1. All-Hex Meshing using Singularity-Restricted Field Yufei Li 1, Yang Liu 2, Weiwei Xu 2, Wenping Wang 1, Baining Guo 2 1. The University of Hong Kong 2. Microsoft Research Asia

2. Motivation All-hex mesh –A 3D volume tessellated entirely by hexahedron elements. Why alll-hex mesh? –Reduced number of elements. –Improved speed and accuracy of physical simulations [Shimada 2006; Shepherd and Johnson 2008]. All-hex mesh Tetrahedral mesh 1/28

3. Motivation Issues –Highly constrained connectivity. –Require much user interaction. Industrial practice –Multiple sweeping [Shepherd et al. 2000]; –Paving and plastering [Staten et al. 2005]; –… Semi-automatic User interaction Semi-automatic User interaction ANSYS software 2/28

4. Motivation Quality criteria for all-hex mesh –Boundary conformity –Feature alignment –Low distortion Goal: automatically generate all-hex meshes with high-quality Feature Alignment Low Distortion All-hex mesh Boundary Conformity 3/28

5. Existing methods: all-hex meshing based on volume parameterization guided by 3D frame field Input volume (tetrahedral mesh) Input volume (tetrahedral mesh) 3D frame field (inside the volume) 3D frame field (inside the volume) Volume parameterization (guided by 3D frame field) Volume parameterization (guided by 3D frame field) All-hex mesh 4/28

6. Input volume (tetrahedral mesh) Input volume (tetrahedral mesh) 3D frame field (inside the volume) 3D frame field (inside the volume) Volume parameterization (guided by 3D frame field) Volume parameterization (guided by 3D frame field) All-hex mesh Hex-dominant mesh [Huang et al. 2011] 5/28 Existing methods: all-hex meshing based on volume parameterization guided by 3D frame field

7. Input volume (tetrahedral mesh) Input volume (tetrahedral mesh) 3D frame field (inside the volume) 3D frame field (inside the volume) Volume parameterization (guided by 3D frame field) Volume parameterization (guided by 3D frame field) All-hex mesh CubeCover [Nieser et al. 2011] Manually designed meta-mesh 6/28 Existing methods: all-hex meshing based on volume parameterization guided by 3D frame field

8. Input volume (tetrahedral mesh) Input volume (tetrahedral mesh) 3D frame field Volume parameterization (guided by 3D frame field) Volume parameterization (guided by 3D frame field) All-hex mesh Hex-dominant mesh Our approach: all-hex meshing framework based on singularity-restricted field (SRF). SRF (singularity-restricted field) SRF (singularity-restricted field) All-hex mesh 3D frame field SRF Major contribution Automatic SRF conversion Major contribution Automatic SRF conversion Key condition 7/28

9. Basics of 3D frame field Discrete setting [Nieser et al. 2011] –3D frame: –Discrete 3D frame field for input tet mesh: a constant 3D frame for each tet. 24 permutations. Chiral Cubical Symmetry Group (24 matrices) Chiral Cubical Symmetry Group (24 matrices) 8/28

10. Basics of 3D frame field FsFs FtFt A pair of arbitrary frames –Difference: a general rotation. –Matching: the permutation that best matches the two frames (24 choices). Matching 9/28

11. Basics of 3D frame field An interior edge –How the frames rotate around it? Identity matrix: regular edge. Non-identity matrix: singular edge (23 types). Singular graph 10/28 Proposition: Any singular edge does not end inside the volume.

12. Singularity-restricted field (SRF) Definition of SRF –A 3D frame field is an SRF if all of its edge types fall into the following subset of rotations: –Ru, Rv, Rw represent the 90 degree rotations around u-, v-, w- coordinate axes, respectively. SRF is necessary for inducing a valid all-hex structure SRF (10 edge types) 3D frame field (24 edge types) [Nieser et al. 2011] 11/28

13. Converting general 3D frame field to singularity-restricted field (SRF) Operations for SRF conversion: –Matching adjustment: tentatively adjust the matching for any triangular face, and check if improper singular edges could be eliminated. –Improper singular edge collapse. SRF (10 edge types ) 3D frame field (24 edge types ) Necessary for all-hex meshing Eliminate the improper singular edges (14 types) 12/28 Geometric operation

14. Improper singular edge collapse (topological operation) –Collapse improper singular edges without introducing new ones; –Preserve the validity of mesh topology during the collapsing process. Converting general 3D frame field to singularity-restricted field (SRF) Collapse improper singular edge e e t s2s2 s1s1 13/28 Our algorithm could eliminate all the improper singular edges, except two extreme cases that do not happen in practice. (See proof in the paper) Key

15. Improper singular edges (in red) are collapsed. Matching adjustment could also smooth the singular graph. SRF Conversion Input frame fieldOutput SRF 14/28

16. SRF (singularity-restricted field) SRF (singularity-restricted field) Input volume (tetrahedral mesh) Input volume (tetrahedral mesh) Volume parameterization (guided by SRF) Volume parameterization (guided by SRF) All-hex mesh A high-quality all-hex meshing framework based on singularity-restricted field (SRF). Input domainParameter domain Gradient field Given SRF Improved CubeCover [Nieser et al. 2011] to solve this mixed-integer problem. Improved CubeCover [Nieser et al. 2011] to solve this mixed-integer problem. Improvement Adaptive rounding Improvement Adaptive rounding 15/28

