Theory
This chapter states the equations CableDyn solves: the kinematics and the two line elements, the constitutive laws, the hydrodynamic and seabed loads, the static and dynamic solution procedures, and the models for attached objects. Units, frames, sign rules, and the tension definition are fixed in Conventions and are not repeated here. The element-level implementation of each path, and the secondary Cosserat path, are described in Solver paths.
Symbols are defined in the section that uses them. A few letters are reused: \(\beta\) is the wave direction, the Syrope fast-stiffness slope, a Newmark parameter, and an eigenvalue parameter of the sagging-cable modes; \(T\) is a tension and, in the wave formulas, a period.
Kinematics and discretisation
A line is a curve \(\mathbf{r}(s, t) \in \mathbb{R}^3\), parametrised by the unstretched arc length \(s \in [0, L]\). It is built from sections joined end to end; each section is meshed into two-node elements of unstretched length \(L_0\), with the local coordinate \(\xi = (s - s_e)/L_0 \in [0, 1]\) on element \(e\). The axial strain and the centreline curvature are the exact large-deformation measures
Neither measure is linearised, so both elements are geometrically exact for arbitrary
displacement and rotation. Which element a section uses is set by the bending stiffness of
its line type: EI = 0 selects the tension element, EI > 0 the bending element.
The EI = 0 tension element
Each node carries its position only (three degrees of freedom). For nodal positions \(\mathbf{r}_a, \mathbf{r}_b\),
The internal force and its exact tangent (material plus geometric stiffness) are
In tension-only mode an element with \(\varepsilon < 0\) carries zero elastic force and
zero material stiffness, so force and tangent remain consistent. The deck and coupled routes
march the dynamics tension-only: a slack chain or rope exerts no elastic force. Axial damping
(BA, below) stays active on a slack element, as in MoorDyn, so a slack element with
\(BA \ne 0\) can carry a damping force while it stretches or shortens. The static initial
condition is the tension-only equilibrium of the same law (see
EI = 0 path). The consistent mass of an element is
with \(\rho_A\) the dry mass per unstretched metre.
The cubic-Hermite bending element
Each node carries its position and a material tangent, \([\mathbf{r},\ \mathbf{m}]\) with \(\mathbf{m} = \mathbf{r}'\), so an element has 12 degrees of freedom and no rotation variables. The centreline is the cubic-Hermite interpolant
which is \(C^1\) across nodes, so curvature is defined pointwise inside every element. The magnitude \(\lvert\mathbf{m}\rvert = 1 + \varepsilon\) carries the axial stretch at a node; only its direction is a bending quantity. The element stores the axial-plus-bending strain energy
Using \(\kappa^2\) directly, rather than \(\kappa\), keeps the energy smooth through the straight state. The stress-free reference is straight, and the bending moment is \(M = EI\,\kappa\); the node outputs recover it from nodal equilibrium (Node tension, curvature and bend moment).
At each Gauss point, \(\mathbf{a}\) and \(\mathbf{b}\) are fixed linear maps of the 12 element DOFs. The gradient and Hessian of the energy density with respect to \((\mathbf{a}, \mathbf{b})\) are written in closed form and chained to the DOFs, giving \(\mathbf{f}_\text{int} = \partial U/\partial \mathbf{q}\) and the symmetric tangent \(\mathbf{K}_t = \partial^2 U/\partial \mathbf{q}^2\) exactly. Automatic differentiation of the same energy is used only in the test suite, as an independent check of the closed form, together with finite differences of \(\mathbf{f}_\text{int}\).
Quadrature. Axial and bending energies use Gauss–Legendre integration, four points each by
default. The two orders are independently selectable from 1 to 6 (options
axial_quadrature_order and bending_quadrature_order); unequal orders give a selective
integration. The consistent mass
\(\mathbf{M}_e = \rho_A \int_0^{L_0} \mathbf{N}^{\mathsf T}\mathbf{N}\,\mathrm{d}s\) uses the same
Hermite shapes and a four-point rule, which is exact for the degree-6 integrand; it is
block-diagonal per Cartesian component. Distributed hydrodynamic loads use three-point Gauss
integration.
Curvature recovery. Curvature outputs are evaluated from the interpolated centreline, not only at nodes or Gauss points. The peak curvature of an element is located by sampling 16 equal intervals and bisecting each at least five and at most twelve times, continuing where the curvature departs from linear variation by more than 5 % of the local maximum. The same search locates the extreme axial resultant \(N = EA(\lvert\mathbf{r}'\rvert - 1)\) inside every element, with a 2 % criterion relative to the largest local magnitude (at least 1 N).
The element follows the geometrically exact Kirchhoff-rod formulation of Boyer et al. (2011) with the \(C^1\) Hermite centreline interpolation of Meier, Popp & Wall (2015), reduced to a torsion-free cable. Torsion enters only through the condensed line term below.
Node tension, curvature and bend moment
Section force. For the energy density \(W(\mathbf{a}, \mathbf{b})\) of the element, the force across a section (exerted by the part of the line beyond \(s\) on the part before it) and the conjugate of the material tangent are
On a stiff-EA cable \(\lvert\mathbf{a}\rvert = 1 + \varepsilon\) with
\(\varepsilon \ll 1\), \(\mathbf{a}\cdot\mathbf{b} = O(\varepsilon)\) and
\(\mathbf{b} = \kappa\,\mathbf{n}\), so that
\(\partial W/\partial\mathbf{a} = (EA\,\varepsilon - 2EI\kappa^2)\,\mathbf{t}\) plus normal
terms, \(\mathbf{m} = EI\,\kappa\,\mathbf{n}\), and
\((\partial W/\partial\mathbf{b})'\cdot\mathbf{t} = EI\,\mathbf{b}'\cdot\mathbf{t} = -EI\kappa^2\).
The axial internal force of the rod, the effective tension that balances the loads, and the
bend moment are therefore
The bending energy measures the curvature per deformed length, and its dependence on the
stretch puts \(-EI\kappa^2\) into the axial force: \(EA(\lvert\mathbf{r}'\rvert - 1)\)
is a formulation quantity, not the tension, and exceeds \(N\) by \(EI\kappa^2\) on any
mesh (161 N on a 1.2 kN tension at the sag bend of the lazy-wave cable of
iea15mw_umaine_mixed_cabledyn.dat, \(\kappa = 0.09\) 1/m). Global equilibrium fixes
\(N\) whatever the formulation: on a suspended run the horizontal component of
\(\mathbf{F}\) is constant and the vertical one changes by the submerged weight, so \(N\)
is what any code converges to, the OrcaFlex effective tension and the MoorDyn segment tension
on a fine mesh among them.
Pointwise oscillation. With a stiff EA the cubic interpolation cannot keep
\(\lvert\mathbf{r}'\rvert\) constant where the line must bend over an element (a nodal
seabed reaction, a section junction): the pointwise axial resultant oscillates about the element
mean by several times the tension, and the element means alternate between neighbours. On that
cable on the example’s former 87-element mesh (10 m on the grounded run, tension 1.2 kN) the
pointwise resultant spans -41.5 to +91.8 kN and the length-weighted element means at the nodes
reach -173 N; at 1 m elements the pointwise field still spans -5.9 to +11.9 kN.
Recovery from nodal equilibrium. The node outputs are recovered from the element end forces, the classical recovery of the finite-element nodal forces. For element \(e\) the end residual
holds only the element’s own distributed loads: consistent submerged weight, the buoyancy lost above the free surface, drag, Froude-Krylov force, structural and added-mass inertia, and the seabed reaction of its end nodes, each nodal reaction spread uniformly over the node’s tributary length (the distributed floor reaction it stands for; without it a line resting on the floor would report the fixed-end moment \(w L_0^2/12\) of its own weight; at a held line end whose end element lies on the floor, the reaction of the end half is the floor-support share of the weight, the share that the static and dynamic residuals carry on the end tangent rows). \(\mathbf{g}_e\) is the generalised force the two end nodes exert on the element: its position rows at the End-B side of the internal arc are the section force \(\mathbf{F}_j^-\) just before node \(j\), minus its rows at the other end the section force \(\mathbf{F}_j^+\) just after it, and its tangent rows are \(\mathbf{m}\) there. An interior node reports
with \(\mathbf{t}_j\) the unit nodal tangent, \(\mathbf{m}_j\) the mean of the two
one-sided values, and at a section junction the larger of \(M_j/EI\) on the two sides. The
recovered moment carries an error that does not depend on \(EI\), of the order of the largest
consistent-load moment \(w L_0^2/12\) of an element (a section junction leaves a jump of that
order between the two one-sided moments), so over a soft section dividing by \(EI\) amplifies
it. A section whose \(EI\) is below \(\max_e(\lvert w_e\rvert L_{0,e}^2)/12\) times 1 m,
the stiffness whose elastic moment at a 1 m bend radius equals that error, takes its curvature
from the exact geometric trace instead, as an element with \(EI = 0\) does (on the 87-element
lazy-wave cable with \(EI\) lowered to 100 N m2 the
recovered curvature reaches 6.9 1/m against a geometric peak of 0.12 1/m; the bend moment of such a
section remains the recovered one, with that error). The
difference of the two one-sided forces is the point load left at the node (seabed friction, an
attachment) plus, where the nodal seabed reaction varies from node to node (the touchdown, a
rounded crease of the bathymetry), the difference between its spread and its nodal form, about
0.094 times its second difference along the line. The seabed reaction spread here is the one the
residual applies, with the crease increment of the penalty potential inside a rounded crease. At
the converged static state the nodal section forces satisfy the discrete
equilibrium exactly. In a dynamic run the recovery is the balance at the output time
\(t_{n+1}\), with the position, velocity and acceleration the step committed there: the
inertia term \(\mathbf{M}_e\mathbf{a}_e\) uses the step’s acceleration
\(\mathbf{a}_{n+1}\) at \(\mathbf{q}_{n+1}\). The integrator balances the generalised-alpha
combination of the \(t_n\) and \(t_{n+1}\) states, so the recovered forces differ from
the integrator’s balance by \(O(\Delta t)\); the recovered forces carry none of the pointwise
oscillation. A line end reports the FairTen/AnchTen end force and its one-sided moment;
the segment tension of an element is the mean of the axial components of its own two end forces.
The recovery costs one element evaluation per element and output row.
The end node tension and the interior node tension are therefore different quantities. The end value is the end reaction, the force on the support or the platform: it holds the end element’s internal force and the end node’s share of the weight, buoyancy, drag and contact, but not inertia, added mass or seabed friction, which stay in the coupled load. The interior value is the recovered section force, built from end forces that include those terms. At rest and in a quasi-static march the two agree; under acceleration the interior values carry the inertia and added-mass terms of the two neighbouring elements, and the end values do not.
On the 87-element lazy wave the interior node tension departs from the equilibrium of the converged line by at most 60 N (1552 N for the element means), 12 N at twice the mesh density and 0.3 N at 1 m elements (516 N); the largest curvature error falls from 64 % to 30 % of the peak at 87 elements and from 2.1 % to 0.8 % at 1 m. The peak curvature at a touchdown meshed coarser than the bending length is limited by the solution itself, not by the recovery.
The condensed torque of a line restrained in torsion is not part of \(\mathbf{g}_e\): it adds
a shear normal to the tangent and does no work on bending rotations. Including its element terms
on torsion_lazy_wave_hangoff_twist.dat (two turns, torque 3.5 kN m) moves the tension by
at most 0.45 N of 9.4 kN and moves the bend moment away from \(EI\kappa\).
OPTIONS OutputRecovery element restores the element definitions: the length-weighted
mean of the element-mean axial forces \(EA(\lvert\mathbf{r}'\rvert - 1)\) at an interior
node, the larger one-sided exact curvature, and the midpoint axial resultant as the segment
tension. The material-strain checks (tensile_safety, the static compression notes) judge
\(EA\varepsilon\) under either setting.
Condensed torsion
A line restrained in torsion at both ends (END CONNECTIONS TorsStiffness, see
Deck format reference (.dat)) adds the twist of an isotropic Kirchhoff rod without adding degrees of
freedom. For an isotropic section with no distributed torque the twisting moment
\(M = GJ\,u_3\), with \(u_3\) the material twist rate, is uniform along the rod
(van der Heijden et al., 2003). The twist field
then condenses to one scalar per line, and the line energy gains
\(\Phi\) is the imposed relative roll of the End B frame with respect to the End A frame
(Pretwist and the motionFile roll column), \(k_A, k_B\) are the optional torsional end
springs (\(1/k = 0\) for Rigid), and \(\Theta(\mathbf{q})\) is the geometric twist of
the centreline: the angle about the End B direction \(\mathbf{d}_B\), right-handed, from the
End B reference normal \(\mathbf{n}_B\) to the End A normal \(\mathbf{n}_A\) carried to End B
by parallel transport. Each end frame \((\mathbf{d}, \mathbf{n})\) turns with the body or vessel
that holds the end. A clamped end has \(\mathbf{d}\) equal to its end tangent; at a
bending-pinned end \(\mathbf{d}\) is the connection direction and the transport starts with
the smallest rotation from \(\mathbf{d}\) to the tangent.
Because the Hermite centreline is \(C^1\), the transport factorises into one holonomy per element and a chain of smallest rotations between the unit node tangents \(\mathbf{t}_k = \mathbf{m}_k/\lvert\mathbf{m}_k\rvert\):
with \(\mathbf{t}_1\) the tangent at the element’s first node, \(\mathbf{a} = \mathbf{r}'\), \(\mathbf{b} = \mathbf{r}''\) at the Gauss points of the bending rule, and \(\chi\) the angle of the smallest-rotation chain \(\mathbf{d}_A \to \mathbf{t}_1 \to \dots \to \mathbf{t}_N \to \mathbf{d}_B\) applied to \(\mathbf{n}_A\). For differentiation, each link of the chain is written with a gauge frozen at the current configuration as a sum of signed solid angles \(\Omega(\mathbf{x}, \mathbf{y}, \mathbf{z}) = 2\operatorname{atan2}(\mathbf{x}\cdot(\mathbf{y}\times\mathbf{z}), 1 + \mathbf{x}\cdot\mathbf{y} + \mathbf{y}\cdot\mathbf{z} + \mathbf{z}\cdot\mathbf{x})\) of geodesic triangles, each of which involves at most two neighbouring tangents. The gradient and Hessian of \(\Theta\) are therefore local, in closed form, and have the half-bandwidth of the cable tangent; the value of \(\Theta\) is taken from the sequential product. This is the discrete parallel transport and holonomy of Bergou et al. (2008) applied to the Hermite centreline. The test suite checks the closed form against automatic differentiation of an independent reference implementation, against finite differences, and for invariance under rigid rotation.
\(\Theta\) is known modulo \(2\pi\). The solver carries its unwrapped value as a state of the line, unwraps each new evaluation to the branch nearest the last accepted value, and accepts no step that changes \(\Theta\) by more than \(\pi/2\); such a step is cut. A link or Gauss point at which the tangent turns by more than 120° within one element or between neighbouring nodes (\(1 + \mathbf{t}_1\cdot\mathbf{t} < 0.5\)) stops with a named error; a mesh that resolves the curvature is far from it.
Residual and tangent. The torsion term adds \(-M\,\nabla\Theta\) to the residual and
to the tangent. The rank-one term is not banded; each Newton step solves the bordered system by Sherman–Morrison on the band factorisation of \(\mathbf{B}\) with two right-hand sides, \(\mathbf{B}\mathbf{y} = -\mathbf{R}\), \(\mathbf{B}\mathbf{z} = \nabla\Theta\), \(\mathbf{x} = \mathbf{y} - \mathbf{z}\,(\nabla\Theta\cdot\mathbf{y})/(C + \nabla\Theta\cdot\mathbf{z})\), and accepts it on a backward-error test, with a shifted factorisation and iterative refinement as the fallback. Dropping the rank-one term would leave a Newton iteration that converges only linearly, at the rate \(\nabla\Theta^{\mathsf T}\mathbf{B}^{-1}\nabla\Theta / C\); on the post-buckled states of the validation (see CableDyn verification and validation) that rate is 0.05 to 0.22. The same term enters the dynamic effective tangent, scaled by \(1 - \alpha_f\) of the force blend, and is frozen together with a reused factorisation.
Statics and stability. The imposed twist is the last static load stage, ramped from the geometric twist of the untwisted equilibrium in steps of at most \(\pi/4\). Each converged stage is tested for stability by the inertia of \(\mathbf{K}\) on the free DOFs, \(\operatorname{neg}(\mathbf{K}) = \operatorname{neg}(\mathbf{B}) - [1 + \nabla\Theta^{\mathsf T}\mathbf{B}^{-1}\nabla\Theta/C < 0]\), with the negative eigenvalues of \(\mathbf{B}\) counted by a banded \(\mathbf{L}\mathbf{D}\mathbf{L}^{\mathsf T}\) factorisation. Above a buckling onset the straight twisted state is a saddle to which Newton converges quadratically; the static solve then descends along the lowest mode with energy acceptance onto the buckled branch, and reports the descent. The onset is that of a clamped–clamped rod under the dead tension \(T\), eq. (33) of van der Heijden et al. (2003); at \(T = 0\) it reduces to \(\tan x = x\), \(x = ML/2EI\), so \(M_{cr} = 8.9868\,EI/L\). At large tension it approaches the localised value \(2\sqrt{EI\,T}\) of an infinite rod from above, reported in the same paper. A bending-pinned end restrained in torsion transmits the torque about the bisector of the connection direction and the tangent (a semi-tangential end, a constant-velocity joint), for which \(\tan x = -x/3\) and \(M_{cr} = 4.9113\,EI/L\) at zero tension, not the \(2\pi EI/L\) of Greenhill’s axial-torque hinge. OrcaFlex’s bending-free end with a twisting spring gives the same value (see CableDyn verification and validation).
Dynamics and loads. The torsion force enters the force-blended residual like the internal
force, so a dynamic deck with torsion requires True alpha_force_blend (the default). Torsion
carries no inertia: the torque follows the imposed twist within the step, without a torsional
wave. This is accurate while the first torsional frequency of the line,
\(\sqrt{GJ/I_p}/(2L)\) with \(I_p\) the polar mass moment per length, is well above the
excitation; it is not for very long or torsionally soft lines. The moment of the torque on the
object at End A is \(M\,\partial\Theta/\partial\boldsymbol{\omega}_A\), the derivative with
respect to a rotation \(\boldsymbol{\omega}_A\) of that object (for a clamped end, of its frame
and the end tangent together), so the virtual work of the body moment is exact; for a straight
line it is the torque along the line axis. The static body equilibrium uses the same moment and
its stiffness \(\mathbf{a}\mathbf{a}^{\mathsf T}/C\), \(\mathbf{a} = \partial\Theta/\partial\boldsymbol{\omega}_A\).
Limits. Seabed friction does not resist twist, so the laid part of a line twists freely (the friction torque of a cable on the seabed can be comparable with the twist torque near touchdown). There is no torque–tension coupling of armoured cables or wire ropes, no anisotropic or pre-twisted section, and no self-contact: a loop that closes on itself is not resolved. The model is that of Meier, Popp & Wall (2014) and Bergou et al. (2008) restricted to the isotropic, torque-free-span case; it is not a new formulation.
Constitutive laws
Axial response is chosen per line type. Stateful laws advance their committed internal state once per converged step, from the converged kinematics, in a single commit that is rolled back together with the positions if any part of the step fails.
Linear elasticity
\(T = EA\,\varepsilon\). On the EI = 0 path the dynamic law is tension-only, as above.
On the Hermite path the axial law is two-sided: the energy
\(\tfrac{1}{2}EA\,\varepsilon^2\) is used for both signs of \(\varepsilon\), which keeps the
bending equations well posed in locally slack regions.
