# Linear Static MITC4 Shell Requirements

## Metadata

- feature_id: `linear-static-mitc4-shell`
- title: `Linear Static MITC4 Shell`
- status: `approved`
- owner_agent: `requirement-agent`
- date: `2026-08-11`
- approval_basis: 사용자와 확정한 선형 정적 범위, `S4`/`S4R` 매핑, 6자유도 외부 계약, drilling 안정화, 자동 director 생성, 결과 및 검증 계약
- current_product_state: `requirements-approved-not-implemented`
- formulation_alignment: `docs/formulations/mitc4-shell-formulation.md`는 선행 draft이며 이 baseline의 6자유도 및 drilling 계약에 맞춘 후속 개정이 필요함
- declared_reference_candidate: 사용자가 `reference/shell/`에 Abaqus `S4R` 모델을 추가했다고 `2026-08-11`에 선언함
- reference_inventory_state: 요구조건 작성 시점의 workspace에서는 `reference/shell/`이 관찰되지 않아 exact artifact inventory는 `needs-reference-artifacts`로 유지함

## Purpose

이 baseline은 승인된 Abaqus `.inp` subset의 4절점 `S4` 또는 `S4R` 요소를
FESA의 단일 MITC4 formulation으로 매핑하고, 소변형·소회전 선형 정적 해석을 수행해
검증 가능한 절점 및 쉘 요소 결과를 authoritative `results.h5`에 기록하는 다음
end-to-end 기능을 정의한다.

FESA는 Abaqus `S4` 또는 `S4R` formulation을 재현한다고 주장하지 않는다. 두 source
element type은 같은 FESA MITC4 formulation으로 매핑하며 source type과 internal
formulation identity를 분리해 보존한다. Abaqus reference comparison은 병진변위
`U1/U2/U3`만 pass/fail에 사용하고 회전 `UR1/UR2/UR3`은 비차단 warning evidence로
사용한다.

이 문서의 승인은 MITC4가 현재 제품에 구현되었다는 뜻이 아니다. Research,
Formulation, Numerical Review, I/O, Reference Model, Implementation Planning 및 이후
검증 gate가 순서대로 완료되어야 제품 기능이 된다.

## Source Basis

- 사용자 승인 결정: 선형 정적 실행 범위, `S4`/`S4R` 공통 MITC4 매핑, 전역 6자유도,
  비물리 drilling 안정화, 자동 두께방향 director, 단일층 등방성 재료, 필수 결과와
  displacement 중심 reference 판정
- `docs/reference-papers/MITC4/`: MITC4 director kinematics, assumed transverse shear,
  5-DOF physical kernel, 6-DOF transformation/stabilization 사례 및 shell benchmark 근거
- `docs/PRD.md`, `docs/ARCHITECTURE.md`, `docs/ADR.md`: end-to-end feature boundary,
  ownership, linear-static lifecycle, deterministic assembly, HDF5, reference immutability 및
  failure atomicity
- `docs/formulations/mitc4-shell-formulation.md`: 후속 정렬이 필요한 선행 draft이며 이
  approved requirements baseline을 변경하는 근거로 사용하지 않음

## In Scope

- 입력 파일당 하나의 `*STEP, *STATIC` 선형 정적 해석
- small displacement와 small rotation
- 4절점 bilinear quadrilateral MITC4 shell
- Abaqus source element type `S4`와 `S4R`의 동일한 FESA MITC4 매핑
- 절점당 전역 자유도 `[UX, UY, UZ, URX, URY, URZ]`
- 3개 병진과 director 접평면 회전 2개로 구성된 물리 MITC4 kernel
- 비물리 local drilling 회전 1개에 대한 작은 scale-aware 수치 안정화
- 막, 굽힘, 횡전단 및 이들의 coupling
- 단일층, 균질 등방성 선형 탄성
- element set별 일정한 양의 두께와 하나의 material을 갖는 `*SHELL SECTION`
- 평면 및 매끄러운 곡면을 근사하는 유효한 사각형 mesh
- midsurface geometry와 element connectivity로부터 초기 두께방향 unit director 자동 생성
- DOF 1~6의 nodal `*BOUNDARY`와 nodal `*CLOAD`
- deterministic assembly와 기존 linear-static partition/factorization/substitution lifecycle
- HDF5 nodal displacement/reaction, shell generalized strain/resultant, in-plane stress, residual 및 energy output
- element invariant, patch, locking, distortion, curved-shell, reference 및 physics verification

