Fluid–Structure Interaction in Hemodynamics: Modeling, Analysis, and Numerical Simulation - Fluid-Structure Interaction and Biomedical Applications - page 118

nonlinearities into the equations, reflecting the geometric nonlinearities of the
moving interface. The transformed gradient, which we denote by r

, will depend
on the fluid-structure interface . Furthermore, by using the chain rule, one can
see that the time derivative of the transformed fluid velocity will have an additional
advection term with the coefficient given by the domain velocity w

WD A t ı A 1 ,

where A t denotes the time derivative of A . Finally, the mapped fluid equations in

F read:

in F .0;T/:

/ u / Dr

F .@ t u C .. u w / r

(2.84)

r u D 0

Here, the notation reflects the dependence of D . u / D

1 u Cr T u / on .

2 .r

Therefore, our problem in ALE formulation reads as follows:

The Coupled Problem in ALE Form defined on F Find u , p, and d such that:

in F .0;T/;

/ u / Dr

F .@ t u C .. u w / r

r u D 0

S @ tt d Dr S in S .0;T/;

9
=

@ t e r D u j R C ;
e r D d ;
K h@ tt C L el ./ C L vis @ t DJ. n /j R C e r C R Se r e r

on .0;T/:

;

As we shall see in Sect. 2.7 , the actual numerical simulations at each time step
are typically performed on the current (fixed) domain F .t/, with only the time-
derivative calculated on F , thereby avoiding the need to calculate the transformed
gradients r

. The corresponding continuous problem in ALE form can be written

as follows:

The Coupled Problem in ALE Form defined on F .t/

Find u , p, and d such that:

in F .t/ .0;T/;

F .@ t u j F C .. u w / r/ u / Dr
r u D 0

S @ tt d Dr S in S .0;T/;

9
=

@ t e r D u j R C ;
e r D d ;
K h@ tt C L el ./ C L vis @ t DJ. n /j R C e r C R Se r e r

on .0;T/:

;