 Solution erg[2] of the financing problem.
Figure 1.9 (a) System 1 with input (N) and output (cm). (b) System 1 dat...
Figure 1.10 (a) Plot of the data in
spring.csv
. (b) System 1 data with regre...
Figure 1.11 Internal mechanics of system 1.
Figure 1.12 (a) Classification of mathematical models between black and whit...
Figure 1.13 Plato's cave allegory: Don't believe that the model is the reali...
Chapter 2
Figure 2.1 Boxplot (a) and jitter plot (b) showing the dependence of tempera...
Figure 2.2 Probability density distributions computed for a normally distrib...
Figure 2.3 Histogram of book software temperature data with estimated probab...
Figure 2.4
pnorm
,
dnorm
, and
qnorm
example for , cf. Listing 2.4.
Figure 2.5 QQ plot of book software weather data temperatures (result of lin...
Figure 2.6 Type I/II error, power, and effect size example. Result of lines ...
Figure 2.7 Result of lines 10–11 of Listing 2.13.
Figure 2.8 (a) Comparison of the regression line Equation 2.54...
Figure 2.9 Result of lines 14–16 of Listing 2.18.
Figure 2.10 Comparison of predicted and measured () values of
DegWilt
usi...
Figure 2.11 Result of lines 6–8 of Listing 2.24.
Figure 2.12 (a) ...
Figure 2.13 Comparison of the regression equation with the data (a) in a con...
Figure 2.14 Velocity data for v1 and v2.
Figure 2.15 (a) Approximation of v1 by a smoothing spline . (b) Zoom into (...
Figure 2.16 Roughness of v1 and v2.
Figure 2.17 (a) Tomography of a Descemet membrane. (b) Representation of the...
Figure 2.18 (a) Graphical interpretation of multiple regression. (b) Artific...
Figure 2.19 Comparison of the neural network Equation 2.88 based on the para...
Figure 2.20 Illustration of overfitting: results of book software Code f74b7...
Figure 2.21 Comparison of a neural network predicting depending on scaled
Figure 2.22 Decision tree, that exactly reproduces all necessary input varia...
Figure 2.23 (a) Score‐plot, (b) biplot, and (c) correlation‐plot for PC1 and...
Figure 2.24 Results from PCA on the mixed
wine
data (scores, correlations, a...
Figure 2.25 Time‐series examples of types A (solid line) and B (dashed line)...
Figure 2.26 Time series of type A (circles) and B (crossed circles) in the s...
Figure 2.27 House painting example: naive experimental design, showing a wal...
Figure 2.28 Completely randomized design for the house painting example, com...
Figure 2.29 Randomized complete block design computed using Listing 2.54 (ha...
Figure 2.30 Latin square design computed using Listing 2.55 (hardness testin...
Figure 2.31 Result of line 3 of Listing 2.60 (two other parameter hypercube ...
Figure 2.32 Examples of design space coverage (span) with a limited sample s...
Chapter 3
Figure 3.1 (a) Body temperature data. (b) Body temperature data (triangles) ...
Figure 3.2 (a) Alarm clock with temperature sensor. (b) Room temperature dat...
Figure 3.3 Simplified models of the alarm clock: (a) Model A and (b) Model B...
Figure 3.4 (a) Comparison of Equation 3.9 (line) with the data of Figure 3.2...
Figure 3.5 (thick line), (thin line), and sensor temperature data. The f...
Figure 3.6
wxMaxima
sessions solving (a) Equation 3.111 and (b) Equation 3.1...
Figure 3.7 (a) Solution of Equations 3.197 and 3.198: exact solution (Equati...
Figure 3.8 Line: numerical solution of Equations 3.213 and...
Figure 3.9 (a) Line: numerical solution of Equations 3.213 and...
Figure 3.10 (a) Line: numerical approximation of based on Equations 3.215,...
Figure 3.11 Example FORTRAN code lines from
odepack.f
used in the
lsoda
comm...
Figure 3.12 (a) Line: numerical solution of Equations 3.213 and 3.214 comput...
Figure 3.13 (a) Line: numerical approximation of based on Equations 3.215,...
Figure 3.14 Line: numerical approximation of based on Equations 3.215, 3.2...
Figure 3.15 (a) Line: numerical approximation of based on Equations 3.215–...
Figure 3.16 (a) Prey (line) and predator (dashed line) population sizes obta...
Figure 3.17 Phase plot of the result in Figure 3.16b, generated using Listin...
Figure 3.18 (a) Phase plot as in Figure 3.17, including several other phase ...
Figure 3.19 Wine fermentation data from [157]. See
fermentation.csv
in the b...
Figure 3.20 Solution of Equations 3.259–3.262 using and the parameters in ...
Figure 3.21 Solution of Equations 3.259–3.262 using the estimated coefficien...
Figure 3.22 Assumed pattern of drug dosage corresponding to Equation 3.269...
Figure 3.23 Drug concentration (a) in the gastrointestinal tract and (b) in ...
Figure 3.24 Drug model as a two‐compartment model.
Figure 3.25 Solutions (a) of the exponential plant growth model, Equation 3....
Figure 3.26 Asparagus spear biomass data asparagus.csv compared with model E...
Chapter 4
Figure 4.1 (a) Cylinder used in
Problem 1
and (b) cube , containing a ...
