A novel formulation for Neumann inflow boundary conditions in biomechanics

Abstract

Neumann boundary conditions prescribing the total momentum flux at inflow boundaries of biomechanical problems are proposed in this study. This approach enables the simultaneous application of velocity/flow rate and pressure curves at inflow boundaries. As the basic numerical method, a residual-based variational multiscale (or stabilized) finite element method is presented. The focus of the numerical examples in this work is on respiratory flows with complete flow reversals. However, the proposed formulation is just as well suited for cardiovascular flow problems with partial retrograde flow. Instabilities, which were reported for such problems in the literature, are resolved by the present approach without requiring the additional consideration of a Lagrange multiplier technique. The suitability of the approach is demonstrated for two respiratory flow examples, a rather simple tube and complex tracheobronchial airways (up to the fourth generation, segmented from end-expiratory CT images). For the latter example, the boundary conditions are generated from mechanical ventilation data obtained from an intensive care unit patient suffering from acute lung injury. For the tube, analytical pressure profiles can be replicated, and for the tracheobronchial airways, a correct distribution of the prescribed total momentum flux at the inflow boundary into velocity and pressure part is observed.

On the continuity of mean total normal stress in geometrical multiscale cardiovascular problems

“…In all the presented cases we impose the continuity of the vessel area at the interfaces between the 3-D geometries and the 1-D arteries through either (15) or (16). Moreover, the continuity of the mean normal stress is always imposed at the coupling nodes between the distal 1-D segments and the corresponding windkessel terminal models.…”

Artificial boundaries and formulations for the incompressible Navier–Stokes equations: applications to air and blood flows

“…Then it leads to difficulties that we will investigate in this paper. Although this basic form is often used (see, e.g., [40,58,74,75,22,62,31,55,1]), some numerical studies (see, e.g., [40,31]) have pointed out that the stability is not guaranteed when dealing with realistic physiological or physical parameters.…”

“…[75,8,6,40,38,35]). Over the last decade, this topic has been a very active field of research and the subject of numerous works (see, e.g., [29,1,56,76,28,55,27,13,66,43,54,73,31]). The work summarized in this review is linked to the numerical simulation of the air flow in the respiratory tract.…”

“…We will see in the theoretical part that the Navier-Stokes equations can be discretized using the total derivative form and the characteristics method, or using the energypreserving form. One can also choose to apply stabilization method to overcome the problem, like the method detailed in [6,20,31].…”

“…In large (or medium size) bronchi, air is commonly modeled as a homogeneous, viscous, Newtonian and incompressible fluid (see, e.g., [1,55,31]). As a mathematical model, we consider therefore the system of partial differential equations involving the Navier-Stokes equations.…”