A predictor-corrector scheme for approximating signed distances using finite element methods

TL;DR

This paper presents a novel finite element predictor-corrector scheme for accurately and robustly computing signed distance functions to complex boundaries in 2D and 3D, improving convergence and handling challenging geometries.

Contribution

The authors introduce a new predictor-corrector finite element method that efficiently computes signed distance functions for complex interfaces, enhancing robustness and accuracy over existing techniques.

Findings

Successfully handles complex geometries like star domains and tori

Demonstrates high accuracy and efficiency in multiple examples

Robustly reinitializes diverse level set functions

Abstract

In this article, we introduce a finite element method designed for the robust computation of approximate signed distance functions to arbitrary boundaries in two and three dimensions. Our method employs a novel prediction-correction approach, involving first the solution of a linear diffusion-based prediction problem, followed by a nonlinear minimization-based correction problem associated with the Eikonal equation. The prediction step efficiently generates a suitable initial guess, significantly facilitating convergence of the nonlinear correction step. A key strength of our approach is its ability to handle complex interfaces and initial level set functions with arbitrary steep or flat regions, a notable challenge for existing techniques. Through several representative examples, including classical geometries and more complex shapes such as star domains and three-dimensional tori, we demonstrate the accuracy, efficiency, and robustness of the method, validating its broad applicability for reinitializing diverse level set functions.