Figure 4.2 Interval corresponding to a small part of a one‐dimensional bod...
Figure 4.3 Periodicity cells of a medium with an anisotropic effective therm...
Figure 4.4 (a) Cube ...
Figure 4.5 Example of a reduction of the computational domain due to symmetr...
Figure 4.6 Solution of Equations 4.72–4.75 for ...
Figure 4.7 (a) Example triangulation of a circle, (b) example of a locally r...
Figure 4.8 Example one‐dimensional “hat” basis functions, and a piecewise li...
Figure 4.9 Spatial discretization with (a) node‐centered finite volumes and ...
Figure 4.10 Sketch for calculating the flow at a boundary edge.
Figure 4.11 Finite volume discretization with labeled nodes and edges .
Figure 4.12 Grid for the numerical calculation with (a) control volumes in t...
Figure 4.13 Solution for the slightly modified problem 1 with...
Figure 4.14 (a)
Salome
window and its module selection box, (b) Box construc...
Figure 4.15 (a)
Salome
showing
Box_1
, (b) Point construction window.
Figure 4.16 (a) Sphere construction window, (b)
Salome
showing
Box_1
in “wir...
Figure 4.17 (a) “Cut Of Objects” window, (b)
Salome
showing
Cut_1
after appl...
Figure 4.18 “Create group” window.
Figure 4.19 Gnuplot output from
foamMonitor
: Residuals of the temperature va...
Figure 4.20 ParaView window showing the ‐normal slice through the cube at t...
Figure 4.21 ParaView window showing the ...
Figure 4.22 ParaView window showing the viewpoint of the isosurfaces for
Figure 4.23 ParaView window showing the viewpoint of the isolines for to...
Figure 4.24 Visualization of the mesh with ParaView: (a) Outside, (b) clip, ...
Figure 4.25 (a) Cross section through mica tape insulation with periodicity ...
Figure 4.26 Water retention curves for a sand (solid line, ...
Figure 4.27 (a) Example location of water pipes under an asparagus ridge and...
Figure 4.28 One‐dimensional deformation of a cylinder.
Figure 4.29 (a) Applanation tonometry measurement procedure and (b) boundari...
Figure 4.30 Three‐dimensional eye model: undeformed (a) and deformed by the ...
Figure 4.31 Effect of scleral rigidity variations on the simulated IOP readi...
Figure 4.32 Industrial cleaning of bottles (
Figure 4.33 Backward‐facing step geometry following [257]. Gray dots indicat...
Figure 4.34 (a) FreeCAD workbenches menu, (b) create polyline, and (c) raw s...
Figure 4.35 (a) Fully constrained backward‐facing step geometry in the FreeC...
Figure 4.36 Toolbar of the CfdOF – Computational fluid dynamics (CFD) workbe...
Figure 4.37 (a) Mesh parameters, (b) first mesh with uniform cell size in th...
Figure 4.38 (a) Physics model selections in CfdOF and (b) refined solver con...
Figure 4.39 Residuals plot in FreeCAD during the solution progress of the ba...
Figure 4.40 Properties of the
OpenFOAMReader1
object in ParaView launched fr...
Figure 4.41
Stream Tracer
visualization focusing on the recirculation zones ...
Figure 4.42 Line integral convolution on a surface representation (Surface L...
Figure 4.43 Line chart and spreadsheet of data for the relative streamwise v...
Figure 4.44 Dimensionless streamline velocity () at different relative dist...
Figure 4.45 Two‐phase flow geometry: A fully water‐filled smaller vessel (le...
Figure 4.46 Mesh with 25 mm base mesh grid resolution.
Figure 4.47 Phase fraction variable
alpha.Air
at the initial time ().
Figure 4.48 Isosurface of
alpha.Air
=0.5 (air‐water interface) at times 0 s, ...
Figure 4.49 Conway's game of life computed using
Conway.r
, that is included ...
Chapter 5
Figure 5.1 Examples of virtual geometries showing (a) CAD software Salome, s...
Figure 5.2 Visualization of static plant models of a cucumber plant in a gre...
Figure 5.3 Schematic overview on two characteristics of leaves, the elevatio...
Figure 5.4 Schematic representation of the ‐phyllotaxis of a cucumber plant...
Figure 5.5 Simulation output of the L‐system presented in 5.12 shown as gene...
Figure 5.6 Cucumbers stand in a modern greenhouse.
Figure 5.7 Differences in light interception measured by using top views o...
Figure 5.8 Simulated data on intercepted light represented by photosynthetic...
Figure 5.9 Schematic representation of a 1D and a 3D approach to model light...
Figure 5.10 Presentation of the main components of the
FvCB
‐model for the ne...
Figure 5.11 Simulated whole‐plant photosynthetic rates of a virtual cucumber...
Figure 5.12 Measured and simulated final internode lengths (a) and correspon...
Figure 5.13 Example of an Arrhenius curve for normalized development rates a...
Figure 5.14 Schematic representation of the core idea of modeling horizontal...
Figure 5.15 Growth duration and abortion rates of fruits at different ranks ...
Figure 5.16 Graphical presentation of a virtual vineyard at day of the yea...
Figure 5.17 Distribution of simulated budburst over time of two years using ...
