Seabed

The Seabed Interaction Model defines how cable and mooring elements interact with the seafloor. This model implements the MoorDyn seabed contact formulation 1 based on the saturated damping friction model, with the addition of a numerical impulse cap to guarantee time-domain stability in the physics engine.

It calculates normal restorative forces based on element penetration, alongside a velocity-regularized directional friction model that evaluates axial and transverse sliding drag independently.

Normal Contact (Soil Penetration)

The normal seabed contact force on a line node is calculated using a linear spring-damper formulation based on absolute soil properties. For a flat seabed, the force is evaluated as:

\[F_n = [(z_b - x_z)k_b - v_zc_b]dl\]

Where \(z_b\) is the depth to the seabed, \(r_3\) is the vertical position of the node, \(k_b\) is the seabed stiffness, \(v_3\) is the vertical velocity, \(c_b\) is the seabed damping, and \(dl\) is the line segment length.

Because our solver evaluates volumetric elements, this continuous distributed load is converted into equivalent nodal forces using a static lever rule based on the element’s burial geometry:

  • Partially Submerged (Triangle Wedge): Force is shifted toward the buried node, with the center of pressure located at \(1/3\) of the submerged length.

  • Fully Submerged (Trapezoid): Load is distributed asymmetrically based on the centroid of the trapezoidal penetration profile.

Directional Tangential Friction

To circumvent numerical instabilities caused by standard Coulomb friction discontinuous flips, the model relies on a saturated damping friction formulation.

The node’s in-plane velocity along the seabed (\(v\)) is broken up into axial (\(v_A\)) and transverse (\(v_T\)) components:

\[v_A = (v \cdot \hat{q})\hat{q}\]
\[v_T = v - v_A\]

Where \(\hat{q}\) is the axial unit vector of the cable element.

The continuous tangential force evaluates a static ramp below the arbitrary ramp-up velocity threshold (\(v_c\)), and saturates to the kinetic limit above it. For example, the transverse component evaluates as:

\[\begin{split}-F_{f_T} = \begin{cases} F_{k_T} & \text{if } |v_T| \ge v_c \\ \left( \mu_{s_T} \frac{F_n}{v_c} \right) v_T & \text{if } |v_T| < v_c \end{cases}\end{split}\]

This linear ramp allows the use of distinct static and kinetic friction coefficients for both the axial and transverse directions without needing an explicitly defined damping constant.

Numerical Impulse Safety Cap: To improve solver robustness beyond standard analytical formulations, the evaluated friction magnitude is mathematically capped by the stopping impulse: \(F_{stop} = (m \cdot v) / dt\). This strictly guarantees that the applied tangential force cannot artificially flip the node’s velocity direction within a single integration time step, eliminating numerical high-frequency chatter.

Parameters

Seabed Stiffness

Defines the linear foundation stiffness of the seabed (\(k_b\)). When a cable element penetrates the seabed (defined by \(z < -m\_waterDepth\)), a normal restorative force is applied based on the penetration volume.

  • Unit: Pa/m

Seabed Damping

Defines the physical viscous damping constant of the seabed soil (\(c_b\)). This treats the seabed as a uniform medium that exerts a resistive pressure against the element’s vertical penetration velocity.

  • Unit: Pa·s/m

Axial Friction Coefficient

The kinetic Coulomb friction coefficient (\(\mu_{k_A}\)) used to calculate drag along the longitudinal axis of the cable. Sliding a cable along its own trench typically yields lower friction.

  • Unit: Dimensionless

Transverse Friction Coefficient

The kinetic Coulomb friction coefficient (\(\mu_{k_T}\)) used to calculate drag perpendicular to the cable axis. Dragging a line sideways generally displaces more soil, yielding higher friction.

  • Unit: Dimensionless

Seabed Static/Kinetic Friction Scale

A multiplier applied to the kinetic friction coefficients to define the static breakout friction coefficients (\(\mu_s = \mu_k \times \text{Scale}\)). If set to 1.0, static and kinetic friction are identical.

  • Unit: Dimensionless

Seabed Friction Break Velocity

Defines the critical arbitrary ramp-up velocity (\(v_c\)) threshold where the friction model transitions from static creeping to fully saturated kinetic sliding. This velocity threshold should be as close to zero as possible.

  • Unit: m/s

  • Note: To port a MoorDyn FricDamp (CV) value, use \(v_c = 1.0 / \text{FricDamp}\).

1

Stein Housner, Ericka Lozon, Bruce Martin, Dorian Brefort, and Matthew Hall. Seabed bathymetry and friction modeling in moordyn. Journal of Physics: Conference Series, 2362(1):012018, 2022. doi:10.1088/1742-6596/2362/1/012018.