Numerical Simulation of Fluid–Structure Interaction Problems with Applications to Flow in Vocal Folds - Fluid-Structure Interaction and Biomedical Applications - page 345

Then we find that
p k C 1 D p k C tq k C k q k C k M 1 G k C 1 M 1

p k C 1

K

1

2 LJ t 2

C.1 / k t M

(5.83)

p k Cq k ;

1 G k M 1

C K

which can be written in the form
I C k M

K p k C 1 D p k C .t C k /q k C k M

1 1 G k C 1 C

C. 1/ k t C 1

2 LJ t 2 M

(5.84)

p k Cq k :

1 G k M 1

C K

where we set for the sake of simplicity
k D LJt 2 1

LJt 2
1 C Ct :

Ct
1 C Ct

D (5.85)

If p k and q k are known, then p k C 1 is obtained from system ( 5.84 ) and afterwards
q k C 1 is computed from ( 5.81 ).
In numerical examples presented in Sects. 5.7.2 and 5.10.2 , the parameters LJ D
1=4 and D 1=2 were used. This choice yields the Newmark method of the second
order.

5.4 Coupled FSI Problems

In the previous sections, the flow and structural problems were considered sepa-
rately. In what follows, we shall be concerned with complete coupled FSI problems.

5.4.1 Coupled Problem of Incompressible Flow and Vocal Fold

Rigid Body Model

The flow model ( 5.9 ), ( 5.4 ) is coupled with the equation of motion ( 5.61 )by
interface conditions. First, the boundary condition ( 5.4 ), (b) must be satisfied on
Q Wt , w he r e t he domain velocity is given by Eq. ( 5.6 ). Further, the aerodynamic
forces F 1 ;F 2 in ( 5.61 ) depend on the flow velocity v and the pressure p.The
forces F 1 and F 2 are computed with the aid of the aerodynamic lift force L and
the aerodynamic torsional moment M as

L.t/
2

1
2l M.t/;

L.t/
2 C

1
2l M.t/;

F 1 .t/ D F 2 .t/ D (5.86)