17. Obstacle 1: degenerate element Element Degeneration Input domainParameter domain Zero volume Fail to trace iso-lines Missing hex elements! Degenerate elements (in red) Why degenerate? Singular edge combination on triangular face. 16/28 c b a

18. SRF (singularity-restricted field) SRF (singularity-restricted field) Input volume (tetrahedral mesh) Input volume (tetrahedral mesh) Volume parameterization (guided by SRF) Volume parameterization (guided by SRF) All-hex mesh All degeneration cases for any triangular face. Handling degenerate elements Preprocessing Topological operations All the degenerate elements (in red) are removed 17/28 See the paper

19. Obstacle 2: flipped element Input domain Parameter domain Flipped Elements Negative volume Erroneous topology of iso-curve network Erroneous topology of iso-curve network Fix the topology Restore a complete all-hex mesh 18/28

20. Comparison with [Huang et al. 2011] SRF by our methodFrame field by [Huang et al. 2011] Red edges are improper edges. More smooth Free of improper singular edges. 19/28

21. Optimized SRF with different frame field initializations 20/28 Singular structure All-hex mesh SRF

22. Comparison with CubeCover [Nieser et al. 2011] J_min [-1,1]: the minimal scaled Jacobian of hexes, bigger is better. Our method: J_min = 0.609CubeCover: J_min = 0.073 Cube-like elements Cube-like elements Distorted elements 21/28

23. Comparison with PolyCube [Gregson et al. 2011] Our method: J_min = 0.351PolyCube: J_min = 0.196 Poor quality due to PolyCube nature 22/28 Distortion Boundary conformity

24. More results by our method J_min = 0.729 J_min = 0.185 J_min = 0.599 23/28 Feature alignment Cube-like elements

25. SRF (singularity-restricted field) SRF (singularity-restricted field) Input volume (tetrahedral mesh) Input volume (tetrahedral mesh) Volume parameterization (guided by SRF) Volume parameterization (guided by SRF) All-hex mesh A high-quality all-hex meshing framework based on singularity-restricted field (SRF). Effective smoothness of 3D frame fields Effective operations for SRF conversion Improved volume parameterization by handling degenerate& flipped elements Contributions Key ingredient 24/28

26. Limitations & Future work No theoretical guarantee that SRF always leads to a valid all-hex structure. Open problem: what is the sufficient condition for all-hex structures? SRF Singularity-restricted field All-hex structure Necessary condition “Almost” but NOT sufficient 25/28

27. Limitations & Future work No theoretical guarantee that SRF always leads to a valid all-hex structure. Singularity mis-alignment: no global control of singularities. Singularity mis-alignment 26/28

28. Limitations & Future work No theoretical guarantee that SRF always leads to a valid all-hex structure. Singularity mis-alignment: no global control of singularities. CANNOT guarantee a degeneracy-free or flip-free volume parameterization.  Shortcoming shared by CubeCover [Nieser et al. 2011]. 27/28

29. Acknowledgements Reviewers for constructive comments. Ulrich Reitebuch, Jin Huang for providing comparison data. Funding agencies: The National Basic Research Program of China (2011CB302400), the Research Grant Council of Hong Kong (718209, 718010, and 718311) 28/28

30. Backup slides

31. Boundary-aligned 3D frame field generation Difference of two frames and FsFs FtFt Optimization –Solved by the L-BFGS method [Liu and Nocedal 1989] Smoothness: closeness from to (24 types of permutations)

32. Frame field initialization Boundary tets –Smooth boundary cross field + surface normals Interior tets –Propagation from boundary tets. –Assigned to be the same as the one of its nearest boundary tet.

33. Frame field guiding Small features, not enough tets User intention

34. Robustness of SRF conversion Test on a random frame field –Initialization: principal-dominant cross-field on the boundary + random frames inside. –Without optimization. SRF conversion 32320 tets 6825 vertices 775 proper singular edges 61 improper singular edges 31930 tets 6766 vertices 753 proper singular edges 0 improper singular edges

35. SRF-Guided Volume Parametrization Definition –The integer grids in induce a hex tessellation of the input volume. Gradient field Given SRF Computation Integer variables: –Boundary faces –Vertices on the singular graph –Adjacent face gaps

36. CANNOT guarantee degeneration- free volume parameterization The triangle has three regular edges (does not belong to the degeneration case in our analysis). Vertices a, b and c are on the singular graph. The triangle has three regular edges (does not belong to the degeneration case in our analysis). Vertices a, b and c are on the singular graph. c b a The triangle might still degenerate due to the integer rounding on vertices a, b and c

37. SRF is not sufficient The topology of the singular graph prohibits the existence of all-hex structures. What is the sufficient condition for all-hex structures? Triangular face

38. The singular graph consists of two spiral and close curves inside a torus volume. The tets mapped to negative volumes in the parameterization are rendered. Fail to retrieve an all-hex mesh. CANNOT guarantee flip-free volume parameterization!

39. Statistics