Supplementary Materials: A reduced-order model of coral transplant unit stability under waves and currents

Authors: Sandy H. S. Herho, Iwan P. Anwar, Faruq Khadami, Karina A. Sujatmiko, Alfita P. Handayani, and Dasapta E. Irawan,

Idealized model of the mechanical failure of coral transplant units under waves, currents and internal solitary waves, written from scratch in Python with no calibration and no external data.

A transplant unit is treated as a planar rigid body resting on a rigid bed, loaded by near-bed wave kinematics through a Morison description and held by two unilateral frictional contacts. The model answers a narrow question that restoration teams face: will this unit stay where it was placed, as deployed and after the coral on it has grown. It says nothing about coral survival, which depends on water quality, sedimentation, bleaching, attachment technique and maintenance.

_{A wave of period 8 s growing to 3 m over 8 m of water. Above, the linear orbital speed over one and a half wavelengths; below, the stretch of bed in the dashed box at true scale, with the five units where the contact solver puts them.}

Waves on a current make a block walk: one tread of the staircase per wave, every one in the same direction.	A two-layer soliton passes, the lower layer running with it and the upper layer returning; the spider steps downstream while it is overhead.

_{Critical wave height over depth and period for a staked rebar table, recomputed as its colonies grow. The white contour is the 2 m design wave; the monsoon reference condition starts inside it and ends outside.}

What is in the model

• Five reference structures: solid block, perforated dome, rebar table, hexagonal spider, tetrahedral frame.
• Six materials, entering the quasi-static criteria only through the submerged weight.
• Forcing: regular and second-order Stokes waves, steady currents, and a two-layer Korteweg-de Vries internal solitary wave.
• Dynamics: Moreau-Jean time-stepping with a projected Gauss-Seidel contact solve, compiled with Numba, with a NumPy reference implementation and two independent event-driven solvers used for verification.
• Diagnostics: closed-form thresholds for sliding, tipping and lift-off, ratchet asymptotics with their relative-velocity correction, critical wave heights over depth and period, and the hold-down force a unit needs to meet a stated safety factor.

Main results

• Sliding precedes tipping if and only if b/z_p > mu, and lift cancels from that comparison exactly.
• Material choice enters the quasi-static thresholds only through the submerged weight; structure controls the mode and the lever arms.
• Waves on a current make a unit walk. The drift per cycle near threshold is (9/2)(mu N/m_x) eps^2/kappa, reduced by the factor 1 - (6/5) Gamma because drag acts on the relative velocity.
• An internal solitary wave at transplant depths is a transient current of a few tens of centimetres per second lasting a few wave periods. It rarely overturns a unit by itself, and it does trigger net displacement in units that are otherwise holding.
• Coral growth erodes the margin it was built for, and mounting height matters only once sliding is prevented.
• The ranking of the five structures by critical wave height is reproduced in 97.5 percent of 4000 samples that vary every uncalibrated coefficient at once, so the ordering is far firmer than the values.

Layout

reefunit/    library: core, units, forcing, kernel, dynamics, reduced,
             statics, ratchet, scenario, draw, plotting, anim, io_utils
scripts/     fig00 ... fig08, anim01 ... anim04, make_reports, run_all
tests/       pytest suite, a fast version of every verification check
outputs/     figures (pdf and png), data (per-panel CSV), reports (txt),
             animations (gif)

Reproducing

pip install -r requirements.txt
python scripts/run_all.py              # figures, animations, reports
python scripts/run_all.py figures      # figures only
python scripts/run_all.py animations   # animations only
python -m pytest tests -q              # verification suite, seconds

Each figure script writes the data behind its panels to outputs/data and its derived numbers to outputs/cache; make_reports.py turns those into the plain-text reports, which hold every number quoted in the manuscript, the verification residuals, the hold-down table, and the limitations.

Figures

fig00	units and the planar load model
fig01	near-bed forcing of the site classes and of a soliton
fig02	failure-mode plane and onset under a ramped current
fig03	critical wave height over depth and period, and by material
fig04	ratchet drift, damping correction, drift rate
fig05	soliton amplitude, margin consumed, displacement per passage
fig06	colony growth against the design wave
fig07	sensitivity to the three weakest assumptions
fig08	verification

Every animation is rendered from the same solvers as the figures, draws the units from their own geometry at true scale, and carries a colour bar in SI units. Nothing in them is exaggerated for effect.

Limitations

Planar motion, a rigid flat bed, non-breaking waves, uncalibrated force coefficients, and a colony model capped at 0.10 m radius. The unit dimensions are a parameter set rather than measurements of particular designs; because the quasi-static thresholds depend on the geometry only through W'/(C_D A) and z_p/b, results can be rescaled to another unit without rerunning anything. These are stated in full in outputs/reports/open_items.txt. Nothing here has been validated against field or laboratory data, so the results are relative and structural rather than predictive.