Figure 5.18 Simulated absorbed photosynthetic radiation, , over time using ...
Figure 5.19 Simulated absorbed photosynthetic radiation, , over time using ...
Figure 5.20 Overview of the key drivers of sunburn in grapevine berries, a h...
Figure 5.21 Observed and simulated budburst data over 1990–2010. Observed da...
Chapter 6
Figure 6.1 Copy and paste button in the book software.
Figure 6.2
quarto1.qmd
(edited in RStudio, (a)) and the compiled version
qua
...
Figure 6.3 Panes A–D in RStudio's graphical user interface.
Guide
Cover
Table of Contents
Title Page
Copyright
Begin Reading
Index
End User License Agreement
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Mathematical Modeling and Simulation
Introduction for Scientists and Engineers
Kai VeltenDominik M. SchmidtKatrin Kahlen
Second Edition
The Authors
Prof. Dr. Kai Velten
Hochschule Geisenheim
Von‐Lade‐Straße 1
Geisenheim
Germany
65366
Dr. Dominik M. Schmidt
Hochschule Geisenheim
Von‐Lade‐Straße 1
Geisenheim
Germany
65366
Prof. Dr. Katrin Kahlen
Hochschule Geisenheim
Von‐Lade‐Straße 1
Geisenheim
Germany
65366
Cover: Simulated virtual Riesling vineyard (Section 5.3.2) and simulated wine fermation flow pattern. (Section 4.13 and Velten, K., & Schmidt, D., (2016). Numerical simulation of bubble flow homogenization in industrial scale wine fermentations. Food and Bioproducts Processing, 100, 102–117.)
All books published by WILEY‐VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
Library of Congress Card No.: applied for
British Library Cataloguing‐in‐Publication Data A catalogue record for this book is available from the British Library.
Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at <http://dnb.d‐nb.de>.
© 2024 WILEY‐VCH GmbH, Boschstraße 12, 69469 Weinheim, Germany
All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.
Print ISBN: 978‐3‐527‐41414‐7
ePDF ISBN: 978‐3‐527‐83939‐1
ePub ISBN: 978‐3‐527‐83940‐7
oBook ISBN: 978‐3‐527‐84960‐4
Preface
The purpose of computing is insight, not numbers.
R.W. Hamming [1]
“Everyone is an artist” was the central message of the famous twentieth‐century artist Joseph Beuys. “Everyone models and simulates” is the central message of this book. Mathematical modeling and simulation is a fundamental method in engineering and science, and it is absolutely valid to say that everybody uses it (even those of us who are not aware of doing so). The question is not whether to use this method or not, but rather how to use it effectively.
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 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, and medicine and agricultural, chemical, electrical, mechanical, and process engineering.
Since the 2009 edition of this book, increasing computer power led to a rapid development in the field of virtualization. This is closely related with mathematical modeling, and we decided to devote a new chapter on this important subject, along with new sections on big data analysis, smoothing splines, financial mathematics, the finite volume method, sample sessions demonstrating the numerical solution of partial differential equations (PDEs) using OpenFOAM, a Python‐based example on time series classification, crash courses on the data analysis software R and the computer algebra system Maxima, and a crash course on Python. The crashcourses refer to a great number of useful code examples and templates in the book software (see below), following a unique concise paragraph‐based design that allows references like “please read §15, §101‐112” in university courses.
The book relies exclusively on free‐of‐charge open‐source software. Comprehensive book software can be obtained at https://www.hs-geisenheim.de/mms/ and provides templates for typical modeling tasks in thousands of code lines. The book software includes GmLinux, an operating system specifically designed for this book with 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.
While our approach applies software to solve most of the mathematical problems, it nevertheless attempts to put the reader mathematically on firm ground as much as possible. Trapdoors and problems that may arise in the modeling process, in the numerical treatment of the models, or in their interpretation are indicated, and the reader is referred to the literature whenever necessary.
The book is organized as follows: Chapter 1 explains the principles of mathematical modeling and simulation, providing definitions and illustrative examples of the important concepts as well as an overview of the main types of mathematical models. After the discussion of data‐based phenomenological models in Chapter 2, the two most important classes of mechanistic models are introduced in Chapter 3 (ordinary differential equations) and Chapter 4 (partial differential equations). Chapter 5 on virtualization explains how the methods presented in the previous chapters can be used to construct and study virtual systems, followed by Chapter 6 with the aforementioned crash courses and details on the book software.
The book is dedicated to our parents; to Birgid, Julia, Theresa, Benedikt, Lukas, Axel, and Ulf (Kai Velten); to Louise, Carlotta, and Cornelius (Dominik M. Schmidt); and to Roman and Till (Katrin Kahlen). We would like to thank our colleagues at Hochschule Geisenheim University, Germany, and our research partners from all around the world for many years of inspiring collaboration and exchange of ideas. Special thanks are due to Christopher Bahr for his contributions to the Virtual Riesling model and for the simulation results used for the cover of this book.
