| # Numerical contract · v1 |
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| 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. |
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| ## Component files |
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| `mesh.npz`: Q1 geometry `points` (Nvertices × 3) and Basix tensor-order hexahedral |
| `cells` (Ncells × 8). These are geometry-node indices, NOT displacement-DOF indices. |
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| `stiffness.npz`, `mass.npz`: SciPy sparse CSR matrices, full solid FE order. |
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| `component.npz` (NumPy arrays, no pickle): |
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| | 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. |
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| 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 |
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| 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. |
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| Data: CC BY 4.0. Source code: Apache-2.0. Reuse must preserve attribution. |
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