Numerical Simulation of Fluid–Structure Interaction Problems with Applications to Flow in Vocal Folds - Fluid-Structure Interaction and Biomedical Applications - page 353

Strong Coupling

w n

i , v h , p h , F n 1 , F n

0. Initialization. Start from the approximations w i ,

2 known at

time t n (as well as on previous time levels).
1. Extrapolation step. Extrapolate aerodynamic forces

F n C 1
i

WD F i .t n / F i .t n C 1 /; i D 1;2:

2. Structural step. Solve ( 5.62 )for w n C 1

1 and w n C 2 .
3. Mesh step. Find the approximation of ALE mapping

A ht n C 1 W h0 7! ht n C 1 at
t n C 1 given either by ( 5.99 )or( 5.106 ), and approximate the domain velocity z n C 1

h

by ( 5.10 ).
4. Fluid step. Solve ( 5.25 ) for the approximations v n C 1
h

, p n C 1
h

on ht n C 1 .

5. Aerodynamic forc es. Usi ng the approximations v n C 1
h

, p n C 1
h

, compute the

aerodynamic forces F n C 1

1 , F n C 1

2 .
6. Structural step. Solve ( 5.62 )for w n C 1
1

w n C 1

and

2 . Compute the difference " WD

P i D 1 j w n C 1

w n C 1

i j.
7. If the difference " is sufficiently small, go to 8 .Else,set w n C 1
1

i

WD w n C 1

1 , w n C 1

WD

2

w n C 1

2 andgoto 3 .
8. n WD n C 1,goto 1 .

In our computations, the strongly coupled scheme was used in order to guarantee
the stability of the computational process. Usually, if the time step is small enough,
only few iterations need to be computed.

5.6.2 Algorithms for Interaction of Fluid and Elastic Structure

In the case of the interaction of fluid and elastic structure we can formulate the
following continuous FSI problem: Our aim is to de termine the domain t , t 2
.0;T , a nd functions v D v.x;t/;p D p.x;t/, x 2 t , t 2 Œ0;T and u D u .X;t/,
X 2 s , t 2 Œ0;T satisfying Eqs. ( 5.9 ), ( 5.66 ), the initial conditions ( 5.3 ), ( 5.67 ),
the boundary conditions ( 5.4 ), ( 5.69 ), and the transmission conditions ( 5.94 ), ( 5.96 ).
Here we consider the following discrete coupled problem at time t n C 1 .
Problem 5.6.2 (Flow Interacting with Elastic Structure). Find the approximate
flow velocity v n C 1

h and pressure p n C h satisfying ( 5.25 ) on the approximation ht n C 1
of the domain t n C 1 , which is defined by the approximation of the ALE mapping
A ht n C 1 given by ( 5.106 ). Find u n C 1

defined by ( 5.76 ) with p k C 1 satisfying ( 5.84 ).

h

The domain velocity z n C 1
h

is given by ( 5.10 ).