An implicit boundary integral method for interfaces evolving by Mullins-Sekerka dynamics

TL;DR

This paper introduces an implicit boundary integral method for simulating Mullins-Sekerka interface dynamics, combining level set and boundary integral techniques to handle complex topological changes efficiently.

Contribution

The paper proposes a novel algorithm that reformulates boundary integrals as volume integrals for implicit interfaces, enabling accurate simulation of Mullins-Sekerka dynamics with topological changes.

Findings

Effective in 2D and 3D simulations

Handles multiply connected and unbounded domains

Demonstrates robustness and accuracy

Abstract

We present an algorithm for computing the nonlinear interface dynamics of the Mullins-Sekerka model for interfaces that are defined implicitly (e.g. by a level set function) using integral equations . The computation of the dynamics involves solving Laplace's equation with Dirichlet boundary conditions on multiply connected and unbounded domains and propagating the interface using a normal velocity obtained from the solution of the PDE at each time step. Our method is based on a simple formulation for implicit interfaces, which rewrites boundary integrals as volume integrals over the entire space. The resulting algorithm thus inherits the benefits of both level set methods and boundary integral methods to simulate the nonlocal front propagation problem with possible topological changes. We present numerical results in both two and three dimensions to demonstrate the effectiveness of the algorithm.