Tensile audit (Hermite path). Option tensile_safety inspects the axial force of the
line after each converged step. The audited force is the one that enters equilibrium, as in
the static branch audit below: the element-mean axial force (the three-point Gauss mean of the
signed resultant \(EA\,(|\mathbf r'| - 1)\)), averaged over each element and its two
neighbours weighted by length (one neighbour at a line end). The pointwise resultant of a
stiff-EA cubic-Hermite element oscillates about that mean, and a single element mean dips
next to a nodal seabed-contact reaction, by amounts that grow with the element length: judged
pointwise, a coarse mesh whose tensions and curvatures are converged reads as compressed (the
Gulf of Mexico 80 m lazy-wave cable of Lozon et al. (2025) under a
3 m, 12 s heave: tens of kN on 48 elements, where the fairlead
tension and the peak curvature are within 1 % of a 1024-element mesh). A compressed span of
three or more elements is never averaged away. A step is flagged when the three-element mean
falls below \(-\epsilon_\text{tol}\,\overline{EA}\), with \(\overline{EA}\) the
length-weighted EA of the same elements and \(\epsilon_\text{tol} = 2\times10^{-6}\) by
default (tensile_strain_tolerance). In error mode the step is rejected; in warn
mode it is committed and the worst event is reported (force, threshold, centre element, time).
The audit is a response diagnostic and does not change the constitutive law.
Axial damping
Axial Kelvin–Voigt damping adds a resultant proportional to the strain rate. On the
EI = 0 path, for an element with tangent \(\mathbf{t}\) and nodal velocities
\(\mathbf{v}_a, \mathbf{v}_b\),
applied along \(\mathbf{t}\) with the MoorDyn sign convention (Hall & Goupee, 2015). On the Hermite path the same law is integrated over the element,
with exact position and velocity Jacobians. A negative BA column value is a target
damping ratio \(\zeta = -BA_\text{input}\) of the axial mode of each segment, resolved per
element as
using the final element length after any mesh refinement. The damping share is included in the reported tension.
Viscoelastic rope (MoorDyn ElasticMod 2 and 3)
The viscoelastic law of Hall, Duong & Lozon (2023) is a four-parameter
series-Kelvin solid: a fast Kelvin–Voigt branch (spring \(EA_D\) parallel to dashpot
\(BA_D\)) in series with a slow Kelvin–Voigt branch (spring \(EA_1\) parallel to dashpot
\(BA_s\)). It reduces to the standard linear solid only when \(BA_D = 0\). The slow
spring is chosen so that the static composite stiffness equals the line-type EA:
Each element carries the slow-branch stretch \(\Delta l_1\) as a state. With \(\Delta l = \ell - L_0\),
The series chain is tension-only: the element carries \(\max(T, 0)\), and while \(T = 0\) the slow branch relaxes under zero load, \(BA_s\,\dot{\Delta l_1} + EA_1\,\Delta l_1 = 0\). At \(T = 0\) the loaded and slack states coincide, so the force and the state are continuous through slack. MoorDyn’s formulation applies the slack condition to the elastic part when \(\Delta l < 0\), so the two formulations differ through slack while \(\Delta l_1 \ne 0\) and agree whenever MoorDyn’s tension is non-negative with \(\Delta l \ge 0\). Inside a time step the state equation is linear in \(\Delta l_1\), so backward Euler eliminates it in closed form,
giving an element load with exact position and velocity Jacobians inside the implicit step. The initial state is the zero-rate partition \(\Delta l_1 = \Delta l\,(EA_D - EA)/EA_D\), at which the tension equals the static \(EA\,\Delta l/L_0\), so the static equilibrium is an exact fixed point of the dynamics.
ElasticMod 3 makes the dynamic stiffness depend on the mean load. For
\(\Delta l_1 > 0\) and \(k = EA/L_0\),
and \(EA_D = \alpha_\text{MBL}\) for \(\Delta l_1 \le 0\). It is evaluated from the
committed state and held over the step. Parameter sets with \(EA_D \le EA\), a stiffness
condition \((EA_D + EA)/(EA_D - EA) > 10^3\), negative dashpots, or \(BA_D + BA_s = 0\)
fail closed. The viscoelastic law applies to EI = 0 lines.
Syrope polyester model
The Syrope model (Falkenberg, Åhjem & Yang, 2017), in its MoorDyn-F/C form, describes polyester from an original working curve (OWC), a strain–tension table \(\varepsilon_\text{owc}(T)\). The strain is split into a fast part with the load-dependent stiffness
and a slow part \(\varepsilon_\text{slow} = \varepsilon_\text{curve}(T) - \varepsilon_\text{fast}(T)\), where \(\varepsilon_\text{curve}\) is the active curve (the OWC or the working curve below). \(T_\text{curve}(\varepsilon_\text{slow})\) denotes the inverse of this slow-strain relation: the mean tension at which the active curve has the slow strain \(\varepsilon_\text{slow}\). Loading beyond the running maximum mean tension \(T_\text{max}\) follows the OWC. Below it the line follows a working curve regenerated from \(T_\text{max}\) with 30 samples between \(\varepsilon_\text{min}\) and \(\varepsilon_\text{max} = \varepsilon_\text{owc}(T_\text{max})\), of linear, quadratic, or exponential shape; for \(x \in [0, 1]\) along the curve the tension is \(T_\text{max}x\), \(T_\text{max}x\,(p_2 x + 1 - p_2)\), or \(T_\text{max}(1 - e^{p_2 x})/(1 - e^{p_2})\), with \(\varepsilon_\text{min} = \varepsilon_0 + p_1(\varepsilon_\text{max} - \varepsilon_0)\) (or \(\varepsilon_\text{max} - T_\text{max}/p_1\) for the linear form with \(p_1 \ge 1\)). The shape parameters are the settings-file keys, \(p_1 = k_1\) and \(p_2 = k_2\). The accepted ranges are \(k_1 \ge 0\) for the linear shape, \(0 \le k_1 < 1\) and \(0 < k_2 \le 1\) for the quadratic shape, and \(0 \le k_1 < 1\) and \(10^{-6} \le k_2 \le 700\) for the exponential shape (beyond 700 the exponential overflows; below \(10^{-6}\) the ratio loses its digits to cancellation).
Each element carries two states: the slow strain \(\varepsilon_\text{slow}\) and
\(T_\text{max}\). With \(T_\text{mean} = T_\text{curve}(\varepsilon_\text{slow})\),
\(K_1 = \alpha + \beta\,T_\text{mean}\), and BA = BA_s|BA_d,
A compressed segment (\(\varepsilon < 0\)) carries no mean tension, only \(BA_s\,\dot\varepsilon_\text{slow}\), and a tension-only line clamps a negative result to zero.
The slow strain is eliminated implicitly in every in-step evaluation: a projected, bracketed backward-Euler solve, \(\varepsilon_\text{slow}^{n+1} = \varepsilon_\text{slow}^{n} + \Delta t\,\dot\varepsilon_\text{slow}(\varepsilon_\text{slow}^{n+1}, \varepsilon, \dot\varepsilon)\) constrained to \(\varepsilon_\text{slow} \ge 0\), gives the slow strain that the force uses. The tangent carries \(\partial\varepsilon_\text{slow}^{n+1}/\partial(\varepsilon, \dot\varepsilon)\) from the implicit-function theorem. The step commits that same slow strain, so the force applied to the nodes equals the reported tension. The working curve is the one regenerated at the committed \(T_\text{max}\). The branch follows the evaluated slow strain: the working curve applies below the OWC slow strain at \(T_\text{max}\), and the OWC applies at or above it, so the recovered mean tension is continuous across the switch. \(T_\text{max}\) then ratchets and regenerates the working curve when the mean tension exceeds it.
Initial condition. The static solve uses the rest curve of the initial load history: the
OWC for a virgin rope, or, with SYROPE IC, the working curve regenerated at
\(T_\text{max,0}\) (zero tension below its zero-tension strain, the OWC above its top
strain \(\varepsilon_\text{OWC}(T_\text{max,0})\)). A single secant axial stiffness of that
curve is iterated to the solved mean strain. Each element then starts with the running maximum
\(\max(T_\text{max,0}, T_e)\) (\(T_e\) for a virgin rope), where \(T_e\) is its
static tension, and with the slow strain whose kernel tension at the static strain and zero
strain rate equals \(T_e\). The dynamic state therefore starts in the static equilibrium,
with no initial transient. Where the static strain lies on the rest curve, that slow strain is
the curve’s rest point. Elsewhere, for example along a sagging line, it creeps at
\((T_e - T_\text{mean})/BA_s\). With fixed line ends the mean tension follows from the
geometry and \(T_\text{max,0}\), so \(T_\text{mean,0}\) does not enter the state. A
value that differs from the static mean tension by more than 1 % is reported on the console.
A strain below the working curve’s zero-tension strain (a slack rope on that history) stops at
initialisation.
The OWC table must cover the motion. An initial strain outside the table, or a state that
leaves it (a slow strain above the last row’s, or a total strain above the last table strain on
the OWC branch), stops with an error rather than extrapolating. The slow strain is well defined
only while the fast spring is stiffer than the working curve: its slope must stay below
\(\alpha + \beta T\) along the whole curve. For a given table and constants this holds
only for some running maxima. For example, the shipped EXP example constants
(\(\alpha = 1.53\times10^8\) N, \(\beta = 23.12\), \(k_1 = 0.2\), \(k_2 = 1.5\))
need \(T_\text{max} \gtrsim 7.4\times10^5\) N. Initialisation scans the table range. A
pretension or Tmax0 outside the admissible range is reported with the offending tension and
the next admissible \(T_\text{max}\).
An optional SYROPE IC row supplies the prior history
\((T_\text{max,0}, T_\text{mean,0})\); see the initial condition above. The model applies
to single-section EI = 0 taut lines; see Capabilities and route selection for the supported combinations.
Hydrodynamic loads
Fluid loads follow the Morison equation (Morison et al., 1950), decomposed into components normal and tangential to the local unit tangent \(\mathbf{t}\). Reference areas and coefficient names are listed in Conventions; \(d\) is the hydrodynamic diameter and \(A = \pi d^2/4\).
Drag
With the relative velocity \(\mathbf{u}_r = \mathbf{u}_f - \mathbf{v}\) split into \(\mathbf{u}_{r,t} = (\mathbf{u}_r\cdot\mathbf{t})\,\mathbf{t}\) and \(\mathbf{u}_{r,n} = \mathbf{u}_r - \mathbf{u}_{r,t}\), the drag per unit length is
Its derivatives with respect to position (through \(\mathbf{t}\)) and velocity enter the implicit tangent exactly.
Added mass and fluid inertia
The added-mass matrix per unit length and the Froude–Krylov plus fluid-inertia force for a fluid acceleration \(\dot{\mathbf{u}}_f = \dot{\mathbf{u}}_{f,n} + \dot{\mathbf{u}}_{f,t}\) are
The fluid acceleration is the local (Eulerian) acceleration of the wave field.
Buoyancy and wetting
Self-weight and buoyancy enter as the submerged weight per unstretched metre,
\(w = (\rho_A - \rho A)\,g\) (positive downward). An EQUIVALENT BUOYANCY row specifies
\(w\) and \(d\) directly and is converted to \(\rho_A = w/g + \rho A\).
On the EI = 0 path, an element crossing the free surface uses its wetted fraction
\(f\), the linear-interpolation fraction of the chord below the local surface elevation.
Drag, added mass, and fluid inertia act on the wetted part (drag with the wet-weighted relative
velocity), and a recovery load \(-(1 - f)\,\tfrac{1}{2}\rho A g L_0\) on each end node removes
the buoyancy of the dry part; the derivatives of \(f\) with respect to the nodal heights are
included in the tangent. On the Hermite path the hydrodynamic loads are integrated over the
deformed element length at three Gauss points, and a Gauss point above the local surface
carries no load. This is exact for fully submerged elements, the regime of lazy-wave cables and
moorings, and approximates the wetted length of a surface-piercing element by the Gauss
sampling. The Hermite buoyancy recovery is exact: the waterline crossings of the cubic
centreline height are located on each element, and the load \(\rho g A\) is integrated in
closed form against the Hermite shape functions over the dry intervals,
where \(\xi_c\) are the crossings and \(z' = \mathrm{d}z/\mathrm{d}\xi\); the tangent is the exact rate at which the crossings move. The surface is the still waterline of the drag configuration, or the element’s sampled elevation with a host-held field. The Hermite static solve uses the same recovery at the deck waterline, \(z = 0\).
The EI = 0 path assembles the added mass at the current configuration, with its configuration derivative in the tangent. The Hermite path evaluates the consistent added mass \(\int \mathbf{N}^{\mathsf T}\mathbf{m}_a\mathbf{N}\,\mathrm{d}s\) at the start of each step and holds it over the step. The spatial gradient of the wave field is omitted from the tangent on both paths; the residual is exact, so the converged state is unaffected.
Waves
Regular waves use linear (Airy) theory (Dean & Dalrymple, 1991). For height \(H\), period \(T\), water depth \(h\), and direction \(\beta\) (the direction of travel, see Conventions), the wavenumber solves the dispersion relation
by monotone bisection in \(kh\) to a relative interval of \(10^{-15}\). With \(\theta = k(x\cos\beta + y\sin\beta) - \omega t\),
where \(u_h\) acts along \((\cos\beta, \sin\beta, 0)\). The accelerations are the time derivatives at a fixed point, \(\dot u_h = \tfrac{H}{2}\omega^2 \frac{\cosh k(z'+h)}{\sinh kh}\sin\theta\) and \(\dot u_z = -\tfrac{H}{2}\omega^2 \frac{\sinh k(z'+h)}{\sinh kh}\cos\theta\). The vertical velocity is exactly zero at the bed.
Wheeler stretching (Wheeler, 1970) evaluates the kinematics at the stretched coordinate
which maps \([-h, \eta]\) onto \([-h, 0]\); points above the instantaneous surface receive zero kinematics. The mapping is undefined when the trough reaches the bed (\(h + \eta \le 0\)); that state fails closed with a named error. Without stretching, \(z' = z\).
Irregular waves are long-crested sums of Airy components, \(\eta = \sum_i a_i\cos\theta_i\), with \(\theta_i = k_i(x\cos\beta + y\sin\beta) - \omega_i t + \phi_i\). The total elevation is summed first and one Wheeler mapping against that total is applied to every component. The built-in JONSWAP spectrum (Hasselmann et al., 1973) is discretised as follows:
200 equal frequency bins on \([0.2\,\omega_p,\ 5\,\omega_p]\), with \(\omega_p = 2\pi/T_p\) and bin width \(\Delta\omega = 4.8\,\omega_p/200\); component \(i\) sits at \(\omega_i = 0.2\,\omega_p + (i - 1 + u_i)\,\Delta\omega\) with \(u_i\) uniform on \((0, 1)\);
spectral shape \(S(\omega) \propto \omega^{-5}\exp\!\left[-\tfrac{5}{4}(\omega_p/\omega)^4\right]\gamma^{r}\), \(r = \exp\!\left[-(\omega/\omega_p - 1)^2/(2\sigma^2)\right]\), with \(\sigma = 0.07\) for \(\omega \le \omega_p\) and \(0.09\) above, \(\gamma \ge 1\);
amplitudes \(a_i = \sqrt{2 S(\omega_i)\,\Delta\omega}\), with \(S\) scaled so that the discrete zeroth moment is \(m_0 = H_s^2/16\);
phases \(\phi_i = 2\pi v_i\) with \(v_i\) uniform on \((0, 1)\).
The draws \(u_i, v_i\) come from the WaveSeed option (default 1) through the
Park–Miller MINSTD generator (multiplier 48271, modulus \(2^{31} - 1\)), so a given
\((H_s, T_p, \gamma, \beta)\) and seed always produce the same sea on every platform, and
different seeds give independent realisations. Because the component frequencies are not
commensurate, the record does not repeat (an equally spaced comb would repeat after
\(2\pi/\Delta\omega \approx 42\,T_p\)). To reproduce a specific external realisation, supply
its component table (amplitude, frequency, phase), for example a WaterKin WaveKinMod 1
elevation history, which is reduced once to its Fourier components on the run time base.
Regular nonlinear waves. A stream wave is Dean’s stream-function wave
(Dean, 1965) solved numerically by the Fourier method of
Rienecker & Fenton (1981). In the frame moving with the celerity
\(c\), with \(X = x' - ct\) and \(Y = z + h\) measured from the bed,
and \(k\), the surface elevations \(\eta_m\) at \(X_m = m\pi/(Nk)\)
(\(m = 0 \ldots N\), crest to trough), \(B_j\), \(\bar U\), \(Q\) and
\(R\) satisfy the kinematic condition \(\psi(X_m, h + \eta_m) = -Q\) and the Bernoulli
condition \(\tfrac12\lvert\nabla\psi\rvert^2 + g\eta_m = R\) on the surface, a zero mean
level, \(\eta_0 - \eta_N = H\), and \(k\bar U T = 2\pi\) (no Eulerian mean current,
\(c = \bar U\)). Newton’s method solves the \(2N + 5\) equations in units of
\(h\) and \(g\), stepping the height up from a linear wave; StreamOrder sets
\(N\) (default 20). The fixed-frame velocity is
\(u = \sum jkB_j \cosh(jkY)/\cosh(jkh)\cos jkX\),
\(w = \sum jkB_j \sinh(jkY)/\cosh(jkh)\sin jkX\), the local acceleration
\(-c\,\partial/\partial X\) of these, the surface elevation the cosine interpolant of the
\(\eta_m\), and the dynamic pressure \(p/\rho + gz = R - \tfrac12\lvert\nabla\psi\rvert^2\).
The field is exact up to the free surface, so no stretching is applied; points above the
surface see still water, and the start-up ramp scales the whole field. Airy and irregular
seas keep the Wheeler stretching above.
Other spectra, spreading and multiple trains. Besides JONSWAP, a spectral train can be ISSC / Pierson–Moskowitz (Bretschneider, Pierson & Moskowitz, 1964), Torsethaugen two-peak (Torsethaugen & Haver, 2004), or Ochi–Hubble two-peak (Ochi & Hubble, 1976). With \(\omega_p = 2\pi/T_p\), the one-sided densities are
The Torsethaugen sea follows DNV-RP-C205 (DNV, 2021): with \(T_f = 6.6\,H_s^{1/3}\), a wind-dominated sea (\(T_p \le T_f\)) has the primary wind sea \(H_1 = r H_s\), \(r = 0.7 + 0.3\exp[-(2\epsilon_l)^2]\), \(\epsilon_l = (T_f - T_p)/(T_f - 2\sqrt{H_s})\), \(\gamma_1 = 35\,(2\pi H_1/g T_p^2)^{0.857}\), and a swell of \(H_2 = \sqrt{1 - r^2}\,H_s\) at \(T_{p2} = T_f + 2\) s; a swell-dominated sea has \(r = 0.6 + 0.4\exp[-(\epsilon_u/0.3)^2]\), \(\epsilon_u = (T_p - T_f)/(25 - T_f)\), \(\gamma_1 = 35\,(2\pi H_s/g T_f^2)^{0.857}(1 + 6\epsilon_u)\) and a wind sea at \(T_{p2} = 6.6\,H_2^{1/3}\) (\(\epsilon\) clamped to [0, 1], \(\gamma_1 \ge 1\), \(\gamma_2 = 1\)). Each partition is \(E_j G_0 A_{\gamma} \tilde\omega^{-4}\exp(-\tilde\omega^{-4})\gamma_j^{r}\) with \(\tilde\omega = \omega T_{pj}/2\pi\), \(E_j = H_j^2 T_{pj}/(32\pi)\), \(G_0 = 3.26\) and \(A_\gamma = (1 + 1.1(\ln\gamma)^{1.19})/\gamma\).
