# Step 2: formulation

## Read First

- `/AGENTS.md`
- `/docs/ABAQUS_SUBROUTINE_AGENT_DESIGN.md`
- `/docs/ARCHITECTURE.md`
- `/docs/ADR.md`
- `/.codex/skills/abaqus-subroutine-formulation/SKILL.md`
- `/docs/requirements/uel-3d-euler-beam.md`
- `/docs/research/uel-3d-euler-beam-research.md`
- `/docs/formulations/README.md`

## Task

Create `/docs/formulations/uel-3d-euler-beam.md`.

Define the implementable finite element formulation for the first 3D Euler-Bernoulli beam `UEL` scope. Include:

- Element topology, node ordering, local axes, and DOF ordering.
- Required material and section parameters.
- Local 12x12 stiffness matrix terms for axial, torsion, bending about local 2, and bending about local 3.
- Transformation matrix from local to global coordinates and orientation-vector validation.
- Internal force/residual relation for static small-displacement analysis.
- `AMATRX` and `RHS` contribution rules by `LFLAGS` for the supported procedure subset.
- Numerical tolerances for zero length, near-parallel orientation vector, non-positive section/material properties, and matrix symmetry.
- State variable policy. If no persistent state is needed, state that `SVARS` is not used except optional diagnostics approved by requirements.
- Output recovery quantities that external CSV validation should compare.

Do not define the full Abaqus ABI signature here unless needed to express formulation responsibilities; the interface step owns the ABI contract.

Do not create or modify Fortran source, tests, reference artifacts, or implementation plans in this step.

## Tests To Write First

- None. This is a formulation artifact step.

## Acceptance Criteria

```bash
python -m unittest discover -s scripts -p "test_*.py"
python scripts/validate_reference_artifacts.py
python scripts/validate_workspace.py
```

The step is complete only when `/docs/formulations/uel-3d-euler-beam.md` has enough matrix, residual, and transformation detail for a later implementation step to code without inventing physics.

## Validation Notes

- Keep solver-result validation as externally generated ODB-extracted CSV evidence.
- Update `/phases/uel-3d-euler-beam/index.json` step 2 with `completed`, `error`, or `blocked` and a concrete summary or reason.

## Forbidden

- Do not add JavaScript/TypeScript/npm fallback.
- Do not run Abaqus analyses from this project.
- Do not generate reference CSVs.
- Do not create Fortran production source.
- Do not break existing tests.