An extrapolative approach to integration over hypersurfaces in the level set framework

TL;DR

This paper introduces a novel kernel-based discretization method within the level set framework for accurately computing integrals over hypersurfaces, including those with corners or cusps, by leveraging kernels with vanishing moments.

Contribution

It proposes a new kernel choice with vanishing moments for integral approximation in the level set method, improving accuracy especially for non-smooth interfaces.

Findings

Exact integral computation for smooth interfaces with sufficient vanishing moments.

Analytical relation between corner severity and neighborhood width.

Numerical validation for piecewise smooth interfaces and singular integrands.

Abstract

We provide a new approach for computing integrals over hypersurfaces in the level set framework. The method is based on the discretization (via simple Riemann sums) of the classical formulation used in the level set framework, with the choice of specific kernels supported on a tubular neighborhood around the interface to approximate the Dirac delta function. The novelty lies in the choice of kernels, specifically its number of vanishing moments, which enables accurate computations of integrals over a class of closed, continuous, piecewise smooth, curves or surfaces; e.g. curves in two dimensions that contain finite number of corners. We prove that for smooth interfaces, if the kernel has enough vanishing moments (related to the dimension of the embedding space), the analytical integral formulation coincides exactly with the integral one wishes to calculate. For curves with corners and cusps, the formulation is not exact but we provide an analytical result relating the severity of the corner or cusp with the width of the tubular neighborhood. We show numerical examples demonstrating the capability of the approach, especially for integrating over piecewise smooth interfaces and for computing integrals where the integrand is only Lipschitz continuous or has an integrable singularity.