An isogeometric finite element formulation for phase transitions on deforming surfaces

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.

arXiv commentfixed typos, extended literature review, added clarifying notes to the text, added supplementary movie files

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• Numerical modelling of phase separation on dynamic surfaces
• An isogeometric finite element formulation for boundary and shell viscoelasticity based on a multiplicative surface deformation split
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• The interplay of geometry and coarsening in multicomponent lipid vesicles under the influence of hydrodynamics
• A fully discrete evolving surface finite element method for the Cahn-Hilliard equation with a regular potential
• Well-posedness and Long-time Behavior of a Bulk-surface Coupled Cahn-Hilliard-diffusion System with Singular Potential for Lipid Raft Formation
• A curvilinear surface ALE formulation for self-evolving Navier-Stokes manifolds - Stabilized finite element formulation
• The irreversible thermodynamics of curved lipid membranes II: Permeability and osmosis
• A curvilinear surface ALE formulation for self-evolving Navier-Stokes manifolds -- General theory and analytical solutions