## Out Of Scope

- Abaqus full compatibility 또는 Abaqus `S4`/`S4R` formulation equivalence 주장
- Abaqus reduced-integration, hourglass-control 또는 finite-membrane-strain 의미의 재현
- `S3`, `S8`, `S8R`, continuum shell, solid-shell 및 axisymmetric shell
- 다중 step과 step 간 load, boundary 또는 state propagation
- 기하비선형, 재료비선형, 좌굴, modal, dynamic, contact 및 thermal analysis 실행
- composite/laminated shell, orthotropic material, layer orientation 및 층별 integration output
- variable/nodal/distributed thickness, section offset 및 thickness stretch DOF
- explicit nodal normal/director input과 user-defined shell orientation
- shared source node에서 불연속 director를 요구하는 sharp fold, hinge 또는 shell-beam joint
- `*DLOAD`, pressure, gravity, body force, edge traction, follower load 및 current-normal load
- drilling 방향의 nodal moment를 물리 하중으로 취급하는 동작
- `S13/S23` section-point stress와 `S33` stress recovery
- Abaqus reaction, stress, strain 또는 section resultant equality를 release pass/fail로 사용하는 비교
- Abaqus, Nastran 또는 다른 reference solver 실행과 reference artifact 생성·수정·복원

## Analysis Definition

- analysis_type: single-step linear static
- kinematics: small displacement, small rotation, fixed initial geometry/director
- element: four-node bilinear quadrilateral MITC4 shell
- source_element_types: Abaqus `S4`, `S4R`
- internal_formulation: one FESA MITC4 formulation independent of source type
- nodal_dofs: global `[UX, UY, UZ, URX, URY, URZ]`
- physical_local_dofs: three translations and two director-tangent rotations per node
- numerical_local_dof: one nonphysical drilling rotation per node
- deformation_modes: membrane, bending, transverse shear, and coupling
- material_model: single-layer homogeneous isotropic linear elasticity from `E` and `nu`
- section_model: one constant positive thickness and one material per assigned element
- boundary_conditions: nodal essential constraints on global DOFs 1 through 6
- loads: nodal concentrated force/moment; drilling-direction moment excluded
- initial_director: deterministic geometry-derived positive-thickness unit vector
- units: user-consistent unit system; no unit system inferred from `.inp`
- authoritative_output: HDF5 `results.h5`
- reference_pass_fail: global translational displacement `U1/U2/U3`
- reference_warning_only: global nodal rotation `UR1/UR2/UR3`