Geisenheim, December 2023
Kai Velten
Dominik M. Schmidt
Katrin Kahlen
1Principles of Mathematical Modeling
We begin this introduction to mathematical modeling and simulation with an explanation of basic concepts and ideas, which includes definitions of terms such as system, model, simulation, and mathematical model, reflections on the objectives of mathematical modeling and simulation, characteristics of “good” mathematical models, and a classification of mathematical models. You may skip this chapter at first reading if you are just interested in a hands‐on application of specific methods explained in the later chapters of the book, such as regression or neural network methods (Chapter 2), differential equations (DEs) (in Chapters 3 and 4), or virtual plants (Chapter 5). Any professional in this field, however, should of course know about the principles of mathematical modeling and simulation. It was emphasized in the preface that everybody uses mathematical models – “even those of us who are not aware of doing so”. You will agree that it is a good idea to have an idea of what one is doing…
Our starting point is the complexity of the problems treated in science and engineering. As will be explained in Section 1.1, the difficulty of problems treated in science and engineering typically originates from the complexity of the systems under consideration, and models provide an adequate tool to break up this complexity and make a problem tractable. After giving general definitions of the terms system, model, and simulation in Section 1.2, we move on toward mathematical models in Section 1.3, where it is explained that mathematics is the natural modeling language in science and engineering. Mathematical models themselves are defined in Section 1.4, followed by a number of example applications and definitions in Sections 1.5 and 1.6. This includes the important distinction between phenomenological and mechanistic models, which has been used as the main organization principle of this book (see Section 1.6.1 and Chapters 2–5). The chapter ends with a classification of mathematical models and Golomb's famous “Don'ts of mathematical modeling” in Sections 1.7 and 1.8.
1.1 A Complex World Needs Models
Generally speaking, engineers and scientists try to understand, develop, or optimize “ systems”. Here, “system” refers to the object of interest, which can be a part of nature (such as a plant cell, an atom, a galaxy, etc.) or an artificial technological system (see Definition 1.2.3). Principally, everybody deals with systems in their everyday life in a way similar to the approach of engineers or scientists. For example, consider the problem of a table that is unstable due to an uneven floor. This is a technical system, and everybody knows what must be done to solve the problem: we just have to put suitable pieces of cardboard under the table legs. Each of us solves an abundant number of problems relating to simple technological systems of this kind during our lifetime. Beyond this, there are a great number of really difficult technical problems that can only be solved by engineers. Characteristic of these more demanding problems is a high complexity of the technical system. We would simply need no engineers if we did not have to deal with complex technical systems such as computer processors, engines, and so on. Similarly, we would not need scientists if processes such as the photosynthesis of plants could be understood as simply as an unstable table. The reason why we have scientists and engineers, virtually their right to exist, is the complexity of nature and the complexity of technological systems.
Note 1.1.1 (The complexity challenge) It is the genuine task of scientists and engineers to deal with complex systems, and to be effective in their work, they most notably need specific methods to deal with complexity.
The general strategy used by engineers or scientists to break up the complexity of their systems is the same strategy that we all use in our everyday life when we are dealing with complex systems: simplification. The idea is just this: if something is complex, make it simpler. Consider an everyday life problem related to a complex system: A car that refuses to start. In this situation, everyone knows that a look at the battery and fuel levels will solve the problem in most cases. Everyone will do this automatically, but to understand the problem‐solving strategy behind this, let us think of an alternative scenario. Assume someone is in this situation for the first time. Assume that “someone” was told how to drive a car, that they have used the car for some time, and now they are for the first time in a situation in which the car does not start. Of course, we also assume that there is no help for miles around! Then, looking under the hood for the first time, our “someone” will realize that the car, which seems simple as long as it works well, is quite a complex system. They will spend a lot of time until they eventually solve the problem, even if we admit that our “someone” is an engineer. The reason why each of us will solve this problem much faster than this “someone” is of course the simple fact that this situation is not new to us. We have experienced this situation before, and from our previous experience we know what is to be done. Conceptually, one can say that we have a simplified picture of the car in our mind similar to Figure 1.1. In the moment when we realize that our car does not start, we do not think of the car as the complex system that it really is, that is, we do not think of this conglomerate of valves, pistons, and all the kind of stuff that can be found under the hood; rather, we have this simplified picture of the car in our mind. We know that this simplified picture is appropriate in this given situation, and it guides us to look at the battery and fuel levels and then solve the problem within a short time.
Figure 1.1 Car as a real system and as a model.
This is exactly the strategy used by engineers or scientists when they deal with complex systems. When an engineer, for example, wants to reduce the fuel consumption of an engine, they will not consider that engine in its entire complexity. Rather, they will use simplified descriptions of that engine, focusing on the machine parts that affect fuel consumption. Similarly, a scientist who wants to understand the process of photosynthesis will use simplified descriptions of a plant focusing on very specific processes within a single plant cell. Anyone who wants to understand complex systems or solve problems related to complex systems needs to apply appropriate simplified descriptions of the system under consideration. This means that anyone who is concerned with complex systems needs models, since simplified descriptions of a system are models of that system by definition.
Note 1.1.2 (Role of models) To break up the complexity of a system under consideration, engineers and scientists use simplified descriptions of that system (i.e. models).
1.2 Systems, Models, Simulations
In 1965, Minsky gave the following general definition of a model [2, 3]:
Definition 1.2.1(Model) To an observer B, an object is a model of an object A to the extent that B can use to answer questions that interest him about A.
