Workshop on Virtual Reality Interaction and Physical Simulation VRIPHYS (2015) F. Jaillet, G. Zachmann, and F. Zara (Editors)
Supplementary Material:
Evaluation of Surface Tension Models for SPH-Based Fluid Animations Using a Benchmark Test
Markus Huber1, Stefan Reinhardt2, Daniel Weiskopf1, and Bernhard Eberhardt2
1VISUS, University of Stuttgart, Germany 2Stuttgart Media University, Germany
1. Overview
In this document, we provide additional images and plots to the main paper. First, images of the equilibrium state of benchmark 1 (drop formation) are shown for all surface tension models presented in Sec. 3 in the main paper, combined with different SPH methods (IISPH [ICS∗14], PCISPH [SP09], and WCSPH [BT07]). Further, individual plots for velocities, surface tension forces, and pressure forces that are shown as aggregated values in Sec. 6 of the main paper, are given. At last, velocity plots for benchmark 2 (liquid crown) applied to the mentioned surface tension models in combination with IISPH and WCSPH are given.
2. Drop Formation 2.1. Using IISPH
(a) [AAT13],γ=1.0 (b) [AAT13],γ=0.8 (c) [AAT13],γ=0.6 (d) [AAT13],γ=0.4 (e) [AAT13],γ=0.2
(f) [BT07],ϕ=0.08 (g) [BT07],ϕ=0.065 (h) [BT07],ϕ=0.05 (i) [BT07],ϕ=0.035 (j) [BT07],ϕ=0.02
(k) [BT07] mod.,ϕ=0.08 (l) [BT07] mod.,ϕ=0.065 (m) [BT07] mod.,ϕ=0.05 (n) [BT07]mod.,ϕ=0.035 (o) [BT07] mod.,ϕ=0.02
(p) [HWZ∗14],κ=1.8 (q) [HWZ∗14],κ=1.4 (r) [HWZ∗14],κ=1.0 (s) [HWZ∗14],κ=0.6 (t) [HWZ∗14],κ=0.2
Figure 1: Snapshots of the equilibrium state of benchmark 1 (drop formation) applied to all surface tension models in com- bination with IISPH. Except for low surface tension coefficients with the model of He et al. [HWZ∗14], a spherical shape is achieved with all combinations of models. With the surface tension model of Becker and Teschner [BT07], deformations of the sphere occur that are resolved with our modifications.
2.2. Using PCISPH
(a) [AAT13],γ=1.0 (b) [AAT13],γ=0.8 (c) [AAT13],γ=0.6 (d) [AAT13],γ=0.4 (e) [AAT13],γ=0.2
(f) [BT07],ϕ=0.08 (g) [BT07],ϕ=0.065 (h) [BT07],ϕ=0.05 (i) [BT07],ϕ=0.035 (j) [BT07],ϕ=0.02
(k) [BT07] mod.,ϕ=0.08 (l) [BT07] mod.,ϕ=0.065 (m) [BT07] mod.,ϕ=0.05 (n) [BT07]mod.,ϕ=0.035 (o) [BT07] mod.,ϕ=0.02
(p) [HWZ∗14],κ=1.8 (q) [HWZ∗14],κ=1.4 (r) [HWZ∗14],κ=1.0 (s) [HWZ∗14],κ=0.6 (t) [HWZ∗14],κ=0.2 Figure 2: Snapshots of the equilibrium state of benchmark 1 (drop formation) applied to all surface tension models in combina- tion with PCISPH. Similar results to the previous combination (using IISPH) are achieved.
