# include # include # include # include int main ( int argc, char *argv[] ); /******************************************************************************/ int main ( int argc, char *argv[] ) /******************************************************************************/ /* Purpose: MAIN is the main program for HEATED_PLATE_OPENMP. Discussion: This code solves the steady state heat equation on a rectangular region. 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] ) 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: This code is distributed under the GNU LGPL license. Modified: 18 October 2011 Author: Original C version by Michael Quinn. This C version by John Burkardt. Reference: Michael Quinn, Parallel Programming in C with MPI and OpenMP, McGraw-Hill, 2004, ISBN13: 978-0071232654, LC: QA76.73.C15.Q55. Local parameters: Local, double DIFF, the norm of the change in the solution from one iteration to the next. Local, double MEAN, the average of the boundary values, used to initialize the values of the solution in the interior. Local, double U[M][N], the solution at the previous iteration. Local, double W[M][N], the solution computed at the latest iteration. */ { # define M 500 # define N 500 double diff; double epsilon = 0.001; int i; int iterations; int iterations_print; int j; double mean; double my_diff; double u[M][N]; double w[M][N]; double wtime; printf ( "\n" ); printf ( "HEATED_PLATE_OPENMP\n" ); printf ( " C/OpenMP version\n" ); printf ( " A program to solve for the steady state temperature distribution\n" ); printf ( " over a rectangular plate.\n" ); printf ( "\n" ); printf ( " Spatial grid of %d by %d points.\n", M, N ); printf ( " The iteration will be repeated until the change is <= %e\n", epsilon ); printf ( " Number of processors available = %d\n", omp_get_num_procs ( ) ); printf ( " Number of threads = %d\n", omp_get_max_threads ( ) ); /* Set the boundary values, which don't change. */ mean = 0.0; #pragma omp parallel shared ( w ) private ( i, j ) { #pragma omp for for ( i = 1; i < M - 1; i++ ) { w[i][0] = 100.0; } #pragma omp for for ( i = 1; i < M - 1; i++ ) { w[i][N-1] = 100.0; } #pragma omp for for ( j = 0; j < N; j++ ) { w[M-1][j] = 100.0; } #pragma omp for for ( j = 0; j < N; j++ ) { w[0][j] = 0.0; } /* Average the boundary values, to come up with a reasonable initial value for the interior. */ #pragma omp for reduction ( + : mean ) for ( i = 1; i < M - 1; i++ ) { mean = mean + w[i][0] + w[i][N-1]; } #pragma omp for reduction ( + : mean ) for ( j = 0; j < N; j++ ) { mean = mean + w[M-1][j] + w[0][j]; } } /* OpenMP note: You cannot normalize MEAN inside the parallel region. It only gets its correct value once you leave the parallel region. So we interrupt the parallel region, set MEAN, and go back in. */ mean = mean / ( double ) ( 2 * M + 2 * N - 4 ); printf ( "\n" ); printf ( " MEAN = %f\n", mean ); /* Initialize the interior solution to the mean value. */ #pragma omp parallel shared ( mean, w ) private ( i, j ) { #pragma omp for for ( i = 1; i < M - 1; i++ ) { for ( j = 1; j < N - 1; j++ ) { w[i][j] = mean; } } } /* iterate until the new solution W differs from the old solution U by no more than EPSILON. */ iterations = 0; iterations_print = 1; printf ( "\n" ); printf ( " Iteration Change\n" ); printf ( "\n" ); wtime = omp_get_wtime ( ); diff = epsilon; while ( epsilon <= diff ) { # pragma omp parallel shared ( u, w ) private ( i, j ) { /* Save the old solution in U. */ # pragma omp for for ( i = 0; i < M; i++ ) { for ( j = 0; j < N; j++ ) { u[i][j] = w[i][j]; } } /* Determine the new estimate of the solution at the interior points. The new solution W is the average of north, south, east and west neighbors. */ # pragma omp for for ( i = 1; i < M - 1; i++ ) { for ( j = 1; j < N - 1; j++ ) { w[i][j] = ( u[i-1][j] + u[i+1][j] + u[i][j-1] + u[i][j+1] ) / 4.0; } } } /* C and C++ cannot compute a maximum as a reduction operation. Therefore, we define a private variable MY_DIFF for each thread. Once they have all computed their values, we use a CRITICAL section to update DIFF. */ diff = 0.0; # pragma omp parallel shared ( diff, u, w ) private ( i, j, my_diff ) { my_diff = 0.0; # pragma omp for for ( i = 1; i < M - 1; i++ ) { for ( j = 1; j < N - 1; j++ ) { if ( my_diff < fabs ( w[i][j] - u[i][j] ) ) { my_diff = fabs ( w[i][j] - u[i][j] ); } } } # pragma omp critical { if ( diff < my_diff ) { diff = my_diff; } } } iterations++; if ( iterations == iterations_print ) { printf ( " %8d %f\n", iterations, diff ); iterations_print = 2 * iterations_print; } } wtime = omp_get_wtime ( ) - wtime; printf ( "\n" ); printf ( " %8d %f\n", iterations, diff ); printf ( "\n" ); printf ( " Error tolerance achieved.\n" ); printf ( " Wallclock time = %f\n", wtime ); /* Terminate. */ printf ( "\n" ); printf ( "HEATED_PLATE_OPENMP:\n" ); printf ( " Normal end of execution.\n" ); return 0; # undef M # undef N }