Structural Analysis of Oil-Spill Booms - Mathematical Modelling and Numerical Simulation of Oil Pollution Problems - page 147

where U denotes the nodal displacements of the finite element mesh, F int are the
equivalent nodal forces representing the internal membrane stress and F ext are the
external loads resulting from the pressures and gravity forces.
The external forces are supposed to be given by F ext , prox defined over the initial
geometry of the membrane mesh and remaining independent of the displacement U .
To achieve the convergence of the Newton-Raphson method we use a threshold on
themaximal nodal displacement component to avoid unrealistic position updating. As
mentioned in the context wemust use an initialization of themembrane position given
by the curvilinear domain solution to have a convergence of the Newton-Raphson
method. The quadratic convergence is observed when U is in the solution vicinity.
The mesh size is defined by using two resolutions. The horizontal resolution
h H follows the curvilinear domain resolution. The vertical resolution h V takes into
account the different parts of the curtain or barrier, the upper leach, the float, the skirt
and the bottom chain. The vertical resolution permits to define the skirt angulation ʸ

along the curtain or barrier. Measure the skirt curvature necessitates three nodes at
least on a same vertical boom section.
The number of iterations required for the Newton-Raphson method convergence
depends on two precisions. The first one is the maximal out of balance between exter-
nal and internal forces. The second one is the maximal nodal coordinate updating.
The values commonly used using S.I. units are 1N for the out of balance of nodal
forces and 10 − 3 m for the nodal displacements.

7.3.4 Operational Usefulness of the Numerical Results

For oil-spill response using floating barrier, the 3D approach permits a complemen-
tary prediction of the pollutant containment with respect to the curvilinear model.
The indicators focus principally on the skirt angulation and mooring tensions at both
end-parts of the boom.
The skirt angulation is a curvilinear function

given by the angle between the
vertical and the skirt section considered as a segment defined with the bottom chain
node and the upper finite element node at the vicinity of the sea surface. The geometry
of the boom

ʸ

is numerically defined in term of the sea water current. Oil-spill model
can use at the water surface the equilibrium geometry of the boom during time.
Depending of the oil containment efficiency of the boom, the oil-spill model can use
a boundary condition for the oil drifting velocity.
The mooring tension at a boom section end-part is defined by the summation
of the nodal reaction forces corresponding to the Dirichlet boundary condition at
this location. The eigenvalues of the stress tensor

ˉ

˃ defined on the membrane at

equilibrium give the principal stresses ˃ min on the tensile structure. Fabric,

leach and bottom chain tensions are defined in term of the boom plan geometry for
the given conditions. The 3D model gives results for an evaluation of contingency
plan and boom design.

˃ max and