# A partition of unity approach to fluid mechanics and fluid-structure interaction

## TL;DR

This paper introduces a novel partition of unity overlapping domain method for fluid-structure interaction problems, achieving optimal convergence and stability without re-meshing, demonstrated on a 2D aortic valve simulation.

## Contribution

The work develops a new overlapping domain approach based on partition of unity, enabling stable, accurate FSI simulations with large deformations without re-meshing.

## Key findings

- Achieves optimal convergence rates for steady and unsteady flow problems.
- Demonstrates stability in fluid-structure interaction simulations.
- Potential for broad application in complex FSI scenarios.

## Abstract

For problems involving large deformations of thin structures, simulating fluid-structure interaction (FSI) remains challenging largely due to the need to balance computational feasibility, efficiency, and solution accuracy. Overlapping domain techniques have been introduced as a way to combine the fluid-solid mesh conformity, seen in moving-mesh methods, without the need for mesh smoothing or re-meshing, which is a core characteristic of fixed mesh approaches. In this work, we introduce a novel overlapping domain method based on a partition of unity approach. Unified function spaces are defined as a weighted sum of fields given on two overlapping meshes. The method is shown to achieve optimal convergence rates and to be stable for steady-state Stokes, Navier-Stokes, and ALE Navier-Stokes problems. Finally, we present results for FSI in the case of a 2D mock aortic valve simulation. These initial results point to the potential applicability of the method to a wide range of FSI applications, enabling boundary layer refinement and large deformations without the need for re-meshing or user-defined stabilization.