Partial Differential Equation Toolbox

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Computation Visualization Programming

Partial Differential Equation Toolbox

For Use with MATLAB

^®

User’s Guide

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Contents 1

Tutorial

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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ii Contents

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

2 Examples

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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iii 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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3 The Graphical User Interface

PDE 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 . . . . 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 . . . 3-38

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4 The Finite Element Method

The 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 . . . 4-29

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vi Contents

5 Reference

Commands 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 . . . 5-20 PDE Coefficients for System Case . . . 5-21 Boundary Condition Dialog Box . . . 5-80 Model M-file . . . 5-81

Index

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1

Tutorial

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 Using Command-Line Functions . . . 1-37 Data Structures and Utility Functions . . . 1-37 Hints and Suggestions for Using Command-Line Function . . . 1-40

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Introduction

This 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?, etc.

What 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, i.e., define 2-D regions, boundary conditions, and PDE coefficients.

• Numerically solve the PDE problem, i.e., 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 users.

The 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 step.

Advanced applications are also possible by downloading the domain geometry, boundary conditions, and mesh description to the MATLAB workspace. From the command line (or M-files) you can call functions from the toolbox to do the hard work, e.g., generate meshes, discretize your problem, perform

interpolation, plot data on unstructured grids, etc., while you retain full control over the global numerical algorithm.

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Introduction

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What Problems Can I Solve?

The basic equation of the PDE Toolbox is the PDE in Ω,

which we shall refer to as the elliptic equation, regardless of whether its coefficients and boundary conditions make the PDE problem elliptic in the mathematical sense. Analogously, we shall use the terms parabolic equation and hyperbolic equation for equations with spatial operators like the one above, and first and second order time derivatives, respectively. Ω is a bounded domain in the plane. c, a, f, and the unknown u are scalar, complex valued functions defined on Ω. c can be a 2-by-2 matrix function on Ω. The toolbox can also handle the parabolic PDE

the hyperbolic PDE

and the eigenvalue problem

where d is a complex valued function on Ω, and λ is an unknown eigenvalue.

For the parabolic and hyperbolic PDE the coefficients c, a, f, and d can depend on time. A nonlinear solver is available for the nonlinear elliptic PDE

where c, a, and f are functions of the unknown solution u. All solvers can handle the system case

You can work with systems of arbitrary dimension from the command line. For the elliptic problem, an adaptive mesh refinement algorithm is implemented.

It can also be used in conjunction with the nonlinear solver. In addition, a fast solver for Poisson’s equation on a rectangular grid is available.

∇

– ⋅(c u∇ )+au = f

∂u

∂t ---

d –∇⋅(c u∇ )+au = f,

2u

∂

∂t² ---

d –∇⋅(c u∇ )+au = ,f

∇

– ⋅(c u∇ )+au = λdu

∇

– ⋅(c u( )∇u)+a u( )u = f u( ),

∇

– ⋅(c₁₁∇u₁)–∇⋅(c₁₂∇u₂)+a₁₁u₁+a₁₂u₂ = f₁

∇

– ⋅(c₂₁∇u₁)–∇⋅(c₂₂∇u₂)+a₂₁u₁+a₂₂u₂ = f₂ .

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The following boundary conditions are defined for scalar u:

• Dirichlet: hu = r on the boundary .

• Generalized Neumann: on .

is the outward unit normal. g, q, h, and r are complex valued functions defined on . (The eigenvalue problem is a homogeneous problem, i.e., g = 0, r = 0.) In the nonlinear case, the coefficients, g, q, h, and r can depend on u, and for the hyperbolic and parabolic PDE, the coefficients can depend on time. For the two-dimensional system case, Dirichlet boundary condition is

the generalized Neumann boundary condition is

and the mixed boundary condition is

where µ is computed such that the Dirichlet boundary condition is satisfied.

Dirichlet boundary conditions are also called essential boundary conditions, and Neumann boundary conditions are also called natural boundary

conditions. See Chapter 4, "The Finite Element Method" for the general system case.

Ω

∂

n⋅(c u∇ )+qu = g ∂Ω n

Ω

∂

h₁₁u₁+h₁₂u₂ = r₁ h₂₁u₁+h₂₂u₂ = r₂,

n⋅(c₂₁∇u₁)+n⋅(c₂₂∇u₂)+q₂₁u₁+q₂₂u₂ = g₂ n⋅(c₁₁∇u₁)+n⋅(c₁₂∇u₂)+q₁₁u₁+q₁₂u₂ = g₁ ,

h₁₁u₁+h₁₂u₂ = r₁

n⋅(c₂₁∇u₁)+n⋅(c₂₂∇u₂)+q₂₁u₁+q₂₂u₂ = g₂+h₁₂µ n⋅(c₁₁∇u₁)+n⋅(c₁₂∇u₂)+q₁₁u₁+q₁₂u₂ = g₁+h₁₁µ ,

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Introduction

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In Which Areas Can the Toolbox Be Used?

