Influence of Blood Rheology and Outflow Boundary Conditions in Numerical Simulations of Cerebral Aneurysms - Mathematical Methods and Models in Biomedicine - page 154

The solution for the boundary nodes of the domain may then be achieved through
resolving a 2 × 2 nonlinear system given by Eqs. ( 18 )and( 19 ) (for more details see

[ 17 ]).
Typically, the inflow condition is a flux or a total pressure, while the outflow
condition is given by W 2 =

b ,
corresponding to a completely absorbing boundary condition at the outflow point.

0, such that there is no incoming characteristic at z =

3D-1D Coupling

To couple the artificial boundary, denoted Γ art from here on, of the 3D fluid Eq. ( 1 )

with the 1D interface point z a of the hyperbolic model ( 11 ), the continuity of the

flow rate and the mean pressure are imposed, for all t

=

> 0 (see for instance [ 7 ])

Q 1 D

···

γ =

( , ) ,

u n d a t (22)

Γ art

1
| Γ

P 1 D

p d γ = ( a , t ) . (23)

| art Γ art

Here u and p denote the 3D velocity vector and pressure, respectively, and Q 1 D
and P 1 D are the 1D flow rate and mean pressure, respectively. The solution of the
coupled problem is approximated in an iterative way, by resorting to a splitting
strategy. This means that each model is solved separately and yields the resultant
information to the other. Thus, at each time step the 3D model returns pointwise
data, which is integrated to obtain the averaged quantities to be provided to the 1D
model as a boundary condition at z a . On the other hand, the 1D model provides

the boundary conditions at the coupling sections of the 3D in terms of average data.
The average data is defective for the 3D problem, since it requires pointwise data
at the coupling interface. Thus, appropriate techniques must be used in order to
prescribe the 1D integrated data onto the 3D model as boundary condition. Precisely,
in this work the coupling is performed by passing the flow rate from the 3D to the
1D model, imposing Eq. ( 22 ) at the coupling point of the 1D model, z

=

a ,and
by imposing the mean pressure, computed by the 1D model, to the 3D problem,
by means of the condition ( 23 ) at the 3D artificial coupling boundary,

=

Γ art .To

prescribe the defective mean pressure on the 3D coupling section, art , the approach

introduced in [ 12 ] is followed, so that the mean pressure is imposed through a
Neumann boundary condition

Γ

) · · ·

σ ( u , p n = P 1 D n , on Γ , ∀ t > 0 . (24) art

The 3D-1D iterative coupling algorithm is carried out explicitly in this work.
At each time step t n , the 3D model provides the flow rate computed at the previous
time step to the 1D model and receives the mean pressure computed from the 1D
model. This is followed by advancing to the next time step (see Fig. 5 ).