## PDE2D Examples

A PDE2D program is created during an interactive session , or a GUI session , and all documentation, including examples, is available on-line. Thus the user does not have to work through a thick user's manual (there is no manual, in fact!) to determine how to set up a problem. All the user has to do is answer a series of questions about the region, partial differential equations and boundary conditions, and select solution method and graphical output options. Extensive error checking is done during the interactive session, and also at execution time.

After all questions have been answered, the FORTRAN90 program created by the driver is compiled, linked to the runtime routines which it calls, and executed. The FORTRAN90 program contains copious documentation and is highly readable (most of the questions asked during the interactive session are repeated in the comments), so the user can make minor changes directly to the program, and does not have to work through another interactive session each time minor modifications to the problem or solution method are desired.

There are 15 prepared example problems, and if the user selects one of these, answers will be provided to each interactive question, appropriate to the example. To see what the interactive sessions look like for these problems, click on one of the following examples:

1. A simply-supported elastic plate equation, with a unit load concentrated at the midpoint of a square.

2. A non-linear, steady-state PDE, solved in an annulus. The initial triangulation is generated automatically, and adaptive grid refinement is illustrated.

3. An eigenvalue problem. The region has a curved interface across which material properties vary abruptly, in this example.

4. The first part of a thermal stress problem. In this part, the temperature distribution in a V-notched block is calculated by solving the time-dependent heat conduction equation, using adaptive time step control.

5. The second part of a thermal stress problem. In this part, the stresses induced in the V-notched block by thermal expansion are calculated using the temperature distribution output by example 4.

6. A 1D time-dependent integro-differential equation for a financial math application. In this problem there is a term involving an integral of the solution, which requires that we use PDE2D's feature for interpolating the solution at the last saved time step, for use in the integral term.

7. A waveguide problem (an eigenvalue problem in which the eigenvalue appears nonlinearly).

8. The Navier-Stokes equations solved for a fluid flowing around a bend in a pipe.

9. A 3D elasticity problem, solved in a torus. This example illustrates the use of user-defined coordinate transformations to handle more general 3D regions.

10. A 3D time-dependent wave equation

11. A 3D eigenvalue problem: the Schrodinger equation in a hydrogen atom. This example illustrates the use of spherical coordinates and periodic boundary conditions.

12. A 3D eigenvalue problem, solved in a composite region consisting of two cylinders of different material properties.

13. The axisymmetric Navier-Stokes equations solved in a non-rectangular channel, using the collocation FEM.

14. A 1D saturated/unsaturated water flow problem

15. A 1D version of the Schrodinger eigenvalue equation of example 11.

To see the FORTRAN90 program generated during the example 9 interactive session, click on: