CGI2021 manuscript No.
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Supplemental Material of ”A Flux-Interpolated Advection Scheme for Fluid Simulation”
First Author · Second Author
1 Implementation Detail
The face subdivision/contouring algorithms may require some implementation effort. Therefore, we provide an example of our implementation (readable single header file within 130 lines of code, no external dependencies) as an auxiliary material licensed under the MIT. In our implementation, a function with intuitive parameters and outputs are provided as in Listing 1. With these codes, our scheme is reproducible with minimal imple- mentation costs. Three examples of subdivided poly- gons are provided in Figure 4.
Listing 1 C++ function for Algorithm 2 and 3 1 structpoint2{floatx[2];}// 2D vector definition 2 structvertex2{// polygon vertex definition 3 point2 xi;// (xi,eta) coordinate 4 point2 x;// (x,z) coordinate 5 };
6 static voidsubdivide(
7 constvertex2∗polygon,// polygon vertices 8 unsigned charcount,// vertex count 9 std::function<void(
10 constvertex2∗polygon,
11 // output fully subdivided polygon vertices 12 unsigned charcount
13 // output fully subdivided polygon vertex count 14 )>output func// fully subdivided polygon
15 // output callback function
16 );
F. Author first adress S. Author second address
2 Additional Results and Discussions
2.1 Momentum Advection
Aside from density, we have also attempted to advect momentum as can be seen in Figure 1. We find that the momentum preserving advection generates visually different motions; however, we did not observe clear im- provements in visual quality. Since the momentum ad- vection using our method add three times more costly than the density advection (three velocity components) we find that our method is not cost effective at least for smoke simulation. This situation may differ for liquid as demonstrated in Lentine et al. [1].
2.2 Negative Mass
We would like to note that our method inevitably gener- ates negative mass during advection due to the diffusion- predictive error corrective scheme. This negative mass may sound nonphysical, but in practice, we found that such a negative mass helps minimize diffusion of the trilinear interpolation kernel. Unlike the MacCormack method where explicit clamping is needed for stabil- ity, we did not experience any stability issue without special treatment. We have rendered our results us- ing Blender, and have also rendered using RenderMan without clamping negative mass in the volume data, and we did not find any visual artifacts. However, we regard that allowing such a negative mass may result in some nonphysical artifacts in a different scenario (e.g., reaction-diffusion simulation) and for this case, our method may not be applicable.
2 First Author, Second Author
Fig. 1 Left: Momentum preserving advection using our flux- interpolated scheme. Right: conventional semi-Lagrangian scheme. Note that in this comparison we did not perform cell-wise volume correction since the free-slip boundary con- dition can be violated by the correction.
0 50 100 150 200 250 300 350 400 Steps
0 5 10 15 20
Negative Mass (%)
Fig. 2 The negative mass ratio over the total mass of Figure 1. Our method inevitably introduces negative mass due to the diffusion correction. However, such a negative mass eventually diffuses and approaches zero near the end of simulation.
References
1. Lentine, M., Aanjaneya, M., Fedkiw, R.: Mass and mo- mentum conservation for fluid simulation. In: Proceedings of the 2011 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, SCA ’11, p. 91ˆa€“100. Asso- ciation for Computing Machinery, New York, NY, USA (2011)
