--- name: fem-formulation-spec description: Use when an approved structural FEM feature needs an implementation-ready mathematical formulation before numerical review or coding. --- # FEM Formulation Specification Translate approved requirements and research into a complete mathematical contract, independent of programming APIs and storage formats. ## Inputs - `docs//requirements.md` - `docs//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//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.