Numerical Simulation of Fluid–Structure Interaction Problems with Applications to Flow in Vocal Folds - Fluid-Structure Interaction and Biomedical Applications - page 328

We approximate the ALE velocity z .t n C 1 / by z n C 1 ,where

3 A t n C 1 .X/ 4 A t n .X/ C A t n 1 .X/

2t

z n C 1 .x/ D

; (5.10)

x D A t n C 1 .X/ 2 t n C 1 :

Further, it is necessary to approximate the ALE derivative of the velocity v at time
t n C 1 . To this end, for i D n;n 1 we set v i

D v i ı A t i ı A 1

t n C 1 , which is defined in
the domain t n C 1 (the symbol ı denotes the composite function). Then, by ( 5.7 ), for
x 2 t n C 1 and X D A 1

t n C 1 .x/, using the second-order backward difference formula,

we can write

D A v
Dt .x;t n C 1 / D

@v.X;t n C 1 /
@t

(5.11)

3v.X;t n C 1 / 4v.X;t n / C v.X;t n 1 /
2t

3v n C 1 .x/ 4v n .x/ C v n 1 .x/
2t

:

This leads to the problem of finding unknown functions v D v n C 1 W ! R 2 and

p D p n C 1 W ! R satisfying the equations

3v 4v n

Cv n 1
2t

C .v z n C 1 / r v r .rv Cr

T v/ Crp D 0;

divv D 0; in ; (5.12)

and the boundary conditions ( 5.4 ) (where condition (a) can be replaced by ( 5.5 )—in
what follows, this eventuality will not be emphasized).

Weak Formulation

The starting point for the finite element discretization of problem ( 5.12 ) with the
boundary conditions ( 5.4 ) is the so-called weak formulation. To this end, we define
the velocity spaces W;X and the pressure space Q:

W D .H 1 .// 2 ; XDfv 2 W Ivj I [ Wt D 0; v 2 j S D 0g; (5.13)

Q D L 2 ./;

(5.14)

where L 2 ./ is the Lebesgue space of square integrable functions over the domain
,andH 1 ./ is the Sobolev space of square integrable functions together with
their first-order derivatives.