Tutorial 3 — A lazy-wave power cable

Goal: find the installed equilibrium of a dynamic power cable with a buoyancy section, locate its sag bend, hog bend, and touchdown, and read curvature and minimum bend radius.

Deck: examples/lozon_gomex80_power_cable.dat (Lozon et al. 2025, Gulf of Mexico, 80 m water depth) · Route: cubic-Hermite finite-EI path, static · Run time: < 1 s

Why a different element

A power cable has bending stiffness that matters: the fatigue- and bend-radius-critical quantities are the curvatures at the sag bend, the hog (arch) bend, and touchdown. Giving a line type EI > 0 switches its line to CableDyn’s cubic-Hermite element, which carries position and tangent at every node and evaluates the exact nonlinear centreline curvature. Chains with EI = 0 in the same deck keep the cable element. See Solver paths.

The deck

--------------------- LINE TYPES ---------------------------------------------
TypeName  Diam   MassDenInAir  EA       BA   EI       Cd_n  Cd_t  Ca_n  Ca_t
bare      0.160  36.70         4.69e8   0.0  1.99e4   1.2   0.1   1.0   0.0
buoy      0.290  59.53         4.69e8   0.0  1.99e4   1.2   0.1   1.0   0.0
--------------------- POINTS ------------------------------------------------
ID  Type     X       Y    Z
1   Coupled  5.0     0.0  -14.0
2   Fixed    125.0   0.0  -80.0
--------------------- LINES -------------------------------------------------
ID  NodeA  NodeB  Outputs
1   1      2      -
--------------------- SECTIONS ----------------------------------------------
LineID  LineType  Length   NumSegs
1       bare      68.114   448
1       buoy      50.000   320
1       bare      52.101   256
--------------------- OPTIONS -----------------------------------------------
80.0     WtrDpth   ...
0.0      TMax      - ... 0 selects static initialization only
True     tensile_safety  - Refine and reject local axial compression in this tensile cable
...
--------------------- OUTPUTS -----------------------------------------------
"FairTen1"  "AnchTen1"  "FairIncl1"  "AnchIncl1"  "Curv1N383"  "BendMom1N383"

(one channel per row in the file). Points to notice:

  • Two line types share EA and EI but differ in outer diameter and mass. bare is heavy in water (36.7 kg/m vs 20.6 kg/m displaced); the buoy section, representing the distributed buoyancy modules as an equivalent cylinder, displaces 67.7 kg/m against 59.5 kg/m of mass and is therefore net buoyant. That is what lifts the arch.

  • One line, three ordered sections, End A at the hang-off (Coupled, 14 m below the surface) and End B at a Fixed seabed termination 120 m away. The cable is longer than the span, so its tail rests on the seabed.

  • The mesh is fine (1024 elements, ~0.17 m) so that the touchdown and the section transitions are resolved. Curv1N383 asks for curvature at node 383, in the sag bend.

  • TMax = 0 makes this a static-only run; tensile_safety refines and rejects a solution with local axial compression. There is no initial-shape input: CableDyn finds the shape.

Run it

New-Item -ItemType Directory -Force results | Out-Null   # already there after the quickstart
.\CableDyn_driver.exe .\examples\lozon_gomex80_power_cable.dat .\results\lazywave80

   Created CableDyn model: 1 line object(s), 2 point(s), 3 section(s) [EI=0: 0, finite-EI: 1].
   Initial conditions: Newton static equilibrium with load continuation completed.
   Fairlead convention: force is on End A toward End B; inclinations are signed below horizontal.
   Line 1 fairlead effective tension:  9.32017E+003 N
      force [Fx, Fy, Fz]: [ 1.32690E+003, -0.00000E+000, -9.22523E+003] N, inclination=   81.815 deg
      line tangent: inclination=   81.601 deg, declination=  171.601 deg, azimuth=    0.000 deg
  CableDyn initialization completed.
CableDyn_driver: converged run written to .\results\lazywave80.out

# CableDyn driver output (finite-EI dynamic; production cubic-Hermite route)
Time(s)  FairTen1       AnchTen1       FairIncl1      AnchIncl1       Curv1N383      BendMom1N383
0.0...   9.3201701E+003 1.3269019E+003 8.1600621E+001 -3.6937724E-001 9.6834389E-002 1.9270043E+003

Read the shape

Plot Z against X and Curvature against ArcLength from results\lazywave80.static.out (Python, pyDatView, or a spreadsheet). The profile gives:

Feature

Arc length

Depth Z

Curvature

hang-off (End A)

0 m

−14.0 m

tension 9.32 kN, 81.6° below horizontal

sag bend (lowest point before the arch)

58.5 m

−64.1 m

0.0968 1/m — minimum bend radius 10.3 m

hog bend (arch crest, buoyancy section)

87.2 m

−52.5 m

0.056 1/m

touchdown region

~131–136 m

−79 to −80 m

0.076 1/m peak

End B (seabed termination)

170.2 m

−80.0 m

tension 1.33 kN

170 of the 1025 nodes rest on the seabed. Curv1N383 in .out is the sag-bend node, and BendMom1N383 = EI × κ = 1.99e4 × 0.0968 = 1927 N·m. The horizontal tension is the same at both ends (1.33 kN): the only horizontal external loads in statics are the end reactions.