## Input Requirements

- **FESA-REQ-LSMITC4-001** — The solver shall accept exactly one `*STEP` containing `*STATIC` and shall reject a second analysis step or a non-linear/non-static procedure with a structured unsupported diagnostic.
- **FESA-REQ-LSMITC4-002** — The semantic mapper shall map both `*ELEMENT, TYPE=S4` and `*ELEMENT, TYPE=S4R` to the same FESA MITC4 formulation without claiming Abaqus formulation equivalence.
- **FESA-REQ-LSMITC4-003** — The model and result metadata shall preserve the source element type separately from the internal `FESA-MITC4` formulation identity.
- **FESA-REQ-LSMITC4-004** — Each accepted shell element shall contain exactly four distinct source nodes in the documented bilinear quadrilateral order, and the original source element identity shall remain stable through diagnostics and results.
- **FESA-REQ-LSMITC4-005** — Each shell node shall expose global DOFs `[UX, UY, UZ, URX, URY, URZ]` in that exact order; equation IDs shall not be stored in Node or Element records.
- **FESA-REQ-LSMITC4-006** — The material subset shall accept homogeneous isotropic `*ELASTIC` data containing `E` and `nu`, require finite `E > 0` and `-1 < nu < 0.5`, and reject temperature/field dependence and unsupported material behavior.
- **FESA-REQ-LSMITC4-007** — The section subset shall accept a single-layer `*SHELL SECTION, MATERIAL=<name>` with one finite constant thickness `t > 0` per section assignment.
- **FESA-REQ-LSMITC4-008** — Every active shell element shall resolve to exactly one shell section and exactly one material; missing, conflicting, duplicate or unresolved assignments shall be model errors.
- **FESA-REQ-LSMITC4-009** — A model may contain multiple element sets, constant-thickness shell sections and isotropic materials, but every individual element shall remain single-layer and homogeneous.
- **FESA-REQ-LSMITC4-010** — Composite data, section offset, distributed/nodal thickness, explicit director/normal input and user-defined material orientation shall be rejected as unsupported rather than ignored.
- **FESA-REQ-LSMITC4-011** — The initial nodal director shall be a dimensionless unit vector in the positive thickness direction; scalar thickness shall remain a separate property and shall not be encoded in the director magnitude.
- **FESA-REQ-LSMITC4-012** — The positive element normal candidate shall be derived deterministically from the source node order and midsurface covariant tangent cross product.
- **FESA-REQ-LSMITC4-013** — At a smooth shared node, consistently oriented incident element normal candidates shall be combined by deterministic area-weighted averaging and normalized to form the common nodal director.
- **FESA-REQ-LSMITC4-014** — Zero/near-zero normal candidates, a zero/near-zero averaged director, opposing incident orientations, and a fold sharper than the approved smooth-patch criterion shall fail model validation.
- **FESA-REQ-LSMITC4-015** — A physical fold or hinge shall be represented with duplicated source nodes so each smooth shell patch owns a separate director; the solver shall not silently average a discontinuous director field.
- **FESA-REQ-LSMITC4-016** — Element geometry validation shall reject duplicate nodes, self-intersection, nonfinite coordinates, degenerate area, and nonpositive or near-singular Jacobians at every formulation-required Gauss and tying location.
- **FESA-REQ-LSMITC4-017** — The solver shall support nodal `*BOUNDARY` targets resolved by source node label or node set for global DOFs 1 through 6, including existing zero and nonzero prescribed-displacement semantics.
- **FESA-REQ-LSMITC4-018** — The solver shall support nodal `*CLOAD` forces on DOFs 1 through 3 and nodal moments on DOFs 4 through 6 after deterministic aggregation in global coordinates.
- **FESA-REQ-LSMITC4-019** — The aggregated nodal moment component parallel to the approved nodal director shall be rejected as `unsupported-drilling-load`; a drilling-direction moment shall not be carried only by numerical stabilization.
- **FESA-REQ-LSMITC4-020** — `*DLOAD`, pressure, gravity, body force, edge traction and follower load shall remain unsupported at parser/CLI level even if a formulation-only equivalent-load kernel is later tested.
- **FESA-REQ-LSMITC4-021** — The accepted parser subset shall include the existing identity-preserving node, set, material, assembly, boundary, load and single-static-step keywords plus `*ELEMENT, TYPE=S4|S4R` and the approved single-layer `*SHELL SECTION` subset.
- **FESA-REQ-LSMITC4-022** — Identity `*PART/*ASSEMBLY/*INSTANCE` wrappers and multiple identity instances shall preserve `SourceEntityId`; instance transforms, nested assembly and dependent/independent mesh semantics shall remain unsupported unless separately approved.
- **FESA-REQ-LSMITC4-023** — Output-request keywords in the existing no-op allowlist shall not alter mandatory FESA results, and unsupported model-affecting keywords outside the allowlist shall be errors.

