Analysis and finite element approximation of a diffuse interface approach to the Stokes--Biot coupling

TL;DR

This work develops and analyzes a diffuse interface (phase-field) approach for Stokes–Biot coupling, reformulating the coupled Stokes and Biot equations on a single computational domain with phase field weights. It establishes well-posedness of the diffuse interface problem in weighted Sobolev spaces, derives finite element error estimates, and quantifies the modelling error between sharp and diffuse interfaces. Two modelling regimes are studied: positive Lipschitz weights achieve higher-order convergence in ε and δ, while power distance weights yield a weaker but robust ε-dependent rate; numerical experiments confirm the theoretical rates and demonstrate applications to complex geometries such as the circle of Willis. Overall the framework enables accurate, mesh-agnostic simulation of fluid-poroelastic interactions in intricate domains with provable convergence properties.

Abstract

We consider the interaction between a poroelastic structure, described using the Biot model in primal form, and a free-flowing fluid, modelled with the time-dependent incompressible Stokes equations. We propose a diffuse interface model in which a phase field function is used to write each integral in the weak formulation of the coupled problem on the entire domain containing both the Stokes and Biot regions. The phase field function continuously transitions from one to zero over a diffuse region of width $\mathcal{O}(\varepsilon)$ around the interface; this allows the equations to be posed uniformly across the domain, and obviates tracking the subdomains or the interface between them. We prove convergence in weighted norms of a finite element discretisation of the diffuse interface model to the continuous diffuse model; here the weight is a power of the distance to the diffuse interface. We in turn prove convergence of the continuous diffuse model to the standard, sharp interface, model. Numerical examples verify the proven error estimates, and illustrate application of the method to fluid flow through a complex network, describing blood circulation in the circle of Willis.

Figures (6)

• Figure 1: Graphical representation of the diffuse interface approach (inspired by Bukac2023), in which $\Omega_F$ and $\Omega_F^\epsilon$ are, respectively, the sharp and diffuse fluid domains. Similarly $\Omega_B$ and $\Omega_B^\epsilon$ are the sharp and diffuse poroelastic domains, and $\Phi_F^\epsilon$ and $\Phi_B^\epsilon$ are the distance functions that define the diffuse fluid and poroelastic domains, respectively. Finally, $\ell_F^\epsilon$ and $\ell_B^\epsilon$ are transitional layers.
• Figure 2: Geometry and the computational mesh used in Example \ref{['sec:ex2a']} for the sharp (left) and the diffuse (right) interface model. The right panel also shows the phase field function used for the diffuse interface model.
• Figure 3: Comparison of the Stokes and Biot pressure (left), the velocity (middle), and the displacement (right) obtained using the sharp interface model (top) and the diffuse interface model (bottom), shown on a cross section of the domain.
• Figure 4: Left: A patient-specific vascular network containing the circle of Willis. Middle: A phase field function for the section of the network considered in our study. Right: A phase field function on a cross section of our computational domain, superimposed with the full vascular network.
• Figure 5: Left: Velocity magnitude superimposed with vectors indicating the flow direction. The marked arteries represent the inlet sections. Right: Flow streamlines coloured by the Stokes pressure.

Theorems & Definitions (27)

• Definition 3.1: weak solution of the sharp interface problem
• Remark 3.2: trace of $\partial_t {\boldsymbol{\eta}}$
• Theorem 3.3: well-posedness of the sharp formulation Avalos2024
• Remark 3.4: geometric configuration
• Proposition 3.5: $\Phi_F^\epsilon \in A_2$
• proof
• Lemma 3.6: diffuse trace inequality Bukac2023
• Definition 3.7: weak solution of the diffuse interface problem
• Theorem 3.8: well-posedness of the diffuse formulation
• proof