Note 1.2.1 (Formal definitions) Note that Definition 1.2.1 is a formal definition in the sense that it operates with terms such as object or observer that are not defined in a strict axiomatic sense similar to the terms used in the definitions of standard mathematical theory. The same remark applies to several other definitions in this book, including the definition of the term mathematical model in Section 1.4. Definitions of this kind are justified for practical reasons, since they allow us to talk about the formally defined terms in a concise way. An example is Definition 2.6.2 in Section 2.6.5, a concise formal definition of the term overfitting, which uses several of the previous formal definitions.
The application of Definition 1.2.1 to the car example is obvious – we just have to identify B with the car driver, A with the car itself, and with the simplified tank/battery description of the car in Figure 1.1.
1.2.1 Teleological Nature of Modeling and Simulation
An important aspect of the above definition is the fact that it includes the purpose of a model, namely, that the model helps us to answer questions and solve problems. This is important because particularly beginners in the field of modeling tend to believe that a good model is one that mimics the part of reality that it pertains to as closely as possible. But as was explained in the previous section, modeling and simulation aim at simplification, rather than at a useless production of complex copies of a complex reality, and hence, the contrary is true.
Note 1.2.2 (The best model) The best model is the simplest model that still serves its purpose, that is, which is still complex enough to help us understand a system and to solve problems. Seen in terms of a simple model, the complexity of a complex system will no longer obstruct our view, and we will virtually be able to look through the complexity of the system at the heart of things.
The entire procedure of modeling and simulation is governed by its purpose of problem‐solving – otherwise, it would be a mere l'art pour l'art. As [4] puts it, “modeling and simulation are always goal‐driven, that is, we should know the purpose of our potential model before we sit down to create it”. It is hence natural to define fundamental concepts such as the term model with a special emphasis on the purpose‐oriented or teleological nature of modeling and simulation. (Note that teleology is a philosophical discipline dealing with aims and purposes, and the term teleology itself originates from the Greek word telos, which means end or purpose [5].) Similar teleological definitions of other fundamental terms, such as system, simulation, and mathematical model, are given below.
1.2.2 Modeling and Simulation Scheme
Conceptually, the investigation of complex systems using models can be divided into the following steps:
Note 1.2.3 (Modeling and simulation scheme)
Definitions
Definition of a problem that is to be solved or of a question that is to be answered
Definition of a system, that is, a part of reality that pertains to this problem or question
Systems Analysis
Identification of parts of the system that are relevant for the problem or question
Modeling
Development of a model of the system based on the results of the system analysis step
Simulation
Application of the model to the problem or question
Derivation of a strategy to solve the problem or answer the question
Validation
Does the strategy derived in the simulation step solve the problem or answer the question for the real system?
The application of this scheme to the examples discussed above is obvious: in the car example, the problem is that the car does not start and the car itself is the system. This is the “definitions” step of the above scheme. The “systems analysis” step identifies the battery and fuel levels as the relevant parts of the system, as explained above. Then, in the “ modeling” step of the scheme, a model consisting of a battery and a tank such as in Figure 1.1 is developed. The application of this model to the given problem in the “ simulation” step of the scheme then leads to the strategy “check battery and fuel level”. This strategy can then be applied to the real car in the “validation” step. If it works, that is, if the car really starts after refilling its battery or tank, we say that the model is valid or validated. If not, we probably need a mechanic who will then look at other parts of the car, that is, who will apply more complex models of the car until the problem is solved.
In a real modeling and simulation project, the systems analysis step of the above scheme can be very time‐consuming. It will usually involve a thorough evaluation of the literature. In many cases, the literature evaluation will show that similar investigations have been performed in the past, and one should of course try to profit from the experiences made by others that are described in the literature. Beyond this, the system analysis step usually involves a lot of discussions and meetings that bring together people from different disciplines who can answer your questions regarding the system. These discussions will usually show that new data are needed for a better understanding of the system and for the validation of the models in the validation step of the above scheme. Hence, the definition of an experimental program is also another typical part of the systems analysis step.
The modeling step will also involve the identification of appropriate software that can solve the equations of the mathematical model. In many cases, it will be possible to use standard software such as the software tools discussed in the next chapters. Beyond this, it may be necessary to write your own software in cases where the mathematical model involves nonstandard equations. An example of this case is the modeling of the press section of paper machines, which involves highly convection‐dominated diffusion equations that cannot be treated by standard software with sufficient precision and hence needs specifically tailored numerical software [6].
In the validation step, the model results will be compared with experimental data. These data may come from the literature or from experiments that have been specifically designed to validate the model. Usually, a model is required to fit the data not only quantitatively but also qualitatively, in the sense that it reproduces the general shape of the data as closely as possible. See Section 3.2.3.4 for an example of a qualitative misfit between a model and data. But, of course, even a model that perfectly fits the data quantitatively and qualitatively may fail the validation step of the above scheme if it cannot be used to solve the problem that is to be solved, which is the most important criterion for a successful validation.
The modeling and simulation scheme (Note 1.2.3) focuses on the essential steps of modeling and simulation, giving a rather simplified picture of what really happens in a concrete modeling and simulation project. For different fields of application, you may find a number of more sophisticated descriptions of the modeling and simulation process in books such as [7–10]. An important thing that you should note is that a real modeling and simulation project will very rarely go straight through the steps of the above scheme; rather, there will be a lot of interaction between the individual steps of the scheme. For example, if the validation step fails, this will bring you back to one of the earlier steps in a loop‐like structure: you may then improve your model formulation, reanalyze the system, or even redefine your problem formulation (if your original problem formulation turns out to be unrealistic).