A spectral train is synthesised like the JONSWAP sea: WaveComponents (default 200) jittered
components over \([0.2\,\omega_{p,\min}, 5\,\omega_{p,\max}]\) of its partition peaks,
random phases, and amplitudes scaled so the train’s discrete \(m_0\) is exactly
\(H_s^2/16\) (\((H_{s1}^2 + H_{s2}^2)/16\) for Ochi–Hubble). A train with a spreading
exponent \(s > 0\) spreads its energy with the cos-2s function
discretised into WaveDirections (default 9) equal-angle bins whose weights are the bin
integrals of \(D\); each direction has its own jittered frequency set, and component
\(i\) travels along its own heading \(\beta_i\). Several trains (wavetrain rows) are
superposed component by component, each with its own heading and seed offset, and the
kinematics of the whole sea follow the irregular-wave conventions above: the total elevation
first, one Wheeler mapping against it, then the component sums. A long-crested JONSWAP train
reproduces the JONSWAP synthesis above for the same seed.
The optional rampTime \(T_r\) multiplies every wave amplitude by
\(r(t) = \tfrac{1}{2}\left[1 - \cos(\pi t/T_r)\right]\) for \(t < T_r\); the fluid
acceleration then includes the ramp rate, \(r\,\dot{\mathbf u} + \dot r\,\mathbf u\). The
current is not ramped, because the static initial condition already includes it.
Currents
A current is uniform or given as a profile of \(N \ge 2\) levels
\((z_j, \mathbf{u}_j)\) with strictly increasing \(z_j\). The velocity is interpolated
linearly between levels and held at the end values outside the profile. A WaterKin
CurrentMod 1 depth table uses the same interpolation. In OpenFAST (maintained by NLR, the National Laboratory of the
Rockies, formerly NREL), the SeaState field supplies waves and current; SeaState’s standard CurrMod = 1 steady current (sub-surface
power law \(U_{ss}\,[(z + h)/h]^{1/7}\), linear near-surface profile, and depth-independent
part) is reproduced so that wave and current contributions can be separated when a WaterKin
file selects them independently. CurrMod = 2 user currents cannot be separated and are
rejected whenever separation is required.
The fluid velocity seen by the drag is the sum of the current and the wave velocity. In a coupled run the fluid velocity, acceleration, and local surface elevation are sampled by the host at the nodes and held over the step.
Seabed contact
Contact acts at the line nodes on the translational DOFs. With the penetration \(g = z_\text{floor}(x, y) - z\) and the nodal stiffness \(k_n\) (Conventions), the normal force is a \(C^1\) penalty law with blend width \(\delta = 10^{-6}\,\mathrm{m}\):
The force and its tangent are continuous through first contact, which avoids an active-set kink in Newton’s method; beyond the blend the law is exactly linear. A normal damper acts only while the node moves into the bed,
where \(\chi\) rises linearly from 0 to 1 across the blend. With the total normal reaction
\(N = f_n^{(k)} + f_n^{(c)}\), seabed friction is a stick-slip spring on the horizontal
motion of every node, on the EI = 0 and the finite-EI lines alike. Node \(i\) holds a
horizontal spring of stiffness \(k_i\), the nodal normal contact stiffness (the shear
stiffness equals the normal stiffness, OrcaFlex’s default), to its anchor
\(\mathbf{a}_i\). With \(\mathbf{d} = \mathbf{x}_i - \mathbf{a}_i\) the horizontal
stretch, the spring sticks and slides:
and at every committed state a sliding spring’s anchor moves to the capacity distance behind the node (return mapping), so the force carries over exactly into the next step; a node that leaves the seabed takes its anchor with it. The anchors are state: they roll back with a failed or subdivided step, join the aggregate snapshot, and are part of the OpenFAST checkpoint. The force and its position and velocity Jacobians (through \(N\)) are exact. Unlike a velocity-regularised Coulomb law, the spring holds its force at rest.
Friction in statics. With seabed friction a static equilibrium is not unique: a grounded run can rest anywhere its friction holds it. Like OrcaFlex, CableDyn resolves this by assuming the position in which the line was originally laid and letting friction act toward it. OrcaFlex (Orcina’s documentation, Line data: statics, https://www.orcina.com/webhelp/OrcaFlex/Content/html/Linedata,Statics.htm, and Friction theory, https://www.orcina.com/webhelp/OrcaFlex/Content/html/Frictiontheory.htm) offers the statics seabed friction policies none, as laid and restart. As laid, with a step 1 statics method other than prescribed, the line is assumed laid starting with its bottom end at its position, in the lay azimuth direction away from it, following the seabed profile, and its top end then moved slowly to its position; the friction targets are spaced along the seabed by the unstretched segment lengths and the as laid tension, so a line with every node on its target carries that tension throughout. Friction applies in statics only to line-seabed contact and only when the policy is not none; each node’s force acts toward its target and is capped at \(\mu R\), and dynamics then starts from the target positions of statics. OrcaFlex warns that a lay azimuth far from the vertical plane through the line ends makes the statics harder to converge. The documentation consulted does not state the defaults of these three data items.
CableDyn implements the as laid and none policies (staticsFriction; restart has no
counterpart, and there is no prescribed starting shape). None is the default, as in MoorDyn:
the static solve is frictionless in still water, and in a deck current the friction springs
below hold the line from its still-water equilibrium (the same line solved without the current
first). It is conservative for anchor loads, since an as-laid run pulled taut is held back by
its friction (below). With asLaid the reference of
node \(i\) is the as-laid position: the line is laid from its lower end (the anchor, or the
fairlead when that is lower) along the azimuth \(\alpha\) away from it (layAzimuth; by
default the horizontal direction of the chord from that end, the vertical plane through the
line ends) on the seabed at the tension \(T_\text{laid}\) (asLaidTension, default 0),
so that the seabed path from that end to node \(i\) is \(s_i = \sum_{e<i} l_{0,e}(1 + T_\text{laid}/EA_e)\) long; on a
bathymetry the horizontal distance follows from integrating
\(\mathrm{d}s = \sqrt{1 + z_{,\alpha}^2}\,\mathrm{d}d\) in sub-steps of at most 1 m. A
laid node sinks into the penalty seabed along its normal by its static penetration, which on a
slope moves it horizontally by \(w_i\,(z_{,x}, z_{,y})/k_i\), with \(w_i\) its nodal
load: its share of the line’s submerged weight, plus the net weight of a discrete attachment
lumped at it (the finite-EI solve first smears an attachment over the two elements at its
node, then rebuilds the reference for the discrete loads before it polishes to them). Where
the slope varies, the laid tension turns at every node and adds
\(-T_\text{laid}\,(\mathbf{t}_{i}^{+} - \mathbf{t}_{i}^{-})\cdot\mathbf{n}_i\) to the
normal load (it presses the line into a crest and lifts it from a hollow), with
\(\mathbf{t}_{i}^{\pm}\) the unit tangents of the laid path on either side of the node and
\(\mathbf{n}_i\) the upward normal; a normal load \(P\) deepens the normal penetration by
\(P/k_i\) and so moves the node horizontally by \(P\,(z_{,x}, z_{,y})/(k_i s)\), with
\(s = \sqrt{1 + z_{,x}^2 + z_{,y}^2}\). The reference moves with the node, so the normal
sink adds no spurious friction deflection (a deflection normal to the seabed is not a friction
deflection, as in OrcaFlex’s in-plane law): the spring of a node laid at rest carries only the
load along the seabed that friction holds there, the slope component of its load (none on a
level seabed). The sink and the turning force depend on the element lengths (a node at a
section change carries a share of either section), so the reference is laid on the mesh of each
static solve, under that mesh’s lumped loads and nodal stiffnesses: a mesh the finite-EI statics
refines (adaptive refinement, the bending-length and stability refinements) or coarsens (the
mesh-sequenced and coarse-current passes) has the reference of its own nodes, and the march
takes it on the final mesh. The static solve then holds every node in
contact with a friction spring toward its reference, capped smoothly at \(\mu N\),
with \(\mathbf{d}\) the horizontal displacement from the reference and the anisotropic
capacity \(\mu(\mathbf d) N\) below when the axial and lateral coefficients differ. This is
the regularised Coulomb law of the stick-slip spring of the dynamics: the same stiffness, the
same capacity, a smooth corner where the dynamics has a sharp one (OrcaFlex ramps the force
linearly to \(\mu R\) over \(D_\text{crit} = \mu R/(k_s a)\)). The springs act in
still water and in a current alike; they are memoryless, so the equilibrium does not depend on
how the loads are brought in. On the finite-EI path their capacities are frozen (with the drag
of a current) in the fixed-point passes and live in the final Newton solve; on the EI = 0
path they are frozen at the line-search base point of the energy step. The EI = 0 solve
tries the as-laid seed first: the nodes up to a lift-off node on their references, the rest
hanging as the suspended catenary to the fairlead, the lift-off node the first at which that
catenary leaves the seabed tangentially (on a line whose fairlead lies below its other end, as
a FAST.Farm shared line may, the reference and this seed are laid from the fairlead). The
finite-EI solve starts its bending-stiffness continuation from the EI = 0 equilibrium with
the same friction, in 3D (only the two ends held) when the lay azimuth leaves the vertical plane
of the chord or the seabed under the laid run slopes across that plane: the seabed normal then
pushes the run out of the plane against its friction, which a solve in the plane would leave
unbalanced at the start of the march. A grounded run on a slope
of angle \(\theta\) then holds where \(\sin\theta \le \mu\) (the horizontal friction
force \(w\tan\theta\) per unit length against the capacity \(\mu w/\cos\theta\);
OrcaFlex’s in-plane law holds while \(\tan\theta \le \mu\), and the two limits differ by
under 1.6 % in \(\mu\) for slopes up to 10°), and a line longer than its span rests where it was laid
instead of sliding down the slope or folding past its anchor. A run pulled taut from its laid
position is held back by friction toward the anchor, so its anchor tension is lower than in a
frictionless static by up to \(\mu w\) times the grounded length (asLaidTension equal
to the run’s tension removes this). In the IEA-15 MW mixed example (850 m chains in 200 m of
water, frictionMu 0.35) asLaid lowers the chains’ anchor tension from 1351.6 to
368.2 kN (FairTen +0.16 %) and the laid power cable’s from 1230 N to 0.36 N. With
staticsFriction none the still-water static solve is
frictionless.
A line longer than the seabed run from its anchor to the point under its fairlead plus its
hang is laid past that point, and its suspended part turns back over the grounded run at the
touchdown. The EI = 0 path solves this fold; on the finite-EI path the bending stiffness
must round it into a hook of the bending length, from which the static solve may not converge
(the solve then fails closed with its route diagnostics): a lay azimuth across the span
(layAzimuth), a shorter line, or the default staticsFriction none with a line that hangs as a
catenary avoid it.
After the static solve the stick-slip anchors of the march start at \(\mathbf{a}_i = \mathbf{x}_i - \mathbf{f}_s/k_i\) for the static spring force \(\mathbf{f}_s\), so the march starts in the static equilibrium (OrcaFlex likewise starts dynamics from the targets of statics) and a line held at rest keeps its static end tensions. Without static friction the anchors are the static node positions and carry no force. Standalone, coupled OpenFAST and C API runs share this model (they build their lines with the same routines).
Anisotropic friction. With frictionMuAxial \(\mu_a\) and frictionMuLateral
\(\mu_n\) (the OrcaFlex axial and normal coefficients) the capacity depends on the slip
direction. With \(\hat{\mathbf a}\) the unit horizontal projection of the local line axis
(the nodal tangent on the finite-EI path, the chord through the neighbouring nodes on the
EI = 0 path), \(\hat{\mathbf n} = \mathbf e_z\times\hat{\mathbf a}\), and
\(d_a = \mathbf d\cdot\hat{\mathbf a}\), \(d_n = \mathbf d\cdot\hat{\mathbf n}\),
so a slip along the line is resisted by \(\mu_a N\), one across it by \(\mu_n N\), and the force stays collinear with \(\mathbf d\). Both laws above use the capacity \(\mu(\mathbf d) N\) in place of \(\mu N\), the return mapping included. The line axis is evaluated at every residual evaluation; the Jacobians carry the dependence of the capacity on the slip direction but omit its dependence on the line axis. A vertical axis (no horizontal projection) takes \(\mu_n\). An equal pair runs the isotropic law unchanged.
A structured bathymetry grid gives \(z_\text{floor}(x, y)\) by bilinear interpolation, clamped to the nearest edge outside the grid, with the creases of that surface rounded. Along every interior grid line the bilinear surface has a crease: its slope across the line jumps by \(J\) (the second difference of the tabulated depths). A node that must rest in a concave crease, a valley along a grid line or the bottom of a pit, takes the normal of one cell or the other and has no static equilibrium there; a slack chain piled at the bottom of a trench and a frictionless grounded run slid sideways into a valley both end there. Each crease is therefore rounded within a band of half-width \(\delta_k\), 5 % of the smaller grid spacing next to line \(k\). Along one axis the interpolant \(g(x)\) becomes
\(h'(0^+) - h'(0^-) = -1\) cancels the slope jump, and \(h\), \(h'\) and \(h''\)
vanish at \(|u| = \delta_k\), so \(\tilde g\) is \(C^2\). Its curvature
\(J_k (\delta_k - |u|)/\delta_k^2\) has the sign of the jump, so a valley stays convex and
gains no spurious pit. The surface is the tensor product of the two rounded interpolants,
\(z_\text{floor} = f + h_x J_x(y) + h_y \big(J_y(x) + h_x M\big)\), with \(f\) the
bilinear surface, \(J_x(y)\) the jumps of the grid rows across the \(x\) line
interpolated linearly along it, \(J_y(x)\) likewise, and \(M\) the mixed second
difference of the grid where two lines cross. Its height, gradient and curvature are
continuous everywhere within the grid. Outside the bands it is the bilinear surface, and
inside the band of one line it moves by at most \(|J|\,\delta/6\); where two bands
cross the two roundings and the mixed term add. A 0.2 slope jump on a 10 m grid moves
it by 1.7 cm, within the static penetration of a chain (w/(kBot Diam), 11 cm for a
120 mm chain at kBot = 1e5). A crease that would move the surface by less than 1 µm is not
rounded: an exactly planar grid and every patch without a crease keep the bilinear surface
exactly. A plane written to two or three decimals has creases of the size of its rounding,
which are rounded by a fraction of a millimetre. The grid edges are not rounded, and outside the grid the
surface stays level. Every height and gradient query uses this one surface: the contact of
the EI = 0 and the Hermite path, statics and dynamics, the friction frame and capacity,
the seabed clearance and touchdown outputs, and the Python post-processing
(cabledyn.Bathymetry). The friction capacity uses the normal force \(f_n^{(k)}(g/s)\);
inside a band it leaves out the curvature term of the contact force below, of the order of
the penetration times the curvature. A Fixed point placed at the
tabulated depth of a creased grid node lies up to \(|J|\,\delta/6\) below the rounded
surface; it rests on the seabed and is placed on the rounded surface.
On a sloped patch the elastic contact is
frictionless and normal to the surface, on the EI = 0 and the Hermite path alike: with
\(s = \sqrt{1 + z_{,x}^2 + z_{,y}^2}\) the force is \(f_n^{(k)}(g/s)\,\mathbf{n}\),
\(\mathbf{n} = (-z_{,x}, -z_{,y}, 1)/s\), and its tangent
\(f_n^{(k)\prime}\,\mathbf{n}\mathbf{n}^{\mathsf T}\) (the twist of a bilinear cell is not
differentiated); on a level floor this is exactly the vertical law above. Inside a crease band,
where the surface curves, the elastic force is the gradient of the penalty potential
\(E(g/s)\) of the normal penetration, \(-f_n^{(k)}\,\partial(g/s)/\partial\mathbf{q}\):
the normal force plus \(f_n^{(k)}\,g\,\nabla s/s^2\), a term of the order of the
penetration times the curvature, and its tangent is the Hessian of that potential (third
derivatives of the surface left out). The static energy minimisation then agrees with the
force up to the twist of the bilinear cell, which is left out of the tangent as on a sloped
patch, and a node held in a rounded valley by its curvature has a consistent stiffness. Its
damping acts on
the velocity along \(\mathbf{n}\), and a friction capacity is \(\mu\) times its
magnitude. The static solve and the dynamics use the same law, so a grounded run on a slope
rests at a static equilibrium of the march; its tension changes along the slope by the weight
component \(w\sin\theta\) per unit length, and a run pushed down the slope into its anchor
is compressed there. On the Hermite path the contact acts on the nodal position DOFs only, not
on the tangent DOFs.
Free and Connect points. A clump weight or a buoy (a Free or Connect point with
Mass or Vol, or a Point3 buoy, whose Body<N> attachment point steps as a free
point with the BODY Mass and Vol) has its own seabed contact, the line law above with
the stiffness \(k_p = k_\text{Bot} A\) and the damping \(c_p = c_\text{Bot} A\) over its
contact footprint
the face area of a cube of the point’s volume, CableDyn’s own area-equivalent of a compact
body (a point carries no shape), with \(\rho_s = 7850\ \mathrm{kg/m^3}\): a clump declared
by its mass alone, its buoyancy left out, takes the volume of solid steel of that mass. A point
with neither (a massless line junction) has no footprint. Without it only the end nodes of the
attached lines held a point on the bed, over their tributary length: a 50 t clump between two
polyester lines (kBot = 1e5) sank 2.8 m into the seabed; on its 3.44 m² footprint it rests
0.95 m deep, within the 1.43 m that its weight alone would press it in, and the lines no longer
hang it from the seabed. The contact acts along the surface normal of a sloped or rounded
bathymetry, in the point static equilibrium and in the monolithic and staggered point steps
alike (in the staggered step its stiffness and damping are linearly implicit, like the end
blocks of the attached lines), so a grounded point starts at rest. A Point3 buoy is not
placed by the static solve (it starts at its deck position) but carries the same contact in
every march, the monolithic and staggered steps and the coupled point step of the OpenFAST
aggregate and the C API. The contact is frictionless: the point routes reject frictionMu. A
point clear of the seabed sees exactly no contact load, and its results are unchanged.
Static equilibrium
Every run starts from a converged Newton equilibrium of
\(\mathbf{R}(\mathbf{q}) = \mathbf{f}_\text{int}(\mathbf{q}) - \mathbf{f}_\text{ext}(\mathbf{q}) = \mathbf{0}\)
on the free DOFs, not from dynamic relaxation. External loads are the submerged weight, the
seabed reaction, end-connection
springs, on the EI = 0 path the steady current, and, on the Hermite path, prescribed nodal
loads. The seabed reaction enters every residual and tangent evaluation; only the load
continuation of the EI = 0 path (step 4 below) scales it during its intermediate stages.
Waves act from the start of the dynamic march, whose initial acceleration is computed
consistently with them.
EI = 0 path
The static-only deck route and the dynamic initial condition (TMax = 0 included) call the
same initialiser, so both report the same equilibrium. An EI = 0 line cannot carry
compression: the accepted state is the tension-only equilibrium, with no element compressed
beyond a \(10^{-6}\) relative round-off band; a slack element reports zero tension.
Seeds, tried in order until one gives an admissible equilibrium: the analytical extensible catenary (grounded when the anchor lies within \(\max(1\,\mathrm{mm}, 10^{-5}\lvert z_\text{floor}\rvert)\) of the seabed, on the seabed plane through the anchor with its slope along the span; otherwise fully suspended); on a bathymetry, a grounded run draped on the actual seabed profile that lifts off tangentially into a suspended catenary; the catenary on the seabed lowered by the static penalty penetration \(w/(k_\text{Bot} d)\) (a soft seabed); for a grounded line longer than span plus rise, the \(H \to 0\) limit (a straight hanging leg with the excess on the seabed: where the seabed keeps falling past the fairlead the excess slides down the frictionless slope and folds back, with zero tension at the fold; otherwise it lies slack); the other catenary variant; and a two-leg polyline of the line length that never compresses the line. With static seabed friction the as-laid seed (friction in statics) is tried first. Seed nodes deeper below the seabed than their static penetration are lifted onto it.