## Architecture and Execution Requirements

- **FESA-REQ-LSMITC4-024** — `Domain` shall own parsed shell elements, material/section assignments, source identity and initial directors; `AnalysisModel` shall expose the active single-step view without copying Domain objects.
- **FESA-REQ-LSMITC4-025** — `DofManager` alone shall own six-DOF node definitions, full/free equation numbering, constraint mappings, shell scatter maps and sparse-pattern ownership.
- **FESA-REQ-LSMITC4-026** — Linear-static `AnalysisState` shall own only the solution, force/residual/reaction, shell recovery and energy rows needed by this procedure and shall not preallocate nonlinear director history, iteration state, velocity or acceleration.
- **FESA-REQ-LSMITC4-027** — Element-local shell calculations shall use independent buffers and global sparse assembly shall preserve stable source-element ordering, deterministic COO ordering and fixed reduction independent of thread count.
- **FESA-REQ-LSMITC4-028** — The linear-static lifecycle shall assemble and partition stiffness, factorize `Kff`, then assemble loads, form `Ff-Kfc*dc`, substitute, reconstruct the full displacement, recover the full residual and only then commit results.
- **FESA-REQ-LSMITC4-029** — Constrained reactions and free-equilibrium evidence shall be recovered from the assembled full residual `K*d-F`; the shell feature shall not define reaction by separately summing recovered element resultants.
- **FESA-REQ-LSMITC4-030** — `S4` and `S4R` source elements with identical geometry, properties, constraints and loads shall execute the same FESA MITC4 numerical path; only preserved source metadata may differ.

## Numerical Formulation Boundary Requirements

- **FESA-REQ-LSMITC4-031** — The physical MITC4 kernel shall use three translations and two director-tangent rotations per node and shall not treat the drilling rotation as a physical strain variable.
- **FESA-REQ-LSMITC4-032** — The six-DOF element embedding shall transform global nodal rotations into two director-tangent components and one director-parallel drilling component using deterministic right-handed orthonormal frames.
- **FESA-REQ-LSMITC4-033** — The drilling contribution shall be symmetric, positive, scale-aware, sufficiently small to remain numerical, and sufficient to remove the nonphysical drilling null modes of otherwise valid models.
- **FESA-REQ-LSMITC4-034** — The drilling scale shall be derived from dimensionally compatible physical rotational stiffness; the implementation shall not apply `10^-3 * min(all Kii)` across mixed translational and rotational diagonal entries without an approved dimensional derivation.
- **FESA-REQ-LSMITC4-035** — Drilling stabilization shall not contribute to physical membrane, bending or transverse-shear generalized strain/resultant or section-point stress; its energy shall be recovered separately.
- **FESA-REQ-LSMITC4-036** — Exact drilling reference scale, dimensionless coefficient and acceptable artificial-energy ratio shall be approved through Research and Numerical Review before Implementation Planning; no implementation-selected default may close this decision silently.
- **FESA-REQ-LSMITC4-037** — Source `S4R` shall not select reduced integration or Abaqus hourglass control; all accepted `S4` and `S4R` inputs shall use the single quadrature and MITC tying contract approved by the FESA formulation.
- **FESA-REQ-LSMITC4-038** — The stabilized element shall retain exactly six physical rigid-body modes within the approved normalized tolerance, preserve stiffness symmetry and coordinate-transformation energy, and have positive energy for every accepted non-rigid physical deformation mode.

