Partial Differential Equation Toolbox For Use with MATLAB® User’s Guide Computer Solutions Europe AB Computation Visualization Programming User’s Guide Computation Visualization Programming Partial Differential Equation Toolbox For Use with MATLAB ® User’s Guide Computer Solutions Europe AB How to Contact The MathWorks: 508-647-7000 Phone 508-647-7001 Fax The MathWorks, Inc. Mail 24 Prime Park Way Natick, MA 01760-1500 http://www.mathworks.com Web ftp.mathworks.com Anonymous FTP server comp.soft-sys.matlab Newsgroup support@mathworks.com Technical support suggest@mathworks.com Product enhancement suggestions bugs@mathworks.com Bug reports doc@mathworks.com Documentation error reports subscribe@mathworks.com Subscribing user registration service@mathworks.com Order status, license renewals, passcodes info@mathworks.com Sales, pricing, and general information Partial Differential Equation Toolbox User’s Guide COPYRIGHT 1984 - 1997 by The MathWorks, Inc. All Rights Reserved. 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Printing History: August 1995 First printing February 1996 Reprint ☎ FAX ✉ @ i 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 M ATLAB Expressions . . . . . . . 1-36 Using PDE Toolbox version 1.0 Model M-files . . . . . . . . . . . . . . 1-36 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 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 iv Contents 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 v 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 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 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 M ATLAB 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 [...]... 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... 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 1-2 Introduction What Problems Can I Solve? The basic equation of the PDE Toolbox is the PDE – ∇ ⋅ ( c ∇u ) + au = f in Ω, which we shall refer to as the elliptic equation, regardless... 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 ∂u d -... implement in the FEM context The solution u(x,t) of the equation ∂u d - – ∇ ⋅ ( c ∇u ) + au = f ∂t can be approximated by u h ( x, t ) = ∑i = 1 U i ( t )φ i( x ) This yields a system of N d ordinary differential equations (ODE) M U + KU = F which you integrate dt using ODE solvers Two time derivatives yield a second order ODE d 2 U + KU = F , etc The toolbox supports problems with one or two time 2 dt... Element Method" for the general system case 1-4 Introduction 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... 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 M ATLAB’s open system philosophy There are no black-box functions, although some functions... tractions by D(x,y) Similar definitions of the equation coefficients are called for in other slab geometry examples and application modes 1-8 Getting Started 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, – ∆u = f The 2-D geometry on which... 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... 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,... 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 . 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. User’s Guide Computation Visualization Programming Partial Differential Equation Toolbox For Use with MATLAB ® User’s Guide Computer Solutions