The PDEs implemented in the toolbox are used as a mathematical model for a wide variety of phenomena in all branches of engineering and science. The following is by no means a complete list of examples:

The elliptic and parabolic equations are used for modeling

• steady and unsteady heat transfer in solids

• flows in porous media and diffusion problems

• electrostatics of dielectric and conductive media

• potential flow

The hyperbolic equation is used for

• transient and harmonic wave propagation in acoustics and electromagnetics

• transverse motions of membranes The eigenvalue problems are used for, e.g.,

• determining natural vibration states in membranes and structural mechanics problems

Last, but not least, the toolbox can be used for educational purposes as a complement to understanding the theory of the Finite Element Method.

How Do I Define a PDE Problem?

The simplest way to define a PDE problem is using the graphical user interface (GUI), implemented in pdetool. There are three modes that correspond to different stages of defining a PDE problem:

• Draw mode, you create Ω, the geometry, using the constructive solid geometry (CSG) model paradigm. A set of solid objects (rectangle, circle, ellipse, and polygon) is provided. You can combine these objects using set formulas.

• In Boundary mode, you specify the boundary conditions. You can have different types of boundary conditions on different boundary segments.

• In PDE mode, you interactively specify the type of PDE and the coefficients c, a, f, and d. You can specify the coefficients for each subdomain

independently. This may ease the specification of, e.g., various material properties in a PDE model.

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How Can I Solve a PDE Problem?

Most problems can be solved from the graphical user interface. There are two major modes that help you solve a problem:

• In Mesh mode, you generate and plot meshes. You can control the parameters of the automated mesh generator.

• In Solve mode, you can invoke and control the nonlinear and adaptive solvers for elliptic problems. For parabolic and hyperbolic problems, you can specify the initial values, and the times for which the output should be generated. For the eigenvalue solver, you can specify the interval in which to search for eigenvalues.

After solving a problem, you can return to the Mesh mode to further refine your mesh and then solve again. You can also employ the adaptive mesh refiner and solver. This option tries to find a mesh that fits the solution.

Can I Use the Toolbox for Nonstandard Problems?

For advanced, nonstandard applications you can transfer the description of domains, boundary conditions etc. to your MATLAB workspace. From there you use the functions of the PDE Toolbox for managing data on unstructured meshes. You have full access to the mesh generators, FEM discretizations of the PDE and boundary conditions, interpolation functions, etc. You can design your own solvers or use FEM to solve subproblems of more complex algorithms.

See also the section “Using Command-Line Functions.”

How Can I Visualize My Results?

From the graphical user interface you can use Plot mode, where you have a wide range of visualization possibilities. You can visualize both inside the pdetool GUI and in separate figures. You can plot three different solution properties at the same time, using color, height, and vector field plots. Surface, mesh, contour, and arrow (quiver) plots are available. For surface plots, you can choose between interpolated and flat rendering schemes. The mesh may be hidden or exposed in all plot types. For parabolic and hyperbolic equations, you can even produce an animated movie of the solution’s time-dependence. All visualization functions are also accessible from the command line.

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Introduction

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Are There Any Applications Already Implemented?

The PDE Toolbox is easy to use in the most common areas due to the

application interfaces. Eight application interfaces are available, in addition to the generic scalar and system (vector valued u) cases:

• Structural Mechanics - Plane Stress

• Structural Mechanics - Plane Strain

• Electrostatics

• Magnetostatics

• AC Power Electromagnetics

• Conductive Media DC

• Heat Transfer

• Diffusion

These interfaces have dialog boxes where the PDE coefficients, boundary conditions, and solution are explained in terms of physical entities. The application interfaces enable you to enter specific parameters, such as Young’s modulus in the structural mechanics problems. Also, visualization of the relevant physical variables is provided.

Several nontrivial examples are included in this manual. Many examples are solved both by using the GUI and in command-line mode.

The toolbox contains a number of demonstration M-files. They illustrate some ways in which you can write your own applications.

Can I Extend the Functionality of the Toolbox?

The PDE Toolbox is written using MATLAB’s open system philosophy. There are no black-box functions, although some functions may not be easy to understand at first glance. The data structures and formats are documented.

You can examine the existing functions and create your own as needed.

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How Can I Solve 3-D Problems by 2-D Models?

The PDE Toolbox solves problems in two space dimensions and time, whereas reality has three space dimensions. The reduction to 2-D is possible when variations in the third space dimension (taken to be z) can be accounted for in the 2-D equation. In some cases, like the plane stress analysis, the material parameters must be modified in the process of dimensionality reduction.

