Partial Differential Equation Toolbox User's Guide

1 ComputationVisualizationProgrammingParti al Differential EquationToolboxFor Use with MATLAB User s GuideComputer Solutions Europe ABHow to Contact The MathWorks:508-647-7000 Phone508-647-7001 FaxThe MathWorks, Prime Park WayNatick, MA 01760-1500 FTP Technical Product enhancement Bug Documentation error Subscribing user Order status, license renewals, Sales, pricing, and general informationPartial Differential Equation Toolbox User s Guide COPYRIGHT 1984 - 1997 by The MathWorks, Inc. All Rights 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.

2 No part of this manual may be photocopied or repro-duced in any form without prior written consent from The MathWorks, GOVERNMENT: If Licensee is acquiring the Programs on behalf of any unit or agency of the 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 , 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 disclo-sure are as set forth in Clause (c)(2) of the , Simulink, Handle Graphics, and Real-Time Workshop are registered trademarks and Stateflow and Target Language Compiler are trademarks of The MathWorks, product or brand names are trademarks or registered trademarks of their respective History: August 1995 First printingFebruary 1996 Reprint%FAX)@iContents1 TutorialIntroduction.

3 1-2 What Does this Toolbox Do? ..1-2 Can I Use the PDE Toolbox ? .. 1-2 What Problems Can I Solve? ..1-3In 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.

4 1-35 Entering Parameter Values as MATLAB Expressions ..1-36 Using PDE Toolbox version Model M-files ..1-36iiContentsUsing 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 .. 1-402 ExamplesExamples of Elliptic Problems .. 2-2 Poisson s Equation on Unit Disk .. 2-2 Using the Graphical User Interface .. 2-2 Using Command-Line Functions .. 2-4A Scattering Problem .. 2-6 Using the Graphical User Interface .. 2-8A 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.

5 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 .. 2-25iiiExamples 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 .. 2-28L-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.

6 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 .. 2-48AC 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 .. 2-61ivContents3 The Graphical User InterfacePDE Toolbox Menus .. 3-3 File Menu .. 3-3 New .. 3-3 Open .. 3-4 Save As .. 3-5 Print .. 3-6 Edit Menu .. 3-7 Paste .. 3-8 Options Menu .. 3-9 Grid Spacing .. 3-10 Axes Limits .. 3-11 Application .. 3-11 Draw Menu .. 3-13 Rotate .. 3-14 Boundary Menu .. 3-15 Specify Boundary Conditions .. 3-16 PDE Menu .. 3-18 PDE Specification.

7 3-19 Mesh Menu .. 3-22 Parameters .. 3-23 Solve Menu .. 3-25 Parameters .. 3-25 Plot Menu .. 3-30 Parameters .. 3-30 Additional Plot Control Options .. 3-34 Window Menu .. 3-37 Help Menu .. 3-37 The Toolbar .. 3-38v4 The Finite Element MethodThe Elliptic Equation .. 4-3 The Elliptic System .. 4-10 The Parabolic Equation .. 4-13 The Hyperbolic Equation .. 4-16 The Eigenvalue Equation .. 4-17 Nonlinear equations .. 4-21 Adaptive Mesh Refinement .. 4-26 The Error Indicator Function .. 4-26 The Mesh Refiner .. 4-27 The Termination Criteria .. 4-28 Fast Solution of Poisson s Equation .. 4-29viContents5 ReferenceCommands Grouped by Function .. 5-3 PDE Algorithms .. 5-3 User Interface Algorithms .. 5-3 Geometry Algorithms .. 5-4 Plot Functions .. 5-4 Utility Algorithms .. 5-5 User Defined Algorithms .. 5-7 Demonstration Programs .. 5-7 PDE Coefficients for Scalar Case.

8 5-20 PDE Coefficients for System Case .. 5-21 Boundary Condition Dialog Box .. 5-80 Model M-file .. 5-81 Index 1 TutorialIntroduction .. 1-2 What Does this Toolbox Do? .. 1-2 Can I Use the PDE Toolbox ? .. 1-2 What Problems Can I Solve? .. 1-3In 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.

9 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 Model M-files ..1-36 Using Command-Line Functions ..1-37 Data Structures and Utility Functions ..1-37 Hints and Suggestions for Using Command-Line Function ..1-401 Tutorial1-2 IntroductionThis section attempts to answer some of the questions you might formulate when you turn the first page: What does this Toolbox do? Can I use it? What problems can I solve?, Does this Toolbox Do?The Partial Differential Equation (PDE) Toolbox provides a powerful and flexible environment for the study and solution of Partial Differential equations in two space dimensions and time. The equations are discretized by the Finite Element Method (FEM). The objectives of the PDE Toolbox are to provide you with tools that: Define a PDE problem, , define 2-D regions, boundary conditions, and PDE coefficients.

10 Numerically solve the PDE problem, , generate unstructured meshes, discretize the equations , and produce an approximation to the solution. Visualize the results. Can I Use the PDE Toolbox ?The PDE Toolbox is designed for both beginners and advanced minimal requirement is that you can formulate a PDE problem on paper (draw the domain, write the boundary conditions, and the PDE). Start MATLAB. At the MATLAB command line type: pdetool This invokes the graphical user interface (GUI), which is a self-contained graphical environment for PDE solving. For common applications you can use the specific physical terms rather than abstract coefficients. Using pdetool requires no knowledge of the mathematics behind the PDE, the numerical schemes, or MATLAB. In Getting Started on page 1-9 we Guide you through an example step by applications are also possible by downloading the domain geometry, boundary conditions, and mesh description to the MATLAB workspace.