## Output Requirements

- **FESA-REQ-LSMITC4-039** — The sole authoritative solver output shall be HDF5 `results.h5`; the writer shall validate and close a temporary candidate before replacing the final path and shall not leave an incomplete final file after failure.
- **FESA-REQ-LSMITC4-040** — Metadata/model output shall include schema and solver versions, source input identity, user-consistent unit label, global/local coordinate conventions, source element type, internal formulation, stable node/element identities, shell section/material identity and initial nodal director.
- **FESA-REQ-LSMITC4-041** — Every node shall have global displacement `[U1,U2,U3,UR1,UR2,UR3]` and full-residual reaction `[RF1,RF2,RF3,RM1,RM2,RM3]` rows in stable source identity order.
- **FESA-REQ-LSMITC4-042** — Every required shell integration location shall output local generalized strain components `[E11,E22,G12,K11,K22,K12,G13,G23]` with membrane/shear strain dimensionless and curvature dimension `1/length`.
- **FESA-REQ-LSMITC4-043** — The same shell locations shall output local section resultant components `[N11,N22,N12,M11,M22,M12,Q13,Q23]`, where `N` and `Q` have dimension `force/length` and `M` has dimension `force` as moment resultant per unit edge length.
- **FESA-REQ-LSMITC4-044** — Each required shell location shall output local in-plane stress `[S11,S22,S12]` at bottom, middle and top section positions with dimension `force/length^2`; `S33` shall be documented as the plane-stress assumption and `S13/S23` point stress shall not be emitted.
- **FESA-REQ-LSMITC4-045** — Result rows shall identify source element, integration/tying or recovery location, natural coordinates, section position, local frame/director and component order without averaging mismatched locations.
- **FESA-REQ-LSMITC4-046** — The output shall include free-DOF residual evidence, total force/moment equilibrium metrics, physical strain energy and drilling stabilization energy with dimension `force*length` and deterministic aggregation order.
- **FESA-REQ-LSMITC4-047** — Abaqus output requests shall neither suppress nor expand the mandatory HDF5 quantity inventory; any deterministic FESA CSV projection shall remain a debugging/review view rather than official solver output.
- **FESA-REQ-LSMITC4-048** — Nonfinite recovery values, inconsistent component/location inventory or failure to finalize required HDF5 rows shall fail the analysis without committing a partial successful state.

## Verification Requirements

- **FESA-REQ-LSMITC4-049** — Every production behavior shall follow project TDD `RED -> GREEN -> VERIFY`, have a related C++ test, and later pass MSVC x64 Debug CMake/CTest with no new warning; this requirements phase shall not run Harness or implementation validation.
- **FESA-REQ-LSMITC4-050** — Element invariant tests shall cover frame orthonormality/handedness, Jacobian sign, stiffness symmetry, coordinate-transformation energy invariance, six physical rigid modes, deformation-mode positivity and deterministic repeatability.
- **FESA-REQ-LSMITC4-051** — Normalized algebraic acceptance shall use `1e-12` for symmetry, frame orthonormality and transformation-energy invariance and `1e-10` for rigid-mode action, linear-system residual and global equilibrium unless Numerical Review approves and documents an evidence-backed scale-aware replacement before Implementation Planning.
- **FESA-REQ-LSMITC4-052** — Patch verification shall independently cover constant membrane strain/stress, pure bending, transverse shear and twist, including sign and component-order checks for generalized strain, resultant and recovered in-plane stress.
- **FESA-REQ-LSMITC4-053** — Locking/convergence verification shall include thin- and thick-shell or plate sequences over documented thickness ratios and mesh refinements and shall demonstrate the approved MITC4 transverse-shear behavior rather than judge one displacement on one mesh.
- **FESA-REQ-LSMITC4-054** — Geometry verification shall include planar, smoothly curved, distorted and warped valid elements plus negative tests for degenerate, inverted, self-intersecting, opposing-normal and sharp-fold topologies.
- **FESA-REQ-LSMITC4-055** — Curved-shell verification shall include at least one nodal-load-compatible pinched-cylinder, hemispherical-shell or equivalently justified benchmark; Scordelis-Lo may be included only with documented equivalent nodal loading inside the approved input subset.
- **FESA-REQ-LSMITC4-056** — Drilling verification shall vary the approved stabilization coefficient around its nominal value, confirm stable equation rank, report physical displacement/resultant sensitivity and enforce the approved drilling-to-physical energy warning criterion.
- **FESA-REQ-LSMITC4-057** — Physics verification shall check load/reaction balance, global moment balance, displacement direction, symmetry, energy positivity, result sign and consistency between assembled residual and recovered shell resultants.

