Fluid–Structure Interaction in Hemodynamics: Modeling, Analysis, and Numerical Simulation - Fluid-Structure Interaction and Biomedical Applications - page 175

where v D u j .t/ means v . z ;t/ D u .R C . z ;t/; z ;t/ on . This is supplemented
with the following boundary conditions:

@ u z
@r . z ;0;t/D

u r . z ;0;t/D 0 on .0;L/; u .0;R;t/D u .L;R;t/D 0;

n in Dp in .t/n in on in ; n out Dp out .t/n out on out ;

and initial conditions: u .t n / D u n ; v .t n / D v n C 1=3 : Then set

u n C 2=3 D u .t n C 1 /; p n C 2=3 D p.t n C 1 /; v n C 2=3 D v .t n C 1 /:

Problem A2(b): The Advection Problem

Solve the fluid and ALE advection sub-problem defined on the fixed domain .t n /,
with the domain velocity w n C 1 just calculated in Problem A1. The displacement of
the structure, the velocity of the thick structure, the velocity of the thin structure,
and the fluid pressure do not change in this step, so that

n C 1 D n C 2=3 ; d n C 1 D d n C 2=3 ;V n C 1 D V n C 2=3 ; v n C 1 D v n C 2=3 ;p n C 1 D p n C 2=3 :

The advection problem reads: Find u such that for t 2 .t n ;t n C 1 /

LJ LJ LJ LJ F C . u n C 2=3

@ u
@t

w n C 1 / r u D 0; in F .t n / .t n ;t n C 1 /;

u D v n C 2=3 ; on .t n ;t n C 1 /;

with the inlet/outlet conditions:

u D u n C 2=3 on n C 2=3

2

jx 2 @ F .t n /;. u n C 2=3 w n C 1 / n <0g;

Dfx 2 R

and initial conditions u .t n / D u n C 2=3 : Then set

u n C 1 D u .t n C 1 /:

Set n D n C 1 and return to Problem A1.

2.7.6 Discretized Scheme in Weak Form

To discretize the problem in time, sub-divide the time interval .0;T/ into N sub-
intervals of width t,andlett n D nt,wheren N. The Backward Euler scheme