Numerical Simulation of Fluid–Structure Interaction Problems with Applications to Flow in Vocal Folds - Fluid-Structure Interaction and Biomedical Applications - page 372

The boundary state w B is defined on the basis of the Dirichlet boundary
conditions and extrapolation:

1
2 I jv I j

w B D . I ; I v I1 ; I v I2 ;c v I .L/

2 / on I ;

C (5.126)

w B D w .L/ on O ;

1

w B D . .L/ ; .L/ z W1 ; .L/ z W2 ;c v .L/ .L/

2 .L/

2 / on Wt :

C j z W j

The approximate solution is defined as w h .t/ 2 S ht such that

Z

X

D A w h .t/
Dt ' h dx C b h . w h .t/;' h / C a h . w h .t/;' h /

K

(5.127) K 2 T ht

C J h . w h .t/;' h / C d h . w h .t/;' h / D ` h . w h .t/;' h /

w h is an approximation of the holds for all ' h 2 S ht ,allt 2 .0;T/,and w h .0/ D

initial state w 0 .

5.9.2 Time Discretization by the BDF Method

Let us construct a partition 0 D t 0 <t 1 <t 2 of the time interval Œ0;T with
t k

w h 2

D kt and time step t. We use the approximations w h .t n / S ht n ,

z .t n / z n ;nD 0;1;::: and introduce the function w h D w h ı A t k ı A 1

t k C 1 ,which
is defined in the domain ht k C 1 . In order to approximate the ALE derivative at time
t k C 1 , we start from its definition ( 5.7 ) and then use the backward difference:

D A w h
Dt .x;t k C 1 / D

@ w h
@t .X;t k C 1 /

.x/ w h .x/
k

w k C 1
h

.X/ w h .X/
k

w k C 1
h

D ; xD A t k C 1 .X/ 2 ht k C 1 :

(5.128)

By the symbol .; / we shall denote the scalar product in L 2 . ht k C 1 /. A possible
full discretization reads

.a/ w k C 1
h

2 S ht k C 1 ; (5.129)

.b/ w k C 1

!

w h
k

h

C b h . w k C 1 ;' h / C a h . w k C 1

;' h ;' h /

h h