Compression-capable Newton at the full load (Armijo line search on \(\tfrac{1}{2}\lVert\mathbf{R}\rVert^2\), Armijo, 1966). A converged state with every element in tension is the tension-only equilibrium and is accepted; a compressed strut or arch is rejected.
Tension-only Newton otherwise: the compressed branch gets a stiffness \(r\,EA\), with \(r\,EA\) of order \(10^{-3}\) and then \(10^{-7}\) of the largest nodal load (too weak to hold an arch), and finally \(r = 0\). Each stage is a Levenberg-Marquardt Newton iteration whose line search descends the potential energy (strain energy, seabed penalty potential, load potential; a steady-current drag is frozen at the base point of each search), so it approaches stable equilibria only; its shift never drops below \(10^{-12}\) of the geometric stiffness scale, which keeps directions without stiffness (a grounded node between slack elements) solvable.
Load continuation in the stages 0.25, 0.5, 0.75, 1.0 is the fallback of each law. Each stage scales the seabed stiffness with its load factor, which keeps a stiff contact well conditioned from a cold seed; the final stage restores the full stiffness, so the converged equilibrium is unchanged.
Convergence. \(\lVert\mathbf{R}\rVert_\infty / s < 10^{-8}\) or \(\lVert\mathbf{R}\rVert_\infty < 10^{-14}\), where \(s\) is the largest of the infinity norms of the external load, the seabed reaction, and any current drag; a residual that stops decreasing inside the round-off floor \(16\,\epsilon\,\max(EA/L_0)\max(\lvert q\rvert, L_0)\) of the nodal forces is converged. A line with no admissible equilibrium fails closed, naming the line, the seeds and stages tried, and the reason.
A steady current (uniform, or a depth profile sampled at each node’s own elevation as in the
march) enters the static equilibrium as Morison drag on the line at rest, so the out-of-plane
offset is part of the initial condition. Connect and Free points are then placed at
their static force balance: a trust-region Newton iteration on the point positions whose
residual sums the attached lines’ end forces (each line re-solved for the moving ends), the
point’s weight and buoyancy, its current drag, and its own seabed contact (see Seabed
contact above). It is the rigid-body equilibrium solve described under Rigid-body initial
condition below, with points as the only unknowns.
A tension-only line on a seabed that does not hold it can reach a folded (hairpin)
equilibrium: a current toward the anchor, or a seabed falling past it, drives the grounded run
beyond the anchor, and the run folds back to it with zero tension at the fold; a slack line can
likewise be laid out and back. The equilibrium is kept, and the driver prints one note per such
line, on every EI = 0 route (static, TMax = 0 and the dynamic initial condition, the
point and multibody systems, and the OpenFAST aggregate, whose line report returns it and whose
OpenFAST initialisation report prints it): the static layout carries the grounded run ... m
past its anchor, where it folds back (a hairpin) when grounded nodes lie more than 1 % of the
horizontal span plus 0.5 m beyond a grounded end whose other end is suspended, measured along
the horizontal chord (the criterion of the finite-EI note below; only for a horizontal span of
at least a tenth of the water depth at that end, or of the height of the other end above it if
that is larger, so that a near-vertical line, whose side away from its other end is not
defined, is left to the next test), or the static layout folds the grounded run back on
itself (a ... deg turn at node ...) when the run turns by more than 150°
at a node whose two neighbours are also grounded (within 5 cm of the floor); the node is the
deck node, numbered as the <J> of the node channels (1 = End A; an EI = 0 line is never
refined, so its deck nodes are its mesh nodes). The note names the
cause: a deck current acting on the line (a current profile that vanishes at the seabed still
pulls the run through the suspended span), a seabed steeper than 0.5° at the fold, or neither
(a slack run). Its remedy follows the deck’s static friction. Without frictionMu: friction
against a current, or friction toward the as-laid position (staticsFriction asLaid) on a
slope, since with the default staticsFriction none the static solve applies friction against
a deck current only. With frictionMu and staticsFriction none on a slope:
staticsFriction asLaid. With friction that already acts and still does not hold the run, or
on a deck whose route rejects frictionMu (Connect/Free points, Point3 buoys) or
staticsFriction asLaid (the multibody march): a shorter line. A run without dtM and
TMax takes the static-only line solve, which carries no seabed friction at all, so there
the note adds that friction needs dtM and TMax (TMax = 0 for the statics alone).
Under staticsFriction asLaid no note is made when the as-laid reference itself (the line
laid on the seabed from its lower end, the anchor or the fairlead when that is lower, along
layAzimuth or by default toward the other end, over its length) reaches past the grounded
end by more than the threshold above: the deck lays the run there, for example from the anchor
away from the fairlead, or from a lower fairlead toward the anchor and beyond it. The same rule
and the same causes and remedies apply to the finite-EI note below. The note never changes a
result. No
example deck and no EI = 0 line of the statics test matrices prints it except the folded
layouts it describes.
Hermite path
A finite-bending deck cable is initialised by the practice of “catenary statics, then full
statics” (continuation cable_statics, the default; sequenced cable_statics runs the
mesh-sequenced route described below first and this route as its fallback):
Exact EI = 0 seed. The multi-segment extensible catenary of the same sections (signed submerged weight, so buoyancy sections are included) is solved through the endpoints, on the declared seabed plane (flat
WtrDpth, or the least-squares plane of a structured bathymetry along the chord) or hung in mid-water when there is no seabed. The solve runs in the vertical plane of the chord, translated to the anchor, so a translated or rotated deck gives the same numbers to round-off. Over a bathymetry that is not a plane along the chord (a surveyed seabed with roughness, bumps or dips), the seed laid on the fitted plane meets the actual bed wherever the bed rises above that plane: every seed node deeper below the actual bed than its static penalty penetration (nodal submerged weight over the nodal contact stiffnesskBot\(\times\)Diam\(\times\) length) is lifted onto the bed, and the continuation below, which queries the full surface, settles the contact. Such a seed is never rejected for its penetration; the equilibrium it leads to is audited like any other. When the only catenary through the endpoints passes below a planar seabed (flat, or a structured bathymetry whose profile under the chord is straight), the line is too long for its span to hang as a catenary. Without buoyant sections the line is a convex curve between the chord and the seabed, so it cannot be longer than the path down to the seabed, along it to below the fairlead and up; a longer line is in the same case. Its bending stiffness can still hold an equilibrium: the excess length stands up as a bow in the vertical plane, carried in compression by the bending stiffness. That state is reached by displacement control, as a line is laid: the anchor is moved back along the chord (by the excess length plus a tenth of the line, then half as far again until the catenary exists), the line is solved there by the steps below, and the anchor is walked back to its position in steps, each a minimisation of the potential energy from a secant predictor (plain Newton in a current, whose drag is not a potential force), the step halved on a failure and doubled after a success. The walked state is polished, audited and checked for stability like any other; the driver then reports its element-mean compression. When the walk ends short of the anchor (the bow snaps through, or the line would have to lie on itself), or the bathymetry does not continue its plane behind the anchor, the run stops and names the geometry: no admissible planar equilibrium clear of the seabed was found, the line being too long for its span to hang as a catenary, with where the walk stopped. A line too long for the plane fitted to a non-planar bathymetry is walked the same way, the moved anchor kept at its height above the bed under it and the seed draped as above; as such a bed can hold the excess length in its dips, a failed walk there proves nothing, and the mesh-sequenced route runs before the run stops. A walked state that the audit below rejects (self-contact, instability) hands over to the mesh-sequenced route, and is named the same way when that finds none; withtensile_safety Truea walked state in compression is rejected by name (a bow held by bending is compressed). The walked arch faces the anchor. When the walk fails or its state is rejected, a line in still water over a flat seabed is also traced to the bow on the other side of the line hanging from the fairlead, which the walk cannot reach without passing the arch through that line. The anchor is placed beyond the fairlead, as far from it as the walk moved it back, where the arch faces away from the real anchor, and walked toward the fairlead. Each step is bounded so as not to pass the point where the grounded run, folded back about its touchdown point, reaches the real anchor (to 2 % of the span), and the walk stops there or at the last state before the arch turns over. The grounded run is folded, the corner at the touchdown relaxed by an energy minimisation, and the anchor walked the rest of the way into place; the state is polished, audited and checked for stability like the walked one. The bow is compressed, so it is not traced withtensile_safety True, nor in a current. On a mesh with elements shorter than \(\sqrt{EI/EA}\) the continuation below has an iteration budget of \(2\times10^{6}\) divided by the element count (a solve that converges there does so within it), and no other route follows a failure: the run stops in seconds and names the mesh.Continuation in
EI. The family \(\lambda\,EI\) is traced from a fraction whose bending length \((\lambda EI/|w|)^{1/3}\) is below a quarter of the element length up to \(\lambda = 1\), every load at full value. Each stage minimises the total potential energy (elastic, weight, buoyancy lost above the surface, seabed penalty, end springs) by a modified Newton iteration: the tangent is shifted until a banded Cholesky factorisation succeeds, so every direction descends toward a stable equilibrium, and the step length follows an Armijo line search on the energy. Close to equilibrium full Newton steps on the complete tangent take over, because the seabed contact and waterline laws are only piecewise smooth (and the curvature of a structured bathymetry grid is not differentiated). Stages that jump further than a small fraction of the line are retried at a smaller step; a first stage that does not converge is restarted at fullEI.Polish and diagnosis on the deck mesh at the static tolerance. A residual floor above it is accepted at \(10^{-6}\) (or, for a line on the seabed, \(10^{-5}\), nodes alternating across the 1 µm contact blend) and reported. An element folded beyond \(L_0\kappa = 0.7\) on the deck mesh is refined locally, from the polished state, until no element exceeds \(L_0\kappa = 0.35\);
adaptive_meshrefines the same way, to its own target. A refinement pass splits into \(2^k\) equal elements every element whose sampled \(L_0\kappa\) exceeds 0.12 (or the requested target, when stricter), that is longer than half its bending length \((EI/|w|)^{1/3}\), or, at a line end, longer than its boundary layer \(\sqrt{EI/T}\), and the elements about a node whose seabed contact differs from both neighbours; \(k\) is the smallest that meets the criterion, at most 4 (16 elements per deck element). The neighbours are then split until no element is more than twice as long as the next one (or than the ratio the deck mesh already has there) and none is left longer than both of its neighbours with fewer splits than either (a single coarser element between finer ones carries a spurious axial oscillation), the state is carried to the new mesh by the Hermite interpolant of each element and polished, and the pass is repeated on the polished state while the target is not met and an element can still be split (at most 12 passes), within 8192 elements: a pass that would exceed them gives up the bending-length splits first, then the grading, and only then the curvature, end-layer and contact splits. Elements the criteria leave alone keep their deck length, so a single coarse section costs only its own refinement (a 462-element lazy wave whose 180 m top section has two elements: 492 elements, where refining the whole line by the factor of its worst element gave 7392). A deck mesh too coarse for the bending length whose continuation or polish fails, or whose folded element the local refinement does not resolve, repeats the route on a mesh refined until the elements are at most half the bending length (when the continuation on that mesh runs out of its iteration budget, on the coarser powers of two down to twice the deck mesh); a rotationally restrained end whose boundary layer \(\sqrt{EI/T}\) is shorter than its element, which shows as a polish that does not converge or a compression at that end, repeats it on a mesh graded geometrically toward that end. The refined mesh is reported.Physical-branch audit. The converged state must be clear of self-contact (no two non-adjacent elements closer than their diameters), unfolded, and stable (positive definite tangent). Element-mean axial compression of a stable bent equilibrium is admissible and is reported; with
tensile_safety Trueit is rejected. That contract is judged, as in the dynamics, on the element-mean axial force averaged over each element and its two neighbours, compared with the tighter of the strain band \(2\times10^{-6}\,EA\) (with theEAof the centre element) and a thousandth of the peak tension (the largest pointwise resultant); the pointwise resultant of a coarse element oscillates about that mean (tens of kN on the Gulf of Mexico 80 m lazy wave at 48 elements) without entering equilibrium, and does not drive the mesh refinement. Before rejecting, the static solve splits the elements whose three-element mean falls below the band (and their neighbours) and solves again, since a coarse element can dip where the line is in tension; it stops, and the solve names the compression, when two passes do not relieve it, when it is the reaction of a held end (the minimum settled on the end node over two passes, which no splitting changes), when the polish on the split mesh does not converge, or at an element budget. The report is judged on element-mean forces: the mean over each element and its two neighbours (length-weighted) beyond the admissible band, or a nodal material axial force (the length-weighted mean of the two adjacent element means) in compression by more than a tenth of the peak tension. The nodal seabed-contact reaction kinks the line at the touchdown node, and the element there can carry a small negative mean between tensile neighbours at any mesh; that dip does not enter equilibrium and is not reported. A failing route falls back to the mesh-sequenced solve below, audited alike. On the Lozon cables the two routes give the same equilibrium to the solver tolerance (positions within 2e-5 m, curvature within 1e-6 1/m).
A deck current loads the line at rest with its steady Morison drag (the relative velocity is the current itself; the element fluid velocity is the mean of the current at its two nodes, as in the dynamics). The drag follows the line and is not a potential force: it is carried along the stiffness continuation frozen at the last accepted configuration, then converged at full stiffness by a fixed-point iteration on the frozen drag and a final Newton solve with the configuration-dependent drag and its Jacobian. If the frozen-drag continuation fails, the still-water equilibrium is found first and the drag is brought in by load steps. A load step that stalls hands over to a pseudo-arclength continuation in the drag fraction \(\lambda\): each corrector solves the bordered system \([K,\,-D;\ t_q^T,\ t_\lambda]\) (\(D\) the drag at full current) by block elimination with two banded solves, so the path can turn at a limit point in \(\lambda\). A current across the vertical plane of the line makes the static solve three-dimensional, as the dynamics are.
Without seabed friction, a current that pushes the grounded run toward its anchored end
harder than the suspended span pulls it back puts that run into compression; a cable cannot
hold it, and the run would have to buckle or slide past its end. There is no tension-only
equilibrium that continues the still-water layout in its plane. With the current in the
vertical plane of the line the problem is symmetric and the planar branch ends in a
pitchfork: the equilibrium beyond it turns the grounded run sideways on the seabed, past its
anchor and back (a lazy wave in a 0.7 to 1.2 m/s current along its plane, for example).
Newton cannot leave the plane of a symmetric problem, so the line is then solved in 3D under
the current with a cross-flow of 1 % of its horizontal speed, and that state is polished at
the exact current and audited like any other. A planar equilibrium in such a current is also
checked with the line free to leave its plane; one that is a saddle there (the grounded run
held in the plane only by the symmetry) is replaced by this 3D solve; when the 3D solve finds
no equilibrium, the planar one is kept and a note says that it is held in the plane of the line
and unstable out of it. Where the planar path
ends depends on the element size (whether a load step or a polish crosses the limit), so the
branch is selected on the deck mesh coarsened by halving every section’s NumSegs as long as
the elements stay within one bending length \((EI/|w|)^{1/3}\): that coarse line goes
through every route above, and its equilibrium is carried to the deck mesh by cubic-Hermite
interpolation along the unstretched arc length and polished there at the exact current, free
to leave the plane, then audited and checked for stability. Decks whose meshes are halvings of
one another select the same coarse mesh, and so the same equilibrium; a residual floor above
the tight tolerance on a fine mesh is accepted at the coupled static tolerance and noted. The
result can be a folded (hairpin) layout:
a very strong current on a frictionless seabed carries the line past its anchor and it folds
back to it, either as a loop lying on the seabed with its two legs side by side or, for a
lazy wave, with the buoyant arch carried beyond the anchor and the line coming back down to it.
Such a layout is a genuine equilibrium: the drag on the legs is held by the tension of both,
the seabed carries the grounded part, and the tangent is positive definite. OrcaFlex statics
seeded with the CableDyn shape converge to the same layout, and a dynamic run from it stays at
rest and returns to it after a disturbance. The standalone driver prints a note when the
grounded run of the static layout lies past its anchor (the static layout carries the
grounded run ... m past its anchor), because the layout differs from the usual one and a
real seabed with friction, or a shorter line, would not reach it. The note names the same
cause and remedy as the EI = 0 fold note above (one rule for both): the finite-EI static
solve also applies frictionMu against a deck current under staticsFriction none, so
frictionMu answers a current and staticsFriction asLaid a slope, and a shorter line
remains where that friction already acts or the route rejects it (the multibody march rejects
asLaid, and a deck with Connect/Free points or Point3 buoys frictionMu).
Under asLaid no note is made when the as-laid reference itself, the line laid from its
lower end along layAzimuth, reaches past the anchor. When the 3D solve also
fails, the solve stops with
the diagnosis (the element-mean compression, where it sits, and its height above the seabed)
and suggests frictionMu; neither the pseudo-arclength continuation of the planar path
nor the mesh-sequenced route, which reach the same planar limit, is then run. With friction the grounded run is held; a lazy wave whose buoyant arch the
current pushes toward the touchdown then goes into compression at the touchdown instead,
which friction cannot hold, and the solve names that. A line that is already compressed
without the current is named as such. A lazy wave whose seabed run points downstream reaches
these limits at a fraction of the current that the same layout holds when the current comes
from the other side.
The free surface uses the partial immersion of the circular section: over the band \(|z| < D/2\) the displaced area, and so the buoyancy lost above the water, follows the circular-segment fraction of the section, which gives a line floating at the surface its waterplane stiffness. The static and dynamic paths share this law.
The fallback and the dynamic Hermite path use
continuation in
EI, and, for a net-buoyant line, continuation in buoyancy, which tracks the smooth branch of the buoyant arch from the seed;the consistent Hermite self-weight, \(wL_0/2\) on each end \(r_z\) and \(\pm wL_0^2/12\) on the end \(m_z\). At a held line end resting on the seabed the floor carries the end element’s weight with this end moment (its normal share \(\cos\theta\,\mathbf{n}\) on a floor of slope \(\theta\) and upward normal \(\mathbf{n}\)), in the share \(3r^2 - 2r^3\) (\(0 \le r \le 1\)) in which the element lies on the floor, \(r\) the elastic contact force of its inner node over half the element’s weight: the nodal contact carries no moment, and the unbalanced end moment would otherwise turn the end tangent of a laid element into the floor, while an element rising from an anchor on the floor (\(r = 0\)) keeps it;
mesh sequencing: the equilibrium is selected on a coarse mesh that preserves section, load, and contact-diameter interfaces, prolonged through the Hermite interpolant, and polished on each finer level and finally on the exact user mesh with the same residual test. Deterministic fallback hierarchies are tried before a direct cold solve; every attempt uses the same physical inputs and tolerance;
a dimensionless per-DOF residual norm, with tangent-DOF residuals divided by the nodal tributary length.
The static tolerance is \(\min(10^{-6},\ \text{tol}_\text{dyn})\), where
\(\text{tol}_\text{dyn}\) is the dynamic Newton tolerance, so the dynamics start from an
equilibrium at least as tight as their own stopping criterion. Two checks keep a numerically
balanced but unresolved state out of the dynamics: an element whose sampled
\(\max(L_0\,\kappa) > 0.7\) is an unresolved fold and is rejected, and, with
adaptive_mesh, exhausting the refinement cap while the curvature or contact topology remains
under-resolved is reported.
Coupled systems of lines, points, and bodies are initialised in sequence: non-line objects are held while each line is solved, then released and the coupled system is solved.
