A Point Creation Strategy for Mesh Generation Using Crystal Lattices As Templates

1. A Point Creation Strategy for Mesh Generation Using Crystal Lattices As Templates Paulo Roma Cavalcanti Ulisses T. Mello Universidade Federal do Rio de Janeiro IBM T. J. Watson Research Center

2. Problem Definition • We need to create auxiliary points to improve the quality of Delaunay based triangulations • Background meshes (ex., regular grid) may be used • Crystals generate perfect tetrahedra in the nature, but they grow inside out with no boundaries • Can we use crystal templates to generate better background meshes?

3. Crystal Lattices • A crystal is a repeating array • There is the pattern of repetition (lattice type) • And what is repeated (unit cell) A hexagonal “foot” lattice

4. Basis (motif) A group of atoms associated to each lattice point Unit cell The smallest component of the crystal, which when stacked together with pure translational repetition reproduces the whole crystal Some Basic Definitions • Crystal structure • The periodic arrangement of atoms in the crystal • Lattice • An infinite array of points in space, in which each point has identical surroundings to all others

5. Crystal Lattice

6. A Single Layer of Graphite • Unit cell is primitive (1 lattice point) but contains two atoms in the basis • Atoms at the corner contribute ¼ to unit cell count • Atoms within the cell contribute 1 to that unit cell

7. Principles of Laves • Followed by metals and inert gases: close packing • Space: space is used most efficiently • Symmetry: highest possible symmetry adopted • Connection: there will be the most possible connections between components

8. Close Packing • A hexagonal array of spheres is a close-packed array • There is no way to pack more spheres into a given area • Packing efficiency = 90.69% • Interstitials = 9.31%

9. 2D Example - A Cross Section of a Geological Model • We used a hexagonal and a cubic lattice to generate auxiliary points for a Delaunay triangulation

10. Triangular Mesh Using a Regular Grid As Background Mesh • This is the result of using the cubic lattice • Most triangles are rectangular isosceles • One 45 and two 90 degree internal angles

11. Triangular Mesh Using a Hexagonal Lattice As Background Mesh • This is the result of using a hexagonal lattice • Most triangles are equilateral • Three 60 degree internal angles

12. Smoothing Applied to the Cubic Lattice • Result of applying a Laplacian smoothing operator to the cubic lattice

13. Smoothing Applied to the Hexagonal Lattice • Result of applying a Laplacian smoothing operator to the hexagonal lattice

14. Histogram of Minimum Angle Distribution

15. Close Packing in 3-D • Two or three layers of spheres • There is no way to pack more spheres into a given volume • Packing efficiency = 74.05% • Interstitials = 25.95%

16. Different Arrangement of Layers

17. 1611 - Johannes Kepler Posed a conjecture No way of packing more spheres than that of a face-centered cubic arrangement 1900 - Hilbert Included this question as the 18th unsolved problem of his list in the international congress of mathematicians - Paris 1988 - Thomas Hales Announced a computer based solution 250 manuscript pages and 3Gb of computer data No one has been able to check A Little Bit of History

18. Actual SMT Image of a Ni Surface Note the hexagonal arrangement of atoms

19. Hexagonal Close-packing (HCP) in Crystals

20. Cubic Close-packing (CCP) in Crystals

21. Non Close-packed (BCC) Adopted By Some Metals Packing efficiency 68%

22. Hexagonal Close-packing With Delaunay • 35 tetrahedra • 20 in the unit cell • 15 in the convex-hull • 8 perfect tetrahedra (22%) 2 atoms in the unit cell at (0,0,0) (2/3,1/3,1/2)

23. Hexagonal Close-packing With Delaunay 8 perfect tetrahedra out of 20 (40%) Top View

24. Geometry of the Hexagonal Close-packing

25. Why Co-circular Vertices Do Not Generate Slivers

26. Triangulation of Multi-domain Models • Multi-domain models have several regions (enclosed 3-D volumes) • We have to determine to which region each point of the background mesh belongs

27. Auxiliary Point Classification • We shoot rays through the model bounding box • Sort the intersections between a ray and the boundary surfaces • Use adjacency information to determine all points between pairs of consecutive intersections • A R*tree speed up the spatial search, avoiding a large number of intersection calculations

28. Effect of the Lattice Type in the Generation of Slivers • Cubic on the faces and into the model - 42 slivers • Hexagonal on the faces and cubic into the model - 13 slivers • Hexagonal on the faces and into the model - NO slivers

29. Triangulation of a Mechanical Part Using a Hexagonal Lattice

30. Histogram of Minimum Dihedral Angles for the Mechanical Part • Cubic lattice • Peaks at: 35.3, 45.0, 54.7 • Hexagonal lattice • Peaks at: 30, 38, 70 • After smoothing it has the most smooth distribution

31. Conclusions • The pattern produced by the appropriate crystal lattice has a superior arrangement of points to the regular grid • Advantages: simplicity, robustness and speed • Also generates perfect tetrahedra and reduces the number of slivers in a Delaunay triangulation

32. Conclusions • The hexagonal lattice produces the best result among the 14 Bravais crystal lattices • Cubic lattices should be avoided due to the possibility of generating a large number of slivers • Using the hexagonal lattice in 2-D, more than 80% of the triangles have internal angles greater than 50 degrees