1.2.4 (Start with simple models!) To find the best model in the sense of Note 1.2.2, start with the simplest possible model and then generate a sequence of increasingly complex model formulations until the last model in the sequence passes the validation step.
1.2.3 Simulation
So far we have given a definition of the term model only. The above modeling and simulation schemes involve other terms, such as system and simulation, which we may view as being implicitly defined by their role in the above scheme. Can this be made more precise? In the literature, you will find a number of different definitions, for example, of the term simulation. These differences can be explained by different interests of the authors. For example, in a book with a focus on the so‐called discrete event simulation, which emphasizes the development of a system over time, simulation is defined as “the imitation of the operation of a real‐world process or system over time” [7]. In general terms, simulation can be defined as follows:
Definition 1.2.2 (Simulation)Simulation is the application of a model with the objective to derive strategies that help solve a problem or answer a question pertaining to a system.
Note that the term simulation originates from the Latin word “simulare”, which means “to pretend”: in a simulation, the model pretends to be the real system. A similar definition has been given by Fritzson [8] who defined simulation as “an experiment performed on a model”. Beyond this, the above definition is a teleological (purpose‐oriented) definition similar to Definition 1.2.1, that is, this definition again emphasizes the fact that simulation is always used to achieve some goal. Although Fritzson's definition is more general, the above definition reflects the real use of simulation in science and engineering more closely.
1.2.4 System
Regarding the term system, you will again find a number of different definitions in the literature, and again some of the differences between these definitions can be explained by the different interests of their authors. For example, [11] defines a system as “a collection of entities, for example, people or machines, that act and interact together toward the accomplishment of some logical end”. According to [12], a system is “a collection of objects and relations between objects”. In the context of mathematical models, we believe it makes sense to think of a “system” in very general terms. Any kind of object can serve as a system here if we have a question relating to that object and if this question can be answered using mathematics. Our view of systems is similar to a definition that has been given by [13] (see also the discussion of this definition in [4]): “A system is whatever is distinguished as a system”. [4] gave another definition of a “system” very close to our view of systems here: “A system is a potential source of data”. This definition emphasizes the fact that a system can be of scientific interest only if there is some communication between the system and the outside world, as will be discussed below in Section 1.3.1. A definition that includes the teleological principle discussed above has been given by Fritzson [8] as follows:
Definition 1.2.3 (System) A system is an object or a collection of objects whose properties we want to study.
1.2.5 Conceptual and Physical Models
The model used in the car example is something that exists in our minds only. We can write it down on paper in a few sentences and/or sketches, but it does not have any physical reality. Models of this kind are called conceptual models[12]. Conceptual models are used by each of us to solve everyday problems such as the car that refuses to start. As K.R. Popper puts it, “all life is problem‐solving”, and conceptual models provide us with an important tool to solve our everyday problems (14). They are also applied by engineers or scientists to simple problems or questions similar to the car example. If their problem or question is complex enough, however, they rely on experiments, and this leads us to other types of models. To see this, let us use the modeling and simulation scheme (Note 1.2.3) to describe a possible procedure followed by an engineer who wants to reduce the fuel consumption of an engine: In this case, the problem is the reduction of fuel consumption, and the system is the engine. Assume that the systems analysis leads the engineer to the conclusion that the fuel injection pump needs to be optimized. Typically, the engineer will then create some experimental setting where they can study the details of the fuel injection process.
Such an experimental setting is then a model in the sense that it will typically be a very simplified version of that engine, that is, it will typically involve only a few parts of the engine that are closely connected with the fuel injection process. In contrast to a conceptual model, however, it is not only an idea in our mind but also a real part of the physical world, and this is why models of this kind are called physical models[12]. The engineer will then use the physical model of the fuel injection process to derive strategies – for example, a new construction of the fuel injection pump – to reduce the engine's fuel consumption, which is the simulation step of the above modeling and simulation scheme. Afterward, in the validation step of the scheme, the potential of these new constructions to reduce fuel consumption will be tested in the engine itself, that is, in the real system. Physical models are applied by scientists in a similar way. For example, let us think of a scientist who wants to understand the photosynthesis process in plants. Similar to an engineer, the scientist will set up a simplified experimental setting – which might be some container with a plant cell culture – in which they can easily observe and measure the important variables, such as , water, light, and so on. For the same reasons as above, anything like this is a physical model. As before, any conclusion drawn from such a physical model corresponds to the simulation step of the above scheme, and the conclusions need to be validated by data obtained from the real system, that is, data obtained from real plants in this case.
Figure 1.2 (a) Communication of a system with the outside world. (b) General form of an experimental dataset.
