# Numerical contract · v1

All stored values use metres, newtons, kilograms, seconds, kelvin and radians.
Local inlet is at the origin, pointing along +x. A straight outlet points +x;
an elbow turns +x to +y. `rolls_rad` rotates each component about its incoming
local x axis before connection. Ports coincide; no extra connector length is added.

## Component files

`mesh.npz`: Q1 geometry `points` (Nvertices × 3) and Basix tensor-order hexahedral
`cells` (Ncells × 8). These are geometry-node indices, NOT displacement-DOF indices.

`stiffness.npz`, `mass.npz`: SciPy sparse CSR matrices, full solid FE order.

`component.npz` (NumPy arrays, no pickle):

| Key | Shape / meaning |
|---|---|
| xyz | Scalar Q2 displacement interpolation coordinates, N × 3 |
| ports | 2 × 3 reference port centres |
| interior, boundary | Scalar vector-DOF indices into full K/M |
| B | Boundary displacement map from 12 port coordinates |
| C | NDOF × 12 static constraint modes |
| T | NDOF × 24 Craig–Bampton basis [C, fixed-interface modes] |
| S | 12 × 12 static condensed stiffness |
| Kr, Mr | 24 × 24 projected stiffness and mass |
| ftemp | NDOF thermal load per kelvin |
| fG | NDOF × 3 full gravity/inertia load per acceleration |
| wT | NDOF particular displacement per kelvin, including port radial dilation |
| wG | NDOF × 3 interior particular displacement per acceleration, zero on ports |
| fT, g | 12 and 12 × 3 condensed thermal/gravity load maps |
| path | 49 × 3 reference centreline samples |
| recovery | 49 × 3 × 12 centreline static recovery |
| thermal_recovery | 49 × 3 particular thermal centreline motion |
| gravity_recovery | 49 × 3 × 3 particular gravity centreline motion |
| mode_recovery | 49 × 3 × 12 fixed-interface modal centreline recovery |

Full vector DOFs are interleaved `[ux0, uy0, uz0, ux1, ...]`.
Port coordinates are `[ux, uy, uz, rx, ry, rz]` at inlet, then outlet.
Fixed-interface modes are mass normalized; their signs are arbitrary.

For a single component in its local frame:

```
S q = fT ΔT + g acceleration + external_port_load
u_full = C q + wT ΔT + wG acceleration
K_dynamic = T.T K T; M_dynamic = T.T M T
```

Particular-load recovery is necessary: `C @ q` alone does not recover a loaded
interior. Port constraints refer to centreline mechanical DOFs; prescribed
radial thermal expansion is superposed even at a mechanically fixed port.

The browser JSONs contain only condensed matrices and centreline recovery, not
full-volume fields. See the dataset for the complete arrays.

## Supports

`node` indexes the chain's N+1 junctions. `fixed` lists constrained DOF components
(0–2 translations, 3–5 rotations). `spring` is six stiffness values (N/m for
translation, N·m/rad for rotation). Stiffness acts in GLOBAL coordinates.

Springs are bilateral linear springs with zero preload at the original geometry.
They are not constant-load hangers or unilateral contact supports.

## Assembly JSONL rows

- `component_ids`, `rolls_rad`, `supports`: complete assembly definition.
- `delta_temperature_K`, `gravity_m_s2`: loads.
- `port_displacements_SI`: 6(N+1) interleaved port coordinates.
- `fixed_reactions_SI`: the assembled residual `K_with_springs q − F`;
  nonzero on fixed DOFs only. Spring force on the pipe is `−k q`, separately
  recoverable from `supports` and `port_displacements_SI`.
- `frequencies_Hz`: first three natural frequencies about the reference state.
- `centerline_m`, `displacement_m`: per-component 49 × 3 samples.
- `free_residual`: relative equilibrium residual on unconstrained DOFs.
- `label_source`: `solid_FE_component_ROM`, deliberately distinct from full FE.

There are 384 planar development cases and 128 held-out spatial cases.
The latter are a prepared topology split, not proof of a learned model's accuracy.

## Geometry and model assumptions

The circular section is polygonally represented by Q1 geometry. Mesh-based mass
is therefore slightly below the analytic circular volume; validation records
this discrepancy. Bending flexibility comes from the solid elbow model; no
empirical piping-code flexibility factor is inserted. Rigid ports approximate
mechanical cross-section behaviour; they are placed on the straight tangent
ends, not directly on the bend. The published continuous-mesh comparisons
measure this approximation only for the tested routes.

Data: CC BY 4.0. Source code: Apache-2.0. Reuse must preserve attribution.