When the problem is such that variation with z is negligible, all z-derivatives drop out and the 2-D equation has exactly the same units and coefficients as in 3-D.

Slab geometries are treated by integration through the thickness. The result is a 2-D equation for the z-averaged solution with the thickness, say D(x,y), multiplied onto all the PDE coefficients, c, a, d, and f, etc. For instance, if you want to compute the stresses in a sheet welded together from plates of different thickness, multiply Young’s modulus E, volume forces, and specified surface tractions by D(x,y). Similar definitions of the equation coefficients are called for in other slab geometry examples and application modes.

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Getting Started

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Getting Started

To get you started, let’s use the graphical user interface (GUI) pdetool, which is a part of the PDE Toolbox, to solve a PDE step by step. The problem that we would like to solve is Poisson’s equation, . The 2-D geometry on which we would like to solve the PDE is quite complex. The boundary conditions are of Dirichlet and Neumann types.

First, invoke MATLAB. To start the GUI, type the command pdetool at the MATLAB prompt. It can take a minute or two for the GUI to start. The GUI looks similar to the figure below, with exception of the grid. Turn on the grid by selecting Grid from the Options menu. Also, enable the “snap-to-grid”

feature by selecting Snap from the Options menu. The “snap-to-grid” feature simplifies aligning the solid objects.

The first step is to draw the geometry on which you want to solve the PDE. The GUI provides four basic types of solid objects: polygons, rectangles, circles, and ellipses. The objects are used to create a Constructive Solid Geometry model (CSG model). Each solid object is assigned a unique label, and by the use of set algebra, the resulting geometry can be made up of a combination of unions, intersections, and set differences. By default, the resulting CSG model is the union of all solid objects.

∆u

– = f

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1-10

To select a solid object, either click on the button with an icon depicting the solid object that you want to use, or select the object by using the Draw pull-down menu. In this case, rectangle/square objects are selected. To draw a rectangle or a square starting at a corner, press the rectangle button without a + sign in the middle. The button with the + sign is used when you want to draw starting at the center. Then, put the cursor at the desired corner, and

click-and-drag using the left mouse button to create a rectangle with the desired side lengths. (Use the right mouse button to create a square.) Notice how the “snap-to-grid” feature forces the rectangle to line up with the grid.

When you release the mouse, the CSG model is updated and redrawn. At this stage, all you have is a rectangle. It is assigned the label R1. If you want to move or resize the rectangle, you can easily do so. Click-and-drag an object to move it, and double-click on an object to open a dialog box, where you can enter exact location coordinates. From the dialog box, you can also alter the label. If you are not satisfied and want to restart, you can delete the rectangle by pressing the Delete key or by selecting Clear from the Edit menu. Next, draw a circle by clicking on the button with the ellipse icon with the + sign, and then click-and-drag in a similar way, using the right mouse button, starting at the circle center.

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Getting Started

1-11 The resulting CSG model is the union of the rectangle R1 and the circle C1,

described by set algebra as R1+C1. The area where the two objects overlap is clearly visible as it is drawn using a darker shade of gray. The object that you just drew — the circle — has a black border, indicating that it is selected. A selected object can be moved, resized, copied, and deleted. You can select more than one object by Shift-clicking on the objects that you want to select. Also, a Select All option is available from the Edit menu.

Finally, add two more objects, a rectangle R2 and a circle C2. The desired CSG model is formed by subtracting the circle C2 from the union of the other three objects. You do this by editing the set formula that by default is the union of all objects: C1+R1+R2+C2. You can type any other valid set formula into Set formula edit field. Click in the edit field and use the keyboard to change the set formula to:

(R1+C1+R2)-C2

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1-12

If you want, you can save this CSG model as an M-file. Use the Save As. . . option from the File menu, and enter a filename of your choice. It’s good practice to continue to save your model at regular intervals using Save. All the additional steps in the process of modeling and solving your PDE are then saved to the same M-file. This concludes the drawing part. You can now define the boundary conditions for the outer boundaries. Enter the Boundary mode by pressing the icon or by selecting Boundary Mode from the Boundary menu. You can now remove subdomain borders and define the boundary conditions.

The gray edge segments are subdomain borders induced by the intersections of the original solid objects. Borders that do not represent borders between, e.g., areas with differing material properties, can be removed. From the Boundary menu, select the Remove All Subdomain Borders option. All borders are then removed from the decomposed geometry.

The boundaries are indicated by colored lines with arrows. The color reflects the type of boundary condition, and the arrow points towards the end of the boundary segment. The direction information is provided for the case when the boundary condition is parameterized along the boundary. The boundary condition can also be a function of x and y, or simply a constant. By default, the boundary condition is of Dirichlet type: u = 0 on the boundary.

