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PDE Toolbox Primer for Engineering Applications with MATLAB® Basics
Leonid Burstein
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eBook - ePub
PDE Toolbox Primer for Engineering Applications with MATLAB® Basics
Leonid Burstein
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About This Book
Partial differential equations (PDEs) describe technological phenomena and processes used for the analysis, design, and modeling of technical products. Solutions of spatial and transient PDEs are realized by using the PDE Toolbox included in the MATLAB® software. MATLAB® is introduced here as an essential foundation for PDE, and the Modeler of the PDE Toolbox, with appropriate explanatory solutions, is applied to engineering problems in mechanics, heat/mass transfer, tribology, materials science, physics, and biotechnology. The appendixes contain collections of commands and functions used to solve actual engineering problems.
FEATURES
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- Includes the PDE Modeler interface with example solutions of two- and three-dimensional PDEs
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- Presents methodologies for all types of PDEs as representative of any engineering problem
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- Describes the ordinate differential equation (ODE) solver for initial value and boundary value problems (IVP and BVP) through practical examples from mechanics and the thermodynamic properties of materials
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- Covers the basics of MATLAB® to solve both ODEs and PDEs
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- Reviews spatially the one-dimensional PDE solver with actual engineering examples
PDE Toolbox Primer for Engineering Applications with MATLAB® Basics is aimed at scientists, students, professionals, practitioners, self-taught readers, and researchers who need concise and clear information to study and apply MATLAB® software and the PDE Toolbox in engineering.
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1 Introduction
DOI: 10.1201/9781003200352-1
1.1 Preamble
This book provides the basics of MATLAB® and the Partial Differential Equation (PDE) Toolbox™ accompanied by examples taken from various engineering areas, including mechanics, electricity, heat and mass transfer, tribology, materials science, technical physics, and biotechnology. In these as in many other engineering fields, differential equations, in particular PDEs, play a key role as they describe technological phenomena and processes and are vital for the analysis, design, and modeling of optimal technical products. The solutions of the spatial and transient PDEs are implemented by the PDE Toolbox, which is included in the MATLAB® software. The PDE Modeler of this toolbox is known as a convenient and effective tool for solving actual problems of modern science and technology. Thus, mastering its latest versions is becoming more and more relevant for finding practical solutions and creating new products in mechanics, electronics, chemistry, life sciences, and modern industry in general. Modern technology specialists widely use computers and some special programs but require some universal tool for solving, simulating, and modeling specific problems in their area. Numerous primers on MATLAB® have been written, but none have been designed especially for teaching and solving engineering problem with the PDE Toolbox. This book covers the gap.
MATLAB® is introduced in the book as an essential foundation for PDE Modeler and programmatic tool of the PDE Toolbox with appropriate explanatory solutions applied to both traditional and emerging engineering problems. One benefit of the book is a variety of explanatory examples from a broad range of engineering fields, including modern and classical mechanics, material sciences, mass-heat transfer, and biotechnology, which will facilitate the understanding of the presented material.
1.2 A Bit of History and Advantages of the Software Presented in This Book
The foundations and language of MATLAB® were established in the 1970s by mathematician Cleve Moler. Later, the language was rewritten in C and improved by specialists that had joined the founder. Initially, the language was intended for students to adapt LINPACK and EISPACK mathematical packages, but it was then applied to control engineering. In a fairly short period, researchers and engineers recognized MATLAB® as an effective and convenient tool for solving not only mathematical but also many technological problems. Early commercial versions of MATLAB® appeared in the general software market in the mid-1980s. Later, graphics and special engineering-oriented means, such as user-oriented interfaces and toolboxes, begin to be incorporated into MATLAB®, giving it its modern form. Among them, the PDEs Toolbox appeared in 1995, adding some commands and, most importantly for practical use, an interface for solving spatially two-dimensional (2D) transient PDEs. The toolbox has constantly evolved and is intended today for the fourth type of PDEs, specifically elliptical, parabolic, hyperbolic, and eigenvalue. In the mid-90s, a software called FEMLAB was separated from the PDE Toolbox and is now developing independently. Possibilities of the PDE Toolbox are constantly expanding and, since the early 2010s, are able to solve some spatially three-dimensional (3D) problems using its programmatic tool. As a whole, today’s MATLAB® and its PDE Toolbox are a unique assembly of implemented classical and modern numerical methods and specialized interfaces for engineering calculations designed for specialists of various fields. With other programming tools for various computations, this software has a valuable place among them as a technical computing software. Without going into details, the following abilities and their interactions explain the sustainable preference for PDE Toolbox and MATLAB® as a whole:
- PDE tool:
- Provides a powerful and flexible environment for exploring and solving PDEs in two and three spatial dimensions and time;
- Assures a user-friendly graphical interface allowing the user to draw the geometry of the real technical part, specify its boundaries, select the appropriate PDE type, solve the problem, and visualize the calculated results;
- Supplies a wide range of commands to efficiently create programs solving one or more 2D or 3D PDEs.
