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

2025-06-21

AI Summary

This work addresses the numerical instability and slow convergence encountered in reinitializing signed distance functions (SDFs) on complex 2D/3D interfaces, where steep or flat initial level-set gradients impede standard methods. We propose a robust finite-element-based approach within a prediction-correction framework: a linear diffusion equation is first solved to generate a high-quality initial SDF, followed by an efficient nonlinear minimization scheme to solve the Eikonal equation for correction. This decoupling enhances convergence and stability while eliminating sensitivity to initial gradient magnitude. Numerical experiments on challenging geometries—including star-shaped domains and 3D tori—demonstrate that the method achieves high accuracy (L² error of order O(h²)), improved efficiency (30–50% reduction in iterations), and strong robustness. The approach is broadly applicable to level-set reconstruction and interface evolution problems in computational geometry and fluid dynamics.

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• Planning, Routing, and Scheduling: Replanning and Plan Repair
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• Search and Optimization: Mixed Discrete/Continuous Search

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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.

Problem

• Robust computation of signed distance functions for arbitrary boundaries
• Handling complex interfaces with steep or flat regions effectively
• Accurate reinitialization of diverse level set functions in 2D/3D

Innovation

• Finite element method for signed distances
• Prediction-correction with diffusion and Eikonal
• Handles complex interfaces and steep regions