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

This paper presents a high-order numerical technique combining the ghost finite element method, the Shifted Boundary Method, and an IMEX time integration scheme to accurately simulate incompressible Navier-Stokes flows on moving domains using unfitted meshes while avoiding costly remeshing.

Original authors: Hridya Dilip, Clarissa Astuto, Armando Coco, Giovanni Russo

Published 2026-03-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hridya Dilip, Clarissa Astuto, Armando Coco, Giovanni Russo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to film a movie of a swimmer doing laps in a pool. In the old days, to get a perfect shot of the swimmer's body moving through the water, you would have to rebuild the entire pool and move the water molecules every single time the swimmer took a step. It would be like trying to sculpt a new sandcastle out of wet sand for every single frame of the movie. It's messy, slow, and incredibly expensive.

This paper introduces a clever new way to film that movie without ever having to rebuild the pool.

Here is the breakdown of their "magic trick" in simple terms:

1. The Problem: The "Moving Obstacle"

The authors want to simulate water (fluid) flowing around objects that move, like a bubble wiggling, a leaf floating, or a swimmer's arm.

The Old Way: You draw a grid (like graph paper) over the water. If an object moves, the grid lines get chopped up and messy. You have to erase the grid and draw a new one every time the object moves. This is called "remeshing," and it's a computational nightmare.
The New Way (Ghost-FEM): Imagine you keep the grid perfectly square and fixed, like a rigid sheet of graph paper. The object moves through the grid, cutting through the squares. Instead of redrawing the grid, the computer just says, "Okay, this square is half-water, half-object. Let's pretend the object isn't there for a split second, calculate what the water would do if it were there, and then use that to fix the math."
- The Analogy: It's like a ghost walking through a wall. The wall (the grid) stays solid, but the ghost (the object) passes through it. The computer calculates the ghost's path by looking at the "ghost points" just outside the wall to figure out how the water should behave right at the edge.

2. The "Shifted Boundary" (Fixing the Rough Edges)

When you cut a circle out of a square grid, the edge looks like a jagged staircase, not a smooth curve. If you just use the staircase, your simulation will be a bit blurry.

The Solution: The authors use a technique called the Shifted Boundary Method.
The Analogy: Imagine you are drawing a circle on a pixelated screen. The pixels make it look blocky. Instead of trying to force the pixels to be round, you tell the computer: "Hey, the real edge is actually a tiny bit further over than this blocky line." The computer then uses a little bit of math (Taylor expansion) to "shift" the boundary to where it should be, smoothing out the jagged edges without actually changing the grid. This keeps the picture sharp and accurate.

3. The "IMEX" Engine (The Time Machine)

Simulating water involves two types of physics:

The Easy Stuff: Water spreading out (diffusion). This is predictable and easy to calculate.
The Hard Stuff: Water crashing into things or swirling (convection). This is chaotic and can break the math if you aren't careful.

The Solution: They use an IMEX (Implicit-Explicit) scheme.
The Analogy: Think of driving a car.
- Explicit (The Gas Pedal): When you accelerate, you react immediately to what's in front of you. This is fast but can make you crash if you go too fast.
- Implicit (The Brakes): When you need to stop, you plan ahead. You calculate where you'll be and brake early. This is slower but very safe.
- The IMEX Trick: The computer uses the "Gas Pedal" approach for the easy parts (fast!) and the "Brakes" approach for the chaotic parts (safe!). This allows them to run the simulation very quickly without the math exploding.

4. The "Extrapolation" (Looking Ahead)

When the object moves, it leaves some grid squares behind and enters new ones. The computer needs to know what the water is doing in those new squares before the object gets there.

The Solution: They use a technique called Aslam Extrapolation.
The Analogy: Imagine a marching band moving across a field. The musicians at the front need to know what the music sounds like before they step onto the grass. The computer "extrapolates" (guesses) the music of the water in the empty space based on the water right next to the object, filling in the blanks so the simulation never stutters.

Why Does This Matter?

The authors built this tool to study surfactants (soapy molecules) on bubbles.

Real World Application: Think of a bubble in a pond, or a drop of oil in water. These bubbles often wobble and change shape. Scientists want to know how soap molecules stick to them and how the water moves around them.
The Goal: Currently, they can simulate the bubble moving and the water reacting (One-way coupling). The next step is to let the water push back on the bubble, changing its shape (Two-way coupling). This is crucial for understanding everything from how pesticides stick to leaves to how oil spills spread in the ocean.

The Bottom Line

This paper is like inventing a new kind of camera that can film a swimmer in a pool without ever having to drain the water or rebuild the pool walls. By keeping the grid fixed and using clever math tricks to "ghost" the moving object through it, they can simulate complex, moving fluids much faster and more accurately than before. It's a huge step forward for understanding how fluids and moving objects interact in the real world.

1. Problem Statement

The paper addresses the numerical simulation of unsteady, incompressible Navier-Stokes equations on moving domains. The primary motivation is to model complex fluid-structure interactions, specifically the flow of surfactants around moving objects (e.g., oscillating bubbles or traps), which is critical for understanding biological and industrial processes like foam stability and agrochemical delivery.

Key Challenges:

Moving Boundaries: Traditional methods often require remeshing as the domain evolves, which is computationally expensive and complex.
Geometric Approximation: Using unfitted meshes (where the boundary cuts through a fixed background grid) introduces geometric errors that can degrade the order of accuracy if not handled correctly.
Stability and Conditioning: Unfitted methods often suffer from "small cut cell" problems, leading to ill-conditioned linear systems.
Nonlinearity: The convective term in the Navier-Stokes equations requires robust time-integration strategies to maintain stability and high-order accuracy.