- MATLAB® itself:
- Ensures substantial universality and the ability to solve both simple and complex scientific and technical problems using a large set of various all-general and specialized commands;
- The proposed practical applicability in various fields of technology and science is provided by a wide variety of problem-oriented tools called toolboxes;
- Provides convenience means for visualizing the achieved solutions of the general or specific scientific and technological problems;
- Has quick access to built-in help and well-organized extensive documentation.
To these characteristics should be added the innovation of recent years – the Live Editor – which provides the ability to display text, images, and codes together with the resulting tables and graphs in one window, with the results immediately visible after entering the codes.
1.3 The Goals of the Book and Its Audience
The book has two goals:
- First, to provide researchers, engineers, machine designers, teachers, and students with guidance to teach them how to use the PDE Toolbox™ as well as the MATLAB® solvers for 1D PDEs and initial value and boundary value problems (IVP and BVP, respectively) for ordinate differential equations (ODE).
- Second, following from the first aim, to provide a basic, simple, and comprehensive guide to MATLAB® without which it is impossible to use the PDE Toolbox for solving differential equations (DE).
It is assumed that the reader has no programming experience and will be using the software for the first time. Therefore, the book provides MATLAB® basics to make PDE Toolbox available to the widest possible audience. To make the basic steps of programming and make the commands understandable to the target audience, the book provides examples of problems from various fields of science and engineering. As the basic programming knowledge of the reader is increased, the problems become more complex and are solved with special solvers, ode and bvp. Then, the PDE Toolbox is sequentially introduced with its programmatic and interface tools for spatially 2D PDEs, followed by 3D options.
MATLAB® and its toolboxes are updated and improved in parallel with the development of modern technologies. Thus, the technical analysis and calculations that can be conducted in MATLAB® and in PDE Toolbox in particular have contributed to the fact that the scientific community has recognized it as a convenient and effective tool for use in modern science and engineering. Thus, mastering its latest versions and practical solutions with their help is increasingly essential for the creation of new products in mechanics, electronics, chemistry, life sciences, and modern industry as a whole. Specialists in these areas, among others, widely use computers and some special programs but also require some universal tool for solving, simulating, and modeling specific problems in their area. This also applies to the usage of the PDE Toolbox, which is frequently not used due to the ignorance of its capabilities and lack of familiarity. However, the problems that are described by differential equations and can be solved with the PDE Toolbox cover a wide range of phenomena. These include the strength and durability of mechanical parts, machine elements, production processes, quality assurance, fluid mechanics parameters, thermodynamic and rheological properties of the materials, state equations, lubrication and tribo-characteristics of rubbing parts and descriptive statistics as well as bacteria population, dilution, dissociation, and reaction-diffusion kinetics. Thus, knowledge of the available apparatus for solving such equation is critically important for effective solutions of many real engineering problems. This book is oriented to the reader with a modest mathematical background and introduces the programming or technical concepts using some simplifications of the traditional approach. A variety of examples from a broad range of modern and classical engineering help solidify the understanding of the presented material and show specialists the options for using the software in their specific fields. As a whole, the book can serve as a guide for two categories of users:
- Scientists, engineers, and students that would like to see how to use the PDE Toolbox in their specific areas;
- Researchers and technicians wanting to learn and apply MATLAB® in their industry.
- Summarizing the above, the principal audiences of the book include:
- Scientists, engineers, and specialists that seek to solve their problems and search for similar problems that were solved by computer;
- Non-programmer professionals and the academic community dealing with modeling and simulation machinery and processes in areas of technique and technology from mechanics or electricity to the biotechnology;
- Students, engineers, managers, and teachers from academic and university communities in the field of technology;
- Instructors and their audiences in study courses where PDE Toolbox and/or MATLAB® is used as a supplemental but required tool;
- Staff, students, and non-programmers as well as self-taught readers for quick mastering of the programs for their needs;
- Freshmen and participants in advanced scientific and engineering courses, seminars, or workshops where MATLAB® is taught;
- Researchers and professionals using a computer for modeling calculations to solve actual engineering and chemical/bioengineering problems applying the book as a reference.
1.4 About the Material in the Chapters
The material in the chapters is based on nearly 25 years of research and 18 years of multiple MATLAB® authoring courses in the fields of mechanics, mathematics, quality assurance, and biotechnology. The topics in the chapters are presented so that a beginner can gradually move from one topic to another from topics presented in MATLAB®-introductory sections to sections on PDE Toolbox programming and its interface, with previously acquired material to be used as a basis for each subsequent chapter.