Dirichlet boundary conditions are indicated by red color. The boundary conditions can also be of a generalized Neumann (blue) or mixed (green) type.

For scalar u, however, all boundary conditions are either of Dirichlet or the generalized Neumann type. You select the boundary conditions that you want to change by clicking to select one boundary segment, by Shift-clicking to select multiple segments, or by using the Edit menu option Select All to select all boundary segments. The selected boundary segments are indicated by black color.

Ω

∂

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Getting Started

1-13 For this problem, change the boundary condition for all the circle arcs. Select them by using the mouse and Shift-click on those boundary segments.

Double-clicking anywhere on the selected boundary segments opens the Boundary Condition dialog box. Here, you select the type of boundary condition, and enter the boundary condition as a MATLAB expression. Change the boundary condition along the selected boundaries to a Neumann condition,

. This means that the solution has a slope of –5 in the normal direction for these boundary segments.

In the Boundary Condition dialog box, select the Neumann condition type, and enter –5 in the edit box for the boundary condition parameter g. To define a pure Neumann condition, leave the q parameter at its default value, 0. When you press the OK button, notice how the selected boundary segments change to blue to indicate Neumann boundary condition.

∂n⁄∂u = –5

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1-14

Next, specify the PDE itself through a dialog box that is accessed by pressing the button with the PDE icon or by selecting PDE Specification. . . from the PDE pull-down menu. In the PDE mode, you can also access the PDE Specification dialog box by double-clicking on a subdomain. That way, different subdomains can have different PDE coefficient values. This problem, however, consists of only one subdomain.

In the dialog box, you can select the type of PDE (elliptic, parabolic, hyperbolic, or eigenmodes) and define the applicable coefficients depending on the PDE type. This problem consists of an elliptic PDE defined by the equation

with c = 1.0, a = 0.0, and f = 10.0.

∇

– ⋅(c∇u)+au = ,f

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Getting Started

1-15 Finally, create the triangular mesh that the PDE Toolbox uses in the Finite

Element Method (FEM) to solve the PDE. The triangular mesh is created and displayed when pressing the button with the icon or by selecting the Mesh menu option Initialize Mesh. If you want a more accurate solution, the mesh can be successively refined by pressing the button with the four triangle icon (the Refine button) or by selecting the Refine Mesh option from the Mesh menu. Using the Jiggle Mesh option, the mesh can be jiggled to improve the triangle quality. Parameters for controlling the jiggling of the mesh, the refinement method, and other mesh generation parameters can be found in a dialog box that is opened by selecting Parameters from the Mesh menu. You can undo any change to the mesh by selecting the Mesh menu option Undo Mesh Change.

Initialize the mesh, then refine it once and finally jiggle it once.

We are now ready to solve the problem. Press the = button or select Solve PDE from the Solve menu to solve the PDE. The solution is then plotted. By default, the plot uses interpolated coloring and a linear color map. A colorbar is also provided to map the different shades to the numerical values of the solution. If you want, the solution can be exported as a vector to the MATLAB main workspace.

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There are many more plot modes available to help you visualize the solution.

Press the button with the 3-D solution icon or select Parameters. . . from the Plot menu to access the dialog box for selection of the different plot options.

Several plot styles are available, and the solution can be plotted in the GUI or in a separate figure as a 3-D plot. Now, select a plot where the color and the height both represent u. Choose interpolated shading and use the continuous (interpolated) height option. The default colormap is the cool colormap; a pop-up menu lets you select from a number of different colormaps. Finally, press the Plot button to plot the solution; press the Done button to save the plot setup as the current default. The solution is plotted as a 3-D plot in a separate figure window.

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Getting Started

1-17 The following solution plot is the result. You can use the mouse to rotate the plot in 3-D. By clicking-and-dragging the axes, the angle from which the solution is viewed can be changed.

This concludes the first example of solving a PDE by using the pdetool GUI.

Many more examples in Chapter 2, "Examples" focus on solving particular problems involving different kinds of PDEs, geometries and boundary conditions and covering a range of different applications.

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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 discrete approximation of the solution. The pdetool graphical user interface provides you with easy-to-use graphical tools to describe complicated domains and generate triangular 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 M-file) and incorporate them into specialized applications.

Below 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 the PDE Toolbox are presented. A more detailed description can be found in Chapter 4,

"The Finite Element Method".

You start by approximating the computational domain with 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 simplest choice but you cannot match values on neighboring triangles. Linear functions come next.

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Basics of The Finite Element Method

1-19 This is like using flat tiles to build a waterproof dome, which is perfectly

possible.

Now you use the basic elliptic equation:

in Ω.

If u_h is the piecewise linear approximation to u, it is not clear what the second derivative term means. Inside each triangle, is a constant (because u_h is flat) and thus the second-order term vanishes. At the edges of the triangles,

is in general discontinuous and a further derivative makes no sense.

