Height-function curvature estimation with arbitrary order on non-uniform Cartesian grids

Abstract

This paper proposes a height-function algorithm to estimate the curvature of two-dimensional curves and three-dimensional surfaces that are defined implicitly on two- and three-dimensional non-uniform Cartesian grids. It relies on the reconstruction of local heights, onto which polynomial height-functions are fitted. The algorithm produces curvature estimates of order $N-1$ anywhere in a stencil of ${(N+1)}^{d-1}$ heights computed from the volume-fraction data available on a d-dimensional non-uniform Cartesian grid. These estimates are of order N at the centre of the stencil when it is symmetric about its main axis. This is confirmed by a comprehensive convergence analysis conducted on the errors associated with the application of the algorithm to a fabricated test-curve and test-surface.