arXiv subjects

Francis R. A. Aznaran

Publications and source records attributed to Francis R. A. Aznaran.

4 recordsLinked to original sources

arXiv2026

The diffuse interface approximation to fluid-structure interaction

We consider a fluid-structure interaction problem in the Eulerian, phase-field formulation. The problem is described using the Navier--Stokes equations for a viscous, incompressible fluid, coupled with the incompressible hyperelasticity system, both written in the Eulerian coordinates. This allows the problem to be written in a unified formulation, using a single field for the fluid and structure velocities. To track the position of the domain, we use a phase-field approach, resulting in a coupled Cahn--Hilliard--Navier--Stokes-type of problem for the diffuse interface fluid-structure interaction. Under certain assumptions, we prove the convergence of the diffuse interface model to the sharp interface fluid-structure interaction problem. To solve the problem numerically, we propose a novel, strongly coupled, second-order partitioned computational method where the system is decoupled into the Cahn--Hilliard problem, the transport problem for the left Cauchy--Green deformation tensor, and the Navier--Stokes problem. The problems are solved iteratively until convergence at each time step. The performance of the method is illustrated on two computational examples.

math.NA↗

arXiv2024

Uniformly $hp$-stable elements for the elasticity complex

For the discretization of symmetric, divergence-conforming stress tensors in continuum mechanics, we prove inf-sup stability bounds which are uniform in polynomial degree and mesh size for the Hu--Zhang finite element in two dimensions. This is achieved via an explicit construction of a bounded right inverse of the divergence operator, with the crucial component being the construction of bounded Poincaré operators for the stress elasticity complex which are polynomial-preserving, in the Bernstein--Gelfand--Gelfand framework of the finite element exterior calculus. We also construct $hp$-bounded projection operators satisfying a commuting diagram property and $hp$-stable Hodge decompositions. Numerical examples are provided.

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arXiv2024

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

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.

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arXiv2022

Finite element methods for multicomponent convection-diffusion

We develop finite element methods for coupling the steady-state Onsager--Stefan--Maxwell equations to compressible Stokes flow. These equations describe multicomponent flow at low Reynolds number, where a mixture of different chemical species within a common thermodynamic phase is transported by convection and molecular diffusion. Developing a variational formulation for discretizing these equations is challenging: the formulation must balance physical relevance of the variables and boundary data, regularity assumptions, tractability of the analysis, enforcement of thermodynamic constraints, ease of discretization, and extensibility to the transient, anisothermal, and non-ideal settings. To resolve these competing goals, we employ two augmentations: the first enforces the mass-average constraint in the Onsager--Stefan--Maxwell equations, while its dual modifies the Stokes momentum equation to enforce symmetry. Remarkably, with these augmentations we achieve a Picard linearization of symmetric saddle point type, despite the equations not possessing a Lagrangian structure. Exploiting the structure of linear irreversible thermodynamics, we prove the inf-sup condition for this linearization, and identify finite element function spaces that automatically inherit well-posedness. We verify our error estimates with a numerical example, and illustrate the application of the method to non-ideal fluids with a simulation of the microfluidic mixing of hydrocarbons.

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