In Python:

import numpy as np
from cabledyn import read_output
p = read_output(r"results\lazywave80.static.out")
s, z, k = p.column("ArcLength"), p.column("Z"), p.column("Curvature")
i = int(np.argmax(k))
print(f"max curvature {k[i]:.4f} 1/m at s = {s[i]:.1f} m -> MBR {1/k[i]:.1f} m")
print("grounded nodes:", int((z <= -79.99).sum()))

max curvature 0.0968 1/m at s = 58.1 m -> MBR 10.3 m
grounded nodes: 170

Check the equilibrium, not only the end tension

A lazy-wave cable can have more than one equilibrium. A plausible fairlead tension does not prove the right branch: always inspect the curvature profile for a smooth sag bend, arch, and touchdown with no kinks at section boundaries, and compare peak curvature with the design minimum bend radius. The three Lozon reference cables agree with OrcaFlex and the published values in CableDyn verification and validation.

Discrete buoyancy modules

lazy_wave_buoyancy_modules.dat replaces the smeared 50 m buoyant section with the bare cable and ten modules at a 5 m pitch, in a 0.5 m/s current. One ATTACHMENTS series row places them:

LineID  ArcLength           Mass    Volume  CdA   Ca   CdAx
1       70.614:5.0:115.614  114.15  0.2297  0.78  1.0  0.2042

Each module carries 5 m worth of the extra mass and volume of the 0.29 m section, with CdA = Cd_n (d_m − d) p and CdAx = π Cd_t (d_m − d) p, so it matches the smeared section in the current. lazy_wave_buoyancy_smeared.dat gives that section as an EQUIVALENT BUOYANCY row on the same mesh, whose 0.156 m elements put a node on every module. The global answer is the same: fairlead tension 9165 N against 9160 N, touchdown at 138.26 m against 138.27 m of arc, arch crest at z = −47.49 m against −47.48 m, and all nodes within 0.035 m. The local bending is not: over the arch the curvature is a sawtooth, 0.084 1/m at a module against 0.061 1/m smeared, and about 20 % lower between modules. The sag-bend (0.085 1/m) and touchdown (0.113 1/m) peaks are unchanged. Use modules when the arch bending matters, and the smeared section for the global response.

Natural periods of the lazy wave

lazy_wave_modes.dat adds 10 nModes to the 80 m cable on its 1024-element mesh. The modes are solved about the static equilibrium with both ends held, the added mass and the linearised seabed contact included, and written to <root>.modes.out. The first periods are 72.8 s (out of plane), 39.9 s (in plane), 31.0 s (out of plane), 20.4 s (in plane, mostly vertical), 19.9 s (out of plane, along the grounded run) and 16.9 s (in plane). Between 512 and 2048 elements the out-of-plane periods and the 20.4 s mode agree to four digits. The in-plane modes that move the touchdown (39.9 s, 16.9 s and 11.7 s) shift by up to 0.5 %, because the linearised seabed contact at the touchdown changes with the node spacing. The run takes about 2 s, most of it in the modal solve.

Exercises

  1. Deeper sites. Run lozon_gomaine200_power_cable.dat (200 m) and lozon_humboldt800_power_cable.dat (800 m). Compare hang-off tension and minimum bend radius; the deep cable is dominated by suspended weight.

  2. Buoyancy sizing. Copy the 80 m deck and increase the buoy mass from 59.53 to 64.00 kg/m (less net buoyancy). Predict first, then check: the arch crest drops from −52.5 m to −64.4 m, the tail lengthens (213 grounded nodes), and the peak curvature rises to 0.110 1/m — the minimum bend radius falls from 10.3 m to 9.1 m.

  3. Mesh. Halve every NumSegs (224/160/128) and rerun. The peak curvature stays at 0.0968 1/m and the hang-off tension changes by less than 0.001 N, so the shipped mesh is converged for these quantities. Do this check on every new cable before trusting a bend-radius result.

Next: Tutorial 4 — Prescribed fairlead motion.