Partial Differential Equation Toolbox User's Guide

Partial Differential Equation Toolbox

For Use with MATLAB?

Computer Solutions Europe AB

User's Guide

Computation Visualization Programming

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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.

U.S. GOVERNMENT: If Licensee is acquiring the Programs on behalf of any unit or agency of the U.S. Government, the following shall apply: (a) For units of the Department of Defense: the Government shall have only the rights specified in the license under which the commercial computer software or commercial software documentation was obtained, as set forth in subparagraph (a) of the Rights in Commercial Computer Software or Commercial Software Documentation Clause at DFARS 227.7202-3, therefore the rights set forth herein shall apply; and (b) For any other unit or agency: NOTICE: Notwithstanding any other lease or license agreement that may pertain to, or accompany the delivery of, the computer software and accompanying documentation, the rights of the Government regarding its use, reproduction, and disclosure are as set forth in Clause 52.227-19 (c)(2) of the FAR.

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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

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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

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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

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