Modal analysis
With nModes each line’s natural frequencies and mode shapes about its static equilibrium
solve the generalised symmetric eigenproblem
over the free degrees of freedom, the two ends held (a fixed anchor and a held or prescribed
fairlead). \(\mathbf K\) is the static tangent stiffness: for an EI = 0 element of rest
length \(l_0\), current length \(l\) along \(\mathbf e\), and tension
\(T = EA(l/l_0 - 1)\) (zero on a slack tension-only element),
\(\mathbf k = (EA/l_0)\,\mathbf e\mathbf e^{\mathsf T} + (T/l)(\mathbf I - \mathbf e\mathbf e^{\mathsf T})\)
assembled as \([[\mathbf k, -\mathbf k], [-\mathbf k, \mathbf k]]\); for a cubic-Hermite
element, the Hessian of its strain energy. The linearised seabed normal contact adds its
stiffness on the contacting nodes. \(\mathbf M\) is the consistent structural mass plus the
Morison added mass at the static shape. Friction, damping, and drag are left out, as in any
undamped modal analysis. \(\mathbf K\) and \(\mathbf M\) are banded (the nodes are
numbered along the line), and only the nModes lowest modes are computed: LAPACK dsbgvx
gives the eigenvalues, inverse iteration with the banded factors of
\(\mathbf K - \omega^2\mathbf M\) gives each shape (M-orthogonal to the shapes before
it, so a repeated frequency gets independent shapes), and the returned \(\omega^2\) is the
Rayleigh quotient of the shape, summed in quadruple precision. That quotient keeps the lowest
frequencies of a long, axially stiff line accurate, where the band reduction alone loses
digits. Memory grows linearly with the line length. The frequencies
\(f = \omega/2\pi\) and the nodal translations of each mode (scaled to a unit largest
nodal displacement) are written to <root>.modes.out. For a taut string of tension
\(T\) the transverse frequencies are \(f_n = (n/2L)\sqrt{T/m}\); for a shallow sagging
cable the out-of-plane and antisymmetric in-plane modes follow the string with the horizontal
tension \(H\), and the symmetric in-plane modes satisfy
\(\tan(\beta/2) = \beta/2 - (4/\lambda^2)(\beta/2)^3\), \(\omega = (\beta/L)\sqrt{H/m}\),
with \(\lambda^2 = (wL/H)^2\,L/(H L_e/EA)\) (Irvine & Caughey, 1974);
the modal analysis reproduces both.
Time integration
Both paths advance
with the generalised-α method (Chung & Hulbert, 1993). There are no
rotation variables, so \(\mathbf{q}\) and \(\ddot{\mathbf{q}}\) share one vector space and
no gyroscopic or rotation-chart terms arise. The parameters follow from the spectral radius
\(\rho_\infty\) (rhoInf, Conventions):
With \(\mathbf{q}_{n+1}\) as the Newton unknown,
and the residual is evaluated at the intermediate states
On the finite-EI path time-dependent loads (waves) are then evaluated at \(t_\alpha\); the
EI = 0 path holds the fluid fields over the step (below). By default both paths blend the
forces instead (alpha_force_blend, default True; False selects the form above),
with the same accuracy and tangent. On the Hermite path the position and tangent-vector DOFs
rotate with the line, and a blended configuration \(\mathbf{q}_\alpha\) would shorten the
tangents of a rotating element, a spurious stretch of order
\(\alpha_f(1 - \alpha_f)(\omega\Delta t)^2 EA\) that raises the mean tension at large time
steps. There \(\mathbf{r}_n\) is the previous step’s converged force, reused unless the state
or the load configuration changed in between. Since it enters every later step, the accepted
state is taken one Newton correction beyond the tolerance (the last factorisation reused), so a
loose Newton tolerance does not bias the response. Both forms converge to the same response; the
force blend removes the rotation-driven bias (see alpha_force_blend in OPTIONS reference and defaults).
On the EI = 0 path the force blend covers the axial force of the elements and every
dynamic load: axial damping, the viscoelastic and Syrope laws, seabed contact, damping and
friction, Morison drag, Froude–Krylov and buoyancy loads. Added mass stays at
\((\mathbf{q}_\alpha, \mathbf{a}_{\alpha_m})\). \(\mathbf{r}_n\) is evaluated afresh at
the committed state of every step, with the viscoelastic and Syrope elements at their committed
internal state (and, after a length change by active line control, at the lengths the committed
state was reached with; see Active line control), so the scheme carries no extra state and a
restart is exact; a re-solve of the
same step from the same state (the junction coupling) reuses it. The fluid fields of an
EI = 0 line (current, wave velocity and acceleration, and the waterline) are held over the
step: before the step the host (the standalone driver, or the coupling host) samples them for
the end of the step, \(t_{n+1}\), at the committed node positions, and every load
evaluation of the step uses them. Both shares of the force blend therefore see the fields of
\(t_{n+1}\), so \(\mathbf{r}_n\) is \(\mathbf{r}(\mathbf{q}_n, \mathbf{v}_n)\) with
those fields rather than with the fields of \(t_n\), and the configuration blend evaluates
the loads with them at \((\mathbf{q}_\alpha, \mathbf{v}_\alpha)\) rather than at
\(t_\alpha\), as it did before the force blend. The finite-EI path instead evaluates its
wave loads at \(t_{n+1}\) for the end share and reuses its \(\mathbf{r}_n\), evaluated at
its own \(t_n\) in the previous step. The configuration blend fails at slack–taut
transitions. A tension-only element is a
one-sided spring, and the residual sets its force at \(\mathbf{q}_\alpha\); the end state
\(\mathbf{q}_{n+1} = (\mathbf{q}_\alpha - \alpha_f\mathbf{q}_n)/(1 - \alpha_f)\) extrapolates
past it. A segment slack at \(t_n\) that carries a tension at \(t_\alpha\) thus ends the
step stretched by \(\alpha_f/(1 - \alpha_f)\) of its former slack on top: with
\(\rho_\infty = 0.4\) a 2 m segment of a light, axially stiff cable (80 kg/m, EA 8e8 N) that
was 0.2 m slack reports about 30 MN. A mass dropped 1 m onto a tension-only spring
(\(\omega = 1000\) rad/s) peaks at 6.9 MN at \(\Delta t\) = 0.1 s with the configuration
blend, against the resolved 0.35 MN; the force blend gives 5.7 kN, the under-resolved impact
dissipated as at \(\rho_\infty = 0\).
Release cap. The force blend alone still creates energy when a taut element releases
within one step. An element with tension \(T_n\), stretch \(e_n = l_n - l_0 > 0\) and
chord \(\mathbf{d}_n = l_n \mathbf{t}_n\) at \(t_n\) keeps \(\alpha_f T_n\) along
\(\mathbf{t}_n\) for the whole step, a constant force. Over the relative displacement
\(\mathbf{d}_{n+1} - \mathbf{d}_n\) of its ends it returns the work \(\alpha_f T_n p\) to
the motion, with the projected shortening \(p = l_n - \mathbf{t}_n \cdot \mathbf{d}_{n+1}\).
This is not the shortening \(e_n - e_{n+1}\) of the element: since
\(p = (e_n - e_{n+1}) + l_{n+1}(1 - \mathbf{t}_n \cdot \mathbf{t}_{n+1})\), a segment that
turns as it releases moves much further against that force than it shortens. The end share
\((1 - \alpha_f) T_{n+1} \mathbf{t}_{n+1}\) returns \(-(1 - \alpha_f) T_{n+1} p_{n+1}\),
with \(p_{n+1} = l_{n+1} - \mathbf{t}_{n+1} \cdot \mathbf{d}_n\). The element released only
its strain energy \(W_n = \tfrac{1}{2} T_n e_n\). An element that ends slack
(\(T_{n+1} = 0\)) therefore creates energy once \(p > e_n/(2\alpha_f)\): along a fixed
chord, a release deeper than \(e_n (1/(2\alpha_f) - 1)\) (0.75 \(e_n\) at
\(\rho_\infty = 0.4\)); a nearly unstretched segment that turns by 90° returns about
\(\alpha_f T_n l_n\), hundreds of times \(W_n\). On a slack light cable this feeds a
sustained rattle of single-step tension spikes. The cap bounds the work of the element’s blended
force by \(W_n\): along \(\mathbf{t}_n\) the element carries the smaller of
\(\alpha_f T_n\) and \((W_n + (1 - \alpha_f) T_{n+1} p_{n+1})/p\), which is the discrete
energy gradient \(W_n/p\) when it ends slack, and the residual and the tangent take the
difference (\(\partial p/\partial \mathbf{d}_{n+1} = -\mathbf{t}_n\)). With
\(T_{n+1} = T_n e_{n+1}/e_n\), the tension of a linear element, the cap fades out as the end
state crosses the rest length, so the residual stays continuous in \(\mathbf{q}_{n+1}\). A
cap on the slack end states alone would jump there, by up to \(\alpha_f T_n\) for a turned
element. The cap does not act on an element that ends above \(\alpha_f/(1 - \alpha_f)\,T_n\)
(0.4 \(T_n\)), nor on one that keeps its direction and ends taut, nor (but for the share of
its dashpot, below) on one slack at \(t_n\): those elements, every element going taut (where the force blend dissipates), and
every line that allows compression keep the force blend. In the slack-cable case above the
internal forces created 2.7e8 J over 20 s with the configuration blend, 6.7e7 J with the force
blend alone, 0.19 J with a cap on \(e_n - e_{n+1}\) (the turn of the releasing segments) and
4e-4 J (round-off) with the projected cap. The fairlead peak is 0.10 MN at
\(\Delta t\) = 2.5 ms, which is not a converged peak. The undamped EI = 0 deck has no
converged snap peak: the peak rises as \(\Delta t\) falls (0.32 to 0.46 MN at microsecond
steps, also with an explicit lumped-mass solver), as single segments rattle, and falls with
mesh refinement at a fixed step; its 20 ms averages converge to 0.083 MN at the fairlead and
0.047 MN at the anchor, the slack-to-taut event, and converged finite-EI runs give 0.093 MN at
the fairlead and 0.057 MN at the anchor (see Modelling workflow, snap loads on slack lines). For
a linear element the
force blend and the cap reduce to the configuration blend, so the linear spectral properties and
the second-order accuracy are unchanged.
The segments whose tension comes from a constitutive law, the viscoelastic and Syrope segments
(whose law enters the step as a dynamic load), are capped in the same way: \(T_n\) is the
law’s tension at the committed state, the force \(\mathbf{r}_n\) carries, with the same
\(e_n\) and \(W_n = \tfrac{1}{2} T_n e_n\). For the viscoelastic law at rest \(W_n\)
is the energy of its two springs, which both carry \(T_n\) and whose stretches add up to
\(e_n\); for the Syrope law, whose loading curve stiffens, it bounds the energy below the
curve from above. A law’s tension is not proportional to the stretch: damping or the lagged
internal state can carry \(T_n\) at a vanishing \(e_n\), where the secant
\(T_n/e_n\) is unbounded and the cap would switch off within a band of width about
\(e_n\), nearly a jump for the Newton iteration. The cap therefore estimates the end
tension as \(\hat T_{n+1} = k\,e_{n+1}\) with \(k = \min(T_n/e_n, k_n)\), \(k_n\)
the law’s axial stiffness at the committed state (\(EA_D/l_0\) for the viscoelastic law,
\(K_1/l_0 = (\alpha + \beta T_\text{mean})/l_0\) for Syrope). In general the blend of an
element returns at most \(W_n + (1 - \alpha_f)(\hat T_{n+1} - T_{n+1})\,p_{n+1}\) over the
step, \(T_{n+1}\) its actual end tension, and never more than without the cap. For an
elastic segment the two tensions coincide and the bound is \(W_n\). A law segment that ends
below \(l_0\) with no law tension returns at most \(W_n\) as well; one that ends above
\(l_0\) with less law tension than the estimate (zero once the slow branch has taken up the
stretch, or below Syrope’s minimum strain) can return up to
\((1 - \alpha_f)\,k\,e_{n+1} p_{n+1}\) more. With the light cable above made lightly damped
viscoelastic (EA 8e8 | 1.6e9 N, BA 1e8 | 1e6 N s) the anchor peaks at 0.085 MN at
\(\Delta t\) = 0.1 s with the cap on its
segments (the snap is not resolved at that step, so the peak depends on round-off: other
compilers and platforms give other values) and at 0.145 MN without it, against 0.082 to
0.095 MN at \(\Delta t\) from 0.5 to 2.5 ms. A taut Syrope line in 4 m
surge that goes slack peaks at 1.64 MN at 0.1 s, against 1.78 MN at 1 and 2.5 ms, with or
without the cap. These figures take every step as given (off ei0_substeps); with the
rotation- and snap-resolved sub-steps of the default (below) the two lines peak at 0.062 MN and
1.81 MN. A single peak of an unresolved snap says little: over seven runs whose heave amplitude
differs by up to \(3\times10^{-4}\) the viscoelastic anchor peaks at 0.056 to 0.079 MN
(mean 0.063 MN) with the sub-steps and at 0.056 to 0.085 MN (mean 0.065 MN) without them.
An elastic segment with axial damping (BA) also keeps \(\alpha_f T_{d,n}\) along
\(\mathbf{t}_n\), \(T_{d,n}\) the tension of its dashpot at \(t_n\). A segment that
extends at \(t_n\) and releases within the step carries that force against the shortening
as well: in the test chain (dashpot tension 1.6 to 62 times the elastic one) the blend of a
segment that turns and releases returned up to 2750 \(W_n\) with only its elastic share
capped.
The dashpot pushes as well as pulls, so its end share
\((1 - \alpha_f) T_{d,n+1}\mathbf{t}_{n+1}\) acts at a slack end state too, and it stores no
energy, so the budget stays \(W_n\). The cap covers the dashpot with a weight \(\phi\)
that is 1 while the end state is at or below \(l_0\) and falls linearly to 0 at
\(l_0 + c\,e_n\), \(c = \alpha_f/(1 - \alpha_f)\) (the end stretch below which the cap
can act on an elastic segment): the kept force \(\alpha_f (T_n + \phi T_{d,n})\) is replaced
by \(N/p\) when that is smaller, with
\(N = \max(W_n + (1 - \alpha_f)(\hat T_{n+1} + \phi T_{d,n+1})\,p_{n+1}, 0)\) and
\(T_{d,n+1}\) the dashpot tension the residual carries at
\((\mathbf{q}_{n+1}, \mathbf{v}_{n+1})\); the tangent carries its dependence on
\(\mathbf{q}_{n+1}\) through \(\mathbf{v}_{n+1}\) (the factor
\(\gamma/(\beta\Delta t)\)). Over a step in which it shortens along \(\mathbf{t}_n\)
(\(p > 0\); the cap leaves the other steps alone) the blend of such a segment then returns
at most \(W_n + (1 - \phi) D\), \(D = \alpha_f T_{d,n} p - (1 - \alpha_f)
T_{d,n+1} p_{n+1}\) the work of its uncapped blended dashpot, except when the end share alone
returns more than \(W_n\) (\(N\) would be negative; the capped kept force is then
zero). At
and below \(l_0\) that is \(W_n\). Above \(l_0 + c\,e_n\) the cap is the elastic one,
so the force-blended dashpot of a taut segment keeps the standard generalised-α treatment, under
which a linear dashpot is dissipative; \(\phi\) is continuous, and so is the capped force.
The viscoelastic law keeps its total tension, dashpots included, as \(T_n\) (above).
A damped segment slack at \(t_n\) (\(T_n = 0\), \(e_n \le 0\)) whose ends separate there keeps \(\alpha_f T_{d,n}\) along \(\mathbf{t}_n\) with no strain energy behind it, and returns \(\alpha_f T_{d,n} p\) when the step turns or reverses its ends. It has no \(W_n\) (and an empty band, \(\phi = 1\)); its budget is the dashpot’s own one-step dissipation \(W_d = T_{d,n}\,|\dot l_n|\,\Delta t\), \(\dot l_n\) the dashpot’s elongation rate at \(t_n\) (the axial rate, less \(l_n \dot l_0/l_0\) while the line pays out), so that \(T_{d,n} = (BA/l_0)\,\dot l_n\) and \(W_d > 0\) whenever \(T_{d,n} \ne 0\). When \(p > 0\) the kept force \(\alpha_f T_{d,n}\) is replaced by \((W_d + N_d)/p\) when that is smaller, with the end share’s credit \(N_d = \max((1 - \alpha_f) T_{d,n+1} p_{n+1}, 0)\), and the tangent takes the difference as above (a compressive \(T_{d,n}\) is capped alike when \(p < 0\)). The kept dashpot share of an element slack at the step start returns at most its own one-step dissipation \(T_{d,n} |\dot l_n| \Delta t\) plus the end share’s credit. The capped force grows without bound as \(p \to 0^+\), so it is continuous there; it acts only where \(\alpha_f |p| > |\dot l_n| \Delta t\), a turn or reversal well beyond the step-start motion whose end share does little work (a reversal along the chord feeds the end share’s dashpot, whose credit covers it). In the test chain the kept share returned up to 46 times \(W_d + N_d\) before the cap covered slack starts. Taut-start and undamped segments are unchanged.
The one-sided seabed contact is not capped. A node that leaves the floor within a step keeps \(\alpha_f\) of its step-start contact force \(N_n\) for the whole step: over a lift-off distance \(d\) it returns at most \(\alpha_f N_n d\) to the motion, against the energy \(\tfrac{1}{2} N_n g_n\) of the penalty spring at the penetration \(g_n\) (millimetres). The excess is the step-start support acting for \(\alpha_f\) of a step longer than it should, an impulse of at most \(\alpha_f N_n \Delta t\) per lift-off that vanishes with \(\Delta t\); the contact law’s sloped-floor normal and the friction anchors, which the step does not see, leave no energy bound as simple as the segments’. Seabed damping and friction are dissipative and keep the plain force blend as well. The effective tangent is
where \(\mathbf{K}_q = \partial(\mathbf{f}_\text{int} - \mathbf{f}_\text{ext})/\partial\mathbf{q}\)
and \(\mathbf{K}_v = \partial(\mathbf{f}_\text{int} - \mathbf{f}_\text{ext})/\partial\mathbf{v}\)
collect the structural, damping, drag, contact, and friction terms, and \(\mathbf{M}\)
includes the added mass. On the EI = 0 path the added-mass contribution also carries its
configuration derivative.
Newton iteration. An EI = 0 step starts from the Newmark predictor
\(\mathbf{q}^{(0)}_{n+1} = \mathbf{q}_n + \Delta t\,\mathbf{v}_n + \Delta t^2(\tfrac{1}{2} - \beta)\,\mathbf{a}_n\)
(the acceleration guess \(\mathbf{a}^{(0)}_{n+1} = \mathbf{0}\)). A Hermite step with the
force blend (the default) starts from the constant-acceleration predictor
\(\mathbf{q}^{(0)}_{n+1} = \mathbf{q}_n + \Delta t\,\mathbf{v}_n + \tfrac{1}{2}\Delta t^2\,\mathbf{a}_n\)
when the committed acceleration balances the committed force \(\mathbf{r}_n\),
\(\lVert\mathbf{M}\mathbf{a}_n + \mathbf{r}_n\rVert \le \tfrac{1}{2}\lVert\mathbf{M}\mathbf{a}_n\rVert\)
over the translational free DOFs, and from the Newmark predictor otherwise (and always with
False alpha_force_blend): while an
under-resolved mode rings, a loosely converged step would keep the extrapolated acceleration,
whereas the zero guess damps it. A failed constant-acceleration attempt is re-solved from the
Newmark predictor. The test uses only the committed state, so a restart is exact.
