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+---
+name: fesa-formulation-spec
+description: Use when drafting FESA FEM formulation specifications, strong or weak forms, shape functions, element equations, numerical integration, Jacobian rules, and output recovery contracts.
+---
+
+# FESA Formulation Spec
+
+Use this skill to turn approved requirements and research evidence into an implementable math-level FEM contract.
+
+## Inputs
+
+Read these first:
+
+- `AGENTS.md`
+- `docs/SOLVER_AGENT_DESIGN.md`
+- `docs/formulations/README.md`
+- `docs/requirements/<feature-id>.md`
+- `docs/research/<feature-id>-research.md`
+
+## Workflow
+
+1. Confirm scope, assumptions, primary variables, DOFs, sign convention, units, and coordinate system.
+2. Write the Strong Form and boundary conditions.
+3. Write the Weak or Variational Form and natural boundary terms.
+4. Define Discretization: interpolation, shape functions, partition of unity, Kronecker delta checks, and nodal layout.
+5. Define Kinematics: strain-displacement relation, B matrix or kinematic operator, strain measure, and deformation assumptions.
+6. Define Constitutive Contract without creating C++ APIs.
+7. Define Element Equations: residual or internal force, external force, stiffness or tangent, mass or damping when applicable, and dimensions.
+8. Define Mapping and Numerical Integration: reference coordinates, isoparametric mapping, Jacobian, derivative transform, determinant checks, Gauss points, and weights.
+9. Define Output Recovery for displacement, reaction, element force, strain, stress, and nodal extrapolation.
+10. List Numerical Risks and downstream tests.
+
+## Output Contract
+
+Produce or revise `docs/formulations/<feature-id>-formulation.md` with:
+
+- Scope and Assumptions
+- Primary Variables and DOFs
+- Strong Form
+- Weak or Variational Form
+- Discretization
+- Kinematics
+- Constitutive Contract
+- Element Equations
+- Mapping and Numerical Integration
+- Output Recovery
+- Algorithm Pseudocode
+- Numerical Risks
+- Open Issues and Downstream Handoff
+
+## Boundaries
+
+- Do not implement C++ code.
+- Do not design C++ APIs.
+- Do not implement parsers.
+- Do not run Abaqus, Nastran, or any reference solver.
+- Do not generate reference CSVs.
+- Do not approve release readiness.
+
+## Quality Gate
+
+- Strong form, weak form, discretization, kinematics, constitutive contract, and element equations are distinct.
+- Shape functions include partition of unity and Kronecker delta checks when applicable.
+- Jacobian, determinant validity, derivative transform, integration rule, and output location are explicit.
+- Missing research or requirements become open issues, not assumptions.
+
+## Handoff
+
+Send the formulation to Numerical Review Agent first. After review, pass implementation-relevant pseudocode and acceptance quantities to Implementation Planning Agent, I/O needs to I/O Definition Agent, and benchmarkable checks to Reference Model Agent.