A 2D Advancing-Front Delaunay Mesh Refinement Algorithm

TL;DR

This paper introduces a generalized 2D advancing-front Delaunay mesh refinement algorithm that produces size-optimal meshes with controlled angles by solving ODEs for segment splitting and using off-center Steiner vertices.

Contribution

It extends Chew's algorithm to generate size-optimal Delaunay meshes with angle bounds, handling constrained PSLGs and local feature size adaptation.

Findings

Produces size-optimal Delaunay meshes with minimum angles less than π/6.

Maintains angle bounds even with small input angles.

Uses ODE-based segment splitting for local feature size adaptation.

Abstract

I present a generalization of Chew's first algorithm for Delaunay mesh refinement. In his algorithm, Chew splits the line segments of the input planar straight line graph (PSLG) into shorter subsegments whose lengths are nearly identical. The constrained Delaunay triangulation of the subsegments is refined based on the length of the radii of the circumcircles of the triangles. This algorithm produces a uniform mesh, whose minimum angle can be at most $\pi/6$. My algorithm generates both truly Delaunay and constrained Delaunay size-optimal meshes. In my algorithm, I split the line segments of the input PSLG such that their lengths are asymptotically proportional to the local feature size (LFS) by solving ordinary differential equations (ODEs) that map points from a closed 1D interval to points on the input line segments in the PSLG. I then refine the Delaunay triangulation (truly or constrained) of the PSLG by inserting off-center Steiner vertices of "skinny" triangles while prioritizing such triangles with shortest edges first. As in Chew's algorithm, I show that the Steiner vertices do not encroach upon any subsegment of the PSLG. The off-center insertion algorithm places Steiner vertices in an advancing front manner such that we obtain a size-optimal Delaunay mesh (truly or constrained) if the desired minimum angle is less than $\pi/6$. In addition, even in the presence of a small angle $\phi < \pi/2$ in the PSLG, the bound on the minimum angle "across" the small angle tends to $\arctan{((\sin{\phi})/(2-\cos(\phi))}$ as the PSLG is progressively refined. Also, the bound on the maximum angle across any small input angle tends to $\pi/2 + \phi/2$ as the PSLG is progressively refined.