A Paraboloid Fitting Technique for Calculating Curvature from Piecewise-Linear Interface Reconstructions on 3D Unstructured Meshes

TL;DR

This paper introduces a paraboloid fitting method to accurately compute interface curvature on 3D unstructured meshes, essential for modeling surface tension in fluid simulations, demonstrating convergence across various mesh types.

Contribution

The paper presents a new paraboloid fitting technique for curvature calculation on unstructured meshes, requiring only local neighbor information and solving small linear systems.

Findings

Converges between first and second order with grid refinement

Works on regular and unstructured meshes

Accurately computes curvature for complex interfaces

Abstract

We present a novel method for calculating interface curvature on 3D unstructured meshes from piecewise-linear interface reconstructions typically generated in the volume of fluid method. Interface curvature is a necessary quantity to calculate in order to model surface tension driven flow. Curvature needs only to be computed in cells containing an interface. The approach requires a stencil containing only neighbors sharing a node with a target cell, and calculates curvature from a least-squares paraboloid fit to the interface reconstructions. This involves solving a 6x6 symmetric linear system in each mixed cell. We present verification tests where we calculate the curvature of a sphere, an ellipsoid, and a sinusoid in a 3D domain on regular Cartesian meshes, distorted hex meshes, and tetrahedral meshes. For both regular and unstructured meshes, we find in all cases the paraboloid fitting method for curvature to converge between first and second order with grid refinement.