HEATED_PLATE
2D Steady State Heat Equation in a Rectangle

HEATED_PLATE is a program which solves the steady state heat equation in a 2D rectangular region, and is intended as a starting point for implementing an OpenMP parallel version.

The final estimate of the solution is written to a file in a format suitable for display by GRID_TO_BMP.

The sequential version of this program needs approximately 18/epsilon iterations to complete.

The physical region, and the boundary conditions, are suggested by this diagram:

                   W = 0
             +------------------+
             |                  |
    W = 100  |                  | W = 100
             |                  |
             +------------------+
                   W = 100
      

The region is covered with a grid of M by N nodes, and an N by N array W is used to record the temperature. The correspondence between array indices and locations in the region is suggested by giving the indices of the four corners:

                  I = 0
          [0][0]-------------[0][N-1]
             |                  |
      J = 0  |                  |  J = N-1
             |                  |
        [M-1][0]-----------[M-1][N-1]
                  I = M-1
      

The steady state solution to the discrete heat equation satisfies the following condition at an interior grid point:

W[Central] = (1/4) * ( W[North] + W[South] + W[East] + W[West] )

Given an approximate solution of the steady state heat equation, a "better" solution is given by replacing each interior point by the average of its 4 neighbors - in other words, by using the condition as an ASSIGNMENT statement:

W[Central] <= (1/4) * ( W[North] + W[South] + W[East] + W[West] )

If this process is repeated often enough, the difference between successive estimates of the solution will go to zero.

This program carries out such an iteration, using a tolerance specified by the user, and writes the final estimate of the solution to a file that can be used for graphic processing.

Usage:

heated_plate epsilon output_file

epsilon

the error tolerance used to halt the iteration. This is an absolute error tolerance, and is applied pointwise. A value of 0.1 might be reasonable for the built in problem.

output_file

the name of the file into which the final estimate of the solution will be written, for possible display by GRID_TO_BMP.

Source Code:

• heated_plate.c, the C source code.
• heated_plate.cc, the C++ source code.
• heated_plate.f, the FORTRAN77 source code.
• heated_plate.f90, the FORTRAN90 source code.

Completed Source Code:

Here are "completed" versions of the programs.
(Access to these files may be restricted until everyone has had a chance to work on the problems!)

• heated_plate_complete.c, the C source code.
• heated_plate_complete.cc, the C++ source code.
• heated_plate_complete.f, the FORTRAN77 source code.
• heated_plate_complete.f90, the FORTRAN90 source code.

Batch Script:

Once you have made a heated_plate executable, you need to copy this script file to the same directory that contains the executable, and issue the command qsub heated_plate.sh. That submits a request to have the executable program run on 4 processors. You can type showq to see if the request has started yet.

• heated_plate.sh, a script to run the program.
• heated_plate_output.txt, the output file.

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