The EI = 0 path uses an Armijo line search on \(\tfrac{1}{2}\lVert\mathbf{G}\rVert^2\)
and converges when \(\lVert\mathbf{G}\rVert_\infty\) relative to the larger of the external
load and the initial residual falls below the relative tolerance, or below the absolute
tolerance. Two near-tolerance cases are also accepted. When the iteration budget is spent or
the line search stalls, a residual within twice the relative tolerance (never looser than
\(\sqrt{\epsilon}\)) is converged. A residual that stops contracting is converged when it is
below \(10^{-4}\) of the force scale and within the round-off floor
\(\epsilon\,\max\lvert q\rvert\,\lVert\mathbf{A}\rVert_\infty\) of the effective operator
\(\mathbf{A}\), which no Newton step can reduce (a finely meshed stiff line at small time
steps sits there). The Hermite path uses a backtracking line search that halves the step until the
residual decreases; its convergence norm divides each translational residual by a force scale
frozen at the predictor and each tangent-DOF residual additionally by the nodal tributary
length. The force scale is the largest inertial or residual entry, bounded below by 1 N, by the
peak element weight, and by the round-off floor
\(16\,\epsilon\,\max(EA/L_0)\,x_\text{ref}/\text{tol}\) of the axial forces
(\(x_\text{ref}\) the largest coordinate magnitude or element length). The
deck defaults are relative tolerance \(10^{-8}\), absolute tolerance \(10^{-14}\), 30
iterations, and 12 backtracks (option dynamic_solver). Option modified_newton reuses the
factorised tangent within a step while each iterate reduces the residual to at most a quarter
of the previous one, and rebuilds it otherwise; it does not relax the tolerance.
A standalone Hermite step on the constant-acceleration branch (so only with the force blend) also starts from the previous step’s factorised tangent, under the same quarter-contraction rule for at most three iterations, and must reach \(0.3\) of the tolerance while on it (a reused factor converges linearly, so this keeps the result as close to the converged solution as a fresh tangent’s last quadratic step). If a step on the reused factor still needs as many factorisations as the last fresh step, the reuse saved nothing: the following steps then factorise fresh, for 1 step after the first such step, 2 after the second in a row, and so on, doubling up to 32 steps. A step that fails on the reused factor is re-solved on a fresh one. Cables run through the coupling aggregate (OpenFAST, FAST.Farm, and mixed mooring-and-cable decks) factorise every step, because the reused factor is not part of the checkpoint state.
Prescribed motion. Prescribed DOFs (a moving fairlead or hang-off) are held at their
target position during the Newton solve; their prescribed velocity and acceleration enter the
free-DOF residual through the mass and stiffness coupling. At initialisation, and after a
non-stepping motionFile boundary commit, the initial acceleration solves
\(\mathbf{M}_{ff}\,\mathbf{a}_f = \mathbf{f}_f - \mathbf{M}_{fp}\,\mathbf{a}_p\), so an
accelerated fairlead loads the adjacent free DOFs through the off-diagonal consistent mass. The
core does not ramp prescribed motion; ramp it from rest over at least one wave period to avoid a
start-up transient.
Adaptive step subdivision. If a step fails to converge, or (Hermite path) converges but
changes the chord of any element (the relative position of its two nodes) by more than 20 % of
its length, turns any nodal tangent by more than 15°, or shrinks a nodal tangent magnitude
\(\lvert\mathbf{m}\rvert\) below 0.01 (a singular parametrisation), the step-start state is restored and the
same interval is re-advanced as
\(n\) substeps of \(\Delta t/n\), with \(n = 4, 16, 64, 256, 1024\) (cap set by
recovery_max_substeps, 4 to 65536). Inside the interval, prescribed position, velocity, and
acceleration follow one quintic (\(C^2\)) trajectory that lands exactly on the target
values at \(t + \Delta t\); host fluid fields and the start-of-interval added mass are held.
Every attempt starts from the complete committed state, including constitutive history. The
host time grid and output times are unchanged, and the run summary reports how many intervals
were subdivided and the largest subdivision used. If the finest subdivision fails, the
interval-start state is restored and the run stops with a named error. A rigid translation of
the line, however fast, changes no chord and no tangent, so it never triggers a subdivision.
EI = 0 lines stepped together as one line system (lines joined at Free/Connect
points or objects, and the OpenFAST and C-API coupled routes) recover a step that stalls by
halving instead: the interval is re-advanced as two half steps, recursively down to
\(\Delta t/64\), with the prescribed motion at each midpoint interpolated linearly between
the interval-start and target position, velocity, and acceleration. If the halves do not
complete the interval, the interval-start state is restored and the full step is re-run, so no
partly subdivided state is ever committed.
Rotation- and snap-resolved sub-steps (ei0_substeps auto, the default). A converged
EI = 0 step is measured against its committed start: the largest chord turn
\(\theta\) of a segment that ends the step above \(10^{-3}\,T_\text{max}\), the
largest step-start tension, and whether a segment slack at the start ends above
\(T_\text{max}/4\) while the axial wave crosses it more than once in the step
(\(\Delta t\,c/l_0 > 1\), \(c\) from the dynamic axial stiffness as in the run audit
below); a step that resolves the axial wave resolves the snap as given. The tensions are those
the run audit takes, the elastic tension of a plain segment and the law’s tension of a
viscoelastic or Syrope one, with the step’s unstretched lengths, and \(T_\text{max}\) is at
least \(10^{-3}\) of the line’s initial tension, so a line slack all over at the step start
does not count barely taut segments. (A floor at the initial tension itself raised the snap
threshold in the slack phases of the Lozon semitaut line below, whose anchor peaks then came
out up to 29 % high.) A
segment slack at the step start counts for the turn: counting only segments taut at both ends
of the step left the Lozon semitaut anchor peaks (below) up to 5 % high on average and up to
25 % high in single runs. When \(\theta\) exceeds ei0_substep_angle
\(\theta_s\) (default 1°), or such a snap occurred, the interval is solved again from the
same committed state, constitutive and friction states included, in
\(n = \max(4, \lceil\theta/\theta_s\rceil)\) equal sub-steps, at most ei0_substep_max
(default 16), with the prescribed motion on the quintic trajectory of the recovery above and
the last sub-step on the target state. The trial step under-turns a line whose rotation it does
not resolve, so the sub-stepped interval is measured again; when its turn needs more sub-steps
it is solved once more with them. If a sub-step does not converge, the plain step’s own
result is kept. A step recovered after a stall (above) and the halves of a line-system step
halved after a failure take their sub-steps as given; when the halves fail too, the full step
is re-run as the nominal step, sub-steps included. The decision uses only the committed state,
the step inputs and the line’s initial tension, so a restart repeats it, and the host time
grid, output and coupling times are unchanged. Every route that steps EI = 0 lines uses it
(standalone lines and line systems, OpenFAST, FAST.Farm and the C API), except the lines
attached to the bodies, rods and points of the monolithic body and point coupling, whose
junction Newton needs the end reaction and condensed end tangent of the nominal step (a line of
such a deck whose ends are all fixed or anchored is sub-stepped); off takes every step as
given, and False alpha_force_blend turns it off unless the deck sets it. A step that
triggers costs the plain step and usually four sub-steps; at most it costs the plain step and
three passes of ei0_substep_max sub-steps, once for each object sub-step where bodies or
rods sub-cycle the lines.
At dtM 0.1 s a taut segment of the Lozon 80 m all-chain mooring turns by up to 6° per step
under 5 m / 7 s heave. Over seven runs whose heave amplitude differs by up to
\(3\times10^{-4}\) (the snap peaks at that step vary by 15 to 20 points from one run to the
next), the fairlead and anchor maxima are then 15 % and 17 % below a 0.25 ms run with the force
blend and 10 % and 18 % below it with the configuration blend; the release cap acts in 0.3 % of
the element steps, never within 0.5 s of a peak, and removing it changes the cycle maxima by less
than 1 %. The peaks follow the algorithmic damping instead: 28 % and 31 % low at
\(\rho_\infty = 0\), 10 % and 40 % high at 0.6. With auto they are 4 % and 8 % low
(1 % high and 4 % low against the reference sampled on the same 0.1 s output grid), and the
anchor maxima of the 200 m semitaut mooring in 10 m / 14 s and 5 m / 7 s surge, 16 % and 26 %
high taken as given, are 4 % high and 2 % low (over thirteen runs 5 % high and 0.3 % low, single
runs from 4 % low to 21 % high).
Two or three sub-steps of a step that turned by 1 to 3° left those anchor peaks 8 % high on
average (up to 22 %); four bring them within 3 %, hence the minimum. Resolving the sub-steps
of a step recovered after a stall as well left them 9 % high, so the recovery takes its sub-steps
as given. Over the 20 motions of the Lozon mooring envelope at dtM 0.1 s the march takes
1.67 times as long as with plain steps, 1.6 to 2.5 times on the 5 m / 7 s and 10 m / 14 s
motions.
Step size. Being implicit, the scheme is not bound by the explicit axial-wave limit. In the
snap-load verification a slack line snapping to about 10 MN integrates stably at
\(\Delta t = 20\,\mathrm{ms}\), well above the explicit limit of 3.2 ms, with the peak
within 1.5% of a 0.5 ms reference (CableDyn verification and validation). The time step must still resolve the
response of interest and, for explicitly integrated points and bodies (below), their own
stability limit. The standalone driver audits every EI = 0 line on the initial state and at
each output step, and reports at the end of the run:
the steps
ei0_substeps autosolved in sub-steps, with the largest count, those that needed more thanei0_substep_max, and those that kept the plain step;the largest turn of the chord of a taut segment in one solver step (the sub-steps of a sub-stepped or recovered step count as steps). Above 1 deg it prints a note (with
autoabove 1.1 times the larger of 1 deg andei0_substep_angle): against a 3.125 ms reference on chain catenaries under 5 m heave, 10 m surge and 15 m sway and on a light laid cable under 10 m sway, the mean and standard deviation of the fairlead tension err by 0.9 % at 1.2 deg per step (sway, standard deviation), and by 0.6 % at 1.4 deg and 1.3 % at 2.9 deg (heave, mean), in proportion with the turn, and at about 6 deg per step snap-load peaks come out 10 to 20 % low (above);segments that go from slack to taut above the initial tension in one step while the axial wave crosses a segment more than once per solver step (\(\Delta t\,c/l_0 > 1\), \(c = \sqrt{EA/m}\) with the dynamic stiffness of the segment, \(EA_D\) of a viscoelastic segment and \(\alpha + \beta T\) of a Syrope one): the snap peaks are then not resolved in time, and the note asks for a confirmation at a smaller
dtM;non-physical tension spikes. The tension of a segment at one step (its elastic tension, or the tension of its viscoelastic or Syrope law at the committed state) is a spike when it exceeds four times its own tension one step before and one step after, ten times the largest tension anywhere on the line one step before and one step after, and the largest tension of the initial state. A physical snap or tension wave persists over the axial transit time and is shared by the neighbouring segments, so no line carries such an isolated single-step peak. The first spike of a line is reported as a warning with its segment, tension and time, at most once per line, and the run summary gives the count. Over the 746 decks of a dynamics robustness campaign it warns in none of the 704 runs that complete. Applied to the same decks’ configuration-blend results it flags 37 of the 49 runs whose segment tensions were 3 to 370 times those of the force blend, and 1 of the 158 runs within 25 % of them. Spikes that persist over several steps (a rattle at \(\rho_\infty\) near 1) are left to the
maxStrainguard and therhoInfnote.
Attached objects and line ends
Deck POINTS, BODIES, and RODS records follow the MoorDyn object model
(Hall, 2020) and are translated into boundary objects of the line system.
Fixed, coupled, and vessel points prescribe the position, velocity, and acceleration of line ends.
Free points (
Point3buoys, clump weights, connectors) have three translational DOFs. Each carries its weight and buoyancy \((\rho V - m)\,g\,\mathbf{e}_z\), point loads, its own seabed contact (withMassorVol, see Seabed contact above), and, while below the surface, lumped Morison terms \(\tfrac{1}{2}\rho\,C_dA\,\lvert\mathbf{u}_r\rvert\mathbf{u}_r + \rho V(1 + C_a)\,\dot{\mathbf{u}}_f\) with added mass \(\rho V C_a\). Points are advanced around the implicit line step with implicit-Euler kinematics, and the attached line ends’ own blocks are linearly implicit: the structural mass diagonal, the Morison added-mass block, and the end-segment elastic tangent and axial damping, and so is the point’s own seabed contact (its stiffness \(k_\text{Bot} A\) and damping \(c_\text{Bot} A\)). A light or massless connector is therefore stable at any \(\Delta t\), including under added mass that exceeds the structural mass, and neither the end segments nor the footprint impose a \(\sqrt{k/m_\text{point}}\) limit, so the sub-step estimate of the explicit object march (below) leaves the points out. The coupling to the first interior node, drag, and seabed contact at the end node stay at the step-start state; the step fails with a named error if a point’s motion becomes non-finite.Rigid6 bodies have six DOFs, a diagonal inertia about the centre of gravity, and the mass matrix about the reference point with the centre-of-gravity offset \(\mathbf{c}\) (coupling blocks \(\pm m[\mathbf{c}]_\times\) and \(-m[\mathbf{c}]_\times^2\), and the terms \(-m\,\boldsymbol{\omega}\times(\boldsymbol{\omega}\times\mathbf{c})\) and \(m\,\boldsymbol{\omega}\times(\mathbf{c}\times(\mathbf{c}\times\boldsymbol{\omega}))\)). The weight acts at the centre of gravity and the buoyancy \(\rho g \phi V\) at the centre of buoyancy. The submerged fraction \(\phi\) is that of an equivalent sphere of volume \(V\) at the centre of buoyancy, \(r_e = (3V/4\pi)^{1/3}\), immersed to \(h = \min(2r_e, \max(0, \eta - z_b + r_e))\): \(\phi = h^2(3r_e - h)/(4r_e^3)\), zero when dry and continuous through the surface (
bodyWetting moordyn: \(\phi = 1\), as in MoorDyn). A body with linear hydrostatic restoring \(C_{33}\), \(C_{44}\), \(C_{55}\) (a surface-piercing hull) keeps \(\phi = 1\); its restoring moment is \(-C_{44}\tau_1\mathbf{h}_1 - C_{55}\tau_2\mathbf{h}_2\), with \(\boldsymbol{\tau}\) the tilt rotation vector of the body axis that was vertical at the reference pose, \(\mathbf{h}_1\) the horizontal direction of the body x axis and \(\mathbf{h}_2 = \mathbf{e}_z\times\mathbf{h}_1\), so it does not depend on the heading. Lumped Morison drag on the relative velocity (also in still water), fluid inertia (omitted withbodyHydro moordyn), and isotropic translational added mass \(\rho\phi V C_a\) act at the reference point, each scaled by \(\phi\), and seabed contact acts over a fixed 1 m² reference area. The line-end forces act at their attachment offsets with lever moments. Accelerations follow the Newton–Euler equations about the reference point; the orientation is advanced on SO(3).Rods are rigid cylinders with a six-DOF rigid state, advanced in the same way (or prescribed, for coupled and vessel rods). Their loads are integrated over the
NumSegssegments of the axis. A cross-section normal to the axis \(\mathbf{q}\) at angle \(\varphi\) to the vertical is wet below the local free surface over the circular segment \(A_w(h_w) = r^2\arccos(-h_w/r) + h_w\sqrt{r^2 - h_w^2}\), \(h_w = (\eta - z)/\sin\varphi\), whose centroid lies \(-\tfrac{2}{3}(r^2 - h_w^2)^{3/2}/A_w\) from the section centre along the in-section up-slope direction. With \(\eta\) linear between the segment end stations, each segment splits into fully wet, partially wet, and dry parts at known positions; the wet part is integrated with two Gauss points and the partially wet part with ten Gauss points in \(\theta\), \(h_w = r\sin\theta\). The buoyancy \(\rho g A_w\) per unit length acts vertically at the wet centroid, so the hydrostatic force and moment are those of the displaced volume of the cylinder, end caps included, at any tilt: a surface-piercing rod carries the exact waterplane restoring (second moment \(\pi d^4/64\)), and the loads are continuous in the submergence. Each station also carries, with its wet fraction \(f = A_w/A\), the drag \(\tfrac{1}{2}\rho d f [C_d\lvert\mathbf{u}_n\rvert\mathbf{u}_n + \pi C_{d,ax} \lvert\mathbf{u}_t\rvert\mathbf{u}_t]\), the fluid inertia, and the directional added mass \(\rho A f [C_a(\mathbf{I} - \mathbf{q}\mathbf{q}^T) + C_{a,ax}\mathbf{q}\mathbf{q}^T]\), assembled at its offset into a 6×6 added-mass matrix about the rod centre (rotational block \(C_a \rho A \int s^2\,\mathrm{d}s\) about the transverse axes). Each wet end carries the MoorDyn end terms: added mass \(\rho C_{a,end} V_\text{end}\mathbf{q}\mathbf{q}^T\) (\(V_\text{end} = \tfrac{2}{3}\pi r^3\)), axial drag \(\tfrac{1}{2}\rho C_{d,end} A\lvert u_q\rvert u_q\mathbf{q}\), axial fluid inertia, and, in deck waves, the linear dynamic pressure \(-(p_\text{dyn} - \rho g\eta^*)A\mathbf{n}_e\) on the end cap (\(\eta^*\) the waterline elevation of a surface-piercing rod, zero for a submerged one), which carries the axial Froude–Krylov force. Seabed contact is distributed over \(\max(20, \text{NumSegs}) + 1\) stations with \(k_\text{Bot}\,d\,\Delta l\).Rods fixed to a body move with it: with \(\mathbf{r}\) from the body reference point to the rod centre and \(\mathbf{T} = [[\mathbf{I}, -[\mathbf{r}]_\times], [\mathbf{0}, \mathbf{I}]]\), the rod’s structural and added mass enter the body mass matrix as \(\mathbf{T}^T\mathbf{M}_c\mathbf{T}\), its loads as \((\mathbf{F}, \mathbf{M} + \mathbf{r}\times\mathbf{F})\), with the velocity terms \(-\mathbf{M}_c[\boldsymbol{\omega}\times(\boldsymbol{\omega}\times\mathbf{r}); \mathbf{0}]\) and the rod’s gyroscopic moment. A pinned rod turns about its fixed End A: with the arm \(\mathbf{r}\) from the pin to the centre its acceleration is \(\mathbf{a} = \boldsymbol{\alpha}\times\mathbf{r} + \boldsymbol{\omega}\times(\boldsymbol{\omega}\times\mathbf{r})\), and the 6×6 equations are reduced to the three rotational unknowns.
A rod pinned to a body keeps its End A on the body’s pin point \(\mathbf{p}\) (body frame) and turns with its own rotation. In reduced coordinates the unknowns are the body twist acceleration \(\mathbf{x}_b\) and the rod’s angular acceleration \(\mathbf{y}\); the rod’s centre twist acceleration is \(\mathbf{X} = \mathbf{G}_b\mathbf{x}_b + \mathbf{G}_c\mathbf{y} + \mathbf{c}\) with \(\mathbf{G}_b = [[\mathbf{I}, -[\mathbf{R}\mathbf{p}]_\times], [\mathbf{0}, \mathbf{0}]]\), \(\mathbf{G}_c = [[-[\mathbf{l}]_\times], [\mathbf{I}]]\) (\(\mathbf{l}\) from the pin to the rod centre) and \(\mathbf{c} = [\boldsymbol{\omega}_b\times(\boldsymbol{\omega}_b\times\mathbf{R}\mathbf{p}) + \boldsymbol{\omega}\times(\boldsymbol{\omega}\times\mathbf{l}); \mathbf{0}]\). The equations are the virtual work of body and rods, \(\mathbf{M}_b\mathbf{x}_b - \mathbf{F}_b + \sum_k\mathbf{G}_{b,k}^T(\mathbf{M}_k\mathbf{X}_k - \mathbf{F}_k) = \mathbf{0}\) and \(\mathbf{G}_{c,k}^T(\mathbf{M}_k\mathbf{X}_k - \mathbf{F}_k) = \mathbf{0}\) for each pinned rod \(k\): the pin passes the rod’s force to the body and no moment, and the rod’s rotation feels the pin acceleration. (MoorDyn adds every body-attached rod, pinned or not, to the body’s equations with its full force, moment and mass; its pinned-rod rotation includes the pin acceleration, taken from the body acceleration the rod’s kinematics were last set with, so the two formulations can differ by that lag for a pinned rod on an accelerating body.)
