Quadrilateral mesh generation II: Meromorphic quartic differentials and Abel–Jacobi condition

Abstract

This work discovers the equivalence relation between quadrilateral meshes and meromorphic quartic differentials. Each quad-mesh induces a conformal structure of the surface, and a meromorphic quartic differential, where the configuration of singular vertices correspond to the configurations of the poles and zeros (divisor) of the meroromorphic differential. Due to Riemann surface theory, the configuration of singularities of a quad-mesh satisfies the Abel-Jacobi condition. Inversely, if a divisor satisfies the Abel-Jacobi condition, then there exists a meromorphic quartic differential whose divisor equals to the given one. Furthermore, if the meromorphic quadric differential is with finite trajectories, then it also induces a a quad-mesh, the poles and zeros of the meromorphic differential correspond to the singular vertices of the quad-mesh. Besides the theoretic proofs, the computational algorithm for verification of Abel-Jacobi condition is also explained in details. Furthermore, constructive algorithm of meromorphic quartic differential on genus zero surfaces is proposed, which is based on the global algebraic representation of meromorphic differentials. Our experimental results demonstrate the efficiency and efficacy of the algorithm. This opens up a novel direction for quad-mesh generation using algebraic geometric approach.

Flux-aligned quad mesh generation in magnetohydrodynamic simulation

“…There is a rich literature on two-dimensional mesh generation. A thorough survey can be found in [Bommes et al(2012)] [ Armstrong et al(2015)] and other recent works [Chen et al(2019)] [ Lei et al(2020)] [Xiao et al(2020)]. For quad-mesh generation, there is a specified requirement in Tokamak experiments [Guillard et al(2018)].…”

Integrable cross-field generation based on imposed singularity configuration -- the 2D manifold case --

“…Among the developed methods for the quad layout generation, a general distinction can be made among the ones which are: computing a seamless global parametrization of the domain where integer iso-values of the parameter fields form the sides [6,7,8], using Riemann geometry [9,10,11], or like in our case, constructing a cross-field structure that will guide the integral lines emanating from singularities [12,13,14,15,16,17].…”

“…It is important to note that the choice of singularity pattern is not arbitrary, though. Moreover, it is under the direct constraint of Abel-Jacobi theory [9,10,11] for valid singularity configurations. Here, the singularity configuration is taken as an input and an integrable isotropic crossfield is computed by solving only two linear systems, Section 3.…”

Analysis-suitable unstructured T-splines: Multiple extraordinary points per face

“…Getting a bilinear quadrilateral mesh for conventional FEA from a trimmed NURBS representation is a nontrivial task. Even though there are several methods that aim at automating this process [6,7,8,9,10,11,12,13], coming up with an algorithm that automatically delivers a non-distorted bilinear quadrilateral mesh suitable for FEA from any trimmed NURBS representation remains an open problem. This problem is particularly hard to automate due to the geometric imperfections (gaps, overlaps, etc.)…”