SCHEMA.md

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.