H-DMD-OMA — Hankel Dynamic Mode Decomposition for Offshore/Marine Applications

Operational modal analysis (OMA) of offshore wind turbine monopiles from sparse sensor response. An OpenSeesPy finite-element time-domain simulation feeds a Hankel Dynamic Mode Decomposition (pyDMD) pipeline that recovers natural frequencies and mode shapes with a Modal Assurance Criterion (MAC) > 0.99997 on the first three modes.

• Robust modal identification from only 4 virtual sensors / 12 acceleration channels, mimicking a sparse real-world instrumentation layout.
• Three methodological additions that make HankelDMD output usable for OMA:
  1. Damping-ratio filter — rejects forced 1P/3P harmonic content (ζ ≈ 0), retaining only physical structural modes.
  2. Delay-block spatial averaging — un-shifts the d replicas in the time-delay embedding and averages them to recover clean m-dimensional mode shapes.
  3. Phase rotation to real mode shapes — aligns the averaged complex mode with the real axis for visualisation and MAC comparison.
• A 1,000-realisation Latin Hypercube population study over material, geometry, wave and wind parameters, reporting per-mode frequency and MAC statistics.
• Full mathematical write-up in docs/report.pdf (source).

OpenSeesPy monopile FE model (apparent fixity, Rayleigh damping,
 Morison wave + turbulent wind + 1P/3P rotor loads)
        │  4 sensors × 3 DOF = 12 acceleration channels
        ▼
FIR anti-alias decimation (×5)  →  Hankel embedding (d = 200)
        ▼
SVD truncation (r = 12)  →  Koopman propagator  →  λ → f, ζ
        ▼
ζ-filter  →  delay-block averaged modes  →  phase-rotated real modes
        ▼
Frequency error + MAC metrics vs. reference eigenmodes

├── config.toml              # environment / sea-state / wind / turbine parameters
├── population.toml          # sampling distributions and sample count
├── docs/
│   ├── report.tex           # full write-up (LaTeX)
│   └── report.pdf
├── examples/
│   ├── example_full.py      # single realisation: simulate → DMD → metrics → figures
│   └── example_population.py# LHS population study over parameter variability
├── figures/                 # generated plots (SVG)
└── src/
    ├── model/               # OpenSeesPy monopile FE model + load models
    ├── simulation/          # transient pipeline + HankelDMD wrapper
    ├── analysis/            # mode matching, MAC, frequency metrics
    ├── population/          # LHS sampling, distributions, parallel runner
    ├── viz/                 # publication-style plotting
    └── config.py            # TOML config loading

Requires Python ≥ 3.10. OpenSeesPy is tested on Linux; other dependencies install from PyPI.

uv sync                                   # or: pip install -r requirements.txt
python examples/example_full.py           # single realisation → figures/ + metrics
python examples/example_population.py     # 1,000-sample LHS study → data/raw/

example_full.py runs the full transient simulation, extracts modes via HankelDMD, prints the frequency/MAC comparison against the reference eigenmodes, and writes publication-style SVGs to figures/.

example_population.py samples structural (E, D, t) and environmental (H_wave, T_wave, U_mean_ref) parameters with Latin Hypercube Sampling and reports per-mode frequency and MAC statistics across the population.

Reference eigenmodes vs. HankelDMD extraction from 12-channel simulated accelerations (50 s at dt = 0.02 s):

Modes 4–6 lie beyond the effective Nyquist frequency (5 Hz after decimation) and are not resolvable from the subsampled data; this is documented in the report.

MIT