An isogeometric finite element formulation for phase transitions on deforming surfaces

TL;DR

This paper develops a comprehensive isogeometric finite element framework for simulating mass-conserving phase transitions on deforming surfaces, coupling the Cahn-Hilliard and Kirchhoff-Love equations on evolving manifolds.

Contribution

It introduces a novel coupled PDE formulation and discretization approach using $C^1$-continuous splines for phase transitions on deforming surfaces.

Findings

Successfully models phase transitions on complex deforming geometries.

Demonstrates numerical stability and accuracy with adaptive time-stepping.

Validates the approach with examples on spheres, tori, and double-tori.

Abstract

This paper presents a general theory and isogeometric finite element implementation for studying mass conserving phase transitions on deforming surfaces. The mathematical problem is governed by two coupled fourth-order nonlinear partial differential equations (PDEs) that live on an evolving two-dimensional manifold. For the phase transitions, the PDE is the Cahn-Hilliard equation for curved surfaces, which can be derived from surface mass balance in the framework of irreversible thermodynamics. For the surface deformation, the PDE is the (vector-valued) Kirchhoff-Love thin shell equation. Both PDEs can be efficiently discretized using $C^1$-continuous interpolations without derivative degrees-of-freedom (dofs). Structured NURBS and unstructured spline spaces with pointwise $C^1$-continuity are utilized for these interpolations. The resulting finite element formulation is discretized in time by the generalized-$\alpha$ scheme with adaptive time-stepping, and it is fully linearized within a monolithic Newton-Raphson approach. A curvilinear surface parameterization is used throughout the formulation to admit general surface shapes and deformations. The behavior of the coupled system is illustrated by several numerical examples exhibiting phase transitions on deforming spheres, tori and double-tori.