Inexact block LU preconditioners for incompressible fluids with flow rate conditions

TL;DR

This work extends the SIMPLE preconditioner to the augmented flow-rate Navier–Stokes problem enforced by Lagrange multipliers, enabling a monolithic, efficient solution of velocity, pressure, and boundary-flow data. It introduces augmented-SIMPLE preconditioners (P_S^aug and its efficient variant aug-aS) and analyzes their algebraic structure, providing practical, diagonal-approximation-based implementations. Numerical experiments on idealized and patient-specific hemodynamics show that aug-aS offers robust, near-parameter-independent convergence, with significant improvements over aug-aS-I and clear advantages over Dirichlet-penalized inflow in accuracy and stability. The results advocate using the Lagrange multipliers formulation for defect flow-rate boundary data in complex CFD applications, particularly in hemodynamics.

Abstract

When studying the dynamics of incompressible fluids in bounded domains the only available data often provide average flow rate conditions on portions of the domain's boundary. In engineering applications a common practice to complete these conditions is to prescribe a Dirichlet condition by assuming a-priori a spatial profile for the velocity field. However, this strongly influence the accuracy of the numerical solution. A more mathematically sound approach is to prescribe the flow rate conditions using Lagrange multipliers, resulting in an augmented weak formulation of the Navier-Stokes problem. In this paper we start from the SIMPLE preconditioner, introduced so far for the standard Navier-Stokes equations, and we derive two preconditioners for the monolithic solution of the augmented problem. This can be useful in complex applications where splitting the computation of the velocity/pressure and Lagrange multipliers numerical solutions can be very expensive. In particular, we investigate the numerical performance of the preconditioners in both idealized and real-life scenarios. Finally, we highlight the advantages of treating flow rate conditions with a Lagrange multipliers approach instead of prescribing a Dirichlet condition.

Figures (9)

• Figure 1: Sketch of a domain $\Omega$, with the partition of its boundary. In this example $m=4$.
• Figure 1: (a) Star-shape domain comprising six cylinders and a sphere ($h\approx0.2\ mm$). (b) Time-evolving flow rates prescribed at the inlet section $\Gamma^{in}$ (inflow) and at each outlet section $\Gamma^{out}_i$, $i=1,\ldots,4$ (outflow). Test I.
• Figure 2: Numerical simulations were run in the star-shape domain (see Figure \ref{['fig::starShape_domain']}) varying the number of Lagrange multipliers. For both preconditioners aug-aS and aug-aS-I we report as a function of the number of Lagrange multipliers: the average-in-time number of GMRES iterations (left) and the total CPU time (right). Test I.
• Figure 3: (a) Carotid arteries computational domain (average mesh size $h\approx0.5\ mm$). (b) Patient-specific flow rate measurements in CCA and ICA, prescribed as inflow at the CCA boundary and as outflow at the ICA boundary. (c)-(d) At the time-instant of peak flow rate in the CCA we report: the pressure distribution (c) and the velocity streamlines, colored by velocity magnitude, (d) obtained in the numerical simulation. Test II.
• Figure 4: (a) Coronary arteries domain comprising three left coronary arteries (LAD, M1 and LCX) and a double bypass originating from the boundary $\Gamma_{in}^1$ ($h\approx0.6\ mm$). (b) Physiological flow rate in left coronary tree digregorio2022prediction, partitioned between the three outlet boundaries. (c)-(d) At the time-instant of peak flow rate in the left coronary tree we report: the pressure distribution (c) and the velocity streamlines, colored by velocity magnitude, (d) obtained in the numerical simulation. Test III.

Theorems & Definitions (4)

• Remark 3.1
• Remark 4.1
• Remark 4.2
• Remark 4.3