## Verification Quantities

- nodal_displacement: required, global six components; `U1/U2/U3` reference pass/fail and `UR1/UR2/UR3` warning-only
- reaction: required, global six components from full residual; internal physics verification
- shell_generalized_strain: required, local eight components at documented locations
- shell_section_resultant: required, local `N/M/Q` eight components at documented locations
- stress: required, local bottom/middle/top `[S11,S22,S12]`; Abaqus equality comparison N/A
- residual: required, free-DOF and normalized global equilibrium evidence
- energy: required, separate physical strain and drilling stabilization energies
- modes_and_invariants: required, six physical rigid modes, symmetry, transformation invariance and positive deformation energy

## Tolerance Policy

- **FESA-REQ-LSMITC4-058** — Abaqus reference pass/fail shall apply only to matched global `U1/U2/U3` rows using `tolerance_c = absolute_floor_c + relative_coefficient_c * reference_scale_c`.
- **FESA-REQ-LSMITC4-059** — `reference_scale_c` shall be computed only from finite Abaqus values in the same model, step/frame, quantity and component group; reference values shall not be zero-clamped and row-specific relative denominators shall not replace the group scale.
- **FESA-REQ-LSMITC4-060** — MITC4 displacement absolute floors and relative coefficients shall not silently inherit the B33 numerical values; Reference Model shall propose dimensioned floors and coefficients from benchmark evidence and Numerical Review shall approve them before reference comparison implementation.
- **FESA-REQ-LSMITC4-061** — Global `UR1/UR2/UR3` rows shall be compared and fully reported but shall not affect pass/fail; errors beyond the approved large-error threshold shall emit deterministic nonblocking warnings and remain visible in known limitations.
- **FESA-REQ-LSMITC4-062** — The UR large-error threshold and drilling-energy warning ratio are `needs-downstream-decision` owned jointly by Reference Model and Numerical Review and shall be fixed before Implementation Planning.
- **FESA-REQ-LSMITC4-063** — Missing, extra, duplicate, nonfinite, schema-mismatched or source-identity-mismatched rows shall fail artifact/schema validation before numeric tolerance evaluation for both U and UR inventories.
- **FESA-REQ-LSMITC4-064** — The comparison report shall record each U/UR row decision, maximum absolute error, component-scale normalized error, RMS error, vector-norm error and worst source row/component; nonblocking UR warnings shall not be omitted from an otherwise passing report.

## Reference Artifact Requirements

The user-declared first candidate is `reference/shell/` and is said to contain an Abaqus
`S4R` model. At requirements-authoring time that directory was not visible in the shared
workspace, so no exact filename, schema, units, generator, step/frame or row inventory is
asserted here. The bundle shall remain read-only when it becomes observable.

