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Duct boundary conditions for incompressible fluid flows: finite element discretizations and parameter estimation in coronary blood flow

Jeremías Garay, David Nolte, Cristóbal Bertoglio

TL;DR

<3-5 sentence high-level summary> The paper introduces the Duct Boundary Condition (DuBC), a MAPDD-based boundary condition that enables geometrically reduced outlets to couple implicitly with full 3D Navier–Stokes simulations using a local, single-parameter boundary term that accounts for both viscous and inertial effects. It develops a fractional-step discretization (CT–DuBC) to accelerate computation while preserving stability, and demonstrates robustness against time-step size in coronary flow simulations. The authors further show that the DuBC parameters (virtual duct lengths) can be estimated from velocity data using a reduced-order unscented Kalman filter, enabling patient-specific calibration from velocity measurements. Overall, the framework offers a stable, efficient route to 3D–0D multiscale hemodynamics in highly ramified networks and supports inverse problems with noisy data.

Abstract

3D-0D coupled flow models are widely used across many application fields but remain challenging to solve. Implicit coupling introduces non-local terms, whereas explicit coupling results in only conditionally stable schemes. Furthermore, incorporating inertial effects alongside viscous resistance enlarges the parameter space, making calibration more difficult. In this work, we propose a new type of boundary condition based on the method of asymptotic partial decomposition of a domain (MAPDD), which we denote as the Duct Boundary Condition (DuBC). This approach enables the incorporation of geometrically reduced domains as a boundary term with only local coupling in the implicit case. Moreover, the DuBC accounts for both viscous and inertial effects simultaneously using a single physical parameter. Additionally, we derive a fractional-step time-marching scheme including the DuBC. We demonstrate the features of the DuBC in coronary artery blood flow simulations, including sequential parameter estimation from noisy velocity data.

Duct boundary conditions for incompressible fluid flows: finite element discretizations and parameter estimation in coronary blood flow

TL;DR

<3-5 sentence high-level summary> The paper introduces the Duct Boundary Condition (DuBC), a MAPDD-based boundary condition that enables geometrically reduced outlets to couple implicitly with full 3D Navier–Stokes simulations using a local, single-parameter boundary term that accounts for both viscous and inertial effects. It develops a fractional-step discretization (CT–DuBC) to accelerate computation while preserving stability, and demonstrates robustness against time-step size in coronary flow simulations. The authors further show that the DuBC parameters (virtual duct lengths) can be estimated from velocity data using a reduced-order unscented Kalman filter, enabling patient-specific calibration from velocity measurements. Overall, the framework offers a stable, efficient route to 3D–0D multiscale hemodynamics in highly ramified networks and supports inverse problems with noisy data.

Abstract

3D-0D coupled flow models are widely used across many application fields but remain challenging to solve. Implicit coupling introduces non-local terms, whereas explicit coupling results in only conditionally stable schemes. Furthermore, incorporating inertial effects alongside viscous resistance enlarges the parameter space, making calibration more difficult. In this work, we propose a new type of boundary condition based on the method of asymptotic partial decomposition of a domain (MAPDD), which we denote as the Duct Boundary Condition (DuBC). This approach enables the incorporation of geometrically reduced domains as a boundary term with only local coupling in the implicit case. Moreover, the DuBC accounts for both viscous and inertial effects simultaneously using a single physical parameter. Additionally, we derive a fractional-step time-marching scheme including the DuBC. We demonstrate the features of the DuBC in coronary artery blood flow simulations, including sequential parameter estimation from noisy velocity data.

Paper Structure

This paper contains 21 sections, 19 equations, 10 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Geometric multiscale fluid flow modeling
  3. Full domain model
  4. The duct boundary condition (DuBC)
  5. Numerical experiments
  6. Setup
  7. Geometry and physical constants
  8. Temporal discretization
  9. Spatial discretization
  10. Dirichlet boundary conditions via penalization
  11. Results
  12. Fractional step discretization including DuBC
  13. Formulation
  14. Numerical experiments
  15. Spatial discretization and details
  16. ...and 6 more sections

Figures (10)

  • Figure 1: Sample domains with $K=2$. (a) Full geometry; (b) Reduced geometry.
  • Figure 2: (a) Left coronary artery's full domain $\Omega_{full}$ used for this study. (b) Reduced geometry $\Omega$ obtained after cutting the branches perpendicularly to their centerline
  • Figure 3: Velocity fields when using different models with increasing simulation time step. All colormaps and arrow sizes are the same
  • Figure 4: Comparison of the relative $L_2$ norms of the velocity obtained with the DuBC model as the simulation time step increases.
  • Figure 5: Velocity fields in the LAD portion of the left coronary artery when using the DuBC with monolithic and CT approaches for three different simulation time steps.
  • ...and 5 more figures

Theorems & Definitions (1)

  • Remark 1