PDE Toolbox Primer for Engineering Applications with MATLAB® Basics

✍ Scribed by Leonid Burstein

• Publisher: CRC Press
• Year: 2022
• Tongue: English
• Leaves: 383
• Edition: 1
• Category: Library

✦ Synopsis

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

• Includes the PDE Modeler interface with example solutions of two- and three-dimensional PDEs

• Presents methodologies for all types of PDEs as representative of any engineering problem

• 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

• Covers the basics of MATLAB® to solve both ODEs and PDEs

• 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.

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Preface
Author
Chapter 1 Introduction
1.1 Preamble
1.2 A Bit of History and Advantages of the Software Presented in This Book
1.3 The Goals of the Book and Its Audience
1.4 About the Material in the Chapters
1.5 Material Arrangement in the Chapter and the Available Program Editors
1.6 MATLAB® and PDE Toolbox Versions Used
in This Book
1.7 The Order of Mastering the Material
Chapter 2 Basics of the Software
2.1 Introduction
2.2 Running the MATLAB®
2.2.1 Desktop, Toolstrip, and Main Windows
2.2.1.1 Toolstrip
2.2.1.2 Command Window
2.2.1.3 Workspace Window
2.2.1.4 Current Folder Window
2.2.2 Simple Calculations and Math Functions
2.2.3 About Online Help and Help Window
2.2.4 Available Toolboxes
2.2.5 Variables and Managing Them
2.2.6 Screen Output Formats
2.2.7 Commands for Output
2.2.8 Application Examples
2.2.8.1 Voltage between Intermediate Points of the Wheatstone Bridge
2.2.8.2 Threaded Bolt: Stiffness Value Estimation
2.2.8.3 Stress State for a Rectangular Plate with a Crack
2.2.8.4 Bravais Lattice Cell Volume
2.3 Vectors, Matrices, and Arrays
2.3.1 Managing Vectors, Matrices, and Arrays
2.3.1.1 Vector Representation
2.3.1.2 Matrices and Arrays Representation
2.3.2 Mathematical Manipulations with Matrices
2.3.2.1 Addition and Subtraction
2.3.2.2 Multiplication
2.3.2.3 Division
2.3.3 Elementwise Operations
2.3.4 Supplementary Commands for Matrix/Array Manipulations
2.3.5 Strings as Variable and Strings as Matrix Elements
2.3.6 About Displaying a Table
2.3.6.1 The disp Command
2.3.6.2 The fprintf Command
2.3.7 Application Examples
2.3.7.1 Cuboid Lattice Cell Volume
2.3.7.2 Table Containing Strings and Numbers
2.3.7.3 Voltage and Current in an RC-type Circuit
2.3.7.4 Momentary Position of the Piston Pin
2.4 Flow Control
2.4.1 Relational and Logical Commands
2.4.1.1 Relational Operators
2.4.1.2 Logical Operators
2.4.1.3 Application Example: Determining Outlying Results in a Sample
2.4.2 The If Statements
2.4.3 Loop Commands
2.5 Application Examples
2.5.1 Currents in an Electrical Circuit
2.5.2 Resistance of Volume of a Material–Bulk Modulus
2.5.3 Compression Piston Ring: Radial Thickness
Chapter 3 Program Managing: Editor and Live Editor
3.1 Introduction
3.2 Scripts and Script Files
3.2.1 Editor and Creating, Saving, and Running a Script File
3.2.1.1 Saving the Script File
3.2.1.2 About the Current Folder
3.2.1.3 Running Created Script File
3.2.2 Input Values to the Program Variables from the Command Window
3.3 User-defined Functions and Function Files
3.3.1 Function Creation
3.3.1.1 Function Definition
3.3.1.2 Help Lines
3.3.1.3 Function Body, Local and Global Variables
3.3.2 About Function File
3.3.3 Running a User-defined Function
3.3.3.1 Comparison of Script and Function Files
3.3.4 Anonymous Function
3.4 Interactive Script and Function Programs: Live Editor
3.4.1 Opening the Live Editor
3.4.2 Creating the Life Script
3.4.3 Additional Information for Using the Live Editor
3.4.3.1 Separate Sections in the Live Program
3.4.3.2 Creating Live Function
3.4.3.3 Converting a Script/Function to a Live Script/Function
3.4.3.4 Text Formatting Options
3.4.3.5 About the Interactive Controls
3.5 Application Examples
3.5.1 Converting Brinell Hardness to Vickers and Rockwell Hardness
3.5.2 Bending Shaft Stress
3.5.3 Dew Point
3.5.4 Live Function for Two-Stage Gear Train Calculations
Chapter 4 Basics of Graphics
4.1 Introduction
4.2 2D Plots Generating and Formatting
4.2.1 One or More Curves on the 2D Plot
4.2.1.1 Using the Hold On/Off Command for Generating Multiple Curves
4.2.2 Plot Formatting Using the Commands and the Plot Tools Editor
4.2.2.1 Formatting with Commands
4.2.2.2 Interactive Plot Formatting
4.2.3 Several Plots in the Same Figure Window
4.3 Three-Dimensional Plots
4.3.1 Generating 3D Lines
4.3.2 Generating Mesh in 3D Coordinates
4.3.3 Generating Surface in Plot
4.3.4 Formatting 3D Plots
4.3.4.1 About Figure Colors
4.3.4.2 About Projections of the 3D Graph
4.4 Supplementary 2D and 3D Plot Commands
4.5 Application Examples
4.5.1 Surface Tension of Water
