附录A 英文原文
PDE toolbox Introduction
Basics of the Finite Element Method：
The solutions of simple PDEs on complicated geometries can rarely be expressed in terms of elementary functions. You are confronted with two problems: First you need to describe a complicated geometry and generate a mesh on it. Then you need to discretize your PDE on the mesh and build an equation for the discreteapproximation of the solution. The pdetool graphical user interface provides you with easy-to-use graphical tools to describe complicated domains and generatetriangular meshes. It also discretizes PDEs, finds discrete solutions and plots results. You can access the mesh structures and the discretization functions directly from the command line (or from a file) and incorporate them into specialized applications.
Here is an overview of the Finite Element Method (FEM). The purpose of this presentation is to get you acquainted with the elementary FEM notions. Here you find the precise equations that are solved and the nature of the discrete solution. Different extensions of the basic equation implemented in Partial Differential Equation Toolbox software are presented. A more detailed description can be found in Chapter 3, “Finite Element Method”．
You start by approximating the computational domain Ωwith a union of simple geometric objects, in this case triangles. The triangles form a mesh and each vertex is called a node. You are in the situation of an architect designing a dome. He has to strike a balance between the ideal rounded forms of the original sketch and the limitations of his simple building-blocks, triangles or quadrilaterals. If the result does not look close enough to a perfect dome, the architect can always improve his work using smaller blocks.
Next you say that your solution should be simple on each triangle. Polynomials are a good choice: they are easy to evaluate and have good approximation properties on small domains. You can ask that the solutions in neighboring triangles connect to each other continuously across the edges. You can still decide how complicated the polynomials can be. Just like an architect, you want them as simple as possible. Constants are the simplestchoice but you cannot match values on neighboring triangles. Linear functions come next. This is like using flat tiles to build a waterproof dome, which is perfectly possible.
A Triangular Mesh (left) and a Continuous Piecewise Linear Function on That Mesh
pdetool GUI
In this section.
“Introduction” on page 1-27 “The Menus” on page 1-29 “The Toolbar” on page 1-30 “The GUI Modes” on page 1-31
“The CSG Model and the Set Formula” on page 1-32 “Creating Rounded Corners” on page 1-33
“Suggested Modeling Method” on page 1-35 “Object Selection Methods” on page 1-39 “Display Additional Information” on page 1-40
“Entering Parameter Values as MATLAB Expressionspage 1-40 ” on
“Using Earlier Version Partial Differential Equation Toolbox Model Fileson page 1-41
”
Introduction
Partial Differential Equation Toolbox software includes a complete graphicaluser interface (GUI), which covers all aspects of the PDE solution process.You start it by typingpdetoolat the MATLAB command line. It may take a while the first time you launchpdetoolduring a MATLAB session. The following figure shows the pdetoolGUI as it looks when you start it.
At the top, the GUI has a pull-down menu bar that you use to control the modeling. It conforms to common pull-down menu standards. Menu item followed by a right arrow lead to a submenu. Menu items followed by an ellipsis lead to a dialog box. Stand-alone menu items lead to direct action. Below the menu bar, a toolbar with icon buttons provide quick and easy access to some of the most important functions.
To the right of the toolbar is a pop-up menu that indicates the currentapplication mode. You can also use it to change the application mode. The upper right part of the GUI also provides the x- and y-coordinates of the current cursor position. It is updated when you move the cursor inside the main axes area in the middle of the GUI. The edit box for the set formula contains the active set formula. In the main axes you draw the 2-D geometry, display the mesh, plot the solution, etc. At the bottom of the GUI, an information line provides information about the current activity. It
can also display help information about the toolbar buttons.
The Menus
There are 11 different pull-down menus in the GUI. See Chapter 2, “Graphical User Interface” for a more detailed description of the menus and the dialog boxes:
?File menu. From the File menu you can Open and Save model files that contain a command sequence that reproduces your modeling session. Youcan also print the current graphics and exit the GUI.
?Edit menu. From the Edit menu you can cut, clear, copy, and paste the solid objects. There is also a Select All option.
?Options menu. The Options menu contains options such as toggling the axis grid, a “-to-gridsnap” feature, and zoom. You can also adjust the axis limits and the grid spacing, select the application mode, and refresh the GUI.
?Draw menu. From the Draw menu you can select the basic solid objects such as circles and polygons. You can then draw objects of the selected type using the mouse. From the Draw menu you can also rotate the solid objects and export the geometry to the MATLAB main workspace.
?Boundary menu. From the Boundary menu you access a dialog box where you define the boundary conditions. Additionally, you can labeedges and subdomains, remove borders between subdomains, and export the decomposed geometry and the boundary conditions to the workspace.
?PDE menu. The PDE menu provides a dialog box for specifying the PDE, and there are menu options for labeling subdomains and exporting PDE coefficients to the workspace.
?Mesh menu. From the Mesh menu you create and modify the triangular mesh. You can initialize, refine, and jiggle the mesh, undo previous mesh changes, label nodes and triangles, display the mesh quality, and export the mesh to the workspace.
?Solve menu. From the Solve menu you solve the PDE. You can also open a dialog box where you can adjust the solve parameters, and you can export the solution to the workspace.
?Plot menu. From the Plot menu you can plot a solution property. A dialog box lets you select which property to plot, which plot style to use andseveral other plot parameters. If you have recorded a movie (animation) of the solution, you can export it to the workspace.
? Window menu. The Window menu lets you select any currently open MATLAB figure window. The selected window is brought to the front
? Help menu. The Help menu provides a brief help window.
The Toolbar
The toolbar underneath the main menu at the top of the GUI contains icon buttons that provide quick and easy access to some of the most important functions.
The five leftmost buttons are draw mode buttons and they represent, fromleft to right:
The draw mode buttons can only be activated one at the time and they all work the same way: single-clicking a button allows you to draw one solid object of the selected type. Double-clicking a button makes it “stick,” and you can then continue to draw solid objects of the selected type until yousingle-click the button to “release” it. Using the right mouse button or Ctrl+click, the drawing is constrained to a square or a circle.
The second group of six buttons includes the following analysis buttons.
The button toggles the zoom function on/off
The GUI Modes
1 Define the geometry (2-D domain). 2 Define the boundary conditions. 3 Define the PDE.
4 Create the triangular mesh. 5 Solve the PDE.
6 Plot the solution and other physical properties calculated from the solution (post processing).
The pdetool GUI is designed in a similar way. You work in six different
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