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.codex/skills/fem-formulation-spec/SKILL.md
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---
name: fem-formulation-spec
description: Use when an approved structural FEM feature needs an implementation-ready mathematical formulation before numerical review or coding.
description: Produce an implementation-ready mathematical formulation for a new or changed structural FEM model.
---
# FEM Formulation Specification
Translate approved requirements and research into a complete mathematical contract, independent of programming APIs and storage formats.
Define only the model required by the assigned feature: assumptions, strong/weak form, kinematics, constitutive update, discretization, DOF/component and sign conventions, residual, consistent tangent where applicable, integration, state update, recovery, and formulation-level invariants. Distinguish small/large deformation and analysis-family assumptions.
## Inputs
- `docs/<feature-id>/requirements.md`
- `docs/<feature-id>/research.md`
- Governing standards and approved analysis assumptions
## Workflow
1. Fix primary variables, degrees of freedom, units, coordinates, signs, configuration, and boundary-condition conventions.
2. Derive governing strong and weak forms with all natural and essential boundary terms.
3. Define discretization, interpolation, kinematics, constitutive update contract, element residual/internal and external forces, and consistent tangent or stiffness.
4. Define mapping, Jacobian validity, quadrature, state/history evolution, constraints, and result recovery that apply to the selected analysis family.
5. State dimensions, invariants, algorithm pseudocode, numerical risks, and tests capable of falsifying each derivation.
Read [analysis-family guidance](references/analysis-families.md) and apply only the sections selected by the requirements. Use [workflow and decision authority](../fem-requirements-baseline/references/workflow-and-decisions.md) for unresolved physics choices and [evidence and handoff](../fem-requirements-baseline/references/evidence-and-handoff.md) for return data.
## Output
Follow [Feature document management](../../../docs/FEATURE_DOCUMENTATION.md): read the Coordinator's
feature context, keep main reports concise, and link necessary owned details in the same feature folder.
Create or revise `docs/<feature-id>/formulation.md` with assumptions; variables; strong/weak forms; discretization; kinematics; constitutive contract; element equations; integration; algorithms; recovery; numerical risks; verification implications; and handoff.
## Evidence
Every equation defines symbols, dimensions, configuration, sign, coordinate basis, and source or derivation. Show linearization when a consistent tangent is required. Link risks and checks to requirement IDs.
## Boundary
Do not design source-level APIs, parsers, storage schemas, or implementation tasks. Do not silently choose physics, stabilization, interpolation, constitutive behavior, or solver semantics outside the approved contract; return that critical choice for decision.
Use approved requirements and evidence; consult [analysis-family guidance](references/analysis-families.md) when specifying that family's formulation. Make physics-changing choices explicit and return unresolved scope or evidence gaps rather than silently choosing them. Include verification implications, but do not prescribe C++ storage, persistent schema, tolerance values, or reference-result edits.