Ryan Viertel, Braxton Osting, Matt Staten

QUAD MESHING, CROSS FIELDS, AND THE GINZBURG-LANDAU THEORY

Cross Field Guided MeshingThere are several cross field guided quad meshing algorithms. One procedure is to (i) find a complex-valued “representation" field that minimizes the Dirichlet energy subject to a boundary constraint, (ii) convert the representation field into a boundary-aligned, smooth cross field, (iii) use separatrices of the cross field to partition the domain into four sided regions, and (iv) mesh each of these four-sided regions using standard techniques.

Representation Map

Original Problem Relaxed Problem

Ginzburg-Landau Theory The problem that is often formulated for cross field design is ill defined because of the pointwise unit norm constraint. This same problem however has been considered before in the Ginzburg-Landau theory from mathematical physics. In the Ginzburg-Landau theory, the problem is relaxed by replacing the unit norm constraint with a second term in the energy functional

Brouwer Degree

d = 2 d = 0

Well Defined Limit of Relaxed Problem

Explicit Formula to Design Field with Fixed Singularities

Merriman-Bence-Osher (MBO) MethodThe MBO method was originally introduced as a method for motion by mean curvature. We have adapted it as a method to minimize the cross field energy. The Method is defined by the iterations:

Asymptotic Behavior of Cross Fields Near Singularities

d = -2 d = 1

Boundary Singularities

Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC., a

wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract

DE-NA0003525.

Partition into Four-Sided Regions

Limit Cycles

Future Work

Wish List

1. High element quality2. Boundary aligned elements3. Block structured mesh4. Prescribed boundary intervals5. Prescribed sizing function6. Guaranteed results7. Produces predictable output

Paver Replacement

Prescribed 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).

Jiang et al. (2015) developed an algorithm to design cross fields with a custom metric. This allows them to take sizing information into account when meshing. The figure below, taken from their paper, illustrates this with the mesh on the right and a visualization of the sizing function on the left. Adapting their approach to apply within our framework is an open problem.

Quad Layout Modification

Satisfying prescribed node locations and preferred element size may require combining partition curves or snapping them to nodes

Boundary Interval Assignment

Items 1 and 2 are checked because the cross field is boundary aligned by design and minimizing the cross field energy leads to isotropic elements. Items 3,6 and 7 are checked because of our conclusions from theorems 5.6 and 5.9. Items 4 and 5 remain open problems

If limit cycles occur or if user specified constraints are not compatible with a mapped mesh, a combinatorial boundary interval assignment problem arises. Hard constraints and the “sum even” condition must be satisfied while balancing any other soft constraints. The algorithm by Mitchell (2000) could be modified to meet this specific challenge.

Summary1. Connection with Ginzburg-Landau Theory2. MBO method for minimizing cross field energy3. Fixed Frame field design method4. Asymptotic Behavior of Singularities5. Cross Field Partitioning Theorem

Acknowledgements• Sandia National Labs• University of Utah• NSF DMS 16-19755• We would like to thank Dan Spirn for the helpful discussions and comments,

and Franck Ledoux for providing access to the GMDS meshing library.

References

• F. Bethuel, H. Brezis, and F. Helein, Ginzburg-Landau Vortices, vol. 13 of Progress in Nonlinear Differential Equations and Their Applications, Birkhauser Boston, (1994).

• T. Jiang, X. Fang, J. Huang, H. Bao, Y. Tong, M. Desbrun, Frame field generation through metric customization, ACM Transactions on Graphics 34 (2015).

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

• R. Viertel, B. Osting, An approach to quad meshing based on harmonic cross-valued maps and the Ginzburg-Landau theory, submitted, arXiv:1708.02316 (2017).