A zero-length rod (
NumSegs0) is a point-like connector: it has no mass, volume or side loads, and its end terms act along an axis it does not have, so it reduces to a massless point at End A (Free,Fixed,Coupledor on a body).
Rigid-body initial condition. Free Rigid6 bodies, free rods and pinned rods start at their
static equilibrium, solved jointly with the Free/Connect points. The unknowns are each
body’s position and a rotation-vector increment on SO(3) (for a free rod only the two rotations
normal to its axis), each pinned rod’s two rotations about its pin, and each point’s position.
A pinned rod’s residual rows are its moment about the pin; its net force acts on its parent body
at the pin (a rod pinned to a fixed point has none, and a CoupledPinned rod in OpenFAST is
balanced about its pin at the platform’s initial pose). A free or pinned rod that no line
attaches to (nor, for a rod pinned to a body, any line of its body) is not an unknown: it keeps
its deck pose and starts at rest there. A deck with pinned rods takes the
central-difference Jacobian of the full residual. A finite-EI line enters with its axial
stiffness and weight only. The residual is the net force and moment of the loads the
dynamic step applies at rest: weight, buoyancy and hydrostatic restoring, steady-current drag,
and seabed contact, plus the end force of every attached line, re-solved to its static
equilibrium for the moved ends. Moments are divided by the object’s length scale (its largest
attachment arm or its half length) and rotations multiplied by it, so forces and moments carry
comparable weight. A trust-region Newton iteration drives the residual below \(10^{-9}\) of
the largest weight or line force; the first trust radius is the first Newton step. Its Jacobian
takes the objects’ own loads by central differences and each line’s contribution analytically,
from the line’s static tangent condensed onto its two end nodes,
\(\mathbf{K}_c = \mathbf{K}_{ee} - \mathbf{K}_{ei}\mathbf{K}_{ii}^{-1}\mathbf{K}_{ie}\)
(axial and geometric stiffness of the tension-only law plus the seabed penalty). Only the lines
attached to a moving object are re-solved. Each re-solve starts from the line’s previous shape
rotated and stretched onto the new chord, and falls back to the cold static solve if that start
does not converge. At a slack line whose condensed stiffness vanishes, the iteration switches to
a central-difference Jacobian of the full residual.
A step is accepted when it lowers the residual, or when the Newton step from the trial point is
at most half the current one. The second test lets an object swing along a curved valley: a
buoy on a stiff taut line moves on a sphere about the anchor, and every straight step
overstretches the line by its sagitta, so the force residual alone would reject any step
longer than a few centimetres. After each accepted step the rotation unknowns are folded into
the object’s orientation and reset to zero, so the analytic Jacobian stays exact however far
the object has turned. A single step turns an object by at most 0.25 rad. A converged
equilibrium must be stable: the symmetric part of the stiffness
\(-\partial\mathbf{R}/\partial\mathbf{x}\) must be positive semi-definite. If the solve fails or ends on an unstable
equilibrium (for example a body upside down or yawed against its lines), it restarts from the
deck pose with the rotations held until the objects balance in force, then releases them. The
solve stops with an error naming the object if the residual stays above \(10^{-6}\) of that
force or the equilibrium it reaches is unstable. The
hydrostatic restoring keeps the deck pose as its reference. deck bodyIC starts from the deck
pose instead, and bodies prescribed by a motionFile keep their prescribed pose.
Rigid-body time integration. Free Rigid6 bodies, Point3 buoys, free and pinned rods and
Free/Connect points are stepped monolithically with their EI = 0 lines
(bodyScheme monolithic, the default; a deck without lines only when it names it): one implicit
generalised-alpha step in which the objects take every load, the line reactions among them, at
the lines’ force level \(t_{n+1-\alpha_f}\), \(\alpha_f = \rho_\infty/(1 + \rho_\infty)\).
The unknowns are the end-of-step accelerations \(\mathbf{x}\) of every such junction (a
body’s \((\mathbf{a}, \boldsymbol{\alpha})\), a pinned rod’s angular acceleration about its
pin, a point’s \(\mathbf{a}\), a Point3 buoy being the point it is attached at). The bodies’
and rods’ inertia sits at the same level (\(\alpha_m = \alpha_f\)), so their positions and
velocities follow Newmark with \(\beta = 1/4\), \(\gamma = 1/2\): they have the principal
roots of the trapezoidal rule, second order and free of numerical dissipation (an undamped
linear body conserves its energy exactly), while the lines keep the dissipation of their
\(\rho_\infty\); the lines’ own \(\alpha_m\) would add a period error of order
\((1 - \rho_\infty)(\omega\Delta t)^2\) to the body modes. The line-end nodes a body or rod
drives follow its Newmark and inertia level inside the line step as well, so their inertia
enters the junction equation consistently. Free/Connect points and Point3 buoys, whose
inertia is largely that of the line ends and whose stiff modes are not sub-stepped, keep the
lines’ own generalised-alpha parameters and dissipation. Rotations are on SO(3):
\(\Delta\boldsymbol{\theta} = \Delta t\,\boldsymbol{\omega}_n + \Delta t^2[(\tfrac{1}{2} - \beta)\boldsymbol{\alpha}_n + \beta\boldsymbol{\alpha}_{n+1}]\),
\(\mathbf{R}_{n+1} = \exp([\Delta\boldsymbol{\theta}]_\times)\mathbf{R}_n\). Each junction’s
equation of motion is evaluated at the generalised-alpha level: positions, velocities and
accelerations at \(t_{n+1-\alpha_f}\) (the orientation on the geodesic between
\(\mathbf{R}_n\) and \(\mathbf{R}_{n+1}\)), with the object’s mass and
added mass, gyroscopic terms, hydrostatics, drag and seabed contact, and the line loads.
The iteration is a multilevel Newton. For a trial \(\mathbf{x}\) every attached line is stepped from its committed state with the resulting end motion prescribed, and returns its reaction at the alpha level, \(\mathbf{f}_e = -\mathbf{R}_E\) (the end rows of its own residual \(\mathbf{M}\mathbf{a}_{\alpha_m} + (1 - \alpha_f)\,\mathbf{r}_{n+1} + \alpha_f\,\mathbf{r}_n\)), so the end-node inertia, drag and added mass enter exactly, and its condensed dynamic end stiffness \(\mathbf{K}_e = \mathbf{S}_{EE} - \mathbf{S}_{EF}\mathbf{S}_{FF}^{-1}\mathbf{S}_{FE}\) of the step’s effective tangent \(\mathbf{S}\), the exact implicit-function derivative of the reaction with the interior re-solved. The junction Jacobian is the objects’ own part (finite differences of each object’s residual, which carry the mass, gyroscopic, hydrodynamic and contact terms and the turning lever arms) plus \(\beta\Delta t^2\,\boldsymbol{\Gamma}^T\mathbf{K}\boldsymbol{\Gamma}\) from the lines, with \(\boldsymbol{\Gamma}\) the map from the junction accelerations to the attachment positions. A Newton step turns an object by at most 0.25 rad. The line re-solves of later iterations start from the previous solve shifted by the linear map \(\mathbf{S}_{FF}^{-1}\mathbf{S}_{FE}\) and reuse its factorisation; the first solve of a step starts from the generalised-alpha predictor corrected the same way. The condensed stiffness is refreshed every twenty steps, after a step that needed more than two iterations, and from the third iteration on. The step converges when every junction residual is within \(10^{-6}\) of its object’s force scale (moments times its size); the predictor itself is accepted only within \(10^{-9}\), so a body at rest does not drift. Typical steps take two iterations. A step that does not converge in twelve is retried as two half steps, to a depth of six, and then stops naming the object with the largest residual.
The scheme is implicit in every term, so it needs no sub-stepping for stability: a moored buoy
and a rod spar released out of balance run at \(\Delta t\) = 0.1 s and 0.5 s. Accuracy is set by
\(\Delta t\) against the modes of interest: the object period error is the trapezoidal
\((\omega\Delta t)^2/12\), the smallest error constant of any unconditionally stable
second-order method (Dahlquist’s second barrier), and twice the \((\omega\Delta t)^2/24\) of the
explicit central difference of the staggered scheme. By default (accuracy bodySubstep) each
coupling step is therefore divided into equal sub-steps with
\(\omega\,\Delta t_\text{sub} \le 0.28 \approx 0.4/\sqrt{2}\) for the stiffest body-mooring
frequency estimate below (bodies and rods), which matches the phase accuracy of the staggered
scheme’s own sub-steps (\(\omega\,\Delta t_\text{sub} \le 0.4\)); none bodySubstep takes the
coupling step as given. The verification tests (CTest monolithic_gates) check: energy of an undamped
linear spring-mass body conserved to \(5\times10^{-10}\) at \(\Delta t\) = 0.1 s for any
\(\rho_\infty\); observed order 2 in positions and fairlead tensions on the moored buoy and in
surge, heave and pitch on the V-SP spar; gyroscopic precession of a torque-free symmetric top
at second order; and an error no larger than the staggered scheme’s at \(\Delta t\) = 0.1 s and
0.05 s on the moored buoy and rod spar. A finite-EI cable pinned to its object is stepped once
per step with the attachment motion of the Newmark predictor, and its end force is held
through the iterations; a cable clamped to its object is re-stepped at every outer iteration
(Cables clamped to bodies and rods, below). Standalone decks with a motionFile, a
FAILURE section, or Coupled/Vessel rods, and the OpenFAST and C-API coupled routes, use the
staggered scheme.
bodyScheme staggered selects the staggered central-difference predictor-corrector
(Newmark \(\gamma = 1/2\), \(\beta = 0\)). The body is first moved to
\(\mathbf{r}_{n+1} = \mathbf{r}_n + \Delta t\,\mathbf{v}_n + \tfrac{1}{2}\Delta t^2\mathbf{a}_n\)
(the rotation likewise on SO(3)). The lines are then stepped to \(t_{n+1}\) with that end
motion. The end-of-step acceleration \(\mathbf{a}_{n+1}\) solves the 6-DOF equation of
motion with the resulting line loads and every body load at \(t_{n+1}\), and the velocity
is the trapezoidal average \(\mathbf{v}_n + \tfrac{1}{2}\Delta t(\mathbf{a}_n + \mathbf{a}_{n+1})\).
Two terms of that solve are implicit, both exact:
the inertia of the attached line-end nodes, which keeps a body lighter than the line inertia it carries free of the partitioned added-mass instability;
the velocity derivative of the body drag and seabed damping.
The line loads therefore include the lines’ own response over the step (waves and drag on the
lines) without a one-step lag. The scheme is second order and adds no numerical damping, so a
moored body’s response converges with \(\Delta t\). The line position and velocity
derivatives at the attachment are not used: they describe the end segment with its neighbouring
node held fixed, and so overstate the stiffness and damping the body actually feels. The position
update is explicit. Each coupled step is therefore divided into equal sub-steps with
\(\omega\,\Delta t_\text{sub} \le 0.4\), where \(\omega\) estimates the stiffest
body-mooring frequency. Each attached line acts as the directional spring
\(k_a\mathbf{t}\mathbf{t}^T + (T/L)(\mathbf{I} - \mathbf{t}\mathbf{t}^T)\) (\(\mathbf{t}\)
the chord direction, \(L\) the chord length, \(k_a = 1/\sum l_0/EA\), \(T\) the end
tension), carried to the reference point and added to the hydrostatic restoring and to the
seabed contact; the inertia is the structural plus added mass. The contact springs are the
penalty springs of the seabed model (\(k_\text{Bot}\,d\,\mathrm{d}l\) at each contact station
of a rod, \(k_\text{Bot}\) over 1 m² at a body’s reference point) that can act within the
coming step. A station at arm \(\mathbf{s}\) from the rod centre moves with
\(\mathbf{v} + \boldsymbol{\omega}\times\mathbf{s}\), and its acceleration is bounded by
\(a_s = |\mathbf{a}| + |\boldsymbol{\alpha}\times\mathbf{s}| + |\boldsymbol{\omega}|^2|\mathbf{s}| + g\)
(\(\mathbf{a}\), \(\boldsymbol{\alpha}\) the committed accelerations, the free fall added
so that a first step without them counts); its reach
\(\rho = \Delta t\,|\mathbf{v} + \boldsymbol{\omega}\times\mathbf{s}| + \tfrac{1}{2}\Delta t^2 a_s\)
bounds its travel in every direction, vertical and horizontal. The station counts when its height
above the highest floor within the horizontal distance \(\rho\) (the depth on a flat seabed;
on a bathymetry grid the highest node of the cells the surrounding box overlaps, raised by
\(0.05\) of their range for the crease rounding) is at most \(\rho\) plus the rod radius.
An object about to land, sliding towards a shallower bed or turning a station onto it is thus
resolved as well as one at rest on the bed. A spring \(k\) at arm \(\mathbf{a}\)
adds \(k\) in heave and \(k a_y^2\), \(k a_x^2\) in roll and pitch. A rod near the
free surface adds its waterplane stiffness in the same way: \(\rho_w g\,d\,\mathrm{d}l\) at each
station whose cross-section can cross the (held or wave) surface within the step, and
\(\rho_w g\,\pi d^2/4\) between each pair of stations whose axis can cross it, which together
bound the elliptical waterplane of any crossing; its inertia is then the structural one, since
the added mass changes as it crosses. A finite-EI cable on a body or rod of the staggered march,
whose end force the march returns explicitly, adds the directional spring above with
\(k_a = 1/\sum l_0/EA\) and its current end tension. The normal seabed damping and the
regularised friction (a damper \(\mu N/\sqrt{v^2 + v_\text{reg}^2}\), \(\mu N/v_\text{reg}\)
at rest) are applied implicitly, the friction as that secant coefficient, so they set no limit,
nor does the seabed contact of a Free/Connect point or a Point3 buoy, which its point
step treats as linearly implicit (see Free points above), so the estimate leaves it out;
the monolithic scheme carries the contact and the hydrostatics in its Jacobian, and its accuracy
sub-steps (above) leave them out. A step that would need more than 256 sub-steps is warned about
once per run. The estimate omits the line-end inertia and the coupling between
the diagonal terms, so it can understate the coupled frequency by about a factor 1.6; stability
alone (the central-difference bound \(\omega\,\Delta t < 2\)) would allow a limit near 1.
The limit of 0.4 is set by accuracy: a rod released out of balance at \(\Delta t = 0.1\) s
follows the fairlead-tension transient of a \(\Delta t = 0.005\) s run within 1.5 %.
The objects’ own terms (hydrostatic restoring, a rod’s waterplane, seabed contact, and the
swing of a pinned rod below, about its pin on every route, with its parent body’s pin
acceleration and the reduced mass \(m m_b/(m + m_b)\) for a rod pinned to a body) enter
whether or not a line is attached: a deck without lines, which marches its
objects by the same explicit scheme, and a staggered deck whose only lines are finite-EI cables
are sub-stepped for them as well.
Both schemes start from the instantaneous acceleration of the initial state.
On the coupled route a Pinned or CoupledPinned rod advances only its rotation in this
way; its centre then follows the pin at \(t_{n+1}\) (fixed, or placed by the host), so End A
drives its lines with the pin motion exactly. The end-of-step angular acceleration solves the
reduced pinned-rod equation with
\(\mathbf{a} = \mathbf{a}_p + \boldsymbol{\alpha}\times\mathbf{r} + \boldsymbol{\omega}\times(\boldsymbol{\omega}\times\mathbf{r})\),
so the pin acceleration \(\mathbf{a}_p\) loads the rotation through the rod’s inertia, and
the implicit drag acts on the velocity change
\(\tfrac{1}{2}\Delta t(\boldsymbol{\alpha}_{n+1} - \boldsymbol{\alpha}_n)\times\mathbf{r}\) of the
pinned centre. The sub-step estimate takes the line springs through their arms from the pin
against the rotational inertia about the pin, and adds the rod’s own swing about the pin, so a
pinned rod that carries no line is still sub-stepped as the compound pendulum it is. Its weight
and buoyancy act vertically at points \(\mathbf{p}_i\) from the pin; the first moment
\(\mathbf{P} = \sum_i f_i\,\mathbf{p}_i\) of these loads turns with the rod, and its length
\(|\mathbf{P}| = |W d_g - B d_b|\) (weight \(W\) and buoyancy \(B\) on the axis at
\(d_g\) and \(d_b\) from the pin) is the swing stiffness at the bottom of the swing,
whatever the current pose. The pin acceleration adds \(|\mathbf{a}_p|\,|\mathbf{G}|\), with
\(|\mathbf{G}|\) the first moment of the rod’s translational inertia (structural and added)
about the pin, \(m d_g\) for a dry rod; for the hanging dry rod this is the effective gravity
bound \(g + |\mathbf{a}_p| \ge |\mathbf{g} - \mathbf{a}_p|\), taken with the larger of the
step-start and end-of-step pin accelerations. The swing stiffness over the smaller rotational
inertia about the pin across the rod axis, about \(3g/(2L)\) for a dry uniform rod of length
\(L\), adds to the line estimate of \(\omega^2\). The buoyancy is that of the wet
length at the evaluated pose; the change of a surface-piercing rod’s waterline with its tilt
(an articulated column) enters through the rod’s waterplane stiffness about the pin (above).
The host places a CoupledPinned pin at \(t_{n+1}\) before the estimate, so
the swing is evaluated with the rod at both its step-start and its end-of-step pin, taking the
larger stiffness and the smaller inertia. The sub-steps move the pin along a straight line; at
the rod’s orientation and with the surface held over the step, every slice’s clearance to the
surface then changes monotonically and in the same sense, so the wetting, the moment
\(W d_g - B d_b\) and the added mass are bounded by their values at the two ends, and a step
whose pin crosses the free surface (a light rod lifted out of the water, or a heavy one lowered
into it) is sub-stepped for the stiffer of its wet states. The seabed contact adds its angular
stiffness about the pin, \(\sum k\,|\mathbf{a}\times\mathbf{e}_z|^2\) over the contact
stations within reach (at both pin poses for a CoupledPinned rod), so a rod whose far end
rests on the bed is sub-stepped for its contact mode. In a sub-cycled step the host pin’s
position,
velocity and acceleration are each interpolated linearly between their step-start values (the End
A motion of the committed rod) and the end-of-step host input, as the host fairleads of the lines
and the Coupled rods are, so points on one platform keep their relative positions under a
translation within the step. This is second order: between exact end values the interpolated
position departs from the host path by at most \(\Delta t^2 \max|\mathbf{a}_p|/8\), the
interpolated velocity is not the derivative of the interpolated position (they differ by
\(O(\Delta t\,|\mathbf{a}_p|)\)), and a pin moving at constant velocity is reproduced exactly.
A first step without a committed angular acceleration (the aggregate and the C API) takes its
starting one with the pin at its step-start pose; under OpenFAST the rod state mirror reloaded
at the initial output call carries the committed accelerations of the initial condition, zero
for the static start, so a start out of equilibrium (bodyIC deck) begins its first step
from rest acceleration, an error of order \(\Delta t\) in that step’s velocity, as for a
free rod.