1.3 Mathematics as a Natural Modeling Language
1.3.1 Input–Output Systems
Any system that is investigated in science or engineering must be observable in the sense that it produces some kind of output that can be measured (a system that would not satisfy this minimum requirement would have to be treated by theologians rather than by scientists or engineers). Note that this observability condition can also be satisfied by systems where nothing can be measured directly, such as black holes, which produce measurable gravitational effects in their surroundings. Most systems investigated in engineering or science also accept some kind of input data, which can then be studied in relation to the output of the system (Figure 1.2a). For example, a scientist who wants to understand photosynthesis will probably construct experiments where the carbohydrate production of a plant is measured at various levels of light, , water supply, and so on. In this case, the plant cell is the system; the light, , and water levels are the input quantities; and the measured carbohydrate production is the output quantity. Or, an engineer who wants to optimize a fuel injection pump will probably change the construction of that pump in various ways and then measure the fuel consumption resulting from these modified constructions. In this case, the fuel injection pump is the system, the construction parameters changed by the engineer are the input parameters, and the resulting fuel consumption is the output quantity.
Note 1.3.1 (Input–output systems) Scientists or engineers investigate “input–output systems”, which transform given input parameters into output parameters.
Note that there are of course situations where scientists are looking at the system itself and not at its input–output relations, for example, when a botanist just wants to describe and classify the anatomy of a newly discovered plant. Typically, however, such purely descriptive studies raise questions about the way in which the system works, and this is when input–output relations come into play. Engineers, on the other hand, are always concerned with input–output relations since they are concerned with technology. The Encyclopedia Britannica defines technology as “the application of scientific knowledge to the practical aims of human life”. These “practical aims” will usually be expressible in terms of a system output, and the tuning of system input toward optimized system output is precisely what engineers typically do, and what is in fact the genuine task of engineering.
1.3.2 General Form of Experimental Data
The experimental procedure described above is used very generally in engineering and in the (empirical) sciences to understand, develop, or optimize systems. It is useful to think of it as a means to explore black boxes. At the beginning of an experimental study, the system under investigation is similar to such a “black box” in the sense that there is some uncertainty about the processes that happen inside the system when the input is transformed into the output. In an extreme case, the experimenter may know only that “something” happens inside the system that transforms input into output, that is, the system may be really a black box. Typically, however, the experimenter will have some hypotheses about the internal processes that they want to prove or disprove in the course of their study. That is, experimenters typically are concerned with systems as gray boxes, which are located somewhere between black and white boxes (more details in Section 1.5).
Depending on the hypothesis that the experimenter wants to investigate, they confront the system with appropriate input quantities, hoping that the outputs produced by the system will help prove or disprove their hypothesis. This is similar to a question‐and‐answer game: the experimenter poses questions to the system, which is the input, and the system answers to these questions in terms of measurable output quantities. The result is a dataset of the general form shown in Figure 1.2b. In rare cases, particularly if one is concerned with very simple systems, the internal processes of the system may already be evident from the dataset itself. Typically, however, this experimental question‐and‐answer game is similar to the questioning of an oracle: we know there is some information about the system in the dataset, but it depends on the application of appropriate ideas and methods if one wants to uncover the information content of the data and, so to speak, shed some light into the black box.
1.3.3 Distinguished Role of Numerical Data
Now what is an appropriate method for the analysis of experimental datasets? To answer this question, it is important to note that in most cases experimental data are numbers and can be quantified. The input and output data of Figure 1.2b will typically consist of columns of numbers. Hence, it is natural to think of a system in mathematical terms. In fact, a system can be naturally seen as a mathematical function that maps given input quantities into output quantities (Figure 1.2a). This means that if one wants to understand the internal mechanics of a system “black box”, that is, if one wants to understand the processes inside the real system that transform input into output, a natural thing to do is to translate all these processes into mathematical operations. If this is done, one arrives at a simplified representation of the real system in mathematical terms. Now remember that a simplified description of a real system (along with a problem we want to solve) is a model by definition (Definition 1.2.1). The representation of a real system in mathematical terms is thus a mathematical model of that system.
Note 1.3.2 (Naturalness of mathematical models) Input–output systems usually generate numerical (or quantifiable) data that can be described naturally in mathematical terms.
This simple idea, that is, the mapping of the internal mechanics of real systems into mathematical operations, has proved to be extremely fruitful to the understanding, optimization, or development of systems in science and engineering. The tremendous success of this idea can only be explained by the naturalness of this approach – mathematical modeling is simply the best and most natural thing one can do if one is concerned with scientific or engineering problems. Looking back at Figure 1.2a, it is evident that mathematical structures emanate from the very heart of science and engineering. Anyone concerned with systems and their input–output relations is also concerned with mathematical problems – regardless of their preferences and regardless of whether they treat the system appropriately using mathematical models or not. The success of their work, however, depends very much on the appropriate use of mathematical models.
1.4 Definition of Mathematical Models
To understand mathematical models, let us start with a general definition. Many different definitions of mathematical models can be found in the literature. The differences between these definitions can usually be explained by the different scientific interests of their authors. For example, Bellomo and Preziosi [15] define a mathematical model to be a set of equations that can be used to compute the time–space evolution of a physical system. Although this definition suffices for the problems treated by Bellomo and Preziosi, it is obvious that it excludes a great number of mathematical models. For example, many economic or sociological problems cannot be treated in a time–space framework or based on equations alone. Thus, a more general definition of mathematical models is needed if one wants to cover all kinds of mathematical models used in science and engineering. Let us start with the following attempt of a definition:
A mathematical model is a set of mathematical statements .
