Lines: Analytic catenary

The analytic catenary representation is primarily designed to facilitate quasi-dynamic mooring analysis in OrcaFlex, in which the mooring line loads are calculated from analytic catenary equations. This may be a reasonable approximation in cases where the inertia and bend stiffness of the mooring lines can be neglected, and where damping (also neglected by the catenary equations) can instead be represented by vessel other damping.

The analytic catenary representation works by solving a set of simple equations to calculate the force applied by the top end of the line to whatever object it is connected to. These equations consider the line as a single continuous object; no discretisation into constituent nodes is required. The analytic catenary representation does not account for dynamic effects: for a given set of input parameters, the equations predict a purely static configuration of the line. The bottom end of the line is referred to as the anchor, and the top end as the fairlead.

At the start of the statics calculation, OrcaFlex will compute a series of analytic catenary solutions for each line that uses this representation. Each solution in the series corresponds to a particular horizontal and vertical separation between the anchor and the fairlead. The anchor is kept fixed, so each solution is determined fully by the position of the top end of the line. These top end positions collectively form a two dimensional solution grid of points in the vertical plane containing the anchor and the fairlead. During a simulation the top end of each line may move through its solution grid as the object to which it is connected moves, as illustrated in the figure below. The force that the top end applies to this object is calculated by interpolation in this grid, based on the instantaneous position of the fairlead.

The line analytic catenary data define the points within the solution grid: each combination of horizontal tension and fairlead vertical offset corresponds to a different solution of the analytic catenary equations. The fairlead positions for these solutions are, collectively, the solution grid. We could equally well, and more directly, define the horizontal separation between the anchor and the fairlead, but instead we choose to specify the horizontal tension for a number of reasons:

• It is easier to know what range of tensions are applicable for the system being modelled than it is to know what values of horizontal separation will be relevant. For instance, if the line top end is attached to a vessel, then the vessel may move a considerable distance during the simulation. The same objection does not apply to the vertical offset, because the vessel is always likely to be near to the sea surface.
• Specifying horizontal tension simplifies the analytic catenary solution; starting from a specified fairlead position would require an extra layer of iteration in the solution process.
• This is the convention used by other software.

Sometimes, no solution to the equations can be found for a particular grid point. In this case OrcaFlex will report a warning and proceed using only the successfully-calculated solutions. However, a minimum requirement is that there is at least one valid solution at each fairlead vertical offset: without this, the statics calculation will terminate with an error message. Often this simply means that an unrealistic fairlead offset value has been specified.

Visualising solutions

As well as the data itself, the analytic catenary page of the line data form also has a View curves button that displays the solutions graphically.

The analytic catenary characteristics window displays curves of horizontal restoring force and fairlead declination against the horizontal offset of the fairlead from the anchor. Curves are presented for any line that uses the analytic catenary representation. Once a line is specified, curves are available for each relevant fairlead vertical offset.

The horizontal restoring forces displayed in the graphs do not necessarily correspond to those that define the solution grid on the line data form. The reason for the discrepancy is that the graphs display the horizontal tension at the fairlead, whereas the solution grid is defined by the horizontal tension at the arc length just prior to the line's final touchdown, closest to the anchor. These tensions can differ if there are additional seabed touchdown points between these two arc lengths.

Calculation

During the calculation, OrcaFlex uses the solution grid to apply a force at the fairlead. To do this, it needs to know the anchor position, $\vec{x}_\textrm{b}$, and the fairlead position, $\vec{x}_\textrm{t}$. From these, it can compute the horizontal separation, $x = \sqrt{|\vec{x}_\textrm{t} - \vec{x}_\textrm{b}|^2 - z^2}$ and the vertical separation, $z = (\vec{x}_\textrm{t} - \vec{x}_\textrm{b}) \cdot \hat{\vec{z}}$, where $\hat{\vec{z}}$ is a unit vector pointing vertically upwards. $(x, z)$ are the coordinates of the fairlead relative to the anchor in their common vertical plane.