What you are looking for is the best approximation of u in the class of continuous piecewise polynomials. Therefore you test the equation for u_h against all possible functions v of that class. Testing means formally to multiply the residual against any function and then integrate, i.e., determine u_h such that

for all possible v. The functions v are usually called test functions.

Partial integration (Green’s formula) yields that u_h should satisfy

−1

−0.5 0

0.5 1

−1

−0.5

0

0.5

1 0

0.2 0.4 0.6 0.8

−1

−0.5 0

0.5 1

−1

−0.5

0

0.5

1 0

0.2 0.4 0.6 0.8

A triangular mesh (left) and a continuous piecewise linear function on that mesh

∇

– ⋅(c u∇ )+au = f

u_h

∇ c u∇ _h

∇

– ⋅(c u∇ _h)+au_h–f

( )v xd

∫

Ω ⁼ ⁰

c u∇ _h ( )∇v

∫

Ω ⁺^au^h^vdx^–

∫

^∂Ωⁿ^⋅⁽^{c u}^∇ ^h^)vds ⁼

∫

^Ω^{fv x}^d ^∀^v ^,

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where is the boundary of and is the outward pointing normal on . Note that the integrals of this formulation are well-defined even if u_h and v are piecewise linear functions.

Boundary conditions are included in the following way. If u_h is known at some boundary points (Dirichlet boundary conditions), we restrict the test functions to v = 0 at those points, and require u_h to attain the desired value at that point.

At all the other points we ask for Neumann boundary conditions, i.e., . The FEM formulation reads: Find u_h such that

where is the part of the boundary with Neumann conditions. The test functions v must be zero on .

Any continuous piecewise linear u_h is represented as a combination , where φ_i are some special piecewise linear basis functions and U_i are scalar coefficients. Choose φ_i like a tent, such that it has the “height” 1 at the node i and the height 0 at all other nodes. For any fixed v, the FEM formulation yields an algebraic equation in the unknowns U_i. You want to determine N unknowns, so you need N different instances of v. What better candidates than v = φ_j, j = 1, 2, . . . , N? You find a linear system KU = F where the matrix K and the right-hand side F contain integrals in terms of the test functions φ_i, φ_j and the coefficients defining the problem: c, a, f, q, and g.

The solution vector U contains the expansion coefficients of u_h, which are also the values of u_h at each node x_i since u_h(x_i) = U_i.

If the exact solution u is smooth, then FEM computes u_h with an error of the same size as that of the linear interpolation. It is possible to estimate the error on each triangle using only u_h and the PDE coefficients (but not the exact solution u, which in general is unknown).

The PDE Toolbox provides functions that assemble K and F. This is done automatically in the graphical user interface, but you also have direct access to the FEM matrices from the command-line function assempde.

To summarize, the FEM approach is to approximate the PDE solution u by a piecewise linear function is expanded in a basis of test-functions φ_i, and the residual is tested against all the basis functions. This procedure yields a linear system KU = F. The components of U are the values of u_h at the nodes.

Ω

∂ Ω n ∂Ω

c∇u_h

( )⋅n+qu_h = g c u∇ _h

( )∇v

∫

Ω ⁺^au^h^vdx⁺

∫

^∂Ω¹^qu^h^vds⁼

∫

^Ω^{fv x}^d ⁺

∫

^∂Ω¹^{gv s}^d ^∀^v,

∂Ω₁

∂Ω ∂Ω– ₁

u_h( )x U_iφ_i( )x

i=1

∑

N

=

u_h⋅u_h

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Basics of The Finite Element Method

1-21 For x inside a triangle, u_h(x) is found by linear interpolation from the nodal

values.

FEM techniques are also used to solve more general problems. Below are some generalizations that you can access both through the graphical user interface and with command-line functions.

• Time-dependent problems are easy to implement in the FEM context. The solution u(x,t) of the equation

can be approximated by . This yields a system of

ordinary differential equations (ODE) which you integrate using ODE solvers. Two time derivatives yield a second order ODE

, etc. The toolbox supports problems with one or two time derivatives (the functions parabolic and hyperbolic).

• Eigenvalue problems: Solve

for the unknowns u and λ (λ is a complex number). Using the FEM discretization, you solve the algebraic eigenvalue problem KU = λ_hMU to find u_h and λ_h as approximations to u and λ. A robust eigenvalue solver is implemented in pdeeig.

• If the coefficients c, a, f, q, or g are functions of u, the PDE is called nonlinear and FEM yields a nonlinear system K(U) U= F(U). You can use iterative methods for solving the nonlinear system. The toolbox provides a nonlinear solver called pdenonlin using a damped Gauss-Newton method.

