A Continuation Method for Computing Constant Mean Curvature Surfaces with Boundary

TL;DR

This paper introduces a novel continuation method to compute constant mean curvature surfaces with fixed boundaries, enabling the exploration of their shapes as parameters like volume vary, with applications in fluid interfaces.

Contribution

The work develops a new numerical approach using arc-length continuation to compute families of CMC surfaces with fixed boundaries, addressing challenges in explicit shape determination.

Findings

Method effectively computes CMC surfaces with varying volume

Algorithm demonstrates high accuracy and robustness

Produces continuous families of physically stable surfaces

Abstract

Defined mathematically as critical points of surface area subject to a volume constraint, constant mean curvatures (CMC) surfaces are idealizations of interfaces occurring between two immiscible fluids. Their behavior elucidates phenomena seen in many microscale systems of applied science and engineering; however, explicitly computing the shapes of CMC surfaces is often impossible, especially when the boundary of the interface is fixed and parameters, such as the volume enclosed by the surface, vary. In this work, we propose a novel method for computing discrete versions of CMC surfaces based on solving a quasilinear, elliptic partial differential equation that is derived from writing the unknown surface as a normal graph over another known CMC surface. The partial differential equation is then solved using an arc-length continuation algorithm, and the resulting algorithm produces a continuous family of CMC surfaces for varying volume whose physical stability is known. In addition to providing details of the algorithm, various test examples are presented to highlight the efficacy, accuracy and robustness of the proposed approach.