Numerical Simulation of Fluid–Structure Interaction Problems with Applications to Flow in Vocal Folds - Fluid-Structure Interaction and Biomedical Applications - page 351

The artificial Young modulus E a and the artificial Poisson ratio a can be derived
from Lamé coefficients a and a in the same way as in (2.11). The problem is
completed by the boundary conditions

D 0; d.X;t/ D u .X;t/; X 2 s W ;

dj I [ O D 0; dj W0 n s W (5.102)

where u is the solution of the elasticity problem treated in Sect. 5.3.2 .
T he solution of the problem ( 5.100 )-( 5.102 ) gives us the ALE mapping of 0
onto t in the form

A t .X/ D X C d.X;t/; X 2 0 ; (5.103)

for each time t.
System ( 5.100 ) is discretized by the conforming piecewise linear finite elements
on the mesh

T h0 used for computing the flow field in the beginning of the
computational process in the polygonal approximation h0 of the domain 0 .We
introduce the finite element spaces

X h Dfd h D .d h1 ;d h2 / 2 C. h0 / 2

I d hi j K 2 P 1 .K/ 8K 2 T h0 ;iD 1;2g; (5.104)
V h Df' h 2 X h I' h .Q/ D 0 for all vertices Q 2 @ 0 g;

and the form

Z

a divd h div' h dX C 2 Z

2

X

a e ij .d h /e ij .' h / dX: B h .d h ;' h / D

h0 h0

i;j D 1

(5.105)

The approximate solution of problem ( 5.100 ), ( 5.102 ) is defined as a function
d h 2 X h satisfying approximately the Dirichlet boundary conditions ( 5.102 )and
the identity

B h .d h ;' h / D 0 8' h 2 V h : (5.106)

T hen the approximate ALE mapping A ht is given by ( 5.103 ) with d WD d h and

X 2 hs :

A ht .X/ D X C d h .X;t/; X 2 h0 : (5.107)

The use of linear finite elements is sufficient, because we only need to know the
movement of the points of the mesh. The domain velocity is approximated by ( 5.10 ).
In our computations we choose the Lamé coefficients a and a as constants
corresponding to the Young modulus and Poisson ratio E a D 10;000 and a D

0:45.