∂u

∂t ---

d –∇⋅(c u∇ )+au = f

u_h(x t, ) U_i( )φt _i( )x

i=1

∑

N

=

Mdt

dU+KU = F

M t²

2

d

d U+KU = F

∇

– ⋅(c∇u)+au = λdu

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• Small triangles are needed only in those parts of the computational domain where the error is large. In many cases the errors are large in a small region and making all triangles small is a waste of computational effort.

Making small triangles only where needed is called adapting the mesh refinement to the solution. An iterative adaptive strategy is the following:

For a given mesh, form and solve the linear system KU = F. Then estimate the error and refine the triangles in which the error is large. The iteration is controlled by adaptmesh and the error is estimated by pdejmps.

Although the basic equation is scalar, systems of equations are also handled by the toolbox. The interactive environment accepts u as a scalar or 2-vector function. In command-line mode, systems of arbitrary size are accepted.

If c ≥ δ > 0 and a ≥ 0, under rather general assumptions on the domain Ω and the boundary conditions, the solution u exists and is unique. The FEM linear system has a unique solution which converges to u as the triangles become smaller. The matrix K and the right-hand side F make sense even when u does not exist or is not unique. It is advisable that you devise checks to problems with questionable solutions.

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Using the Graphical User Interface

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Using the Graphical User Interface

The PDE Toolbox Graphical User Interface

The PDE Toolbox includes a complete graphical user interface (GUI), which covers all aspects of the PDE solution process. You start it by typing:

pdetool

at the MATLAB command line. It may take a while the first time you launch pdetool during a MATLAB session. The figure below shows the pdetool GUI as it looks when you have started 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 items 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.

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To the right of the toolbar is a pop-up menu that indicates the current application 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. For a more detailed description of the menus and the dialog boxes, see Chapter 3, "The Graphical User Interface".

• File menu. From the File menu you can Open and Save model M-files that contain a command sequence that reproduces your modeling session. You can 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 “snap-to-grid” 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 label edges 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.

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• 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 and several 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 left-most buttons are Draw mode buttons and they represent, from left to right:

• Draw a rectangle/square starting at a corner.

• Draw a rectangle/square starting at the center.

• Draw an ellipse/circle starting at the perimeter.

• Draw an ellipse/circle starting at the center.

• Draw a polygon. Click-and-drag to create polygon sides. You can close the polygon by pressing the right mouse button. Clicking at the starting vertex also closes the polygon.

The Draw mode buttons can only be activated one at the time and they all work the same way: single-clicking on a button allows you to draw one solid object of the selected type. Double-clicking on a button makes it “stick,” and you can then continue to draw solid objects of the selected type until you

single-click on the button to “release” it. Using the right mouse button or Control-click, the drawing is constrained to a square or a circle.

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The second group of six buttons includes the following buttons:

• : Enters the Boundary mode.

• PDE: Opens the PDE Specification dialog box.

• Initializes the triangular mesh.

• The refine button (the four triangle icon): Refines the triangular mesh.

• =: Solves the PDE.

• The 3-D solution plot icon: Opens the Plot Selection dialog box.

The right-most button with the magnifying glass toggles the zoom function on/off.

The GUI Modes

The PDE solving process can be divided into several steps:

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 modes, each corresponding to one of the steps in the PDE solving process:

• In Draw mode, you can create the 2-D geometry using the constructive solid geometry (CSG) model paradigm. A set of solid objects (rectangle, circle, ellipse, and polygon) is provided. These objects can be combined using set formulas in a flexible way.

• In Boundary mode, you can specify the boundary conditions. You can have different types of boundary conditions on different boundaries. In this mode, the original shapes of the solid objects constitute borders between

subdomains of the model. Such borders can be eliminated in this mode.

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• In PDE mode, you can interactively specify the type of PDE problem, and the PDE coefficients. You can specify the coefficients for each subdomain independently. This makes it easy to specify, e.g., various material properties in a PDE model.

• In Mesh mode, you can control the automated mesh generation and plot the mesh.

• In Solve mode, you can invoke and control the nonlinear and adaptive solver for elliptic problems. For parabolic and hyperbolic PDE problems, you can specify the initial values, and the times for which the output should be generated. For the eigenvalue solver, you can specify the interval in which to search for eigenvalues.

• In Plot mode there is wide range of visualization possibilities. You can visualize both in the pdetool GUI and in a separate figure window. You can visualize three different solution properties at the same time, using color, height, and vector field plots. There are surface, mesh, contour, and arrow (quiver) plots available. For parabolic and hyperbolic equations, you can animate the solution as it changes with time.

The CSG Model and the Set Formula

The PDE Toolbox uses the Constructive Solid Geometry (CSG) model paradigm for the modeling. You can draw solid objects that can overlap. There are four types of solid objects:

• Circle object — represents the set of points inside and on a circle.

