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

Abstract

We provide a new approach for computing integrals over hypersurfaces in the level set framework. In particular, this new approach is able to compute high order approximations of line or surface integrals in the case where the curve or surface has singularities such as corners. The method is based on the discretization (via simple Riemann sums) of the usual line or surface integral formulation used in the level set framework. This integral formulation involves an approximate Dirac delta function supported on a tubular neighborhood around the interface and is an approximation of the line or surface integral one wishes to compute. The novelty of this work is the choice of kernels used to approximate the Dirac delta function. We prove that for smooth interfaces, if the kernel has enough vanishing moments (related to the dimension of the space), the analytical integral formulation coincides exactly with the integral one wishes to calculate. For curves with singularities, the formulation is not exact but we provide an analytical result relating the severity of the singularity (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 has an integrable singularity.