The host receives the rod’s net load less its inertia at the pin, and no moment. It is the direct feedthrough of the pin acceleration at the rod’s current state: the pin position, velocity and acceleration are those the host last set (re-derived from the rod’s End A motion, so that a checkpoint restart reproduces the force between steps bit for bit), the rotation, the angular velocity and the held fluid are the committed ones, and the angular acceleration is solved again for the pin’s current acceleration. A CalcOutput at an off-step pin acceleration therefore returns the rod’s effective inertia about the pin. The attached lines keep their committed interior state, accelerations included, as in every output evaluation, and their end loads are taken at the end accelerations of the re-solved rotation: the solve moves the end-node inertia from the end accelerations the system holds with the lines’ coupled mass block \(\mathbf{M}_{cc}\) (structural and added mass; the step’s corrector instead lets the line interior follow, \(\mathbf{M}_{cc} - \mathbf{M}_{cf}\mathbf{M}_{ff}^{-1}\mathbf{M}_{fc}\)). At a held interior the end loads are linear in the end accelerations, so the force does not depend on the end accelerations the line loads were evaluated with. MoorDyn-F’s coupled pinned rod keeps the same pin acceleration term in its rotation equation and returns the same force: its output call evaluates the rod’s state derivative at the current input first (its lumped line mass has no interior coupling).
Mixed topologies. One step marches every object of a deck together: bodies with their fixed
and pinned rods, free and pinned rods, the EI = 0 lines with their Free/Connect
points, and the finite-EI cables. In the monolithic scheme the points are junctions of the
Newton iteration above. In the staggered scheme they are integrated on their summed end loads
within the line step. A cable’s End A is driven with its attachment’s predicted motion and its
End B held; its end force enters the object’s equation as a point load at the attachment
(pinned: force only). The cable end-node inertia is not implicit, unlike the EI = 0 line
ends.
Cables clamped to bodies and rods. A Rigid or spring end connection at a cable’s End A
on a body point or a rod end fixes the connection direction in the object,
\(\mathbf{d} = \mathbf{R}_j\mathbf{d}_0\), so the end tangent follows the object’s
rotation; only the direction of \(\mathbf{m}\) is prescribed and its magnitude (the axial
stretch) stays free. The cable returns to the object its end force at the attachment and the
connection moment \(\mathbf{M}_e = \mathbf{m}\times\mathbf{g}_m\), with
\(\mathbf{g}_m\) the constraint (or spring) force on the tangent coordinates. In the
monolithic step the cable is re-stepped from the step entry at every outer iteration with the
iterate’s end-of-step motion, orientation, angular velocity and angular acceleration, and its
end force and moment load the object at the force level
\((1-\alpha_f)\,\mathbf{f}_{n+1} + \alpha_f\,\mathbf{f}_n\). Its condensed dynamic end
stiffness with respect to the object’s translation and rotation enters the Newton matrix by
differences of the cable step. On a rod end the direction must lie along the rod axis, so the
moment has no component about the axis, which carries no degree of freedom. The static solve
carries the cable first without bending; passes then add the difference between the cubic-Hermite
cable’s end force and that surrogate, and the Hermite connection moment, at the previous
pass’s pose, until these loads change by less than \(10^{-7}\) of the force scale. At the
fixed point the object balances the Hermite cable exactly.
End connections (Hermite lines). A line end may carry an isotropic rotational spring with
potential \(U = \tfrac{1}{2}k\,\theta^2\), where \(\theta\) is the angle between the end
tangent \(\mathbf{m}\) and the reference direction \(\mathbf{d}_0\). The spring restrains
only the direction of \(\mathbf{m}\); its magnitude, which carries the axial stretch, is
free. \(k = 0\) is a pinned end. A Rigid end is an exact two-coordinate constraint
that keeps \(\mathbf{m}\) on the ray of \(\mathbf{d}_0\), not a large stiffness. At a coupled
end, \(\mathbf{d}_0\) rotates with the supporting body and the connection moment is returned
to it. The connection moment equals the bending moment of the line at its end node.
Prescribed vessel motion. In the standalone driver a rigid vessel can carry the coupled
points. With reference point \(\mathbf{r}(t)\) (vesselRef at the reference pose) and
rotation \(\mathbf{R}(t)\) from vessel to global axes, a point at the vessel-frame offset
\(\mathbf{p}\) has
with \(\boldsymbol{\omega}\) and \(\boldsymbol{\alpha}\) the angular velocity and
acceleration in global axes. These are the prescribed position, velocity and acceleration of the
line ends at the point. An end connection at such a point uses
\(\mathbf{d}_0(t) = \mathbf{R}(t)\,\mathbf{d}_0\) and, for a Rigid end, the exact rates
\(\dot{\mathbf{d}}_0 = \boldsymbol{\omega}\times\mathbf{d}_0\) and
\(\ddot{\mathbf{d}}_0 = \boldsymbol{\alpha}\times\mathbf{d}_0 + \boldsymbol{\omega}\times\dot{\mathbf{d}}_0\)
in the prescribed tangent direction; the tangent magnitude stays a free axial degree of freedom.
Euler angles follow the OrcaFlex vessel convention
\(\mathbf{R} = \mathbf{R}_z(\psi)\mathbf{R}_y(\theta)\mathbf{R}_x(\phi)\) (roll
\(\phi\), pitch \(\theta\), yaw \(\psi\)), for which
and \(\boldsymbol{\alpha}\) is its exact time derivative.
With a displacement RAO table the vessel motion is the linear response to the deck sea. A wave component of amplitude \(a_i\) and frequency \(\omega_i\) has the elevation \(a_i\cos(\omega_i t - \varepsilon_i)\) at the reference point, where \(\varepsilon_i = k_i(x_r\cos\beta + y_r\sin\beta) + \varphi_i\) carries the component’s phase \(\varphi_i\) and the reference-point position. Degree of freedom \(j\) then moves as
with RAO amplitude \(A_j\) and phase lag \(P_j\) relative to the wave crest at the
reference point, the relative heading \(\beta\) equal to the component’s own direction of
travel (so each component of a spread or multi-train sea takes the RAO at its heading), and the
rampTime factor \(r(t)\). The complex RAO \(A_j e^{-iP_j}\) is interpolated linearly
in period and heading inside the table; headings are taken modulo 360°, but a heading beyond the
last tabulated one (between 345° and 360° in a 0–345° table, for example) is rejected rather than
interpolated across the wrap to the first. The velocities and accelerations are the exact time
derivatives of
\(\xi_j\), and for an irregular sea the motion spectrum is
\(S_{\xi_j}(\omega) = A_j^2(\omega)\,S_\eta(\omega)\) component by component.
Discrete attachments (Hermite lines). A buoyancy module or clump of mass \(m\), displaced volume \(V\), normal and axial drag areas \(C_dA\) and \(C_dA_x\), and added-mass coefficient \(C_a\) is lumped at a node \(j\) of the cable. It adds the constant mass \(m + C_a\rho V\) to the three translational DOFs of the node and the nodal load
with the fluid velocity \(\mathbf{u}\) and acceleration \(\dot{\mathbf{u}}\) at the node and the relative velocity \(\mathbf{w} = \mathbf{u} - \dot{\mathbf{r}}_j\) split normal and along the nodal tangent \(\mathbf{m}_j/\lvert\mathbf{m}_j\rvert\) (the line’s Morison law with \(C_dA = C_{d,n} d\) and \(C_dA_x = \pi C_{d,t} d\)). The drag enters the velocity tangent and, through \(\mathbf{m}_j\), the position tangent exactly. The statics start from the equilibrium with the net weight smeared over the node’s two elements and continue to the lumped load in warm-started Newton solves; in a deck current the drag at rest is added and the final solve is repeated with the drag of its own solution until the nodes stop moving. The load moves in eighths; a step whose damped Newton solve does not converge is repeated from the last converged state with pseudo-transient continuation and then with a residual-merit line search, and is halved (down to 1/1024) when neither converges. A halved step doubles again (up to 1/8) after two consecutive converged steps of its size; each time a doubled step fails, the number of converged steps a doubling waits for doubles, so a homotopy that converges steadily at one size and fails at twice it keeps advancing and tries the larger size about ten times at most. At most 64 load steps may fail in all. A discrete state reached this way passes the kink, tensile, physical-branch and stability checks of the smeared state before it is accepted, the kink limit \(L_0\kappa \le 0.7\) applied away from the attachment nodes: a lumped load turns the line by a slope jump \(\theta\) at its node, which the \(C^1\) elements beside the node carry as \(L_0\kappa \approx 2\theta\) at any mesh size. The smeared state carries no attachment drag, and on a frictional seabed, whose regularised friction force turns from its sticking stiffness to its sliding capacity over a short slip, the Newton update for a cross-flow drag can overshoot and diverge from the first step (a lazy wave with 25 modules in a 0.3 m/s cross current on 2 m elements, even in steps of 1/64), which the pseudo-transient regularisation avoids. As the pitch \(p\) of identical modules shrinks, the discrete equilibrium converges to the smeared section at \(O(p^2)\).
Line failure. A FAILURE row detaches a line end from its point at a prescribed time or
when the end tension reaches a threshold, whichever occurs first. The end is transferred onto a
pre-allocated reserve free point that then moves with the attached line ends. On a Rigid6 deck
the bodies keep their staggered step with the remaining lines, and the reserve point is
integrated on its line-end loads within the same line step.
Active line control. A CONTROL row assigns a control channel to a line. The command
acts on the segment at the line’s fairlead-side end, the end on a host-driven point (coupled,
vessel, turbine or body point, or an end of a coupled or vessel rod); when both or neither ends
are host-driven, it acts on the segment at the NodeB written in the deck (MoorDyn-F’s
segment \(N\) at AttachB). On an EI = 0 line the commanded \(\Delta L\) and
\(\Delta\dot L\) change the unstretched length and its rate of that end element,
\(L_0 = L_{0,\text{base}} + \Delta L\); the structural mass and the point mass shares are
updated consistently, while BA keeps its initial resolution. Commands outside MoorDyn-F’s
bounds, \(-\bar L/2 \le \Delta L \le \bar L\) with \(\bar L\) the line’s unstretched
length over its number of segments (UnstrLen/N, the base length of the controlled segment
on a line of one section), are rejected with the same comparisons MoorDyn-F makes, on both line
kinds; so is a command that would make the controlled segment non-positive, which only a line
of several sections allows.
On an EI = 0 line the length is applied before a step and held over it, and the
committed state keeps the lengths and rates it was reached with, \(L_{0,c}\) and
\(\dot L_{0,c}\), together with its acceleration. Each share of the generalised-\(\alpha\)
residual takes the lengths of its own time level, and the step commits the new lengths with the
new state. With the force blend (the default) the step-start force
\(\mathbf{r}(\mathbf{q}_n, \mathbf{v}_n)\) is the committed state’s: its internal force, axial
damping, viscoelastic and Syrope laws, seabed contact, damping and friction, Morison drag,
Froude–Krylov and buoyancy loads are evaluated with \(L_{0,c}\) and \(\dot L_{0,c}\), with
the distributed external load and the nodal seabed coefficients those lengths give, and so is
every step-start quantity of the release cap: the stretch \(e_n = l_n - L_{0,c}\), the
tension \(T_n\) (the elastic one, or that of the viscoelastic or Syrope law at the committed
state), \(W_n\), the secant \(T_n/e_n\), the law stiffness \(k_n\) (\(EA_D/L_{0,c}\),
\(K_1/L_{0,c}\)) and the dashpot tension \(T_{d,n}\) (with \(\dot L_{0,c}\)).
\(\mathbf{r}(\mathbf{q}_{n+1}, \mathbf{v}_{n+1})\) and the cap’s end-state quantities take the new
lengths and rates: the stretch \(e_{n+1} = l_{n+1} - L_0\), the estimate
\(\hat T_{n+1} = \min(T_n/e_n, k_n)\,e_{n+1}\) and \(T_{d,n+1}\); the projections \(p\) and
\(p_{n+1}\) are geometric. The dashpot weight \(\phi\) measures the end stretch against the new
length, so its band runs from \(L_0\) to \(L_0 + c\,e_n\), the width set by the step-start
stretch as without a change. For an elastic element the secant estimate then differs from its end
tension \(EA\,e_{n+1}/L_0\) by the factor \(L_0/L_{0,c}\), that is by the relative length change
of the step. A haul-in can make \(p_{n+1}\) more negative than without a change; the
capped force is then bounded below by zero, and the cap’s quick inactivity test uses the bound
\(N \ge W_n - (1-\alpha_f)\,T_n (l_n - L_0)^2/(4 e_n)\). The external load of the step,
constant over it, is \((1-\alpha_f)\,\mathbf{f}_\text{ext}(L_0) + \alpha_f\,\mathbf{f}_\text{ext}(L_{0,c})\).
The inertia, which the force blend keeps at the \(\alpha\) level (the structural mass applied
to \(\mathbf{a}_{n+1-\alpha_m}\) and the added mass at \(\mathbf{q}_\alpha\)), takes the
blended lengths \((1-\alpha_f) L_0 + \alpha_f L_{0,c}\) of the changed elements, as
\(\mathbf{q}_\alpha\) blends the configuration. With the configuration blend
(False alpha_force_blend) the residual evaluates the internal force, the damping and the
hydrodynamic loads at the blended configuration \(\mathbf{q}_{n+1-\alpha_f}\); for an element
whose length changed it also blends the lengths, \((1-\alpha_f) L_0 + \alpha_f L_{0,c}\) (and
the rates likewise, the structural and added mass included), so that the configuration and the
lengths are blended consistently and the new length belongs to \(t_{n+1}\); the external load
and the seabed coefficients are then the new lengths’. Evaluating the committed side with the
new lengths (the step-start force, or a configuration blended without its lengths), or
re-deriving the committed acceleration under the new length, would add the force of the length
change to the committed side of the blend, weighted by \(\alpha_m\) and \(\alpha_f\); on a
pay-out or haul-in ramp the tension would then lag the quasi-static value by a fixed fraction of
its change per step, at any \(\rho_\infty\) (on a taut line paid out at a tension change of
1 N per step, 2.6 N with the step-start force at the new lengths, against 1.8 mN, the start-up
transient that \(\rho_\infty = 0\) shows as well). An acceleration re-derived between the
command and the step (a host state update or a boundary commit) belongs to the committed state,
so it is solved with \(L_{0,c}\) and \(\dot L_{0,c}\), which stay the committed lengths and
rates, and evaluated as the step evaluates its committed side: with the force blend at the
external load and seabed coefficients of \(L_{0,c}\) as well, with the configuration blend at
the present ones.
A step that the rotation- and snap-resolved sub-steps (ei0_substeps auto, above) solve
again spreads a pending length change over its sub-steps. The committed lengths and rates are
part of the state each pass starts from; sub-step \(k\) of \(n\) ends at the lengths and
rates interpolated linearly in time between \(L_{0,c}\) and \(L_0\) at \(k/n\), with
the external load and seabed coefficients rebuilt at them, and the last on the new ones bit for
bit. Each sub-step treats its start and end lengths as the step above treats \(L_{0,c}\) and
\(L_0\), so the first starts at \(L_{0,c}\), and the trigger takes the step-start
tensions at \(L_{0,c}\). A constant command rate makes the interpolated lengths those the
rate integrates to: on a line turning by 2.3° per step whose end segment is paid out at
0.05 m/s, every step solved in four sub-steps, the march equals one taken as given at a quarter
of the step to 3e-8 N, where applying the whole change in the first sub-step departs from it by
22 N (2.2 times the tension change per step). A step recovered after a stall and the halves of a
line-system step halved after a failure, which take their sub-steps as given, apply the whole
change in their first part.
On a finite-EI cable, whose fairlead is End A, the command acts on the cable elements that make
up the End-A segment of the deck mesh, of initial length \(S\). Each element scales by
\(\lambda = (S + \Delta L)/S\), \(L_{0,e} = \lambda L_{0,e,\text{base}}\), with rate
\(\dot L_{0,e} = \Delta\dot L\, L_{0,e,\text{base}}/S\). The length is applied before a
step and held over it. The
consistent mass, the attachment masses, the tributary lengths and the nodal seabed coefficients
are rebuilt from the new lengths in the order initialisation used, so the model is the one a
build with those lengths would give; the mass of the new lengths acts over the step and its rate
term \(\dot{\mathbf{M}}\mathbf{v}\) is not modelled (as in MoorDyn), and the committed
acceleration is kept as the history of the preceding step. The committed state keeps the lengths
and rates it was reached with: in the force-blended residual the committed force
\(\mathbf{F}_n = \mathbf{R}(\mathbf{q}_n, \mathbf{v}_n, t_n)\) is the force at those
lengths, and only \(\mathbf{R}(\mathbf{q}_{n+1}, \mathbf{v}_{n+1}, t_{n+1})\) takes the new
ones; an acceleration re-derived between the command and the step (a boundary commit) is solved
at those lengths too, since it belongs to the committed state. The configuration-blended form
(False alpha_force_blend) has no such treatment and is refused for a controlled cable. Evaluating \(\mathbf{F}_n\) at the new lengths would add the force of the length change
to the committed state, which the blend weighs by \(\alpha_f\); on a pay-out ramp the
tension would then lag the quasi-static value by a fixed fraction of its change per step. A step
that follows a change starts its Newton iteration from the Newmark predictor, not from the
constant-acceleration one, since the change is a step in the load. The axial damping acts on
the rate of the strain \(\varepsilon = |\mathbf{r}_{,\xi}|/L_0 - 1\) along the motion and
the length change. The tangent terms of \(\mathbf{r}_{,\xi}\) carry the
factor \(L_0\) and cancel, so with \(\mathbf{p} = H'_1 \mathbf{r}_1 + H'_3 \mathbf{r}_2\)
the end-position part,
and a segment paid out at constant strain carries no damping force. The position Jacobian of the damping force gains \(-BA\, H'_a H'_b\, (\dot L_0/L_0^2)\, \mathbf{t}\mathbf{t}^\mathsf{T}\) on the end-position columns.
Tension and coupled loads
An interior node tension is the segment tension, the average of the two adjacent element
tensions, on an EI = 0 line, and the effective tension recovered from nodal equilibrium
(Node tension, curvature and bend moment) on a Hermite line, including axial damping and
constitutive-state contributions.
FairTen, AnchTen and the end-node tension channels report the line-end force instead
(Conventions, Output files and channels). By the convention in Conventions the reported
tension is the effective tension. The
load returned to a coupled object is the reaction of the cable at the coupled node,
\(-(\mathbf{M}\mathbf{a} + \mathbf{f}_\text{int} - \mathbf{f}_\text{ext})\) at the prescribed
DOFs, so it includes the line’s inertia and hydrodynamic reaction there. Its derivatives with
respect to coupled position, velocity, and acceleration are analytic, with the free DOFs
statically condensed; the acceleration derivative is
\(-(\mathbf{M}_{cc} - \mathbf{M}_{cf}\mathbf{M}_{ff}^{-1}\mathbf{M}_{fc})\) including added mass
(Coupling boundary).
Linear algebra
All global operators are stored and factorised in LAPACK band form (DGBTRF/DGBSV), with
Dirichlet conditions applied in-band and workspaces allocated once, so a dynamic step performs
no heap allocation. The Hermite static energy minimisation also uses the banded Cholesky
factorisation DPBTRF/DPBTRS, whose success is the positive-definiteness test of its
shifted tangent. The Hermite operator has half-bandwidth \(k_l = k_u = 11\) (6 DOFs per
node, 12 per element). The EI = 0 bandwidth is computed from the element connectivity after
removing prescribed DOFs; for a line numbered node by node it is 5. A dense-assembly reference
path packs the assembled matrix into band storage and solves it with DGBSV.