# An Immersed Interface Method for Discrete Surfaces

## TL;DR

This paper presents an immersed interface method that achieves higher-order accuracy for fluid-structure interaction problems using only a C0 interface representation, effectively resolving stress discontinuities without requiring smooth interface geometry.

## Contribution

It introduces a novel immersed interface method that attains second-order accuracy with simple interface representations, expanding applicability to complex geometries without needing analytic interface data.

## Key findings

- Achieves 2nd-order global convergence in velocity and nearly 2nd-order in Eulerian velocity.
- Attains 1st-order global convergence in pressure with 2nd-order local convergence.
- Successfully simulates flow in complex anatomical geometries.

## Abstract

Fluid-structure systems occur in a range of scientific and engineering applications. The immersed boundary(IB) method is a widely recognized and effective modeling paradigm for simulating fluid-structure interaction(FSI) in such systems, but a difficulty of the IB formulation is that the pressure and viscous stress are generally discontinuous at the interface. The conventional IB method regularizes these discontinuities, which typically yields low-order accuracy at these interfaces. The immersed interface method(IIM) is an IB-like approach to FSI that sharply imposes stress jump conditions, enabling higher-order accuracy, but prior applications of the IIM have been largely restricted to methods that rely on smooth representations of the interface geometry. This paper introduces an IIM that uses only a C0 representation of the interface,such as those provided by standard nodal Lagrangian FE methods. Verification examples for models with prescribed motion demonstrate that the method sharply resolves stress discontinuities along the IB while avoiding the need for analytic information of the interface geometry. We demonstrate that only the lowest-order jump conditions for the pressure and velocity gradient are required to realize global 2nd-order accuracy. Specifically,we show 2nd-order global convergence rate along with nearly 2nd-order local convergence in the Eulerian velocity, and between 1st-and 2nd-order global convergence rates along with 1st-order local convergence for the Eulerian pressure. We also show 2nd-order local convergence in the interfacial displacement and velocity along with 1st-order local convergence in the fluid traction. As a demonstration of the method's ability to tackle complex geometries,this approach is also used to simulate flow in an anatomical model of the inferior vena cava.