UC Davis

We present a full pipeline for producing all-quadrilateral, high-quality, low-singularity 2D surfacemeshes, that strictly adhere to input boundary geometry while being guided by a prescribed size function. The meshes are built for the simulation of groundwater flow, where input geometries are highly irregular due to following e.g. rivers and mountain ranges, but are sampled at somewhat regular intervals. The size of the produced elements are guided by a size function, coarsening or refining the mesh, allowing local control over the trade-off between simulation speed and accuracy. The irregular boundaries set groundwater flow apart from typical quad mesh generation applications, where boundaries are typically smooth and able to be subdivided. Early testing with existing field-aligned methods showed significant drawbacks to using the boundaries directly, so we developed a tool to splice a non-conforming but reasonably-sized mesh into the fixed boundary, allowing arbitrary mesh generation techniques to be applied to the problem of groundwater flow. This left the problem of generating size-driven meshes, where a minimal number of singularities is added to affect a change in size of the produced quads. Existing methods tended to focus on quads with uniform size, or with size determined by the input geometry. By treating the target size function as a Riemannian metric, we are able to exploit the curvature of the Levi-Civita connection to apply surface meshing techniques to our problem of size-aware planar meshing. We build off of the approach in Globally Optimal Direction Fields to first produce a cross field that approximates the desired growth, from which a quad mesh can be extracted. When analyzing the resulting cross fields, we found it difficult to estimate how well they matched the desired size function without simply meshing them and measuring the resulting quads. Attempts to quantify the growth in size implied by a cross field led to an analysis of anisotropy, size functions that are not uniform in all directions. We were able to demonstrate that harmonic cross fields admit isotropic size functions, while general cross fields still admit orthogonal anisotropic size functions. This gives us a method to quantify how accurately a cross field matches the desired size function, independent of any size inaccuracy that arises from the meshing process. Finally, we use a combination of existing techniques to extract a quad mesh from the cross field, which can then be stitched to the original boundary. We conclude with simple Laplacian smoothing as a postprocessing step to improve the quality of the resulting meshes.