Conduction on a Steel Strip.

The Problem.
 We compute the temperature over time of a thin square steel strip with a heat source at its center. Initially, the steel strip is at 0 degrees. The heat source within the steel strip generates heat with increasing intensity as time increases.

The PDE is given by

Ut = L(U) + f(x,y,t) on R, t > 0

where
R : -3.14 <= x,y <= 3.14

and
f(x,y,t) = 0.2*t*(sin(x)+sin(y))
L(U) = Uxx + Uyy

We assign boundary condition on R:    U(x,y,t) = 0.2*t*(sin(x)+sin(y))    with initial condition:    U(x,y,0) = 0.0. Click here to see a physical description of the problem.

We solve this parabolic problem on a 25 x 25 uniform grid, using PDELab's 5-Point Star inside a time loop, where the time variable is discretized. We take advantage of the symbolic manipulation available in PDELab: MAXIMA is accessed to discretize the time variable via Crank-Nicolson, and then GENCRAY generates the mix of PDELab language and fortran code required for the problem definition. The linear system is solved using the direct Band GE. The PDELab output can be animated in with the OutputTool using Visual2D. The solution U, as well as its derivatives Ux and Uy are saved at every iteration of the time loop.

Use PDELab to ...

Why use PDELab?

Define the Problem


Specify the Solution

Execute the Problem
Step 13: We are now ready to enter the ExecuteTool environment, where the PDELab language program will be processed. A fortran main program will be generated from the .e file. It will be compiled and linked with the PDELab module libraries and the standard PDELab I/O libraries. Click on the ExecuteTool button in the Session toolkit.

Step 14: The PDELab .e file is loaded automatically. Click on the Run button The trace of the process for compiling and linking the PDELab executable appears in the trace window. When the compiling and linking process is complete, the executable program is run and the solution and trance files are generated and placed in your user directory. Click on Quit to exit the ExecuteTool.

Visualize the Solution
Step 17: The solution output has been generated during execution. You should find the output file containing the PDELab format data in the location specified by the save segment. If no path was specified, this file is located in the execution directory. The file contains the computed values of the solution at each finite difference point for each time step.

Click on the OutputTool in the Session toolkit. Select PDELab Solution in the File menu. Select the output file that was generated by your program when the OutputTool requests the filename. The Select Solution Box is displayed after the file is loaded. The selection box contains a list of the computed solutions, one for each time step. You can choose any or all of the solutions. If more than one solution is selected, PDELab will animate the solution sets.


Snapshots from the animation of thermal behavior of steel strip with heat source.