# Extending fields in a level set method by solving a biharmonic equation

## TL;DR

This paper introduces a biharmonic equation-based method for extending fields in level set methods, which implicitly handles interfaces and ensures smooth, accurate extensions with simple implementation and symmetry preservation.

## Contribution

It presents a novel biharmonic extension approach that implicitly manages interfaces and improves smoothness and accuracy in level set methods.

## Key findings

- Effective at producing smooth, accurate extensions near interfaces
- Handles symmetry and periodicity naturally
- Simple implementation with efficient solvers

## Abstract

We present an approach for computing extensions of velocities or other fields in level set methods by solving a biharmonic equation. The approach differs from other commonly used approaches to velocity extension because it deals with the interface fully implicitly through the level set function. No explicit properties of the interface, such as its location or the velocity on the interface, are required in computing the extension. These features lead to a particularly simple implementation using either a sparse direct solver or a matrix-free conjugate gradient solver. Furthermore, we propose a fast Poisson preconditioner that can be used to accelerate the convergence of the latter. We demonstrate the biharmonic extension on a number of test problems that serve to illustrate its effectiveness at producing smooth and accurate extensions near interfaces. A further feature of the method is the natural way in which it deals with symmetry and periodicity, ensuring through its construction that the extension field also respects these symmetries.