High order ghost-FEM for incompressible Navier-Stokes equations on moving domains

TL;DR

This paper introduces a high-order ghost finite element method for simulating incompressible Navier-Stokes flows on moving domains, combining unfitted mesh techniques, high-order boundary condition approximation, and stability-enhancing strategies.

Contribution

The paper presents a novel high-order ghost-FEM approach with an IMEX scheme and Shifted Boundary Method for accurate, stable simulations of fluid flows on moving, unfitted meshes.

Findings

Validated accuracy through numerical experiments

Achieved high-order boundary condition approximation

Demonstrated stability with eigenvalue-based parameter tuning

Abstract

We develop a new numerical technique for approximating solutions of the Navier-Stokes equations on moving domains. The method aims at simulating an incompressible fluid past an object whose motion is assigned a priori using a level-set function. The proposed approach relies on a space discretization based on the ghost finite element method (ghost-FEM), which allows computations on unfitted meshes and avoids costly remeshing as the domain evolves in time. Time integration is performed using an IMplicit-EXplicit (IMEX) scheme to address the nonlinearity of the convective term, ensuring high-order accuracy for incompressible flows. The error introduced by the geometrical approximation is handled using the Shifted Boundary Method, which allows higher order approximations of boundary conditions on unfitted meshes. Dirichlet boundary conditions are imposed weakly by means of Nitsche's method. The associated stabilization parameter is chosen by solving a generalized eigenvalue problem, ensuring stability and accuracy of the numerical scheme. We present a series of numerical experiments designed to validate the accuracy of the proposed method, as well as comparisons with established benchmark problems involving moving boundaries.