Partial Differential Equation Toolbox User's Guide
Partial Differential Equation Toolbox
For Use with MATLAB?
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User's Guide
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Partial Differential Equation Toolbox User's Guide
? COPYRIGHT 1984 - 1997 by The MathWorks, Inc. All Rights Reserved.
The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or reproduced in any form without prior written consent from The MathWorks, Inc.
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Printing History: August 1995 First printing February 1996 Reprint
Contents
Tutorial
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2 What Does this Toolbox Do? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2 Can I Use the PDE Toolbox? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2 What Problems Can I Solve? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3 In Which Areas Can the Toolbox Be Used? . . . . . . . . . . . . . . . . . 1-5 How Do I Define a PDE Problem? . . . . . . . . . . . . . . . . . . . . . . . . 1-5 How Can I Solve a PDE Problem? . . . . . . . . . . . . . . . . . . . . . . . . 1-6 Can I Use the Toolbox for Nonstandard Problems? . . . . . . . . . . 1-6 How Can I Visualize My Results? . . . . . . . . . . . . . . . . . . . . . . . . 1-6 Are There Any Applications Already Implemented? . . . . . . . . . . 1-7 Can I Extend the Functionality of the Toolbox? . . . . . . . . . . . . . 1-7 How Can I Solve 3-D Problems by 2-D Models? . . . . . . . . . . . . . 1-8
Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-9
Basics of The Finite Element Method . . . . . . . . . . . . . . . . . . .1-18
Using the Graphical User Interface . . . . . . . . . . . . . . . . . . . . .1-23 The PDE Toolbox Graphical User Interface . . . . . . . . . . . . . . .1-23 The Menus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-24 The Toolbar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-25 The GUI Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-26 The CSG Model and the Set Formula . . . . . . . . . . . . . . . . . . . .1-27 Creating Rounded Corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-28 Suggested Modeling Method . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-31 Object Selection Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-35 Display Additional Information . . . . . . . . . . . . . . . . . . . . . . . . .1-35 Entering Parameter Values as MATLAB Expressions . . . . . . .1-36 Using PDE Toolbox version 1.0 Model M-files . . . . . . . . . . . . . .1-36
i
Using Command-Line Functions . . . . . . . . . . . . . . . . . . . . . . . 1-37 Data Structures and Utility Functions . . . . . . . . . . . . . . . . . . . 1-37 Constructive Solid Geometry Model . . . . . . . . . . . . . . . . . . . 1-38 Decomposed Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-39 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-39 Equation Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-39 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-39 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-40 Post Processing and Presentation . . . . . . . . . . . . . . . . . . . . . 1-40 Hints and Suggestions for Using Command-Line Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-40
Examples
2
Examples of Elliptic Problems . . . . . . . . . . . . . . . . . . . . . . . . . 2-2 Poisson's Equation on Unit Disk . . . . . . . . . . . . . . . . . . . . . . . . . 2-2 Using the Graphical User Interface . . . . . . . . . . . . . . . . . . . . 2-2 Using Command-Line Functions . . . . . . . . . . . . . . . . . . . . . . 2-4 A Scattering Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6 Using the Graphical User Interface . . . . . . . . . . . . . . . . . . . . 2-8 A Minimal Surface Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-10 Using the Graphical User Interface . . . . . . . . . . . . . . . . . . . 2-10 Using Command-Line Functions . . . . . . . . . . . . . . . . . . . . . 2-11 Domain Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12
Examples of Parabolic Problems . . . . . . . . . . . . . . . . . . . . . . . 2-16 The Heat Equation: A Heated Metal Block . . . . . . . . . . . . . . . . 2-16 Using the Graphical User Interface . . . . . . . . . . . . . . . . . . . 2-17 Using Command-Line Functions . . . . . . . . . . . . . . . . . . . . . 2-19 Heat Distribution in Radioactive Rod . . . . . . . . . . . . . . . . . . . . 2-21 Using the Graphical User Interface . . . . . . . . . . . . . . . . . . . 2-22
Examples of Hyperbolic Problems . . . . . . . . . . . . . . . . . . . . . 2-23 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-23 Using the Graphical User Interface . . . . . . . . . . . . . . . . . . . 2-23 Using Command-Line Functions . . . . . . . . . . . . . . . . . . . . . 2-25
ii Contents
Examples of Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . 2-27 Eigenvalues and Eigenfunctions for the L-Shaped Membrane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-27 Using the Graphical User Interface . . . . . . . . . . . . . . . . . . . 2-27 Using Command-Line Functions . . . . . . . . . . . . . . . . . . . . . 2-28 L-Shaped Membrane with Rounded Corner . . . . . . . . . . . . . . . 2-31 Eigenvalues and Eigenmodes of a Square . . . . . . . . . . . . . . . . 2-32 Using the Graphical User Interface . . . . . . . . . . . . . . . . . . . 2-33 Using Command-Line Functions . . . . . . . . . . . . . . . . . . . . . 2-33
Application Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-35 The Application Modes and the GUI . . . . . . . . . . . . . . . . . . . . . 2-35 Structural Mechanics - Plane Stress . . . . . . . . . . . . . . . . . . . . . 2-36 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-39 Using the Graphical User Interface . . . . . . . . . . . . . . . . . . . 2-39 Structural Mechanics - Plane Strain . . . . . . . . . . . . . . . . . . . . 2-41 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-43 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-44 Using the Graphical User Interface . . . . . . . . . . . . . . . . . . . 2-44 Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-46 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-47 Using the Graphical User Interface . . . . . . . . . . . . . . . . . . . 2-48 AC Power Electromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-51 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-52 Using the Graphical User Interface . . . . . . . . . . . . . . . . . . . 2-53 Conductive Media DC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-55 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-55 Using the Graphical User Interface . . . . . . . . . . . . . . . . . . . 2-56 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-57 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-58 Using the Graphical User Interface . . . . . . . . . . . . . . . . . . . 2-59 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-61
iii
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