# Integrable Frame Fields for Quad Mesh Generation

## Outline

In the context of computational engineering sciences, generation of meshes is a necessary step to discretize the domain. In certain applications where vector fields are simulated such as in computational fluid dynamics, a desired feature for the meshes is to conform to the metric of the solution: i.e. alignment with its principal directions and scaling according to the anisotropy of the metric at each point. The quad mesh generation problem for arbitrary domains is usually complex, involving continuous variables describing the local geometry and integer variables describing the topology of the domain. Thus state of the art methods for quad mesh generation decompose the problem in two main steps: 1) cross-field optimization and 2) parametrization. The cross field describes locally the alignment of parametrization functions and its matchings with neighboring charts. The parametrization step computes the surface coordinate functions by integrating a globally combed matching of the cross field on a subdomain of disc-like topology. Unfortunately the coupling of the cross-field with the parametrization functions is rather loose, unless the field is curl-free. This means that the parametrization will deviate from the guidance cross-field in areas where the curl of the gradient field is non-zero. Thus, our current research problem is the optimization integrability of gradient cross field such that its integration is as exact as possible.