# 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.