- **FESA-REQ-LSMITC4-065** — Reference Model shall inventory `reference/shell/` without creating, renaming, rewriting or repairing files and shall classify a missing/unobservable candidate as `needs-reference-artifacts` rather than fabricate its content.
- **FESA-REQ-LSMITC4-066** — A usable new reference bundle shall contain `model.inp`, `metadata.json` and `<model-id>_displacements.csv`, or shall receive an explicitly approved legacy-alias contract that records its exact existing filenames without rename.
- **FESA-REQ-LSMITC4-067** — Reference metadata shall record Abaqus generator/version, creation provenance, source element type, model/step/frame identity, user unit system, global coordinate convention, section/material/thickness data, CSV schema and tolerance policy.
- **FESA-REQ-LSMITC4-068** — Artifact validation shall confirm an approved single `*STEP, *STATIC` model, expected `S4` or `S4R` source type, supported keyword subset, unique finite displacement rows and exact source-node/component identity before comparison.
- **FESA-REQ-LSMITC4-069** — End-to-end mapping evidence shall cover at least one `S4` input and at least one `S4R` input; the declared `reference/shell/` S4R model may satisfy only the S4R side after inventory validation, so separate S4 evidence remains required.
- **FESA-REQ-LSMITC4-070** — Reference verification shall compare FESA HDF5 global nodal displacement rows directly against Abaqus displacement CSV rows by model, step/frame, source node and component identity; a FESA-generated CSV view shall not become the authoritative comparison source.
- **FESA-REQ-LSMITC4-071** — Abaqus reaction, stress, strain and shell force/moment output may be retained as review evidence if present but shall not change the approved `U1/U2/U3` pass/fail boundary or become an undeclared equality gate.
- **FESA-REQ-LSMITC4-072** — FESA agents shall not execute Abaqus or another reference solver and shall not generate, modify, restore or normalize reference artifacts during requirements, research, formulation, implementation or verification unless a later phase explicitly authorizes that operation.

## Requirement Verification Matrix

Each range row below applies its verification method and acceptance criteria to every
individual requirement ID in that inclusive range; the ranges cover `001` through `072`
without gaps or overlap.

