Diamond-Kite Adaptive Quadrilateral Meshing

TL;DR

This paper introduces a family of adaptive quadrilateral meshes based on diamond, kite, and rhombus shapes, which efficiently adapt to local size functions while maintaining optimal element count and invariance under smoothing.

Contribution

It presents a novel mesh generation method using diamond and kite shapes that adapts to local size functions with near-minimal element count and dual circle packing properties.

Findings

Meshes are within a constant factor of minimal element count.

Meshes are invariant under Laplacian smoothing.

Dual meshes are well-centered and adapted to size functions.

Abstract

We describe a family of quadrilateral meshes based on diamonds, rhombi with 60 and 120 degree angles, and kites with 60, 90, and 120 degree angles, that can be adapted to a local size function by local subdivision operations. Our meshes use a number of elements that is within a constant factor of the minimum possible for any mesh of bounded aspect ratio elements, graded by the same local size function, and is invariant under Laplacian smoothing. The vertices of our meshes form the centers of the circles in a pair of dual circle packings. The same vertex placement algorithm but a different mesh topology gives a pair of dual well-centered meshes adapted to the given size function.