2. Methodology

The authors propose a novel numerical technique combining several advanced components:

A. Spatial Discretization: Ghost Finite Element Method (Ghost-FEM)

Unfitted Mesh: The domain is embedded in a fixed, regular Cartesian background mesh. The active mesh consists of cells intersecting the physical domain.
Ghost Nodes: To handle cut cells (cells partially inside the domain), the method employs a "snapping-back-to-grid" approach. Ghost nodes (outside the physical domain) are defined and their values are extrapolated from internal nodes to ensure the system remains well-posed without the need for complex stabilization techniques like those in CutFEM.
Finite Element Spaces: The method uses $Q_2/Q_1$ elements (biquadratic velocity, bilinear pressure), which are inf-sup stable, providing third-order accuracy for velocity and second-order for pressure.

B. Handling Geometric Errors: Shifted Boundary Method (SBM)

Since the boundary $\Gamma$ is approximated by a polygonal boundary $\Gamma_h$ on the grid, direct application of boundary conditions leads to accuracy loss.
The authors use the Shifted Boundary Method, which applies boundary conditions on the approximate boundary $\Gamma_h$ but corrects them using Taylor expansions. This allows the method to maintain high-order accuracy (specifically targeting third-order) despite the geometric approximation.
Nitsche's Method: Dirichlet boundary conditions are imposed weakly using Nitsche's method to avoid the need for Lagrange multipliers.

C. Stabilization and Parameter Selection

Penalization Parameter ( $\lambda$ ): A critical stabilization parameter is introduced to enforce boundary conditions. The authors propose a novel approach to select $\lambda$ by solving a generalized eigenvalue problem ( $Kv = \Lambda Mv$ ). This ensures the bilinear form is coercive and the system remains stable without being overly ill-conditioned.
Saddle-Point Stabilization: To address the singularity arising from the pressure-velocity coupling, a small diagonal perturbation ( $\epsilon I$ ) is added to the system matrix.

D. Temporal Discretization: IMEX Schemes

The authors utilize Implicit-Explicit (IMEX) schemes to handle the nonlinearity of the convective term (explicit) and the stiffness of the viscous and pressure terms (implicit).
They implement two-stage (2nd order) and four-stage (3rd order) IMEX schemes based on Butcher tableaux, allowing for flexible high-order time integration.

E. Domain Motion and Extrapolation

One-Way Coupling: The motion of the domain is prescribed a priori (the fluid does not yet affect the object's motion).
Aslam's Extrapolation: To solve equations on the new active domain $\Omega^{n+1}_{act}$ , the solution from the previous time step (defined on $\Omega^n_{act}$ ) must be extended to new nodes. The authors adapt Aslam's technique (originally for finite differences) to the finite element framework. This involves solving a series of hyperbolic PDEs in a fictitious time to perform quadratic extrapolation of the solution into a narrow band outside the current domain, preserving the order of accuracy.

3. Key Contributions

High-Order Ghost-FEM for Moving Domains: The first application of high-order Ghost-FEM combined with IMEX time integration for incompressible Navier-Stokes equations on moving domains.
Robust Stabilization Strategy: A systematic method for choosing the Nitsche stabilization parameter via a generalized eigenvalue problem, ensuring stability for high-order discretizations.
Adaptation of Aslam Extrapolation: Successfully extending Aslam's quadratic extrapolation technique to the finite element context to handle domain evolution without remeshing.
Validation of Accuracy: Rigorous proof that the method achieves the expected theoretical convergence rates (2nd order in $H^1$ and 3rd order in $L^2$ for velocity) on both stationary and moving geometries.

4. Numerical Results and Validation

The paper validates the method through a series of benchmarks:

Accuracy Tests (Stationary & Moving):
- Using the Kim and Moin problem (exact solution available), the method demonstrated 2nd-order convergence in $H^1$ and 3rd-order convergence in $L^2$ for the 4-stage IMEX scheme on both fixed and rotating domains (ellipses and flowers).
Flow Past a Disk (Re = 100):
- Simulated the classic benchmark of flow past a circular cylinder.
- Results for drag/lift coefficients ( $C_D, C_L$ ) and Strouhal number ($St$) matched established reference values (e.g., from [50]) with high precision, confirming the method captures unsteady vortex shedding correctly.
Driven Cavity with Obstacle:
- Simulated flow in a cavity with a stationary flower-shaped obstacle. The results correctly captured the inhibition of corner vortices and the splitting of the central vortex at low Reynolds numbers, consistent with literature.
Rotating Flower-Shaped Object:
- Demonstrated the method's ability to handle complex rotating geometries. Observed distinct vortex formation patterns depending on the Reynolds number (diffusion-dominated vs. convection-dominated).
Oscillating Bubbles:
- Simulated bubbles undergoing vertical oscillation and ellipsoidal deformation at low Reynolds numbers ($Re=0.1$). The results correctly reproduced periodic vortex formation and shedding patterns observed in experimental studies.

5. Significance and Future Work

Significance: This work provides a robust, high-order, and mesh-free (in the sense of body-fitting) framework for simulating complex fluid-structure interactions. It eliminates the need for costly remeshing while maintaining high accuracy, making it suitable for problems involving large deformations or complex topological changes.
Current Limitation: The current implementation uses one-way coupling (the object moves independently of the fluid forces).
Future Directions: The authors aim to extend the method to two-way coupling, where the fluid dynamics influence the motion of the object (e.g., a vibrating bubble). This will enable the simulation of multi-bubble systems where the motion of one bubble induces oscillations in another through the fluid field, directly addressing the surfactant adsorption kinetics problem mentioned in the introduction.

Implementation Note: The code was implemented using Gridap (a Julia-based open-source library), leveraging its ability to handle embedded finite element methods and weak formulations efficiently.