Each solution in the grid can be labelled by its planar coordinates, $(x_i, y_i)$, for some index $i$. Let $f$ denote some property of an analytic catenary, such as the force at the fairlead. Then $f(x_i, z_i)$ gives the value of $f$ associated with solution $i$. During an OrcaFlex calculation, we wish to compute $f(x, z)$ for arbitrary values of $x$ and $z$. In general, however, the fairlead will not lie at a grid point. We therefore adopt the following procedure:

• If $(x, z)$ lies within the solution grid, then bilinear interpolation is used. It is important to note that the solution grid is not regular: there may be a different spacing of $x$ coordinates at each fairlead offset $z$. Therefore, let the bounding $z$ coordinates be the previous value $z_{p}$ and the next $z$ value, $z_{n}$. Associated with each of these $z$ is a pair of bounding $x$ coordinates, finally yielding grid points with coordinates $(x_{p_{p}}, z_{p})$, $(x_{p_{n}}, z_{p})$ and $(x_{n_{p}}, z_{n})$, $(x_{n_{n}}, z_{n})$. Our property $f(x, z)$ is then calculated as \begin{align} f(x, z) =& \left( \frac{z_{n} - z}{z_{n} - z_{p}} \right) \left[ \left( \frac{x_{p_{n}} - x}{x_{p_{n}} - x_{p_{p}}} \right) f(x_{p_{p}}, z_{p}) + \left( \frac{x - x_{p_{p}}}{x_{p_{n}} - x_{p_{p}}} \right) f(x_{p_{n}}, z_{p}) \right] \nonumber \\ +& \left( \frac{z - z_{p}}{z_{n} - z_{p}} \right) \left[ \left( \frac{x_{n_{n}} - x}{x_{n_{n}} - x_{n_{p}}} \right) f(x_{n_{p}}, z_{n}) + \left( \frac{x - x_{n_{p}}}{x_{n_{n}} - x_{n_{p}}} \right) f(x_{n_{n}}, z_{n}) \right] \end{align}

• If $(x, z)$ is outside of the solution grid, then extrapolation is used. This is the natural continuation of the above interpolation scheme to the exterior of the grid, save for one exception: the horizontal tension at the fairlead is never allowed to drop below zero. Extrapolation continues only up to the point at which the tension reaches zero, say $(x_0, z_0)$; for any $(x, z)$ values beyond this point, we assign $f(x, z) = f(x_0, z_0)$.

  OrcaFlex will display a warning if extrapolation has been used. This means that the top end of the line has gone outside of the grid of valid solutions, so any results should be treated with extreme caution. It may be that the grid of points needs to be extended to fully encompass the fairlead motion.

This interpolation/extrapolation scheme is used to compute both the horizontal, $F_h(x, z)$, and vertical, $F_v(x, z)$, components of the force applied at the fairlead. These are converted to a force vector, $\vec{F}$, applied in the plane of the solution: $\vec{F} = F_h \, \hat{\vec{h}} + F_v \,\hat{\vec{z}}$, where $\hat{\vec{h}}$ is a horizontal unit vector pointing from $\vec{x}_b$ to $\vec{x}_t$.

Scope and limitations

The analytic catenary representation will not be suitable for some lines. In this section, we present in broad terms the circumstances in which the analytic catenary will provide a good representation of the line, and those when it will not.

The analytic catenary representation accounts for the following aspects of the line:

• Weight
• Buoyancy
• Multiple line sections
• Contents
• Axial stiffness
• Contact with a flat seabed
• Axial seabed friction
• Clump attachments

Important aspects not accounted for by the analytic catenary representation include

• Inertia
• Drag
• Added mass
• Bend stiffness
• Torsional stiffness
• End connection stiffnesses
• Pressure effects

As described above, the analytic catenary representation works essentially in terms of static solutions. Forces are derived during the dynamic simulation by interpolating into the solution grid with the instantaneous position of the fairlead. This method is particularly appropriate to applications modelling mooring lines, which have very little bend stiffness or inertia compared to the vessel they are attached to. The absence of drag/damping, however, can result in motions that are much larger than would otherwise be predicted; to account for this, an equivalent amount of damping must be added to the moored object. In the case of a vessel, this can be achieved using the low frequency vessel other damping data.

Flat seabeds are fully accounted for, including those with non-zero slope. In solving the analytic catenary equations, the seabed is represented as a single line, $z = m x + c$, in the plane of the solution (i.e. the vertical plane containing the anchor and fairlead), where $m$ and $c$ are constants. $c$ is minus the height of the anchor above the seabed; $m$ is the effective slope of the seabed. The slope can be set explicitly on the line data form or, by setting the data to '~', inferred from the positions of the line ends when the model is in reset state.

Seabed friction is also accounted for. Although the model used is slightly different to that used by the finite element representation, it is similar to models used by other analytic catenary software packages.

The analytic catenary solver itself has known limitations, which are presented on the theory page.

Results

A reduced set of results is available for analytic catenary lines.