Parallel IB timings

git clone https://github.com/sonwell/ib.cu.git

Results

The tables below summarize the results of a simulation in a triply periodic 16 μm × 16 μm × 16 μm domain with a single red blood cell in mock shear flow with flow rate ɣ̇ = 1000 s^-1. The red blood cell was constructed from 864 data sites and 8832 sample sites.

Strong scaling

Results of strong scaling tests for IB spread and interpolate. Computation time for Lagrangian forces are also reported for comparison.
`p`	cells	interpolate `N` = 20000	forces `N` = 10000	spread `N` = 10000
64	1	0.01080 ± 0.00004	0.00241 ± 0.00002	0.11888 ± 0.00055
128	1	0.00586 ± 0.00002	0.00243 ± 0.00002	0.06535 ± 0.00023
256	1	0.00341 ± 0.00002	0.00241 ± 0.00002	0.03912 ± 0.00012
512	1	0.00179 ± 0.00001	0.00243 ± 0.00002	0.02649 ± 0.00014
1024	1	0.00093 ± 0.00001	0.00241 ± 0.00002	0.01924 ± 0.00011
2048	1	0.00094 ± 0.00001	0.00243 ± 0.00002	0.01627 ± 0.00012
4096	1	0.00093 ± 0.00001	0.00241 ± 0.00002	0.01455 ± 0.00009

Results of strong scaling for IB interpolate and spread with one red blood cell.

Results of strong scaling tests for IB spread and interpolate. Computation time for Lagrangian forces are also reported for comparison.
`p`	cells	interpolate `N` = 20000	forces `N` = 10000	spread `N` = 10000
64	4	0.04150 ± 0.00013	0.00289 ± 0.00010	0.40148 ± 0.00146
128	4	0.02251 ± 0.00003	0.00286 ± 0.00003	0.20628 ± 0.00060
256	4	0.01172 ± 0.00002	0.00286 ± 0.00003	0.11208 ± 0.00031
512	4	0.00603 ± 0.00002	0.00286 ± 0.00003	0.06313 ± 0.00014
1024	4	0.00349 ± 0.00002	0.00286 ± 0.00003	0.03931 ± 0.00011
2048	4	0.00181 ± 0.00001	0.00286 ± 0.00003	0.02785 ± 0.00011
4096	4	0.00103 ± 0.00001	0.00288 ± 0.00002	0.02168 ± 0.00009

Results of strong scaling for IB interpolate and spread with four red blood cells.

Weak scaling

Results of weak scaling tests for IB spread and interpolate.
`p`	cells	interpolate `N` = 20000	forces `N` = 10000	spread `N` = 10000
64	1	0.01079 ± 0.00004	0.00242 ± 0.00002	0.11881 ± 0.00055
128	2	0.01165 ± 0.00003	0.00249 ± 0.00002	0.11219 ± 0.00051
256	4	0.01171 ± 0.00003	0.00287 ± 0.00003	0.11214 ± 0.00036
512	8	0.01199 ± 0.00003	0.00532 ± 0.00008	0.11354 ± 0.00047

Weak scaling results.

Results of `n` randomly placed IB points in a 16 μm × 16 μm × 16 μm periodic domain.
`p`	`n`	interpolate `N` = 20000	forces `N` = 10000	spread `N` = 10000
128	2¹⁶	0.43930 ± 0.00019	0.00026 ± 0.00031	0.47632 ± 0.00142
256	2¹⁷	0.44918 ± 0.00056	0.00024 ± 0.00002	0.46503 ± 0.00026
512	2¹⁸	0.45072 ± 0.00061	0.00029 ± 0.00001	0.44533 ± 0.00028
1024	2¹⁹	0.45442 ± 0.00049	0.00055 ± 0.00000	0.43561 ± 0.00024

Convergence study

Results of a convergence study for a single RBC in mock periodic flow for the ℓ₂ and max norms. Errors are reported for `t` = 0.4 ms against the result of a simulation with grid refinement of 128, 2210 data sites, and 8832 sample sites, `u_f`.
`refinement`	`h`	`n_d`	`n_s`	‖`u_n` - `u_2n`‖₂	order₂	‖`u_n` - `u_2n`‖_∞	order_∞
16	1.00 μm	31	132	0.000789		0.006200
32	0.50 μm	132	546	0.000331	1.253194	0.004700	0.399607
64	0.25 μm	546	2210	0.000111	1.576272	0.002486	0.918834

Convergence study, against a simulation with grid refinement of 128, n_d = 2210, and _s = 8832, for ℓ₂ and max norms. The short-dashed line represents first order convergence and the long-dashed line represents second order convergence. The error converges at approximately order 1.4 in the ℓ₂ norm and approximately order 0.7 in the max norm.

Dependence on background grid

Dependence of the algorithm on background grid
`n`	`h` (μm)	interpolate `N` = 500(`n`/16)²	forces `N` = 250(`n`/16)²	spread `N` = 250(`n`/16)²
16	1.000	0.00100 ± 0.00001	0.00252 ± 0.00014	0.00694 ± 0.00057
32	0.500	0.00108 ± 0.00001	0.00247 ± 0.00006	0.00677 ± 0.00036
64	0.250	0.00111 ± 0.00001	0.00282 ± 0.00002	0.01501 ± 0.00007
128	0.125	0.00113 ± 0.00002	0.00282 ± 0.00003	0.05514 ± 0.00019

Comparison to serial code

Comparison to serial code.
`p`	`n`	interpolate `N` = 20000	spread `N` = 10000
1 (CPU)	2¹⁶	1.40735 ± 0.06325	1.65417 ± 0.05096
12 (CPU)	2¹⁶	0.12494 ± 0.00218	0.18054 ± 0.00369
128 (GPU)	2¹⁶	0.43930 ± 0.00019	0.47632 ± 0.00142
* (GPU)	2¹⁶	0.01738 ± 0.00009	0.02369 ± 0.00018