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6.4 Numerical Results
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Figure 6.6.2 Comparison of the bistatic scattering coefficients for PEC rough surface obtained by the single code and parallel code. 8 x 8,\;;, h = 0.1,\0 incident angle = 40
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Figure 6.6.3 Comparison of the bistatic scattering coefficients for lossy dielectric rough surface obtained by the single code and parallel code. 8 x 8,\;;, h = 0.2'\0 incident angle = 30 , Er = 45 + i30.
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CPU time (Sec.) Surface size (A~ ) 8x8 16 x 16 32x 32 64x64 32 x 32 64x64 Number of unknowns 8,192 32,768 131,072 524,288 (one CG term) (one CG term) Sequential code 518 3020
Speedup factor 7.85 13.73
Parallel code 66 220 1,120 6,045 9.2 37.9
382.48 4381.74
41.57 115.61
Table 6.6.1 Comparison of CPU time for PEC rough surface h
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Surface size (~ )
Number of unknowns
rms height ("-0) 0.05 0.10 0.15 0.20 0.25 0.30 0.05 0.10 0.05 0.10 Sequential code 49,368 77,643 114,385 161,313 207,095 275,485 270,204 469,831
CPU time (Sec.) Speedup factor 9.77 11.57 11.55 11.79 11.92 12.23 13.20 13.43 -------------
Parallel code 5,051 6,712 9,906 13,682 17,371 22,517 20,477 34,981 133,798 249,460
16 x 16 32 x 32
393,216 1,572,864
Table 6.6.2 Comparison of CPU time for lossy dielectric rough surface
45 + i30.
when the number of unknowns increases. For the surfaces with the same size, the CPU time increases with h because of the slow convergence of the CGM for rougher surfaces. For example, it only needs 2 iterations (n = 1) with 40 CG terms in the first iteration and 1 CG term in the second iteration for the 8 x 8>'~ surface with h = 0.05>'0' In contrast, it needs 3 iterations (n = 2) with 100 CG terms in the first iteration, 41 CG terms in the second iteration and 4 CG terms in the last iteration when h increases to 0.3>'0' The speedup factors become large when the CPU time increases. It is shown that the best speedup factor is around 13.7 for the PEC surface with 32,768 surface unknowns and h = 0.1>.0' and 13.4 for the lossy dielectric surface with 0.4 million surface unknowns for the same h. For the PEC surfaces, CPU times of the parallel code are obtained when the strong matrix is computed once
6.4 Numerical Results
Total time Communication time Near interaction calculation time Far interaction calculation time Emissivity calculation time
4584.8 s 120.8 s 2918.6 s 289.4 s 1225.7 s
Table 6.6.3 CPU time distribution for the case of 8 x 8'\~ surface with h
and stored. When the number of unknowns exceeds 131,000, we recalculate the strong matrix in the sequential code when necessary. In the last two rows of Table 6.6.1, we give the comparison of CPU time for one CG term when the surface sizes are 32 x 32 and 64 x 64>'~, respectively. The strong matrix is recalculated in the sequential code. One can see that the speedup factor can be up to 115.6 for 1 CG term. To complete the conjugate gradient solution of sequential code will be prohibitive long. For dielectric surfaces, when the number of unknowns exceeds 1.5 millions, the code on a single processor cannot be run and therefore parallel computation is the only alternative. In the 1.5 million unknown example, the computer code requires about 78% of the 4.0 GBytes of RAM. We should emphasize that the CPU time comparisons shown in Table 6.6.1 and 6.6.2 are based on the collocation method. For the lossy dielectric surface, it is found that near-field integration is necessary. Although this near-field integration requires additional CPU time, the resulting matrix is better conditioned so that the number of CG terms may be reduced. For examples, with near-field integration, the CPU time for the 8 x 8 and 16 x 16>'~ surface with h = 0.05>'0 reduces from 5,051 to 4,585 and 20,474 to 19,085 seconds, respectively. It is expected that similar speed-up factors are obtained when replacing the collocation method with the nearfield integration. Table 6.6.3 shows the CPU time distribution for a 8 x 8>'~ surface with near-field integration, where h = 0.05>'0' The communication time is only a small part of the total CPU time. Most of the CPU time is spent on calculating the sparse matrix and sparse-matrix-vector multiplication. Communication time only attributes to 2.6% of the total CPU time while that of the far interaction calculation based on FFT is 6.3%. Table 6.6.4 shows the comparison of the computed brightness temperature versus surface size and rms height, and the comparison between the results obtained with and without the near-field integration. The incident angle for the 8 x 8>'~ surfaces is 30 while that of the remaining ones is 50 . The result of an infinite flat surface with the same permittivity is also given for reference. It should be pointed out that the brightness temperatures of the rough surfaces are obtained from one realization. Different realizations