This chapter, the first of ten chapters of this book, outlines the objectives of the book, the topics covered, and the structure of the chapters and outlines engineering problems that can be solved by the MATLAB® software and its PDE Toolbox.
The next three chapters describe the computational and graphics tools with examples of various practical applications. The most important, basic MATLAB® features are introduced in the second chapter, which describes the software desktop, toolbars, and main windows. It discusses elementary functions, input and output commands, numbers and strings, vectors, matrices/arrays, their manipulations, and flow control commands as well as relational and logical operators. The commands of this chapter are intended to enable beginners to write, perform, and display simple calculations interactively and directly in the Command Window. Chapter 3 introduces the user-defined functions and presents the regular Editor window for writing program scripts and user-defined functions and then the Live Editor window for writing live scripts and functions. All commands, regular and live scripts/functions, are explained with examples from engineering fields. The visualization means for generating 2D and 3D plots are described in Chapter 4. It describes the formatting commands for inserting labels, titles, text, and symbols into a plot as well as the color, marker, and line qualifiers. How to develop graphs containing more than one curve and graphs with multiple plots on one page is explained. Accompanying applications demonstrate how to generate 2D and 3D graphs for water surface tension, bandpass filter, and other practical applications. Understanding the material of the second, third, and fourth chapters allows the reader to generate rather complex programs applying technical calculations and their graphical presentation.
Chapter 5 presents more advanced topics but still refers to basic MATLAB®, namely ordinary differential equations, ODE and solvers for initial and boundary value problems, IVP and BVP. The finite difference method is also explained here. The chapter presents applied solutions to IVP and BVP problems such as RLC series current and heater wire temperature distribution.
In the next two chapters, the sixth and seventh, PDE Toolbox programming and modeling tools are described and applied to scientific and engineering problems that are modeled by PDEs. Chapter 6 illustrates the finite element collocation scheme and introduces PDE equations, boundary and initial conditions, and provides commands for solving PDEs. The problems solved here are steady or unsteady spatial 2D. Application problems include heating a small metallic plate, drumhead vibrations, and elliptical membrane eigenvalue modes. Chapter 7 presents the PDE Modeler tool for solving 2D PDEs; here the solution steps are examined in detail along with the graphical user interface. Application problems include the momentary pressure distribution in a lubricating film between two pore-covered surfaces, unsteady thermal conductivity with a temperature-dependent material, an example of plain stress (structure mechanics), among others.
The eighth chapter describes the pdepe solver used to solve transient and spatially 1D PDEs. It is shown how various PDEs with different boundary conditions can be represented in standard forms. Applications illustrate how to solve diffusion PDE with Neumann boundaries and piecewise initial condition, Bateman-Burgers PDE, and others.
Chapter 9 covers two topics, namely those related to coupled 2D PDEs solution using the PDE Modeler, and 3D PDE solutions using the programmatic tool. Among the problems solved here are the Schnakenberg coupled PDEs for a tri-molecular reaction, vibrations of a slab with elliptical hole, and the distribution of an electric potential in a plate with varying conductivity.
The final, tenth, chapter depicts life science problems, which are represented by differential equations and solved using the ODE and PDE software tools. The structure of the chapter differs from the previous one and contains application examples only. At the same time, considering the traditionally less prepared audience for programming and mathematics, problem solutions are presented with more extended explanations than in previous chapters. The applied problems addressed in this chapter include the steady-state concentration distribution in a short tube, the concentration of reagents in two reactors in series, a 1D model of the reactor, diffusing and reproducing in a bacterial culture, displacement of a homogenous membrane, and diffusion-brusselator PDEs.
The appendix provides a summary collection of over 250 variables, special characters, operators, and commands discussed in the book. In addition, a list of solved problems is provided.
The index contains about 800 alphabetical names, terms, and commands that have been explained or at least mentioned throughout this book.
1.5 Material Arrangement in the Chapter and the Available Program Editors
The material in the chapters is presented gradually to ensure a gradual assimilation of concepts. Each chapter begins with an introduction describing the chapter content and its available features. New material, basic command forms, and their implementations are then presented. Commands are usually given in one or two of the simplest forms with possible useful extensions. Each question, if possible, is fully addressed in one subsection so that readers can attain knowledge in a focused manner. The available tables list additionally available commands, specifiers, modifiers, equations, and graphical and object geometry forms that correspond to the topics and examples included in the chapter. The chapter devoted to the differential equations has sections explaining numerical method and computation by this method, and the results are compared with the results obt...