1. QUAD MESHING, CROSS FIELDS, AND THE GINZBURG-LANDAU THEORY
Ryan Viertela,b, Braxton Ostinga
Sept 29 – Oct 1, 2017
The 3rd Annual Meeting of SIAM Central States Section
Colorado State University
Fort Collins, CO
a University of Utah
b Sandia National Laboratories

2. Introduction

3. Classical Quad Meshing Methods
Pattern Based
Unstructured - Paving

4. Basic Cross Field Meshing Algorithm (Kowalski et al. 2013)

5. 2D Cross Field Meshing Algorithm

6. 2D Cross Field Meshing Algorithm

7. The Representation Map

8. 2D Cross Field Meshing Algorithm

9. 2D Cross Field Meshing Algorithm

10. 2D Cross Field Meshing Algorithm

11. 2D Cross Field Meshing Algorithm

12. 2D Cross Field Meshing Algorithm

13. 2D Cross Field Meshing Algorithm

14. Cross Field Singularities
Index = +1/4
Index = 0
Index = -1/4

15. 2D Cross Field Meshing Algorithm

16. 2D Cross Field Meshing Algorithm

17. Connection to Ginzburg-Landau Theory

18. Ginzburg-Landau Functional
Original problem:
Relaxed problem:

19. Results of Ginzburg-Landau Theory and Applications to Cross Fields

20. Brouwer Degree Let be the boundary condition on the domain G.
Let be the Brouwer degree.
d = 2
d = 0

21. Canonical Harmonic Map
Harmonic vector field defined everywhere except a finite number of points
All vectors are unit vectors
Unique for a given boundary condition and configuration of singularities

22. Well Defined Limit of Relaxed Problem
This gives us a generalized sense in which to understand the energy minimization problem

23. Explicit Formula to Design Field with Fixed Singularities

24. Application: New Cross Field Design Method

25. Merriman-Bence-Osher (MBO) Method

26. Merriman-Bence-Osher (MBO) Method
Original Method
Introduced as a method for motion by mean curvature
Minimizes an two-well potential energy analogous to the complex GL energy
New Application to Frame Fields
Iterative method to minimize cross field energy:

27. MBO Method

28. Asymptotic Behavior of Cross Fields Near Singularities

29. Riemann Surface and Streamlines

30. Separatrices of a Singularity

31. Boundary Singularities

32. Partition into four-sided regions

33. Meshing

34. Limit Cycles

35. Future Research

36. Candidate: Cross field guided methods
Wish List
High element quality
Boundary aligned elements
Block Structured mesh
Prescribed size map
Conform to pre meshed boundary
Guaranteed results
Produces predictable output

37. Size Map
T. Jiang, X. Fang, J. Huang, H. Bao, Y. Tong, M. Desbrun, Frame field generation through metric customization, ACM Transactions on Graphics 34 (2015).
We intend to adapt the work of Jiang et al. to the MBO method for designing cross fields

38. Partition Modification

39. Boundary Interval Assignment
S. A. Mitchell, High fidelity interval assignment, International Journal of Computational Geometry & Applications 10 (2000) 399–415.

40. Summary of Contributions
Connection with Ginzburg-Landau Theory
MBO method for minimizing cross field energy
Fixed cross field design method
Asymptotic analysis of cross fields near singularities
Cross Field Partitioning Theorem
Paper: An approach to quad meshing based on harmonic cross-valued maps and the Ginzburg-Landau theory
Future Work
Software Implementation
Extend each step to work on manifolds
Design cross fields with custom metric
Robust singularity graph partitioning algorithm
Theory
Manifolds
Numerical Analysis

41. Acknowledgements Sandia National Labs University of Utah
NSF DMS
Matt Staten
Braxton Osting

42. Extra Slides

43. Implication for Cross Fields: Strange Minimizer

44. Cross Fields Automatically Generate Good Meshes in 2D
Kowalski et al. 2013