4.5.2 A Bandpass Filter
4.5.3 Vibrational Modes of a Rectangular Membrane
4.5.4 Flow in a Rectangular Channel
Chapter 5 ODE Solvers for Initial and Boundary Value Problems
5.1 Introduction
5.2 ODE Solver for IVP
5.2.1 About Numerical Methods for Solving DEs
5.2.2 ODE Solver Commands
5.2.2.1 Steps to Solve ODE
5.2.2.2 ODE Solver Commands with Additional Parameters or with Anonymous Function
5.2.3 Supplemental Commands of the ODE Solver
5.3 Solver for Boundary Value Problem of ODEs
5.3.1 The bvp4c Command
5.3.2 Solution Steps with bvp4c
5.3.3 About the bvp5c Command
5.4 Application Examples for IVP of the ODEs
5.4.1 Current in a Series RLC Circuit
5.4.2 Relative Strengths of Military Forces
5.5 Application Examples for BVP
5.5.1 Temperature Distribution in a Wire Heater
5.5.2 Uniformly Loaded Beam Hinged at Both Ends
Chapter 6 Partial Differential Equations and Programmatic Tool of the PDE Toolbox
6.1 Introduction
6.2 About Numerical Methods Used for Solution of the PDEs
6.3 PDE Tool and Types of Differential Equations That Can Be Solved
6.4 Programmatic Tool of the PDE Toolbox
6.4.1 Solution Steps and Commands for Their Implementation
6.4.1.1 Command for Step 1
6.4.1.2 Commands for Step 2
6.4.1.3 Command for Step 3
6.4.1.4 Command for Step 4
6.4.1.5 Commands for Step 5
6.4.1.6 Commands for Step 6
6.4.1.7 Commands for Step 7
6.4.2 Elliptic PDE, Programmatic Solution, Example
6.4.3 Eigenvalue PDE with a Programmatic Solution Example
6.5 Application Examples
6.5.1 Heating a Small Metallic Plate: Transient Heat Transfer
6.5.2 Drumhead Vibrations
6.5.3 Elliptic Membrane Eigenvalue Problem
Chapter 7 Solving Two-Dimensional Partial Differential Equations with PDE Modeler
7.1 Introduction
7.2 PDE Toolbox Interface
7.2.1 Solution Steps in the PDE Modeler
7.2.2 Export Defined Solution and Generated Mesh to the Workspace
7.2.3 From the Triangular to Rectangular Grid
7.3 Drawing Two-Dimensional Objects
7.4 Application Examples
7.4.1 Momentary Pressure Distribution in a Lubricating Film between Two Surfaces Covered by Hemispherical Pores
7.4.2 Unsteady Thermal Conduction with Temperature-Dependent Property of a Material
7.4.3 “Structural Mechanics, Plane Stress” Application Example
7.4.4 T-shaped Membrane
Chapter 8 Solving One-Dimensional Partial Differential Equations
8.1 Introduction
8.2 Standard Forms for 1D PDE and Initial and Boundary Conditions
8.2.1 One-Dimensional PDE in Standard Form
8.2.2 Proper Form of the Initial Conditions
8.2.3 Proper Form of the Boundary Conditions
8.2.4 About Finite Difference Methods for Solving PDEs
8.3 The pdepe Command, Its Input and Output
8.3.1 The Steps of Solution by Example
8.3.2 Passing Additional Parameters to the PDE, IC, and PC Functions
8.4 Application Examples
8.4.1 Transient 1D Diffusion Equation with Neumann Boundaries and Piecewise IC
8.4.2 Action Potential Model with Diffusion Term
8.4.3 Pipe Flow
8.4.4 Bateman-Burgers PDE
Chapter 9 Coupled 2D PDE Solutions and 3D PDE Solutions
9.1 Introduction
9.2 Solving a Set of Two PDEs Using PDE Modeler
9.2.1 Steps to Solve a Set of Two 2D PDEs with PDE Modeler
9.2.2 About the Export Solution and Mesh for a Set of Two PDEs
9.3 3D PDEs and Steps to Its Solution
9.3.1 Steps 1 and 2: Create a PDE Model Object and 3D Geometry
9.3.2 Steps 3 and 4: BC, IC, and PDE Coefficients
9.3.3 Steps 5 and 6: Creating a Tetrahedral Mesh and Perform the Solution
9.3.4 Step 7. Presentation of the Solution
9.3.5 Steady-State 3D Heat Equation
9.3.6 Unsteady 3D Heat Equation
9.4 Supplementary Commands of the 2D/3D PDE Programmatic Tool
9.5 Application Examples
9.5.1 Tri-molecular Reaction: The Schnakenberg Coupled 2D PDEs
9.5.2 Vibrations of a Slab with an Elliptical Hole
9.5.3 Electric Potential, Plate with Varying Conductivity
Chapter 10 Toward Solving ODE and PDE Problems in the Life Sciences
10.1 Introduction
10.2 Solving ODEs with ode- and bvp- Solvers
10.2.1 Rate Equation for Biomolecular Reaction
10.2.2 Steady-State Concentration Distribution in a Short Tube
10.2.3 A Model of Enzyme Kinetics
10.2.4 Two Reactors in Series
10.3 Solving 1D PDEs Using the pdepe Command
10.3.1 A 1D Reactor Model
10.3.2 Initial Stage of Tumor Growth
10.3.3 Bacterial Culture, Diffusing and Reproducing
10.3.4 A Model of the Reaction-Diffusion System
10.4 Solving 2D PDEs Using the PDE Toolbox
10.4.1 Steady-State Concentration Distribution in a Reactor
10.4.2 Displacement of Homogenous Membrane
10.4.3 Diffusion-Brusselator PDEs for Reagent Concentrations
Appendix A: Special Characters, Predefined Variables,
Operators, and Commands Used in PDE ToolboxTM and
MATLAB® Programming
Appendix B: List of Examples, Problems, and Applications Considered in the Book
References
Index