Engineering Simulation and its Applications: Algorithms and Numerical Methods

By Xin-She Yang

Engineering Simulation and its Applications: Algorithms and Numerical Methods covers the essential quantitative methods needed for engineering simulations, introducing optimization techniques that can be used in the design of systems to minimize cost and maximize efficiency. This book serves as a reference and textbook for courses such as engineering simulation, design optimization, mathematical modeling, numerical methods, data analysis, and engineering management. Diverse coverage of the various subject areas within the field puts the essential topics into a single book for easy access for graduates and senior undergraduates. It also serves as a reference book for lecturers and industrial practitioners.

• Introduces all essential algorithms and numerical methods
• Balances theory and numerical techniques
• Provides numerous worked examples

• Language: English
• Publisher: Academic Press
• Release date: Feb 1, 2024
• ISBN: 9780443140853

Xin-She Yang

Xin-She Yang obtained his DPhil in Applied Mathematics from the University of Oxford. He then worked at Cambridge University and National Physical Laboratory (UK) as a Senior Research Scientist. He is currently a Reader in Modelling and Simulation at Middlesex University London, Fellow of the Institute of Mathematics and its Application (IMA) and a Book Series Co-Editor of the Springer Tracts in Nature-Inspired Computing. He has published more than 25 books and more than 400 peer-reviewed research publications with over 82000 citations, and he has been on the prestigious list of highly cited researchers (Web of Sciences) for seven consecutive years (2016-2022).

Book preview

Chapter One: Introduction to engineering simulation

Abstract

This chapter introduces the fundamentals of engineering simulation, which starts with different types of mathematical models based on differential equations and integral equations. Continuous models, discrete models, and statistical models are explained, followed by a brief introduction to various numerical models. Relevant software tools for engineering simulation are also outlined.

Keywords

Engineering simulation; Continuous model; Discrete model; Mathematical model; Numerical model; Statistical model; Software

Engineering simulation is a vast area with many different topics and applications. This chapter introduces the fundamentals of engineering simulation with the emphasis on mathematical models and numerical models. Relevant software tools will also be discussed.

1.1 Introduction

Simulation and optimization can be paramount in many applications such as engineering design, business management, manufacturing engineering, product design, and industrial applications. Obviously, quantitative models are required to analyze the behavior and characteristics of a system and to make any possible predictions. Such quantitative models are often written as mathematical models or numerical models.

The analysis of mathematical models requires a diverse range of mathematical methods, including calculus, ordinary differential equations (ODEs), partial differential equations (PDEs), Fourier transforms, Laplace transforms, statistical techniques, and many others. Numerical solutions of mathematical models require a class of numerical techniques, including numerical integration, finite difference methods, finite element methods, finite volume methods, Monte Carlo simulation, statistical sampling, and many others.

With a good mathematical model for modeling and simulating an engineering system, it may be necessary to carry out some optimization. The aims of optimization can be anything – to minimize the energy consumption, costs, or environmental impact, and to maximize the profit, output, performance, sustainability, or efficiency. It is no exaggeration to say that optimization is everywhere, from engineering design to business planning. Because resources, time, and money are always limited in real-world applications, we have to find solutions to optimally use these valuable resources under various constraints, ideally in a long-term, sustainable way. Mathematical optimization or mathematical programming is the study of such planning and design problems using mathematical tools and numerical algorithms. For example, linear programming is one of powerful tools for solving optimization problems related to supply chains, logistics, resource allocation, scheduling, manufacturing, and other applications. In addition, engineering optimization is a well-established area of research with many algorithms, techniques, and software tools.

1.2 What is a model?

A model is a representation or an approximation to a system or part of a system in a quantitative way. Such models are often called mathematical models or numerical models if they are represented on a computer. See Fig. 1.1.

•  Models can be mathematical models, statistical models, numerical models, geometric models, and others.

•  Simulation requires the use of a numerical simulator, or a solver, implemented in a programming language or as a software package.

•  Algorithms are essential to both numerical simulation and optimization.

Figure 1.1 Components and procedure of modeling, simulation, and optimization.

Modeling and simulation are two important components for engineering simulation tasks. However, it is rarely possible to get a mathematical model correctly in the first place, and modifications are usually needed. In addition, the parameters of the model under consideration usually need to be optimized.

•  Optimization changes the design parameters or decision variables so as to make the design objective (also called the cost function) optimally (either maximum or minimum, depending on the actual design requirements).

•  Modification of the model may be carried out iteratively. The results should be compared with experiments or real data, and further modifications of the model may be necessary, so as to explain the data more accurately.

Even with the most accurate models and most efficient algorithms, the interpretation of the analytical solution and/or numerical solutions requires care so as to extract and explain the results in a meaningful way.

All models are wrong, but some models are useful.

— George Box (British statistician)

Approximate solutions to correct models worth much more than exact solutions to wrong models.

— John Tukey (American mathematician)

There are different types of model. Loosely speaking, we can divide the models into four types: mathematical models, numerical model, geometry models, and others. Models can be static (not change with time), dynamic (vary with time), stochastic (models in terms of some expectation/average of model parameters), or mixed.

1.  Mathematical models can take many forms, including continuous, discrete, and stochastic forms.

•  Continuous models: Continuous models usually use ordinary differential equations (ODEs), partial differential equations (PDEs), and even a set of both ODEs and PDEs.

•  Discrete models: Queueing models simulate discrete events, such as the arrivals of customers at a bank and the queueing process in a supermarket.

•  Stochastic models: Some models are probabilistic or statistical models, such as regression models and neural network models. Monte Carlo methods can also be used to simulate stochastic models and discrete models.

2.  Numerical models: The representations of mathematical models as computer models often require discretization of continuous models. Numerical methods are often discrete, typically in terms of vectors, matrices, discrete systems such as a set of discrete ODEs or PDEs, computer files, and other forms.

3.  Geometric models: Some models are represented as geometric models, such as the 3D objects or models in Solidworks and AutoCAD, mesh models in finite element simulation, surface/mesh models in animation, and others.

4.  Other types of models: Some models cannot be explicitly written as one of the above models, but they can still simulate and represent the characteristics and behavior of systems under consideration. Such models can take various forms such as data, mixture of models and data, descriptive (both qualitative and quantitative), implicit models, and others.

This books mainly concerns mathematical models in both continuous and discrete forms. We will also introduce various numerical methods and algorithms for solving models based on differential equations.

1.3 Mathematical models

Most mathematical models can be expressed as algebraic equations, ODEs, PDEs, a set of ODEs or PDEs, or some probabilistic relationships. Let us briefly introduce them here, and we will provide more details in late chapters.

1.3.1 Algebraic equations

Many physical laws can be expressed as algebraic relationships. For example, Newton's second law is a good example, though strictly speaking, we should express it in the vector form

(1.1)

because both force F and acceleration a are vectors. Another example is Ohm's law.

Example 1.1

Ohm's law relates the current I through a conductor with resistance R and voltage V applied across the conductor:

or

Here the main assumption is that the resistance does not vary with V or I. Obviously, in case of R depending on I, we can still write this as an equivalent approximate relationship. In fact, R usually depends on the temperature T, and the flow of current will generate heat (and thus vary the temperature).

Almost all the physical and chemical laws we learned in school are expressed as algebraic equations.

1.3.2 ODE models

A simple model for modeling the charging process of a smartphone battery with voltage is

(1.2)

where is the charger voltage (typically, volts), and is the time constant. Here R is the resistance of the battery, whereas the battery storage is loosely modeled as the capacitor C, as shown in Fig.