Analysis and application of a lower envelope method for sharp-interface multiphase problems

TL;DR

This paper introduces a lower envelope method (LEM) for tracking interfaces in multiphase problems, providing theoretical analysis and a numerical algorithm with applications to shape optimization and inverse conductivity problems.

Contribution

The paper presents a novel lower envelope method for multiphase interface tracking, including theoretical properties and a practical numerical algorithm.

Findings

Level set method is a special case of LEM.

The LEM-based algorithm effectively solves multiphase shape optimization.

Numerical results demonstrate the method's applicability to inverse conductivity problems.

Abstract

We introduce and analyze a lower envelope method (LEM) for the tracking of interfaces motion in multiphase problems. The main idea of the method is to define the phases as the regions where the lower envelope of a set of functions coincides with exactly one of the functions. We show that a variety of complex lower-dimensional interfaces naturally appear in the process. The phases evolution is then achieved by solving a set of transport equations. In the first part of the paper, we show several theoretical properties, give conditions to obtain a well-posed behaviour, and show that the level set method is a particular case of the LEM. In the second part, we propose a LEM-based numerical algorithm for multiphase shape optimization problems. We apply this algorithm to an inverse conductivity problem with three phases and present several numerical results.