• Polygon object — represents the set of points inside and on a polygon given by a set of line segments.

• Rectangle object — represents the set of points inside and on a rectangle.

• Ellipse object — represents the set of points inside and on an ellipse. The ellipse can be rotated.

Each solid object is automatically given a unique name by the GUI. The default names are C1, C2, C3, etc., for circles; P1, P2, P3, etc. for polygons; R1, R2, R3, etc., for rectangles; E1, E2, E3, etc., for ellipses. Squares, although just a special case of rectangles, are named SQ1, SQ2, SQ3, etc. The name is

displayed on the solid object itself. You can use any unique name, as long as it contains no blanks. In Draw mode you can alter the names and the geometries of the objects by double-clicking on them. This opens a dialog box where you can

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edit the name and the geometry. The following figure shows an object dialog box for a circle.

You can use the name of the object to refer to the corresponding set of points in a set formula. The operators +, *, and – are used to form the set of points Ω in the plane over which the differential equation is solved. The operators +, the set union operator, and *, the set intersection operator, have the same precedence. The operator –, the set difference operator, has higher precedence.

The precedence can be controlled by using parentheses. The resulting

geometrical model, Ω, is the set of points for which the set formula evaluates to true. By default, it is the union of all solid objects. We often refer to the area Ω as the decomposed geometry.

Creating Rounded Corners

As an example of how to use the set formula, let’s model a plate with rounded corners (fillets).

Start the GUI and turn on the grid and the “snap-to-grid” feature using the Options menu. Also, change the grid spacing to -1.5:0.1:1.5 for the x-axis and -1:0.1:1 for the y-axis.

Select Rectangle/square from the Draw menu or press the button with the rectangle icon. Then draw a rectangle with a width of 2 and a height of 1 using the mouse, starting at (-1,0.5). To get the round corners, add circles, one in each corner. The circles should have a radius of 0.2 and centers at a distance that is 0.2 units from the left/right and lower/upper rectangle boundaries ((-0.8,-0.3), (-0.8,0.3), (0.8,-0.3), and (0.8,0.3)). To draw several circles, double-click on the button for drawing ellipses/circles (centered). Then draw the circles using the right mouse button or Control-click starting at the circle centers. Finally, at each of the rectangle corners, draw four small squares with a side of 0.1.

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1-29 The figure below shows the complete drawing.

Now you have to edit the set formula. To get the rounded corners, subtract the small squares from the rectangle and then add the circles. As a set formula, this is expressed as

R1-(SQ1+SQ2+SQ3+SQ4)+C1+C2+C3+C4

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Enter the set formula into the edit box at the top of the GUI. Then enter the Boundary mode by pressing the button or by selecting the Boundary Mode option from the Boundary menu. The CSG model is now decomposed using the set formula, and you get a rectangle with rounded corners, as shown below.

Because of the intersection of the solid objects used in the initial CSG model, a number of subdomain borders remain. They are drawn using gray lines. If this is a model of, e.g., a homogeneous plate, you can remove them. Select the Remove All Subdomain Borders option from the Boundary menu. The subdomain borders are removed and the model of the plate is now complete.

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Suggested Modeling Method

Although the PDE Toolbox offers you a great deal of flexibility in the ways that you can approach the problems and interact with the toolbox functions, there is a suggested method of choice for modeling and solving your PDE problems using the pdetool GUI. There are also a number of shortcuts that you can use in certain situations.

NOTE: There are platform-dependent keyboard accelerators available for many of the most common pdetool GUI activities. Learning to use the accelerator keys may improve the efficiency of your pdetool sessions.

The basic flow of actions is indicated by the way the graphical push buttons and the menus are ordered from left to right. You work your way from left to right in the process of modeling, defining, and solving your PDE problem using the pdetool GUI.

• When you start, pdetool is in a Draw mode, where you can use the four basic solid objects to draw your Constructive Solid Geometry (CSG) model.

You can also edit the set formula. The solid objects are selected using the five left-most push buttons (or from the Draw menu).

• To the right of the Draw mode buttons you find push buttons through which you can access all the functions that you need to define and solve the PDE problem: define boundary conditions, design the triangular mesh, solve the PDE, and plot the solution.

The following sequence of actions covers all the steps of a normal pdetool session:

1 Use pdetool as a drawing tool to make a drawing of the 2-D geometry on which you want to solve your PDE. Make use of the four basic solid objects and the grid and the “snap-to-grid” feature. The GUI starts in the Draw mode, and you can select the type of object that you want to use by clicking on the corresponding button or by using the Draw pull-down menu.

Combine the solid objects and the set algebra to build the desired CSG model.