2.3. Using WCSPH
(a) [AAT13],γ=1.0 (b) [AAT13],γ=0.8 (c) [AAT13],γ=0.6 (d) [AAT13],γ=0.4 (e) [AAT13],γ=0.2
(f) [BT07],ϕ=0.08 (g) [BT07],ϕ=0.065 (h) [BT07],ϕ=0.05 (i) [BT07],ϕ=0.035 (j) [BT07],ϕ=0.02
(k) [BT07] mod.,ϕ=0.08 (l) [BT07] mod.,ϕ=0.065 (m) [BT07] mod.,ϕ=0.05 (n) [BT07]mod.,ϕ=0.035 (o) [BT07] mod.,ϕ=0.02
(p) [HWZ∗14],κ=1.8 (q) [HWZ∗14],κ=1.4 (r) [HWZ∗14],κ=1.0 (s) [HWZ∗14],κ=0.6 (t) [HWZ∗14],κ=0.2
Figure 3: Snapshots of the equilibrium state of benchmark 1 (drop formation) applied to all surface tension models in combina- tion with WCSPH. With very low values for the surface tension coefficient, no spherical shape is formed. The results achieved with the model of He et al. [HWZ∗14] shows almost no difference in combination with different SPH methods.
3. Measurements Benchmark Test 1 (Drop Formation) 3.1. Velocities
0 200 400 600 800 1000
t
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Avg.velocities
IISPH + [AAT13]
γ=0.2 γ=0.4 γ=0.6 γ=0.8 γ=1.0
0 200 400 600 800 1000
t
0.00 0.05 0.10 0.15 0.20 0.25
Avg.velocities
PCISPH + [AAT13]
γ=0.2 γ=0.4 γ=0.6 γ=0.8 γ=1.0
0 200 400 600 800 1000
t
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Avg.velocities
WCSPH + [AAT13]
γ=0.2 γ=0.4 γ=0.6 γ=0.8 γ=1.0
0 200 400 600 800 1000
t
0.00 0.02 0.04 0.06 0.08 0.10 0.12
Avg.velocities
IISPH + [BT07]
ϕ=0.02 ϕ=0.035 ϕ=0.05 ϕ=0.065 ϕ=0.08
0 200 400 600 800 1000
t
0.00 0.02 0.04 0.06 0.08 0.10
Avg.velocities
PCISPH + [BT07]
ϕ=0.02 ϕ=0.035 ϕ=0.05 ϕ=0.065 ϕ=0.08
0 200 400 600 800 1000
t
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
Avg.velocities
WCSPH + [BT07]
ϕ=0.02 ϕ=0.035 ϕ=0.05 ϕ=0.065 ϕ=0.08
0 200 400 600 800 1000
t
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Avg.velocities
IISPH + ours
ϕ=0.02 ϕ=0.035 ϕ=0.05 ϕ=0.065 ϕ=0.08
0 200 400 600 800 1000
t
0.00 0.01 0.02 0.03 0.04 0.05 0.06
Avg.velocities
PCISPH + ours
ϕ=0.02 ϕ=0.035 ϕ=0.05 ϕ=0.065 ϕ=0.08
0 200 400 600 800 1000
t
0.00 0.01 0.02 0.03 0.04 0.05 0.06
Avg.velocities
WCSPH + ours
ϕ=0.02 ϕ=0.035 ϕ=0.05 ϕ=0.065 ϕ=0.08
0 200 400 600 800 1000
t
0.000 0.002 0.004 0.006 0.008 0.010 0.012
Avg.velocities
IISPH + [HWZ*14]
κ=0.2 κ=0.6 κ=1.0 κ=1.4 κ=1.8
0 200 400 600 800 1000
t
0.000 0.002 0.004 0.006 0.008 0.010 0.012
Avg.velocities
PCISPH + [HWZ*14]
