Fluid–Structure Interaction in Hemodynamics: Modeling, Analysis, and Numerical Simulation - Fluid-Structure Interaction and Biomedical Applications - page 162

with

r

t

u N D 0; v N D .. u r / N / j ; N D .d N / j ;

(2.173)

u N .0;:/ D u 0 ;.0;:/ N D 0 ;v N .0;:/ D v 0 :

Here u N , v N ,and V N are the piecewise linear functions defined in ( 2.162 ), t is
the shift in time by t to the left, defined in ( 2.151 ),

t N is the transformed
gradient via the ALE mapping A t N ,definedin( 2.110 ), and v N , u N , v N , N , d N ,
and V N are defined in ( 2.148 ).
Using the convergence results obtained for the approximate solutions in
Sect. 2.6.6 , and the convergence results just obtained for the test functions q N ,
we can pass to the limit directly in all the terms except in the term that contains
@ t u N . To deal with this term we notice that, since q N are smooth on sub-intervals
.jt;.j C 1/t/, we can use integration by parts on these sub-intervals to obtain:
Z T

r

Z

.1 C t N /@ t u N q N

0 F

Z .j C 1/t Z

N 1

X

.1 C j N /@ t u N q N

D

jt F

j D 0

Z .j C 1/t Z

N 1

X

.1 C t N / u N @ t q N D

jt F

j D 0

Z

C j / u j C 1
N

.1 C j C 1 j C 1 C q N ..j C 1/t/

F

Z

.1 C j / u j N q N .jtC/ : (2.174)

F

Here, we have denoted by q N ..j C 1/t/ and q N .jtC/ the limits from the left
and right, respectively, of q N at the appropriate points.
The integral involving @ t q N can be simplified by recalling that q N D q ı A N ,
where N are constant on each sub-interval .jt;.j C 1/t/. Thus, by the chain
rule, we see that @ t q N D @ t q on .jt;.j C 1/t/. After summing over all j D
0;:::;N 1 we obtain

Z .j C 1/t Z Z T Z

N 1

X

.1 C t N / u N @ t q N D .1 C t N / u N @ t q :

jt F 0 F

j D 0

To deal with the last two terms in ( 2.174 ) we calculate