# heated_plate_openmp

heated_plate_openmp, a FORTRAN90 code which uses the OpenMP application program interface by employing an iteration that solves the 2D steady state heat equation.

The sequential version of this program needs approximately 18/eps 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] )
```
where "Central" is the index of the grid point, "North" is the index of its immediate neighbor to the "north", and so on.

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.

### Licensing:

The computer code and data files described and made available on this web page are distributed under the MIT license

### Languages:

heated_plate_openmp is available in a C version and a C++ version and a FORTRAN90 version.

### Related Data and Programs:

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FFT_OPENMP, a FORTRAN90 code which demonstrates the computation of a Fast Fourier Transform in parallel, using OpenMP.

HEATED_PLATE, a FORTRAN90 code which solves the steady (time independent) heat equation in a 2D rectangular region, and is intended as a starting point for implementing an OpenMP parallel version.

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### Reference:

1. Peter Arbenz, Wesley Petersen,
Introduction to Parallel Computing - A practical guide with examples in C,
Oxford University Press,
ISBN: 0-19-851576-6,
LC: QA76.58.P47.
2. Rohit Chandra, Leonardo Dagum, Dave Kohr, Dror Maydan, Jeff McDonald, Ramesh Menon,
Parallel Programming in OpenMP,
Morgan Kaufmann, 2001,
ISBN: 1-55860-671-8,
LC: QA76.642.P32.
3. Barbara Chapman, Gabriele Jost, Ruud vanderPas, David Kuck,
Using OpenMP: Portable Shared Memory Parallel Processing,
MIT Press, 2007,
ISBN13: 978-0262533027,
LC: QA76.642.C49.
4. OpenMP Architecture Review Board,
OpenMP Application Program Interface,
Version 3.0,
May 2008.

### Source code:

Last revised on 13 July 2020.