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

Introduction

Classical Quad Meshing Methods Pattern Based Unstructured - Paving

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

2D Cross Field Meshing Algorithm

2D Cross Field Meshing Algorithm

The Representation Map

2D Cross Field Meshing Algorithm

2D Cross Field Meshing Algorithm

2D Cross Field Meshing Algorithm

2D Cross Field Meshing Algorithm

2D Cross Field Meshing Algorithm

2D Cross Field Meshing Algorithm

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

2D Cross Field Meshing Algorithm

2D Cross Field Meshing Algorithm

Connection to Ginzburg-Landau Theory

Ginzburg-Landau Functional Original problem: Relaxed problem:

Results of Ginzburg-Landau Theory and Applications to Cross Fields

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

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

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

Explicit Formula to Design Field with Fixed Singularities

Application: New Cross Field Design Method

Merriman-Bence-Osher (MBO) Method

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:

MBO Method

Asymptotic Behavior of Cross Fields Near Singularities

Riemann Surface and Streamlines

Separatrices of a Singularity

Boundary Singularities

Partition into four-sided regions

Meshing

Limit Cycles

Future Research

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

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

Partition Modification

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

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

Acknowledgements Sandia National Labs University of Utah NSF DMS 16-19755 Matt Staten Braxton Osting

Extra Slides

Implication for Cross Fields: Strange Minimizer

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