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2 Save the geometry to a model file. The model file is an M-file, so if you want to continue working using the same geometry at your next PDE Toolbox session, simply type the name of the model file at the MATLAB prompt. The pdetool GUI then starts with the model file’s solid geometry loaded. If you save the PDE problem at a later stage of the solution process, the model file also contains commands to recreate the boundary conditions, the PDE coefficients, and the mesh.

3 Move to the next step in the PDE solving process by pressing the button.

The outer boundaries of the decomposed geometry are displayed with the default boundary condition indicated. If the outer boundaries do not match the geometry of your problem, re-enter the Draw mode. You can then correct your CSG model by adding, removing or altering any of the solid objects, or change the set formula used to evaluate the CSG model.

NOTE: The set formula can only be edited while you are in the Draw mode.

If the drawing process resulted in any unwanted subdomain borders, remove them by using the Remove Subdomain Border or Remove All Subdomain Borders option from the Boundary menu.

You can now define your problem’s boundary conditions by selecting the boundary to change and open a dialog box by double-clicking on the boundary or by using the Specify Boundary Conditions. . . option from the Boundary menu.

4 Initialize the triangular mesh. Press the button or use the corresponding Mesh menu option Initialize Mesh. Normally, the mesh algorithm’s default parameters generate a good mesh. If necessary, they can be accessed using the Parameters. . . menu item.

5 If you need a finer mesh, the mesh can be refined by pressing the Refine button. Pressing the button several times causes a successive refinement of the mesh. The cost of a very fine mesh is a significant increase in the number of points where the PDE is solved and, consequently, a significant increase in the time required to compute the solution. Don’t refine unless it is required to achieve the desired accuracy. For each refinement, the number of triangles increases by a factor of four. A better way to increase the

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1-33 accuracy of the solution to elliptic PDE problems is to use the adaptive

solver, which refines the mesh in the areas where the estimated error of the solution is largest. See the entry on adaptmesh in Chapter 5, "Reference" for an example of how the adaptive solver can solve a Laplace equation with an accuracy that requires more than 10 times as many triangles when regular refinement is used.

6 Specify the PDE from the PDE Specification dialog box. You can access that dialog box using the PDE button or the PDE Specification. . . menu item from the PDE menu.

NOTE: This step can be performed at any time prior to solving the PDE since it is independent of the CSG model and the boundaries. If the PDE coefficients are material dependent, they are entered in the PDE mode by double-clicking on the different subdomains.

7 Solve the PDE by pressing the = button or by selecting Solve PDE from the Solve menu. If you don’t want an automatic plot of the solution, or if you want to change the way the solution is presented, you can do that from the Plot Selection dialog box prior to solving the PDE. You open the Plot Selection dialog box by pressing the button with the 3-D solution plot icon or by selecting the Parameters. . . menu item from the Plot menu.

8 Now, from here you can choose one of several alternatives:

• Export the solution and/or the mesh to MATLAB’s main workspace for further analysis.

• Visualize other properties of the solution.

• Change the PDE and recompute the solution.

• Change the mesh and recompute the solution. If you select Initialize Mesh, the mesh is initialized; if you select Refine Mesh, the current mesh is refined. From the Mesh menu, you can also jiggle the mesh and undo previous mesh changes.

• Change the boundary conditions. To return to the mode where you can se- lect boundaries, use the button or the Boundary Mode option from the Boundary menu.

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• Change the CSG model. You can re-enter the draw mode by selecting Draw Mode from the Draw menu or by clicking on one of the Draw mode icons to add another solid object. Back in the Draw mode, you are able to add, change, or delete solid objects and also to alter the set formula.

In addition to the recommended path of actions, there are a number of shortcuts, which allow you to skip over one or more steps. In general, the pdetool GUI adds the necessary steps automatically.

• If you haven’t yet defined a CSG model, and leave the Draw mode with an empty model, pdetool creates an L-shaped geometry with the default boundary condition and then proceeds to the action called for, performing all the steps necessary.

• If you are in the Draw mode and press the ∆ button to initialize the mesh, pdetool first decomposes the geometry using the current set formula and assigns the default boundary condition to the outer boundaries. After that, an initial mesh is created.

• If you press the refine button to refine the mesh before the mesh has been initialized, pdetool first initializes the mesh (and decomposes the geometry, if you were still in the Draw mode).

• If you press the = button to solve the PDE and you have not yet created a mesh, pdetool initializes a mesh before solving the PDE.

• If you select a plot type and choose to plot the solution, pdetool checks to see if there is a solution to the current PDE available. If not, pdetool first solves the current PDE. The solution is then displayed using the selected plot options.

• If you haven’t defined your PDE, pdetool solves the default PDE, which is Poisson’s equation:

-∆ u= 10

(This corresponds to the generic elliptic PDE with c = 1, a = 0, and f = 10.) For the different application modes, different default PDE settings apply.