Certainly, this definition covers all kinds of mathematical models used in science and engineering as required. But there is a problem with this definition. For example, a simple mathematical statement such as would be a mathematical model in the sense of this definition. In the sense of Minsky's definition of a model (Definition 1.2.1), however, such a statement is not a model as long as it lacks any connection with some system and with a question we have relating to that system. The above attempt of a definition is incomplete since it pertains to the word “mathematical” of “mathematical model” only, without any reference to purposes or goals. Following the philosophy of the teleological definitions of the terms model, simulation, and system in Section 1.2, let us define instead:
Definition 1.4.1 (Mathematical model) A mathematical model is a triplet where is a system, is a question relating to , and is a set of mathematical statements which can be used to answer .
Note that this is again a formal definition in the sense of Note 1.2.1 in Section 1.2. Again, it is justified by the mere fact that it helps us to understand the nature of mathematical models and that it allows us to talk about mathematical models in a concise way. A similar definition was given by Bender [16]: “A mathematical model is an abstract, simplified, mathematical construct related to a part of reality and created for a particular purpose”. Note that Definition 1.4.1 is not restricted to physical systems. It covers psychological models as well that may deal with essentially metaphysical quantities, such as thoughts, intentions, feelings, and so on. Even mathematics itself is covered by the above definition. Suppose, for example, that is the set of natural numbers and our question relating to is whether there are infinitely many prime numbers or not. Then, a set is a mathematical model in the sense of Definition 1.4.1 if contains the statement “There are infinitely many prime numbers” along with other statements which prove this statement. In this sense, the entire mathematical theory can be viewed as a collection of mathematical models.
The notation in Definition 1.4.1 emphasizes the chronological order in which the constituents of a mathematical model usually appear. Typically, a system is given first, then there is a question regarding that system, and only then a mathematical model is developed. Each of the constituents of the triplet is an indispensable part of the whole. Regarding , this is obvious, but and are important as well. Without , we would not be able to formulate a question ; without a question , there would be virtually “nothing to do” for the mathematical model; and without and , the remaining would be no more than “l'art pour l'art”. The formula , for example, is such a purely mathematical “l'art pour l'art” statement as long as we do not connect it with a system and a question. It becomes a mathematical model only when we define a system and a question relating to it. For example, viewed as an expression of the exponential growth period of plants (Section 3.10.4), is a mathematical model which can be used to answer questions regarding plant growth. One can say it is a genuine property of mathematical models to be more than “l'art pour l'art”, and this is exactly the intention behind the notation in Definition 2.3.1. Note that the definition of mathematical models by Bellomo and Preziosi [15] discussed above appears as a special case of Definition 1.4.1 if we restrict to physical systems, to equations, and only allow questions , which refer to the space–time evolution of .
Note 1.4.1 (More than “l'art pour l'art”) The system and the question relating to the system are indispensable parts of a mathematical model. It is a genuine property of mathematical models to be more than mathematical “l'art pour l'art”.
Let us look at another famous example that shows the importance of . Suppose we want to predict the behavior of some mechanical system . Then the appropriate mathematical model depends on the problem we want to solve, that is, on the question . If is asking for the behavior of at moderate velocities, classical (Newtonian) mechanics can be used, that is, {equations of Newtonian mechanics}. If, on the other hand, is asking for the behavior of at velocities close to the speed of light, then we have to set {equations of relativistic mechanics} instead.
1.5 Examples and Some More Definitions
Generally speaking, one can say we are concerned with mathematical models in the sense of Definition 1.4.1 whenever we perform computations in our everyday life, or whenever we apply the mathematics we have learned in schools and universities. Since everybody computes in their everyday life, everybody uses mathematical models, and this is why it was valid to say that “everyone models and simulates” in the preface of this book. Let us look at a few examples of mathematical models now, which will lead us to the definitions of some further important concepts.
Note 1.5.1 (Everyone models and simulates) Mathematical models in the sense of Definition 1.4.1 appear whenever we perform computations in our everyday life.
Suppose we want to know the mean age of some group of people. Then, we apply a mathematical model , where is that group of people, asks for their mean age, and is the mean value formula . Or, suppose we want to know the mass of some substance in the cylindrical tank of Figure 1.3, given a constant concentration of the substance in that tank. Then, a multiplication of the tank volume with gives the mass of the substance, that is,
Figure 1.3 Tank problem.
This means we apply a model , where is the tank, asks for the mass of the substance, and is Equation 1.1. An example involving more than simple algebraic operations is obtained if we assume that the concentrationin the tank in Figure 1.3 depends on the height coordinate, . In that case, Equation 1.1 turns into
This involves an integral, that is, we have entered the realms of calculus now.
Note 1.5.2 (Notational convention) Variables such as and in Equation 1.1, which are used without further specification, are always assumed to be real numbers, and functions such as in Equation 1.2 are always assumed to be real functions with suitable ranges and domains of definition (such as in the above example) unless otherwise stated.
In many mathematical models involving calculus, the question asks for the optimization of some quantity. Suppose, for example, we want to minimize the material consumption of a cylindrical tin having a volume of 1 l. In this case,
can be used to solve the problem. Denoting by and the radius and height of the tin, the first statement in Equation 1.3
