HFES: A Height Function Method with Explicit Input and Signed Output for High-Order Estimations of Curvature and unit Vectors of Planar Curves

Abstract

The author proposes a height function (HF) method for estimating curvature and unit vectors of planar curves. The new method is named HFES since its signature consists of explicit input parameters (instead of implicit volume fractions) and signed output estimates. Due to this signature and its formal description, HFES has a number of appealing advantages. First, the convergence rate of HFES can be 4, 6, or a greater even integer. Second, local arcs can be rotated to further reduce leading truncation errors. Third, the sampling size of estimation is completely independent from the Eulerian grid size of the main flow. Fourth, signs of the signed output can be conveniently chosen by orientations of input Jordan curves, e.g., surface tension of a tracked material can be determined in a simple and worry-free manner. Last, HFES is applicable both to numerical simulation of multiphase flows and shape analysis in computer imaging. Furthermore, the author analyzes the effect of input perturbations upon output estimates, derives expressions of the best attainable accuracy afforded by a given input on uniform grids, and explains how to achieve this best accuracy with a concrete example. Results of numerical experiments with both exact and inexact input demonstrate that HFES can be much more accurate than previous HF methods. For the vortex-shear test of a circular disk, the eighth-order HFES coupled with the recent cubic iPAM method is millions of times more accurate than a fourth-order HF method coupled with volume-of-fluid methods! This drastic accuracy improvement is mainly due to the purely explicit spline representation of the curve and the rotation of local arcs.