κ=0.2 κ=0.6 κ=1.0 κ=1.4 κ=1.8
0 200 400 600 800 1000
t
0.000 0.002 0.004 0.006 0.008 0.010 0.012
Avg.velocities
WCSPH + [HWZ*14]
κ=0.2 κ=0.6 κ=1.0 κ=1.4 κ=1.8
3.2. Surface tension forces
0 50 100 150 200
t
0.0 0.5 1.0 1.5 2.0 2.5
Avg.surfacetensionforces
IISPH + [AAT13]
γ=0.2 γ=0.4 γ=0.6 γ=0.8 γ=1.0
0 50 100 150 200
t
0.0 0.5 1.0 1.5 2.0 2.5
Avg.surfacetensionforces
PCISPH + [AAT13]
γ=0.2 γ=0.4 γ=0.6 γ=0.8 γ=1.0
0 50 100 150 200
t
0.0 0.5 1.0 1.5 2.0 2.5
Avg.surfacetensionforces
WCSPH + [AAT13]
γ=0.2 γ=0.4 γ=0.6 γ=0.8 γ=1.0
0 50 100 150 200
t
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Avg.surfacetensionforces
IISPH + [BT07]
ϕ=0.02 ϕ=0.035 ϕ=0.05 ϕ=0.065 ϕ=0.08
0 50 100 150 200
t
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Avg.surfacetensionforces
PCISPH + [BT07]
ϕ=0.02 ϕ=0.035 ϕ=0.05 ϕ=0.065 ϕ=0.08
0 50 100 150 200
t
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Avg.surfacetensionforces
WCSPH + [BT07]
ϕ=0.02 ϕ=0.035 ϕ=0.05 ϕ=0.065 ϕ=0.08
0 50 100 150 200
t
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Avg.surfacetensionforces
IISPH + ours
ϕ=0.02 ϕ=0.035 ϕ=0.05 ϕ=0.065 ϕ=0.08
0 50 100 150 200
t
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Avg.surfacetensionforces
PCISPH + ours
ϕ=0.02 ϕ=0.035 ϕ=0.05 ϕ=0.065 ϕ=0.08
0 50 100 150 200
t
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Avg.surfacetensionforces
WCSPH + ours
ϕ=0.02 ϕ=0.035 ϕ=0.05 ϕ=0.065 ϕ=0.08
0 50 100 150 200
t
0.000 0.005 0.010 0.015 0.020 0.025
Avg.surfacetensionforces
IISPH + [HWZ*14]
κ=0.2 κ=0.6 κ=1.0 κ=1.4 κ=1.8
0 50 100 150 200
t
0.000 0.005 0.010 0.015 0.020 0.025
Avg.surfacetensionforces
PCISPH + [HWZ*14]
κ=0.2 κ=0.6 κ=1.0 κ=1.4 κ=1.8
0 50 100 150 200
t
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014
Avg.surfacetensionforces
WCSPH + [HWZ*14]
κ=0.2 κ=0.6 κ=1.0 κ=1.4 κ=1.8
Figure 5: Temporal plots of average surface tension forces of all combinations of surface tension models with different SPH models for benchmark test 1 (drop formation). Surface tension forces are given inNand timetis given ins. The equilibrium state of surface tension forces is reached considerably earlier than the respective velocities. Hence, the plots cover 200 frames as opposed to 1000 with the velocities to see all the details in oscillations.
