An open-source computational framework for immersed fluid-structure interaction modeling using FEBio and MFEM

This paper presents a novel open-source immersed fluid-structure interaction framework that synergistically couples the high-performance, GPU-ready MFEM library with the biomechanics-focused FEBio solver to enable robust, scalable simulations of complex biological systems like heart valves.

The Gist

Imagine you are trying to simulate how a heart valve works. It's a bit like trying to film a dancer (the valve) moving inside a swimming pool (the blood).

In the past, computer scientists tried to solve this by building a digital "net" (a mesh) that perfectly wrapped around the dancer. As the dancer moved, the net had to stretch, twist, and reshape itself in real-time. If the dancer spun too fast or touched another dancer (like heart valve leaflets touching), the net would get tangled, stretch too thin, or rip apart. The computer would then have to stop, untangle the mess, build a new net, and start over. This is slow, expensive, and often fails when the movement gets too wild.

This paper introduces a new, smarter way to do it.

Here is the breakdown of their solution using simple analogies:

1. The "Ghost in the Pool" Approach (Immersed Method)

Instead of wrapping a net around the dancer, imagine the dancer is swimming in a giant, rigid grid of water that never changes shape. The dancer is "immersed" in this grid.

• The Old Way: The water grid tries to hug the dancer's skin.
• The New Way: The water grid stays fixed and square. The dancer swims through it. When the dancer moves, the computer just calculates how the dancer pushes the water through the grid squares. If the dancer touches another dancer, the computer handles the collision without ever needing to rebuild the grid.

2. The "Dream Team" Partnership

The authors built a software framework that connects two very different, highly specialized tools to work together like a dream team:

• Team Member A: MFEM (The High-Speed Engine)

  • What it does: It's a super-fast, parallel-processing engine built by scientists at Lawrence Livermore National Lab. Think of it as a Formula 1 race car. It is designed to run on massive supercomputers and even graphics cards (GPUs) to calculate how the water (fluid) moves incredibly fast.
  • Why they need it: Blood flow is complex and needs serious computing power.

• Team Member B: FEBio (The Biomechanics Expert)

  • What it does: It's a specialized tool built by experts at the University of Utah for understanding how soft tissues (like heart muscle or valve leaflets) stretch, squish, and react to stress. Think of it as a master tailor who knows exactly how fabric behaves when pulled or twisted.
  • Why they need it: Heart valves aren't rigid metal; they are soft, floppy tissue that stretches and folds. You need a tailor, not just a race car, to model that.

The Innovation: They created a "plugin" (a translator) that lets the Race Car (MFEM) talk to the Tailor (FEBio) instantly. The Race Car handles the water, the Tailor handles the tissue, and they agree on the physics at the boundary where they meet.

3. Why This Matters for Kids with Heart Defects

The paper specifically mentions children with congenital heart defects.

• The Problem: Children grow. If you put a metal heart valve in a child, it doesn't grow with them. They have to have surgery again and again to get bigger valves, which is dangerous and traumatic.
• The Goal: Doctors want to fix the child's own valve so it can grow with them. But to do that, they need to know exactly how the valve will stretch and stress over years.
• The Solution: This new software allows doctors to simulate a child's specific heart valve, seeing not just how blood flows through it, but exactly how the tissue stretches and where it might tear or calcify. This helps them plan the perfect repair before they ever cut into the patient.

4. The "Open Source" Promise

Usually, this kind of super-complex software is locked behind expensive paywalls or kept secret by big companies.

• The Paper's Gift: The authors made this entire framework open-source. This means it's free for anyone (researchers, doctors, students) to download, use, and improve. They are handing the keys to the kingdom to the whole scientific community so everyone can build better heart models faster.

Summary Analogy

Imagine you are trying to predict how a piece of wet clay (the heart valve) moves in a river (the blood).

• Old Software: You try to mold a plastic cage around the clay. Every time the clay squishes, you have to melt the plastic cage, reshape it, and try again. It's frustrating and slow.
• This New Software: You drop the clay into a river of digital water that is already mapped out. You have a Super-Computer calculating the water speed and a Clay Expert calculating how the clay squishes. They talk to each other instantly. The clay moves freely through the water without breaking the map. And best of all, you can give a copy of this setup to anyone in the world for free.

This framework bridges the gap between high-speed computing and detailed biological modeling, opening the door to better treatments for heart disease.

Technical Summary

1. Problem Statement

Fluid-structure interaction (FSI) simulations in biomechanics, particularly for cardiovascular applications like heart valve dynamics, present significant computational challenges.

• Limitations of Traditional Methods: Standard Arbitrary Lagrangian-Eulerian (ALE) methods, which use moving meshes that conform to the solid boundary, struggle with large structural deformations, topological changes (e.g., leaflet contact/coaptation), and complex geometries. These issues often necessitate frequent, expensive remeshing and can lead to mesh distortion or failure.
• Limitations of Existing Immersed Methods: While immersed boundary methods (IB) avoid mesh distortion by embedding solids in a fixed background fluid, existing open-source solvers often lack sophisticated solid mechanics capabilities. They frequently rely on simplified structural models (e.g., rigid or reduced-order models) that fail to capture critical tissue-level stresses, strains, and contact mechanics required for studying disease progression, calcification, and device-tissue interactions.
• Computational Cost: High-fidelity FSI simulations are computationally expensive, and there is a lack of open-source tools that combine advanced biomechanical modeling with high-performance computing (HPC) infrastructure (e.g., GPU acceleration and distributed memory parallelization).

