SCHEMA.md

# Morph Rods v0.1: mechanics and data contract

## Scope and units

Planar unshearable, extensible, conservative elastic rods, clamped at s=0.
All coordinates are divided by reference length L; forces by EA; moments by EA L;
curvature is kappa L; potential energy by EA L. `chi = EI/(EA L^2)`.
Positive fx denotes tension; positive fy points upward. No gravity, contact,
friction, inertia, material hysteresis, or physical-time evolution is included.

The generated 1D data use this project's angle finite-element implementation in
NumPy/SciPy. They are NOT 16,200 AgentFEM solid-FE runs. Three separately packaged
2D cases use native AgentFEM/FEniCSx. The hinge chain is a separate reference.

## Energy

Let a=fx+2h, g=4h delta, theta be the rod tangent angle, and
f(theta)=a cos(theta)+fy sin(theta). Axial strain is eliminated exactly as
epsilon=f(theta). With prescribed intrinsic curvature kappa0(s), the reduced
energy is, up to an additive control-dependent constant:

    Pi = integral[ chi/2 (theta' - kappa0)^2
                   - f(theta) - f(theta)^2/2 + g cos(theta) ] ds
         - moment * theta(1).

For h=0 this is a conventional extensible elastica with eigen-curvature. For
kappa0=0 and moment=fy=0, the microforce term reproduces the homogenized Model (i)
of Ye, Piccolroaz & Bigoni, DOI 10.1016/j.jmps.2026.106870. The theory belongs to
those authors. Our additions are the implementation, checks, data organization,
surrogate baselines, and bounded inverse-design example.

Theta uses linear FE nodes; strain and potential terms use midpoint quadrature.
The clamp angle is eliminated. The default is 48 elements, with 96-element paired
checks. Three constant intrinsic-curvature regions partition the reference length
equally. They describe idealized actuation, not a calibrated pneumatic actuator.

For the paper's original chain, theta of the first cell is fixed and n-1 hinge
rotations remain; kr=chi*n. Do not interpret chain cell count as continuum FE
resolution. The paper reference uses n=15,30,60,120 in validation.

## Fields

- `scene_id`, `split`: a physical parameter draw and its fixed partition.
- `path_id`, `step`, `path_parameter`: ordered equilibrium samples, NOT seconds.
- `family`: end_load, intrinsic_curvature, or microforce.
- `controls`: chi, fx, fy, moment, h, delta, k1, k2, k3.
- `z`: 48 free angle DOFs; append theta(0)=0 at the front.
- `angle`, `strain`: 48 element-midpoint values.
- `centerline`: 65 [x/L,y/L] points at uniformly spaced reference coordinate s/L.
- `tip`: endpoint [x/L,y/L].
- `energy`: reduced total potential with the convention above. Do not interpret
  its difference across changed controls as dissipated work.
- `residual`: infinity norm of the reduced free-angle energy gradient.
- `lambda_min_over_chi`: smallest generalized Hessian eigenvalue divided by chi.
  Angular mass is the same midpoint quadrature matrix, not physical inertia.
- `stability`: stable if eigenvalue>1e-4, unstable if < -1e-4, critical otherwise.
  This is LOCAL stability under fixed dead loads and the stated clamp, in this
  discretized conservative model; it is not a hardware safety margin.
- `branch`: coarse descriptor up/down by tip y, or straight if max|z|<1e-5.
  This is not a universal enumeration of every nonlinear equilibrium branch.
- `seed_source`, `solver_trace`, `fallback_nfev`: numerical initialization,
  residual/step evidence, optional root-solver work count.
  Discarded wrong-branch/failed seed candidates are separately retained in
  validation/discarded_seed_attempts.jsonl.gz, not counted as accepted data.
- `previous_id`, `continuous_stable_pair`: adjacent accepted stable samples with
  max angle-DOF change<0.2 rad. This heuristic does NOT certify the unsampled
  interval or prove reachability from the unloaded state.

## Sampling and partitions

216 scenes, 72 per family: 60 ID + 12 stiffness-OOD. Each scene has three ordered
sweeps and 25 samples per sweep, giving 648 sweeps and 16,200 states. Reversible
sweeps and unloaded endpoints may repeat shapes: these are NOT 16,200 independent
scenarios or unique geometries.

Each family uses 42 training, 9 validation, 9 test and 12 OOD scenes. All mirrored
branches, all sweeps, and paired resolutions stay with their parent scene.
Parameter-OOD increases chi beyond the training interval; it does not establish
geometry, path, or discretization OOD capability. Test is held out from fitting
and checkpoint selection, but is not a permanently sealed benchmark.

ID chi: log-uniform [1e-4,10^-2.5]; OOD chi: log-uniform [10^-2.5,10^-2.1].
End loads: fx/chi in [0,2], fy/chi in [-2,2], moment/chi in [-1,1].
Curvature controls: each ki in [-2.4,2.4].
Microforce scenes: h/chi in [4,30], delta in [.05,.35], p starts at pcr-.2chi
and decreases to pcr-c chi with c in [2,8]. Here p=fx and pcr is the first
continuum bifurcation root. Up/down/straight solutions are deliberately seeded.
The unstable straight states are retained, not followed as a physical motion.

No initial-imperfection distribution, subcritical branch atlas, snap dynamics,
self-contact validation, or hardware data are provided in v0.1.

## Native 2D evidence

Three Figure 9 cases: Q2 plane-stress Saint-Venant--Kirchhoff solid, nu=0,
EA=1, thickness=sqrt(12 chi), plus the paper's orientation potential. This is an
equivalent finite-thickness realization, not a resolved hinge microstructure.
`U` is solved downward, not reflected for plotting. PVD/VTU carries Q2 displacement
and DG0 MISES/microforce potential. XDMF/HDF5 carries P1-interpolated U and DG0 fields
on a single geometry per case. Exact serial Q2 coefficients, coordinates and cell
DOF maps are retained in scientific/p2_coefficients.h5. These coefficients require
the supplied element convention/case to interpret; they are not a universal restart.
Case indices 0,1,2 are independent loads, NOT a transient trajectory.

## Baselines

Ordinary and branch-conditioned MLP: two hidden layers of width 96 with tanh,
48 angle outputs, Adam, seed14, 401 epochs; checkpoint selected by validation
angle MSE. Only training data define input normalization. Branch conditioning is
supplied information, not predicted by a hidden classifier. The ordinary baseline
is intentionally ambiguous on mirrored multi-solution inputs: its worse result
demonstrates the missing-input problem, not architectural superiority.
The branch-conditioned 1-nearest-neighbor baseline receives the same branch label.
Raw neural estimates require mechanical correction before equilibrium/stability
claims. The correction can change a predicted branch; check the returned state.

## Reproduction

    python tests/validate.py
    python src/generate.py
    python src/train.py
    python tests/evaluate_release.py
    python src/inverse.py

Native FE requires AgentFEM/FEniCSx separately:

    agentfem run --project cases/fig9 --name downward

Dependencies and measured versions are in requirements.txt and reports/environment.json.