Reduced Unified Continuum for Vascular FSI

• Reduced Unified Continuum (RUC) is a computational framework for vascular FSI that couples blood flow and thin-walled vessel dynamics on a fixed Eulerian mesh.
• It leverages variational multiscale discretization and membrane reduction to achieve efficient, second-order accuracy in simulating hemodynamics.
• The framework employs a robust block-preconditioned Newton–Krylov solver, validated against analytical, in vitro, and patient-specific models for clinical relevance.

The Reduced Unified Continuum (RUC) formulation is a computational framework for monolithic vascular fluid–structure interaction (FSI) that enforces strong coupling between blood flow and thin-walled vessel dynamics directly on a fixed Eulerian mesh. Developed by direct reduction of the unified continuum arbitrary Lagrangian–Eulerian (ALE) FSI formalism, the RUC achieves efficient, second-order accurate simulations of hemodynamics and wall motion by combining variational multiscale discretization, a membrane reduction for the vessel wall, and a block-preconditioned Newton–Krylov solver. The RUC has been rigorously validated against classical analytical theory, in vitro 4D-flow MRI, and large-scale patient-specific vascular models (Lan et al., 2021, Lan et al., 2022, Lan et al., 2022).

1. Thermodynamic and Mathematical Foundations

The RUC originates from the unified continuum ALE-VMS formulation for FSI. The theoretical basis is established through a Gibbs free energy functional, which yields the coupled incompressible Navier–Stokes equations for the fluid and a hyperelastic (or linear elastic) elastodynamics model for the solid (Lan et al., 2021). The ALE kinematics employs a mapping $\bm x = \bm\chi(\bm X, t)$ from reference to current configuration, with fluid and mesh velocities $\bm v^f$, $\bm w$. The full variational form is discretized by residual-based variational multiscale (RBVMS) stabilization in both subdomains, yielding consistent, parameter-free stabilization for all conservation laws.

The transition to RUC is achieved through three critical modeling simplifications targeting vascular applications:

• Infinitesimal strain: Wall kinematics are linearized, rendering reference and current configurations identical and freezing mesh motion ($\bm w = 0$).
• Thin-walled reduction: The vessel wall $\Omega^s$ is parameterized as a shell of prescribed thickness $h^s$ over the fluid–solid interface $\Gamma_I$, enabling volume integrals to collapse to surface integrals.
• Membrane approximation: Wall displacement depends solely on surface coordinates $(\xi,\eta)$, with transverse stress vanishing, producing a 2D membrane system for $\bm u^w(\bm x, t)$ on $\Gamma_I$.

These reductions eliminate bulk solid degrees of freedom and directly embed wall inertia and stiffness into the fluid domain's equations via surface delta functions, enabling direct coupling to the fluid DOFs on the boundary (Lan et al., 2021, Lan et al., 2022).

2. Governing Equations and Discretization

The Eulerian RUC formulation expresses the system unknowns as $\bm v^f$0 defined on $\bm v^f$1. The strong form is:

$\bm v^f$2

The weak, semi-discrete form requires finding $\bm v^f$3 in suitable finite element spaces such that, for all admissible test functions, the only modification to the standard fluid RBVMS discretization is the incorporation of membrane inertia and stress as boundary integrals over $\bm v^f$4.

Spatial discretization utilizes high-order (typically quadratic/P2) tetrahedral elements for $\bm v^f$5 and $\bm v^f$6 in $\bm v^f$7, and quadratic surface triangles for $\bm v^f$8 on $\bm v^f$9. This yields $\bm w$0 $\bm w$1 spatial accuracy for velocity and wall-shear stress, in contrast to $\bm w$2 for linear elements, and is particularly advantageous when clinical metrics such as pressure or shear are of direct interest (Lan et al., 2021).

3. Temporal Integration and Solver Algorithms

Temporal discretization employs the generalized-$\bm w$3 scheme, providing second-order accuracy uniformly in all primary variables and permitting user control of algorithmic damping via the spectral radius $\bm w$4. Critically, both velocity and pressure are collocated at the intermediate time $\bm w$5 enforcING second-order accuracy for $\bm w$6—an improvement over conventional approaches in which $\bm w$7 is typically updated only at $\bm w$8, resulting in first-order temporal error for pressure (Lan et al., 2021, Lan et al., 2022).

At each time step a fully-implicit Newton–Raphson solve is performed. The tangent matrix exhibits block $\bm w$9 structure in the variables $\bm w = 0$0, but can be algebraically reduced to a $\bm w = 0$1 system by exploiting the linear kinematic constraint, yielding a Schur complement system in the fluid variables (Lan et al., 2022).

