Partial Differential Equation Toolbox

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 the MATLAB 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.

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.

How Can I Visualize My Results?

Getting Started