A three-dimensional multi-phase-field vesicles model and its practical finite difference solver

Authors: Yutong Wu, Zecheng Qiu, Junxiang Yang

This paper presents a three-dimensional multi-phase-field vesicles model and discusses its practical finite difference solver. It is currently published in Computer Physics Communications (CPC).

Abstract: This paper presents a numerical model for simulating the dynamics of multiple interacting vesicles using a multi-phase-field framework. We use 𝑁 phase-field variables, each possibly containing multiple disconnected vesicles, and enforce volume and surface-area constraint per variable. Their evolution is governed by the variational derivatives of a total energy functional encompassing bending elasticity, surface area and volume conservation, and inter-vesicle repulsion. A semi-implicit finite difference scheme is developed to discretize the system, achieving numerical stability and efficiency. Extensive three-dimensional simulations demonstrate the method’s capability to maintain physical constraints and accurately capture complex vesicle deformations and interactions across various configurations. The simulation code corresponding to Sections 4.3.4 and 4.3.5 (Figs. 10 & 11) in this paper can be accessed at https://github.com/aaron-z-chiu/multiple-vesicles.

Keywords: Phase-field method; Vesicle dynamics; Multi-phase modeling; Numerical simulation

Highlights:

• New phase-field model is developed for multiple vesicles.
• The model satisfies the global constraints preserving properties.
• A practical and easy finite difference method is presented.
• The simulation code is shared for the interested readers.

Role & Responsibilities:

• Implemented a hybrid numerical solver for 3D fluid vesicle dynamics in \textbf{C++}, integrating phase-field models into an existing simulation framework.
• Applied a semi-implicit finite difference scheme to evolve phase-field equations, ensuring rigorous numerical stability and energy conservation.
• Optimized memory management and data storage strategies, significantly reducing computational overhead for multi-vesicle interaction simulations.

Recommanded BibTEX:

@ARTICLE{Wu2026,
	author = {Wu, Yutong and Qiu, Zecheng and Yang, Junxiang},
	title = {A three-dimensional multi-phase-field vesicles model and its practical finite difference solver},
	year = {2026},
	journal = {Computer Physics Communications},
	volume = {321},
	doi = {10.1016/j.cpc.2026.110053},
	type = {Article}
}

Phase-field simulation of 8 spherical vesicles with 2 phase-field variables.

Phase-field simulation of 3 stacked vesicles with distinct phase-field variables.

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