2. Methodology

The authors present a novel, open-source framework that couples two mature finite element libraries: MFEM (a scalable, GPU-ready library from Lawrence Livermore National Laboratory) and FEBio (a nonlinear finite element solver for biomechanics from the University of Utah and Columbia University).

Mathematical Formulation

The framework utilizes an Immersed FSI approach based on the Fictitious Domain (FD) method with Distributed Lagrange Multipliers (DLM).

• Domain: The solid is immersed in a fixed background fluid domain. Artificial fluid exists within the solid region, and its work is subtracted from the solid equations.
• Coupling:
  1. Traction Continuity: Naturally satisfied by the variational formulation.
  2. Velocity Matching: Enforced via a distributed Lagrange multiplier ($\lambda$) over the overlapping region, eliminating the need for a moving interface mesh.
• Fluid Solver: Solves the incompressible Navier-Stokes equations using a Variational Multiscale (VMS) stabilization. This ensures accuracy on under-resolved grids (common in immersed methods) and handles advection-dominated flows. A modified stabilization coefficient ($s$) is introduced to handle large pressure jumps across thin immersed structures (like heart valves).
• Solid Solver: Models hyperelastic and viscoelastic solids using FEBio's constitutive models. The formulation accounts for the density difference between solid and fluid and subtracts the artificial fluid's viscous stress.
• Constraint Enforcement: The Lagrange multiplier space is chosen such that basis functions are delta functions at solid nodes. This allows the Lagrange multiplier to be eliminated analytically, reducing the system size to that of a fluid-only problem.
• Time Integration: A fully implicit, monolithic scheme using the generalized-$\alpha$ method is employed for robust coupling of strongly coupled interactions.

Software Architecture (The MFEMiFSI Plugin)

The implementation is a plugin for FEBio that leverages MFEM as a backend for the fluid solver and linear algebra.

• Monolithic Coupling: Unlike partitioned approaches, the fluid and solid equations are solved simultaneously to ensure stability (avoiding added-mass instabilities).
• Modularity: The plugin uses a modular design (MFEMImmersedElasticDomain) allowing users to extend the framework with different solid formulations (e.g., mean dilation method for incompressibility) without modifying core code.
• Parallelism: Utilizes MFEM's distributed-memory parallelization (MPI) and supports future GPU acceleration.
• FSI Constraint Matrix: A specialized parallel algorithm (using FindPtsGSLIB and hypre) is implemented to compute the constraint matrix ($C_\lambda$) that maps solid nodes to fluid elements ("force spreading").

3. Key Contributions

1. Novel Coupling: The first open-source framework to strategically couple MFEM's high-performance parallel computing capabilities with FEBio's sophisticated biomechanical modeling (hyperelasticity, contact, complex constitutive laws).
2. Robust Immersed Method: Implementation of a fictitious domain/DLM method with VMS stabilization and specific modifications for handling large pressure jumps across thin structures.
3. Monolithic Solver: A fully implicit, monolithic solver that eliminates Lagrange multipliers to reduce system size while maintaining stability for strongly coupled cardiovascular problems.
4. Extensibility: A plugin architecture that allows the community to easily add new physics (e.g., electromechanics, transport) and element technologies.
5. Open-Source Availability: The code is publicly available via the FEBio Plugin Repository and GitHub, filling a critical gap in open-source cardiovascular simulation tools.

4. Results and Validation

The framework was validated through a series of numerical test problems:

• Immersed Annular Solid: A benchmark problem with an analytical solution demonstrated optimal convergence rates (order 2.0 for velocity in $L_2$, 1.5 for pressure in $L_2$) on uniform Cartesian grids.
• Idealized Heart Valve (Open/Closed):
  • Open Valve: Showed convergence of leaflet tip displacement to a reference ALE-FSI solution.
  • Closed Valve: Demonstrated the framework's ability to capture pressure discontinuities across a beam. Crucially, it validated the necessity of the modified stabilization parameter ($s$) to accurately represent hydrostatic states (zero velocity) during valve closure.
• Oscillating Flexible Leaflet: Simulated large, complex deformations driven by fluid forces in a pulsatile flow, a scenario difficult for ALE methods due to remeshing requirements. Results agreed well with reference solutions.
• Free-Falling Sphere: Validated 3D accuracy by simulating a sphere reaching terminal velocity. The numerical solution converged to the analytical solution with a 3.02% error on the finest mesh.
• 3D Semilunar Heart Valve: A full simulation of a tri-leaflet valve in an idealized aortic root. The framework successfully captured:
  • Complex vortex dynamics and jet formation/breakdown.
  • Large deformations and contact (coaptation) of leaflets.
  • Nonsymmetric stress distributions in the tissue, highlighting the importance of full 3D FSI for tissue mechanics analysis.

5. Significance

This work addresses a critical need in computational biomechanics by providing a tool that bridges the gap between high-fidelity fluid dynamics and advanced tissue mechanics.

• Clinical Relevance: It enables patient-specific simulations for optimizing native valve repairs and designing prosthetic devices, where understanding tissue-level stress (not just hemodynamics) is vital for predicting durability and failure.
• Scalability: By leveraging MFEM, the framework is ready for exascale computing and GPU acceleration, making high-fidelity simulations of complex cardiovascular problems computationally feasible.
• Community Impact: As an open-source, modular platform, it lowers the barrier to entry for researchers to perform advanced FSI studies, fostering innovation in cardiovascular device design and surgical planning.

Future Work: The authors plan to incorporate contact models to prevent body intersections, implement adaptive mesh refinement (AMR) for interface resolution, and fully integrate GPU acceleration and non-Newtonian fluid models.