Fluid–Structure Interaction in Hemodynamics: Modeling, Analysis, and Numerical Simulation - Fluid-Structure Interaction and Biomedical Applications - page 122

problem for the linearly elastic thick structure in S , with the boundary condition
at the lateral boundary given by a PDE that determines the motion of the lateral
boundary. The motion of the lateral boundary in this sub-problem is driven by the
normal component of the first Piola-Kirchhoff stress tensor S , and by the initial data
for the velocity of the thin structure, which is given by the trace of the fluid velocity,
just calculated in the previous time step. In this step we also calculate the domain
velocity w , which is given by the time-derivative of the ALE mapping, associated
with Problem A1.

Problem A2 W FLUID
@ t u C .. u w / r

/ u Dr

; in F

u D 0; in F
u D v e r ; on
K h@ t v C L vis v DJ n e r on

r

Here u is the value of u from the previous time step, and w , which is the domain
velocity (the time derivative of the ALE mapping), is obtained from the just
calculated Problem A1. Furthermore, r

is the transformed gradient, which is based
on the value of from the previous time step. The initial data for u is given from the
previous time step, while the initial data for the trace of the fluid velocity v is given
by the just calculated velocity of the thin structure @ t .
This concludes our description of the general framework based on the Lie
splitting scheme for solving the class of FSI problems ( 2.58 )-( 2.63 ) with multiple
structural layers.
Before we continue, several remarks are in order:

• The splitting works as well when the thin structure is purely elastic, i.e., when

L vis D 0.

• Switching the order of solution (fluid step first, structure second) works as well.

The corresponding algorithm is explicitly shown below in the corresponding
block-diagram.

• The symmetrized Lie splitting obtained by solving the structure problem,

followed by the fluid problem, and then the structure problem, increases the
accuracy of the scheme to second-order in time.

• A version of Strang splitting for this problem was performed by Lukacova et al.

in [ 95 , 96 ] achieving second-order accuracy in time.

• Adding additional modules to capture different physics in a given multi-physics

problem can be accomplished in a similar way. See [ 23 ] for an application of
this scheme to an FSI problem with multiple poroelastic structural layers. Also,
see [ 122 ] for an application of this scheme to an FSI between a vascular device
called stent, elastic arterial wall, and the flow of an incompressible, viscous fluid.

• A modification of this scheme to achieve higher accuracy within the class of first-

order schemes was introduced in [ 20 , 21 ]. Details of this modified scheme, called
the Kinematically Coupled LJ-scheme, are presented next.