3.3. Pressure forces
0 50 100 150 200
t
0 2 4 6 8 10 12 14 16 18
Avg.pressureforces
IISPH + [AAT13]
γ=0.2 γ=0.4 γ=0.6 γ=0.8 γ=1.0
0 50 100 150 200
t
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Avg.pressureforces
PCISPH + [AAT13]
γ=0.2 γ=0.4 γ=0.6 γ=0.8 γ=1.0
0 50 100 150 200
t
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
Avg.pressureforces
WCSPH + [AAT13]
γ=0.2 γ=0.4 γ=0.6 γ=0.8 γ=1.0
0 50 100 150 200
t
0.0 0.5 1.0 1.5 2.0 2.5
Avg.pressureforces
IISPH + [BT07]
ϕ=0.02 ϕ=0.035 ϕ=0.05 ϕ=0.065 ϕ=0.08
0 50 100 150 200
t
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
Avg.pressureforces
PCISPH + [BT07]
ϕ=0.02 ϕ=0.035 ϕ=0.05 ϕ=0.065 ϕ=0.08
0 50 100 150 200
t
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Avg.pressureforces
WCSPH + [BT07]
ϕ=0.02 ϕ=0.035 ϕ=0.05 ϕ=0.065 ϕ=0.08
0 50 100 150 200
t
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Avg.pressureforces
IISPH + ours
ϕ=0.02 ϕ=0.035 ϕ=0.05 ϕ=0.065 ϕ=0.08
0 50 100 150 200
t
0.00 0.02 0.04 0.06 0.08 0.10 0.12
Avg.pressureforces
PCISPH + ours
ϕ=0.02 ϕ=0.035 ϕ=0.05 ϕ=0.065 ϕ=0.08
0 50 100 150 200
t
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Avg.pressureforces
WCSPH + ours
ϕ=0.02 ϕ=0.035 ϕ=0.05 ϕ=0.065 ϕ=0.08
0 50 100 150 200
t
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035
Avg.pressureforces
IISPH + [HWZ*14]
κ=0.2 κ=0.6 κ=1.0 κ=1.4 κ=1.8
0 50 100 150 200
t
0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040
Avg.pressureforces
PCISPH + [HWZ*14]
κ=0.2 κ=0.6 κ=1.0 κ=1.4 κ=1.8
0 50 100 150 200
t
0.00 0.01 0.02 0.03 0.04 0.05
Avg.pressureforces
WCSPH + [HWZ*14]
κ=0.2 κ=0.6 κ=1.0 κ=1.4 κ=1.8
Figure 6: Temporal plots of average pressure forces of all combinations of surface tension models with different SPH models for benchmark test 1 (drop formation). Pressure forces are given inNand timetis given ins. The equilibrium state of pressure
4. Measurements Benchmark Test 2 (Liquid Crown) 4.1. Velocities
0 200 400 600 800 1000
t
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Avg.velocities
IISPH + [AAT13]
γ=0.2 γ=1.0
0 200 400 600 800 1000
t
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Avg.velocities
IISPH + [BT07]
ϕ=0.02 ϕ=0.08
0 200 400 600 800 1000
t
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Avg.velocities
IISPH + [HWZ*14]
κ=0.2 κ=1.8
0 200 400 600 800 1000
t
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Avg.velocities
WCSPH + [AAT13]
γ=0.2 γ=1.0
0 200 400 600 800 1000
t
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Avg.velocities
WCSPH + [BT07]
ϕ=0.02 ϕ=0.08
0 200 400 600 800 1000
t
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Avg.velocities
WCSPH + [HWZ*14]
κ=0.2 κ=1.8
0 200 400 600 800 1000
t
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Avg.velocities
IISPH
No surface tension
Figure 7: Temporal plots of average particle velocities for a set of combination of surface tension models with SPH models for benchmark test 2 (liquid crown). Only small differences can be noticed using varying surface tension parameters. However, there are big differences in the resulting animations (see main paper and accompanying video).
References
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[BT07] BECKERM., TESCHNERM.: Weakly compressible SPH for free surface flows. InACM SIGGRAPH/Eurographics Symposium on Computer Animation(2007), pp. 209–218.doi:10.2312/SCA/SCA07/209-218.1,2,3,4,5
[HWZ∗14] HEX., WANGH., ZHANGF., WANGH., WANGG., ZHOUK.: Robust simulation of sparsely sampled thin features in SPH- based free surface flows.ACM Transactions on Graphics 34, 1 (2014), 7:1–7:9.doi:10.1145/2682630.2,3,4,5
[ICS∗14] IHMSENM., CORNELISJ., SOLENTHALERB., HORVATHC., TESCHNERM.: Implicit incompressible SPH.IEEE Transactions on Visualization and Computer Graphics 20, 3 (2014), 426–435.doi:10.1109/TVCG.2013.105.1
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