The nested block preconditioning technique for the incompressible Navier-Stokes equations with emphasis on hemodynamic simulations

TL;DR

This paper introduces a nested block preconditioning method for solving the incompressible Navier-Stokes equations, improving robustness and efficiency in complex hemodynamic simulations involving multiscale and multiphysics coupling.

Contribution

A novel three-level nested block preconditioner is developed to enhance the solution of coupled Navier-Stokes equations with boundary conditions and reduced models.

Findings

Demonstrated robustness and efficiency in benchmark problems.

Achieved good parallel scalability in simulations.

Validated effectiveness in patient-specific hemodynamic cases.

Abstract

We develop a novel iterative solution method for the incompressible Navier-Stokes equations with boundary conditions coupled with reduced models. The iterative algorithm is designed based on the variational multiscale formulation and the generalized-$\alpha$ scheme. The spatiotemporal discretization leads to a block structure of the resulting consistent tangent matrix in the Newton-Raphson procedure. As a generalization of the conventional block preconditioners, a three-level nested block preconditioner is introduced to attain a better representation of the Schur complement, which plays a key role in the overall algorithm robustness and efficiency. This approach provides a flexible, algorithmic way to handle the Schur complement for problems involving multiscale and multiphysics coupling. The solution method is implemented and benchmarked against experimental data from the nozzle challenge problem issued by the US Food and Drug Administration. The robustness, efficiency, and parallel scalability of the proposed technique are then examined in several settings, including moderately high Reynolds number flows and physiological flows with strong resistance effect due to coupled downstream vasculature models. Two patient-specific hemodynamic simulations, covering systemic and pulmonary flows, are performed to further corroborate the efficacy of the proposed methodology.