Skip to main content
main-content

Über dieses Buch

This textbook provides a step-by-step approach to numerical methods in engineering modelling. The authors provide a consistent treatment of the topic, from the ground up, to reinforce for students that numerical methods are a set of mathematical modelling tools which allow engineers to represent real-world systems and compute features of these systems with a predictable error rate. Each method presented addresses a specific type of problem, namely root-finding, optimization, integral, derivative, initial value problem, or boundary value problem, and each one encompasses a set of algorithms to solve the problem given some information and to a known error bound. The authors demonstrate that after developing a proper model and understanding of the engineering situation they are working on, engineers can break down a model into a set of specific mathematical problems, and then implement the appropriate numerical methods to solve these problems.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Modelling and Errors

Abstract
As an engineer, be it in the public or private sector, working for an employer or as an independent contractor, your job will basically boil down to this: you need to solve the problem you are given in the most efficient manner possible. If your solution is less efficient than another, you will ultimately have to pay a price for this inefficiency. This price may take many forms. It could be financial, in the form of unnecessary expenses. It could take the form of less competitive products and a reduced market share. It could be the added workload due to problems stemming from the inefficiencies of your work. It could be an intangible but very real injury to your professional reputation, which will be tarnished by being associated to inefficient work. Or it could take many other forms, all negative to you.
Richard Khoury, Douglas Wilhelm Harder

Chapter 2. Numerical Representation

Abstract
The numerical system used in the Western World today is a place-value base-10 system inherited from India through the intermediary of Arabic trade; this is why the numbers are often called Arabic numerals or more correctly Indo-Arabic numerals. However, this is not the only numerical system possible. For centuries, the Western World used the Roman system instead, which is a base-10 additive-value system (digits of a number are summed and subtracted from each other to get the value represented), and that system is still in use today, notably in names and titles. Other civilizations experimented with other bases: some precolonial Australian cultures used a base-5 system, while base-20 systems arose independently in Africa and in pre-Columbian America, and the ancient Babylonians used a base-60 counting system. Even today, despite the prevalence of the base-10 system, systems in other bases continue to be used every day: degrees, minutes, and seconds are counted in the base-60 system inherited from Babylonian astrologers, and base-12 is used to count hours in the day and months (or zodiacs) in the year.
Richard Khoury, Douglas Wilhelm Harder

Chapter 3. Iteration

Abstract
This chapter and the following four chapters introduce five basic mathematical modelling tools: iteration, linear algebra, Taylor series, interpolation, and bracketing. While they can be used as simple modelling tools on their own, their main function is to provide the basic building blocks from which numerical methods and more complex models will be built.
Richard Khoury, Douglas Wilhelm Harder

Chapter 4. Linear Algebra

Abstract
The first of the five mathematical modelling tools, introduced in the previous chapter, is iteration. The second is solving systems of linear algebraic equations and is the topic of this chapter. A system of linear algebraic equations is any set of n equations with n unknown variables x 0, …, x n−1:
$$ \begin{array}{c}{m}_{0,0}{x}_0+{m}_{0,1}{x}_1+\cdots +{m}_{0,n-1}{x}_{n-1}={b}_0\\ {}{m}_{1,0}{x}_0+{m}_{1,1}{x}_1+\cdots +{m}_{1,n-1}{x}_{n-1}={b}_1\\ {}\vdots \\ {}{m}_{n-1,0}{x}_0+{m}_{n-1,1}{x}_1+\cdots +{m}_{n-1,n-1}{x}_{n-1}={b}_{n-1}\end{array} $$
where the \( n\times n \) values m 0,0, …, m n−1,n−1 are known coefficient values that multiply the variables, and the n values b 0, …, b n−1 are the known result of each equation. A system of that form can arise in many ways in engineering practice. For example, it would be the result of taking measurements of a dynamic system at n different times. It also results from taking measurements of a static system at n different internal points. Consider, for example, the simple electrical circuit in Fig. 4.1. Four internal nodes have been identified in it. If one wants to model this circuit, for example, to be able to predict the voltage flowing between two nodes, the corresponding energy losses, or other of its properties, it is first necessary to model the voltages at each node using Kirchhoff’s current law. Removing units and with appropriate scaling, this gives the following set of four equations and four unknown variables:
$$ \begin{array}{cc}\hfill \mathrm{Node}\ 1:\hfill & \hfill \frac{v_1-0}{120}+\frac{v_1-{v}_2}{240}=-0.01\hfill \\ {}\hfill \mathrm{Node}\ 2:\hfill & \hfill \frac{v_2-0}{320}+\frac{v_2-{v}_1}{240}+\frac{v_2-{v}_3}{180}+\frac{v_2-{v}_4}{200}=0\hfill \\ {}\hfill \mathrm{Node}\ 3:\hfill & \hfill \frac{v_3-0}{160}+\frac{v_3-{v}_2}{180}+\frac{v_3-{v}_4}{360}=0\hfill \\ {}\hfill \mathrm{Node}\ 4:\hfill & \hfill \frac{v_4-{v}_2}{200}+\frac{v_4-{v}_3}{360}=0.01\hfill \end{array} $$
which is a system of four linear algebraic equations of the same form as Eq. (4.1).
Richard Khoury, Douglas Wilhelm Harder

