number of master points, and the conservative load transfer algorithm [26] is applied to compute the nodal
4
force
f
S
j
i
at each slave node. Substituting eq. (2) and eq. (3) into eq. (4) leads to

δW
F
=
n
M
X
i
=1
n
i
X
j
=1
f
S
j
i
·
δ
u
M
i
+
δ
θ
M
i
× R
(
θ
M
i
)
d
j
i
,
=
n
M
X
i
=1
n
i
X
j
=1
f
S
j
i
·
δ
u
M
i
+
R
(
θ
M
i
)
d
j
i
×
f
S
j
i
·
δ
θ
M
i
,
=
n
M
X
i
=1
f
M
i
·
δ
u
M
i
+
p
M
i
·
δ
θ
M
i
,
(5)
Finally, noting that eq. (5) is the virtual work
δW
S
of the onedimensional beam structure, the energy
is conservative in the decoupled masterslave procedure.
Moreover, the masterslave kinematics does not
require the matched master point
M
i
to be coincident with a beam node.
In such cases the conservative
load transferred algorithm in [26] needs to be applied. Although we focus on cable subsystems with circular
crosssections in the present work, this procedure can also be generalized for more complicated surfaces, such
as bridges [32], turbine blades and flexible aircraft.
4. Embedded computational framework
The aforementioned cablesubsystem modeling approaches are particularly suitable for utilization in an
Eulerian framework equipped with an Embedded or Immersed boundary method (EBMs or IBMs) [33, 34, 35,
36, 20, 18, 19, 21, 22, 37], also known as the fictitious domain method [38] and the Cartesian method [39, 40].
These methods effectively handle FSI applications featuring large structural deformations and/or topologi
cal changes [41, 21, 42], and therefore are an attractive option for cable subsystems characterized by large
lengthtodiameter ratios which generally undergo large deformations. Arbitrary EulerianLagrangian (ALE)
methods [43, 44] incorporating mesh motion algorithms or remeshing techniques [32, 10, 8] have also been
applied to explore cable subsystem dynamics. In the present work, we focus mainly on the Eulerian frame
work. However, the cable subsystem modeling approaches discussed above can be readily incorporated into
the ALE framework for problems with small or moderate displacements.
4.1. Governing equation
Let Ω
F
denote the fixed fluid domain in the Eulerian computational framework, the conservative form
of the NavierStokes equations can be written as
∂
W
∂t
+
∇ · F
(
W
) =
∇ · G
(
V
,
∇
V
)
,
in Ω
F
(6)
where
V
and
W
are the vectors of the primitive and conservative variables describing the fluid state, respec
tively.
F
(
W
) and
G
(
V
,
∇
V
) are respectively the inviscid and viscous flux tensor functions. Specifically,
V
=
ρ
v
p
,
W
=
ρ
ρ
v
E
,
F
(
W
) =
ρ
v
ρ
v
⊗
v
+
p
I
(
E
+
p
)
v
,
and
G
(
V
,
∇
V
) =
0
τ
τ
·
v

q
,
(7)
where
ρ
,
v
,
p
, and
E
denote the density, velocity, static pressure, and total energy per unit volume of the
fluid, respectively. The velocity and total energy per unit volume are given by:
v
= (
v
1
, v
2
, v
3
)
T
and
E
=
ρe
+
1
2
ρ
(
v
2
1
+
v
2
2
+
v
2
3
)
,
(8)
5
where
e
denotes the specific (i.e., per unit of mass) internal energy. In the physical inviscid flux tensor
F
,
I ∈
R
3
×
3
is the identity matrix. In the viscous flux tensor