SCIENTIA SINICA Informationis, Volume 51 , Issue 2 : 263(2021) https://doi.org/10.1360/SSI-2019-0277

## Controllable curl-correction of 3D frame fields

• AcceptedApr 18, 2020
• PublishedJan 25, 2021
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### References

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• Figure 1

(Color online) Frame alignment. The six frames are equivalent under cubic-symmetric rotational group, and there are other 18 frames

• Figure 2

(Color online) Local frame alignment on the vertices of a tetrahedron

• Figure 3

(Color online) Curl-correction optimization of 3D frame fields. (a) is the input 3D orthogonal frame field; protectłinebreak (b) is the final frame field; (c)$\sim$(e) are the optimized scalar fields in the three directions

• Figure 4

(Color online) The curl-correction results of the hollow cylinder model with different constraints. The first column shows the analysis of the original frame field. The columns from 2 to 4 show results under different constraints $[-0.2,0.2],[-0.5,0.5],[-2,2]$. The first row is the optimized curl energies, the second row is the parametric errors, the third row corresponds to the scalar fields in the circumferential direction, and the fourth row shows the meshes generated by PGP3D [6]. (a) Without curl correction; (b) $[-0.2,0.2]$; (c) $[-0.5,0.5]$; (d) $[-2,2]$

• Figure 5

(Color online) The curl-correction results of the sphere model with different constraints. The original input field is a boundary-aligned cross-frame field and $[a,b]$ represents the range of $\tilde{s}$. The first row shows the distribution of the curl energies, the second row shows the parameterization errors, the third row visualizes the average value of the optimized scalar fields (the meshes are demonstrated partly by cutting), and the fourth row shows hex elements of the hex-dominant meshes generated by PGP3D, and the fifth row demonstrates the meshes partly by cutting. (a) Without curl correction; protectłinebreak (b) $[-0.2,0.2]$; (c) $[-0.5,0.5]$; (d) $[-2,2]$

• Figure 6

(Color online) The distributions of curl energies and parametric errors of the hollow cylinder model. (a) Non curl correction; (b) $[-0.2,0.2]$; (c) $[-0.5,0.5]$; (d) $[-2,2]$

• Figure 7

(Color online) The distributions of the curl energies and parametric errors of the sphere model. (a) Non curl correction; (b) $[-0.2,0.2]$; (c) $[-0.5,0.5]$; (d) $[-2,2]$

• Figure 8

(Color online) The results of the hollow cylinder model and the sphere model (demonstrated partly by cutting) with $\tilde{s}_{{\rm~min}}=-2,~\tilde{s}_{\rm~max}=2$ under different smoothness controls. Considering the smoothness term makes the resulting scalar fields smoother, so are the element size of the resulting meshes. (a) and (c) $w_s=0.01$, (b) and (d) $w_s=0$

• Figure 9

(Color online) Hex-dominant meshes guided by the curl-corrected frame fields. After curl-correcting, the volume ratio of the hex elements increases considerably. The first row shows the hex-dominant meshes generated by PGP3D, and the second row shows all the hex elements of the meshes. (a), (c) Without curl correction; (b), (d) $[-2,2]$

• Table 1   Statistics
 Model $n_t$ $[\tilde{s}_{\text{min}},\tilde{s}_{\text{max}}]$ $T_{\text{cc}}$ (s) $r_H$ (%) $n_H$ $w_s$ $s_g$ hole_cylinder 31791 (Non curl correction) – 45.05 13870 0.01 0.04 hole_cylinder 31791 $[-0.2,~0.2]$ 7 61.25 16460 0.01 0.04 hole_cylinder 31791 $[-0.5,~0.5]$ 7 84.32 19805 0.01 0.04 hole_cylinder 31791 $[-2,~2]$ 11 98.49 28986 0.01 0.04 hole_cylinder 31791 $[-2,~2]$ 12 97.79 28736 0 0.04 sphere 265280 (Non curl correction) – 9.02 11857 0.01 0.03 sphere 265280 $[-0.2,~0.2]$ 120 45.91 60383 0.01 0.03 sphere 265280 $[-0.5,~0.5]$ 221 83.93 112063 0.01 0.03 sphere 265280 $[-2,~2]$ 122 92.82 244418 0.01 0.03 sphere 265280 $[-2,~2]$ 153 92.42 347428 0 0.03 bunny 169281 (Non curl correction) – 40.78 17032 0.01 0.005 bunny 169281 $[-2,~2]$ 130 66.41 28036 0.01 0.005 bone 145842 (Non curl correction) – 44.07 3384 0.01 0.3 bone 145842 $[-2,~2]$ 49 67.09 7203 0.01 0.3

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