A predictor–multi-corrector algorithm is employed:

• Predictor step initializes $\bm w = 0$2,
• Newton-like corrector iterations update acceleration and pressure from the linearized tangent, followed by algebraic updates of velocity and displacement.

4. Block Preconditioning and Computational Performance

The RUC linear system is preconditioned by a three-level nested block strategy to effectively represent the Schur complement:

• Level 1: Algebraic multigrid (AMG) or ILU for the momentum (fluid + membrane inertia) block,
• Level 2: Schur complement action computed as required for pressure solves with momentum block applications,
• Level 3: AMG on an approximate Poisson-type operator for the pressure Schur complement.

This hierarchical preconditioning sustains robust, mesh-independent Krylov convergence, particularly in simulations involving stiff membrane walls or high wall inhomogeneity (Lan et al., 2021, Lan et al., 2022).

Parallel implementations using MPI demonstrate near-optimal strong scaling up to at least 512 cores with $\bm w = 0$3 efficiency. Typical wall-clock time for three cardiac cycles in a $\bm w = 0$4 tetrahedra mesh is approximately 9 hours on 128 cores, with computational expense only $\bm w = 0$5 that of a rigid-wall CFD simulation, compared to $\bm w = 0$6 for standard FSI formulations (Lan et al., 2022).

5. Verification and Experimental Validation

Verification of RUC against analytic Womersley solutions in straight pipes, with both rigid and deformable walls, confirms second- and third-order convergence in space and time for velocity and wall displacement. Errors in representative benchmarks are $\bm w = 0$7 for $\bm w = 0$8 velocity and wall displacement, and RUC recovers expected axisymmetry and pressure fidelity superior to the coupled momentum method (Lan et al., 2021, Lan et al., 2022).

Experimental validation against in vitro 4D-flow MRI in patient-specific compliant aortic phantoms demonstrates quantitative agreement in mean and peak systolic pressure, lumen area change, and pulse wave velocity within $\bm w = 0$9 of measured data. Simulated early-systolic arch velocities differ by $\Omega^s$0 in $\Omega^s$1 norm versus MRI. Qualitative agreement in late-systolic flow structures—including reverse flow and vortex formation—is robust. Mass conservation is maintained to within $\Omega^s$2 mL/s per cycle (Lan et al., 2022).

6. Clinical Modeling and Applications

RUC supports efficient and accurate modeling strategies essential for clinical vascular FSI:

• Wall thickness prescription: Centerline-based methodologies provide uniform thickness-to-radius ratios, improving anatomical fidelity relative to Laplacian-based approaches.
• Tissue prestress: Fixed-point prestressing aligns the initial stress field to image-derived diastolic conditions, producing correct static equilibrium and physiologically realistic baseline states.
• Physiological boundary conditions: In-plane wall cap rollers, viscoelastic external tissue supports (modeled via Robin integrals), and RCR-type outflow models are integrated natively.

Applications include assessment of pulse wave velocity, local vessel compliance, and wall motion in aneurysms, dissections, and congenital heart disease, as well as inverse estimation of wall properties from dynamic MRI data. In the study of patient-specific abdominal aortic aneurysms, RUC captures the dependence of pressure wave amplitude, wall displacement, and area change on material properties, and shows that time-averaged wall shear stress (TAWSS) and oscillatory shear index (OSI) are only weakly sensitive to wall compliance for typical clinical geometries and flows (Lan et al., 2022).

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Summary Table: Core Algorithmic Elements in RUC

Component	Methodology	Key Benefit
Thermodynamic basis	Gibbs free energy functional	Unified FSI derivation; consistency
Wall model simplification	Infinitesimal strain, thin-wall, membrane	Surface reduction, efficient coupling
Spatial discretization	Quadratic (P2) tetrahedra; RBVMS	Higher-order spatial accuracy
Temporal discretization	Generalized-$\Omega^s$3, collocation at $\Omega^s$4	Second-order uniform; pressure accuracy
Solver strategy	Predictor–multi-corrector, Newton–Krylov solver	Robust convergence; block structure
Preconditioning	Three-level nested block (AMG, Schur, Poisson)	Mesh, core-count independence
Clinical features	Prestress, in-plane BCs, RCR outlets	Realistic patient-specific FSI

Each component is critical for the stability, accuracy, and clinical applicability of the RUC framework (Lan et al., 2021, Lan et al., 2022, Lan et al., 2022).