can use under-relaxation of either node-position
x
fluid=solid
x
j
k;fluid
¼ x
k
x
j
k;solid
þð1 x
k
Þx
j
k1;fluid
ð5Þ
or load values t
fluid=solid
t
j
k;solid
¼ x
k
t
j
k;fluid
þð1 x
k
Þt
j
k1;solid
ð6Þ
A precise description of the coupling flow chart and
engine (Coupling settings (A), Time-step settings (B),
and Time-step-looping (C)) can be found in Appendi-
ces A and B.
Interface Equilibrium and Relaxation
A major feature within the coupling concept is built
by the convergence control (Energy analysis (D)) and
the adaptive determination of relaxation factors
(Relaxation engine (E)) which are discussed subse-
quently. Regarding the overall multi-physics system
the quantities exch anged at the interface must not
contribute to the conservation equations (RE
k
j
= 0).
RE
j
k
¼ E
j
k;fluid
þ E
j
k;solid
¼
Z
X
t
j
k;fluid
_
x
j
k;fluid
þ t
j
k;solid
_
x
j
k;solid
dX
ð7Þ
The discrete looping procedure needs to en sure the
artificial energy difference RE
k
j
must fall under a cer-
tain threshold in order to satisfy the coupling condi-
tions defined in Eqs. (3)and(4) and to finalize the
current time-step. As required by the energy analysis, a
coupling residual ([
]
k
j
) can now be formulated as
½
j
k
:¼
j
k1;fluid
j
k;solid
ð8Þ
for load ð ¼ tÞ and position relaxation ð ¼ xÞ,
respectively. Using Eq. (8)anAitken convergence
acceleration
22
determines the relaxation factor x
k
for
Eqs. (5)and(6) for k > 1 and x
1
= x
start
:
x
k
¼ x
k1
1 þ
½
j
k1
½
j
k
T
½
j
k
½
j
k1
½
j
k
2
"# !
ð9Þ
Hydro-Elastic Evaluation Methods
The evaluation procedures used within this work are
based on classical streamline visualization techniques.
However, in order to visualize the complex inner-
ventricular flow field additional methods need to be
introduced. Furthermore, newly defined dimens ionless
FSI numbers help to understand cardiovascular fluid–
structure interaction effects.
Vortex Visualization (k
2
-Method)
Three-dimensional iso-surface representation can be
obtained by the so-called k
2
method. The underlying
concept relates areas of minimum pressure with vortex
structures due to flow field rotation. Using the sym-
metric (S) and anti-symmetric part (A) of the velocity
gradient tensor rv one can show with Eq. (2) that
q(SS + AA) = p. A sufficient criterion for
minimum pressure requires the left-hand side to be
negative which is true if the second invariant k
2
< 0 with
k
1
< k
2
< k
3
.
13
General Flow Field Quantification
Using the absolute flow simulation results, dimen-
sionless characteristic flow numbers help to judge, and
FIGURE 2. Implicit FSI Coupling (for KaHMo FSI)
8
: Iterative solution of fluid and solid mechanics (implemented with fluent user
libraries).
Partitioned Fluid–Solid Coupling for Cardiovascular Blood Flow 1429