# A theoretical and experimental investigation of a family of immersed finite element methods

## TL;DR

This paper provides a comprehensive theoretical and experimental analysis of immersed finite element methods, comparing explicit, implicit, and one-field fictitious domain approaches, highlighting their relationships, robustness, and efficiency in simulating fluid-structure interactions.

## Contribution

It demonstrates that the one-field FDM is more robust and efficient than explicit IFEM, and closely approximates implicit IFEM results without iterative steps.

## Key findings

- One-field FDM is more robust than explicit IFEM.
- One-field FDM can simulate wider parameter ranges with larger time steps.
- Results of one-field FDM closely match implicit IFEM outcomes.

## Abstract

In this article we consider the widely used immersed finite element method (IFEM), in both explicit and implicit form, and its relationship to our more recent one-field fictitious domain method (FDM). We review and extend the formulation of these methods, based upon an operator splitting scheme, in order to demonstrate that both the explicit IFEM and the one-field FDM can be regarded as particular linearizations of the fully implicit IFEM. However, the one-field FDM can be shown to be more robust than the explicit IFEM and can simulate a wider range of solid parameters with a relatively large time step. In addition, it can produce results almost identical to the implicit IFEM but without iteration inside each time step. We study the effect on these methods of variations in viscosity and density of fluid and solid materials. The advantages of the one-field FDM within the IFEM framework are illustrated through a selection of parameter sets for two benchmark cases.