Unequal Subcycled to Non-Subcycled Region Ratio

Over the next two problems, we increase the region ratio to $2:1$ ( having $12294$ nodes, $28341$ edges, and $17381$ volumetric elements ) and $4.5:1$ ( having $30889$ nodes, $81148$ edges, and $52020$ volumetric elements ). We plot the displacement profiles of a random node for each of the region ratios in Figures 4.3 and 4.4, respectively. The corresponding timing results are presented in Tables 4.2 and 4.3. For the $4.5:1$ region ratio case, the $m = 20$ result is not presented because the solution became unstable late in the simulation, possibly due to lack of local computer memory.

As we can see from the two figures, the displacements for each of the subcycling parameters are very close to the reference solution; in fact, the differences are nearly imperceptible at certain points in the simulation. In addition, the time savings for both cases increases as the subcycling parameter increases; with the biggest increase occurring for the lower subcycling parameter transitions. For example, for the $2:1$ region ratio results, the increase from $m = 4$ to $m = 10$ generates a savings of $28\%$, while from $m = 10$ to $m = 16$ it is only $4\%$. The results for the $4.5:1$ region ratio fair slightly better with savings of $32\%$ and $6\%$, for both subcycling parameter transitions.

Table 4.2: Timing results (in $s$) for a problem with an equal region ratio ($2:1$) with subcycling parameters of ${\it m} = 1, 4, 10, 16$ and $20$.

Subroutine	Reference Case [s], m = 1	m = 4	m = 10	m = 16	m = 20
$R^\textrm{co}$	438.42	634.97	499.39	487.99	473.49
$R^\textrm{in}$	1561.98	893.63	483.02	376.92	339.56
$Main$	704.72	994.00	742.12	736.87	700.87
$Total$	2853.85	2693.81	1894.97	1774.19	1682.16
$\%$ $Total$ $Savings$		6%	34%	38%	41%

Table 4.3: Timing results (in $s$) for a problem with an equal region ratio ($4.5:1$) with subcycling parameters of ${\it m} = 1, 4, 10$ and $16$.

Subroutine	Reference Case [s], m = 1	m = 4	m = 10	m = 16
$R^\textrm{co}$	600.59	886.33	703.14	653.98
$R^\textrm{in}$	4215.44	2670.78	1264.39	911.16
$Main$	1119.75	1115.83	1114.14	1116.31
$Total$	7930.09	7577.06	5072.81	4604.52
$\%$ $Total$ $Savings$		4%	36%	42%

Figure 4.3: Nodal displacements of a random node for the $2:1$ region ratio with subcycling parameters of ${\it m} = 1, 4, 10, 16$ and $20$.
Figure 4.4: Nodal displacements of a random node for the $4.5:1$ region ratio with subcycling parameters of ${\it m} = 1, 4, 10$ and $16$.