146
F. Muttin
element, the Young modulus and the cable straight section [13]. This section is a
normal section of the floating barrier which is composed of three kinds of materials:
the curtain fabric, the upper aerial leach and the subsea chain.
The approximation of the continuous equilibrium equation permits to define a
discrete problem. The curvilinear derivative operator on the internal force
d
ds F int (s)
is transformed into a vectorial difference at each node between neighbouring element
tensions. A weighted summation of the external force F ext (s) is made at each node
over its two adjacent elements. The nodal displacement vector is denoted U. We
use the same notations F int and F ext after the transformations from the continuous
to the discrete formulations. They depend on the discrete problem unknown U, i.e.,
the equivalent internal force vector F int (U) and external force vector F ext (U). The
non-linear discrete equation is written in term of the residual vector R(U)
R(U) = F int (U) − F ext (U) = 0.
(7.8)
The equation is solved using the Newton-Raphson method. The method uses the
derivative
d
dU R(U). The derivative of the internal force vector is defined on each
element by a stiffness matrix. The external force vector derivative is not taken into
account here.
Remark 7.3 The current pressure p(s) is defined by a classical hydrodynamic rule
p(s) =
1
2
ρ w C v(s)
2
,
(7.9)
where ρ w is the water density, C the drag coefficient and v(s) the current velocity.
The atmospheric wind action on the aerial part of the barrier can be defined by
using a similar rule than those for the current pressure by considering the air density
ρ air and the wind velocity taken at the altitude of the float.
The waves generated by the wind act on the subsea part of the barrier. A wave
pressure can be added to the current pressure by using an incident velocity v w supplementary to the current velocity v(s). The velocity v w can be defined by using the
square root of the significant wave height.
7.2.4 Operational Point of View of the Numerical Results
From the operational point of view the 2D model gives the boom tensions t(A) and
t(B) at both end-points A and B of a boom section. The direction of the mooring
tensions T (A) and T (B) are also given by the numerical model. The hydrodynamic
current can vary in time depending on the tide stages. As a consequence boom tension
and boom tangent direction can be time-dependent.
F. Muttin
element, the Young modulus and the cable straight section [13]. This section is a
normal section of the floating barrier which is composed of three kinds of materials:
the curtain fabric, the upper aerial leach and the subsea chain.
The approximation of the continuous equilibrium equation permits to define a
discrete problem. The curvilinear derivative operator on the internal force
d
ds F int (s)
is transformed into a vectorial difference at each node between neighbouring element
tensions. A weighted summation of the external force F ext (s) is made at each node
over its two adjacent elements. The nodal displacement vector is denoted U. We
use the same notations F int and F ext after the transformations from the continuous
to the discrete formulations. They depend on the discrete problem unknown U, i.e.,
the equivalent internal force vector F int (U) and external force vector F ext (U). The
non-linear discrete equation is written in term of the residual vector R(U)
R(U) = F int (U) − F ext (U) = 0.
(7.8)
The equation is solved using the Newton-Raphson method. The method uses the
derivative
d
dU R(U). The derivative of the internal force vector is defined on each
element by a stiffness matrix. The external force vector derivative is not taken into
account here.
Remark 7.3 The current pressure p(s) is defined by a classical hydrodynamic rule
p(s) =
1
2
ρ w C v(s)
2
,
(7.9)
where ρ w is the water density, C the drag coefficient and v(s) the current velocity.
The atmospheric wind action on the aerial part of the barrier can be defined by
using a similar rule than those for the current pressure by considering the air density
ρ air and the wind velocity taken at the altitude of the float.
The waves generated by the wind act on the subsea part of the barrier. A wave
pressure can be added to the current pressure by using an incident velocity v w supplementary to the current velocity v(s). The velocity v w can be defined by using the
square root of the significant wave height.
7.2.4 Operational Point of View of the Numerical Results
From the operational point of view the 2D model gives the boom tensions t(A) and
t(B) at both end-points A and B of a boom section. The direction of the mooring
tensions T (A) and T (B) are also given by the numerical model. The hydrodynamic
current can vary in time depending on the tide stages. As a consequence boom tension
and boom tangent direction can be time-dependent.