Chapter 5. Taylor Series

Abstract
It is known that, zooming-in close enough to a curve, it will start to look like a straight line. This can be tested easily by using any graphic software to draw a curve, and then zooming into a smaller and smaller region of it. It is also the reason why the Earth appears flat to us; it is of course spherical, but humans on its surface see a small portion up close so that it appears like a plane. This leads to the intuition for the third mathematical and modelling tool in this book: it is possible to represent a high-order polynomial (such as a curve or a sphere) with a lower-order polynomial (such as a line or a plain), at least over a small region. The mathematical tool that allows this is called the Taylor series. And, since the straight line mentioned in the first intuitive example is actually the tangent (or first derivative) of the curve, it should come as no surprise that this Taylor series will make heavy use of derivatives of the functions being modelled.
Richard Khoury, Douglas Wilhelm Harder

Chapter 6. Interpolation, Regression, and Extrapolation

Abstract
Oftentimes, practicing engineers are required to develop new models of existing undocumented systems they need to understand. These could be, for example, man-made legacy systems for which documentation is outdated or missing, or natural systems that have never been properly studied. In all cases, there are no design documents or theoretical resources available to guide the modelling. The only option available is to take discrete measurements of the system and to discover the underlying mathematical function that generates these points. This chapter will present a set of mathematical and modelling tools that can perform this task.
Richard Khoury, Douglas Wilhelm Harder

Chapter 7. Bracketing

Abstract
Consider the problem of searching for the word “lemniscate” in a dictionary of thousands of pages. It would be unthinkable to find the word by reading the dictionary systematically page by page. However, since words are sorted alphabetically from A to Z in the dictionary, it is easy to open the dictionary at a random page and determine whether the letter L is before or after that page. This single step will greatly reduce the number of pages to search through. Next, select a page at random in the portion of the dictionary the word is known to be in, and determine whether the word is before or after that second point. Once a point starting with the letter L is reached, the following letter E is considered the section kept is the one containing LE, then LEM, and so on until sufficient precision is achieved (namely that the page containing the word is found).
Richard Khoury, Douglas Wilhelm Harder

Chapter 8. Root-Finding

Abstract
The of a continuous multidimensional function f(x) is any point x = x r for which the function f(r) = 0. Algorithms that discover the value of a function’s root are called algorithms, and they constitute the first numerical method presented in this book.
Richard Khoury, Douglas Wilhelm Harder

Chapter 9. Optimization

Abstract
One major challenge in engineering practice is often the need to design systems that must perform as well as possible given certain constraints. Working without constraints would be easy: when a system can be designed with no restrictions on cost, size, or components used, imagination is the only limit on what can be built. But when constraints are in place, as they always will be in practice, then not only must engineering designs respect them, but the difference between a good and a bad design will be which one can get the most done within the stated constraints.
Richard Khoury, Douglas Wilhelm Harder

Chapter 10. Differentiation

Abstract
Differentiation and its complement operation, integration, allow engineers to measure and quantify change. Measuring change is essential in engineering practice, which often deals with systems that are changing in some way, by moving, growing, filling up, decaying, discharging, or otherwise increasing or decreasing in some way.
Richard Khoury, Douglas Wilhelm Harder

Chapter 11. Integration

Abstract
Chapter 10 has already introduced the need for differentiation and integration to quantify change in engineering systems. Differentiation measures the rate of change of a parameter, and integration conversely measures the changing value of a given parameter. Chapter 10 demonstrated how important modelling change was to insure that engineering models accurately reflected reality.
Richard Khoury, Douglas Wilhelm Harder

Chapter 12. Initial Value Problems

Abstract
Consider a simple RC circuit such as the one shown in Fig. 12.1. Kirchhoff’s law states that this circuit can be modelled by the following equation:
$$ \frac{dV}{dt}=-\frac{V(t)}{RC} $$
This model would be easy to use if the voltage and the values of the resistor and capacitor are known. But what if the voltage is not known or measurable over time, and only the initial conditions of the system are known? That is to say, only the initial value of the voltage and of its derivative, along with the resistor and capacitor value, are known.
Richard Khoury, Douglas Wilhelm Harder

Chapter 13. Boundary Value Problems

Abstract
Consider a simple circuit loop with an inductor of 1 H, a resistor of 10 Ω and a capacitor of 0.25 F, as shown in Fig. 13.1. This circuit can be modelled by the following ODE:
$$ {I}^{(2)}(t)+10{I}^{(1)}(t)+4I(t)= \cos (t) $$
This model would be easy to use if the initial value of the current and its derivative was known. But what if the rate of change of the current is not known, and the only information available is the current in the circuit the moment the system was turned on and when it was shut down? That is to say, only the initial and final values of the current are known, while its derivatives are unknown.
Richard Khoury, Douglas Wilhelm Harder

Backmatter

Weitere Informationen

Premium Partner

    Marktübersichten

    Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen. 

    Bildnachweise