Shape theory

Elastic solids

Elastic solids may be penetrated by lines, 3D buoys, and 6D buoys. Other objects are not affected at all by elastic solids. Penetration generates a reaction force consisting of a resistance force and a friction force. If explicit integration is used in the dynamic analysis then, in addition, a damping force is applied.

Objects affected

Line interaction occurs when one of the line nodes penetrates the solid. The reaction force is calculated from the penetration of the outer surface of the line (based on the line type outer contact diameter), and the force is applied at that point.

A 3D buoy interacts when the buoy origin penetrates the solid. The reaction forces are calculated from the penetration of the buoy origin, and are applied at the buoy origin.

A 6D buoy interacts when any of its vertices penetrates the shape. Each penetrating vertex experiences its own reaction and friction forces, based on the penetration of the vertex and displacement of the vertex from its friction target position, and the forces are applied at that vertex. This gives a model where each vertex behaves rather like a pad, such as the landing pad on a lunar module.

In all cases, the lateral friction force is calculated using the friction coefficient data and following the usual friction model.

Penetration model

Consider an object which penetrates the surface of the elastic solid. Denote by $\vec{p}_\mathrm{o}$ the position of the penetrating object and by $\vec{p}_\mathrm{s}$ the closest point on the surface of the solid to $\vec{p}_\mathrm{o}$. Note that if the penetrating object has non-zero contact diameter (e.g. a line node) then this closest point may be on an edge or corner of the shape.

The reaction force on the penetrating object, $\vec{f}_\mathrm{R}$, is then \begin{equation} \vec{f}_\mathrm{R} = r(d)\ a\ \vec{n} \end{equation} where

$d = \lvert\vec{p}_\mathrm{o} {-} \vec{p}_\mathrm{s}\rvert=$ depth of penetration

$r(d)=$ reaction force per unit contact area at penetration $d$

$a=$ contact area for the 3D buoy, 6D buoy or line

$\vec{n}=$ unit vector in direction $\vec{p}_\mathrm{o} {-} \vec{p}_\mathrm{s}$

For simple linear stiffness, $r(d)$ is given by $kd$, where $k$ is the stiffness of the shape material. For nonlinear stiffness, $r(d)$ may be defined by a variable data table.

Damping force

If explicit integration is used then a reaction damping force $\vec{f}_\mathrm{Dn}$ is also applied when the object is travelling into the solid. This damping force is in the same outward direction $\vec{n}$, and is given by \begin{equation} \vec{f}_\mathrm{Dn} = \begin{cases} 2 \lambda (m\ k_\textrm{nom} a)^{1/2}\ v_\mathrm{n} \vec{n} & \text{for $v_\mathrm{n}\gt0$} \\ 0 & \text{for $v_\mathrm{n}\leq0$} \end{cases} \end{equation} where

$\lambda=$ percentage of critical damping / 100

$m=$ mass of the penetrating object

$k_\textrm{nom}=$ nominal normal stiffness

$v_\mathrm{n}=$ component of object velocity into the solid.

For linear stiffness, $k_\textrm{nom} = k$. For nonlinear stiffness, $k_\textrm{nom}$ is taken to be the stiffness corresponding to a penetration of zero.

Trapped water

Inside a trapped water shape the fluid translational velocity and acceleration are calculated on the assumption that the trapped water moves and rotates with the shape. So if the trapped water shape is fixed or anchored, then no fluid motion occurs inside the shape. But if the shape is connected to a moving vessel, for example, then the trapped water is assumed to move and rotate with the vessel.

Note: If the shape intersects the water surface then the surface is assumed to pass through the shape unaltered. Thus a wave in the open sea also appears inside the shape. We make this assumption because of the difficulty in predicting realistically how the surface will behave inside the trapped water volume.
For example, a moonpool with an open connection at the bottom will suppress most of the wave and current action. However there will be some flow in and out of the moonpool, depending on the size of the opening to the sea, pressure difference effects and the local geometry. The surface elevation in the moonpool therefore does respond to the wave outside, but it is attenuated to some extent and lags behind the surface outside.