| id range | requirement theme | category | source | priority | verification method | acceptance criteria | tolerance/decision owner | downstream agents | status |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| `001` | Single linear-static step | analysis | User approval; PRD/ADR linear-static lifecycle | must | Parser/semantic negative and CLI integration tests | One supported static step runs; other procedures or a second step fail deterministically | Exact procedure inventory | I/O Definition; Implementation Planning | approved |
| `002-004` | `S4`/`S4R` mapping and identity | input/model | User approval | must | Parser/semantic/HDF5 metadata tests | Both source types map to one MITC4 path and preserve distinct source metadata and four-node identity | Exact element/type identity | I/O Definition; Implementation Planning | approved |
| `005` | Six global nodal DOFs | model | User approval; project DOF convention | must | DofManager and HDF5 schema tests | Exact component order and no distributed equation ownership | Exact ordering | Formulation; I/O Definition; Implementation Planning | approved |
| `006-010` | Isotropic material and single-layer section | input/model | User approval | must | Parser, mapping and validation tests | Valid `E,nu,t` resolve once per element; excluded section/material meanings fail | Exact inequalities; finite values | Research; I/O Definition; Implementation Planning | approved |
| `011-016` | Initial director and geometry validity | geometry | User approval; MITC director kinematics | must | Geometry/unit/property-based tests | Deterministic unit directors for smooth valid meshes; every invalid topology fails | Smooth-angle and scale-aware Jacobian thresholds: Research + Numerical Review | Research; Formulation; Numerical Review; I/O Definition | approved-with-downstream-decision |
| `017-020` | Boundary and nodal-load subset | input/load | User approval | must | Parser/semantic/load tests | Global BC/CLOAD works; director-parallel moment and distributed loads fail | Drilling projection threshold: Numerical Review + I/O Definition | Formulation; I/O Definition; Implementation Planning | approved-with-downstream-decision |
| `021-023` | Parser subset, wrappers and no-op policy | input | User approval; ADR-003/013/018 | must | Parser diagnostic and semantic identity tests | Only approved meanings affect Domain; excluded meanings fail closed | Exact keyword/diagnostic inventory | I/O Definition; Implementation Planning | approved |
| `024-030` | Ownership, deterministic assembly and linear-static execution | architecture/execution | PRD; ADR-004/007/008/009/016/017 | must | Unit, orchestration and repeated-thread-count tests | Ownership boundaries, event order, residual reaction and deterministic bytes/rows match | `1e-12` deterministic numeric target where applicable | Numerical Review; Implementation Planning | approved |
| `031-038` | 5-DOF physics embedded in 6-DOF with drilling stabilization | numerical boundary | User approval; MITC literature and thesis 6-DOF discussion | must | Formulation review, element invariant, rank and energy tests | Physical outputs exclude drilling; stable system retains six physical rigid modes | Scale/coefficient/energy ratio: Research + Numerical Review | Research; Formulation; Numerical Review | approved-with-downstream-decision |
| `039-048` | Mandatory HDF5 output and failure atomicity | output | User approval; ADR-005/016/018 | must | Recovery, schema, identity, nonfinite and atomicity tests | Every quantity/location/unit/identity exists; failure commits no partial success | Exact component/location inventory; I/O Definition owns schema | Formulation; I/O Definition; Implementation Planning | approved |
| `049-057` | TDD, invariants, patch, locking, geometry, curved shell and physics | verification | User approval; shell benchmark evidence; project process | must | CTest evidence, analytical/patch/convergence reports and physics review | Every verification family has passing evidence before release | `1e-12` symmetry/frame; `1e-10` rigid/residual; convergence targets by Numerical Review | Research; Numerical Review; Reference Model; Implementation Planning | approved-with-downstream-decision |
| `058-060` | Translational displacement pass/fail tolerance | tolerance | User approval | must | Comparator unit/integration tests and report review | Every matched U row uses the approved mixed tolerance without clamp/omission | Floors/coefficient: Reference Model + Numerical Review | Reference Model; Reference Verification | approved-with-downstream-decision |
| `061-062` | Rotational warning and drilling-energy warning | tolerance/warning | User approval | must | Comparator/diagnostic tests and report review | UR never changes pass/fail; large error and energy ratio produce deterministic visible warnings | Thresholds: Reference Model + Numerical Review | Numerical Review; Reference Model; Reference Verification | approved-with-downstream-decision |
| `063-064` | Row/schema failure and report completeness | reference verification | User approval; ADR-005/014/018 | must | Negative comparator and report-schema tests | Invalid inventory fails before numeric comparison; all U/UR metrics remain visible | No ignored invalid rows | I/O Definition; Reference Verification | approved |
| `065-068` | `reference/shell` inventory and artifact validity | reference | User declaration; project artifact policy | must | Read-only inventory and schema/provenance review | Candidate becomes usable only after exact files, S4R type, provenance and row schema are validated | Current state `needs-reference-artifacts` | Reference Model | needs-reference-artifacts |
| `069-071` | S4/S4R coverage and displacement-only comparison | reference | User approval | must | Portfolio coverage and HDF5-to-CSV comparison | At least one model per source type; only U is blocking and UR is warning-only | Requirements `058-064` | Reference Model; Reference Verification; Physics Evaluation | approved-with-downstream-decision |
| `072` | Reference solver/artifact immutability | governance | User/project policy; ADR-010 | must | Process audit and Git diff | No unapproved execution or artifact mutation | Exact zero mutations | All downstream agents | approved |

## Open Questions and Required Downstream Decisions

No additional user scope decision is pending. The following technical values must remain visible
and be approved by their named downstream owners before Implementation Planning:

1. Research and Numerical Review shall define a dimensionally consistent drilling reference stiffness, dimensionless coefficient range and nominal value.
2. Numerical Review shall approve the maximum drilling-to-physical strain-energy warning ratio and the displacement/resultant sensitivity criterion for coefficient variation.
3. Research and Numerical Review shall define the smooth-patch director angle criterion and scale-aware degenerate/Jacobian thresholds without an arbitrary `max(1, ...)` clamp.
4. Reference Model and Numerical Review shall define MITC4 U component absolute floors and relative coefficients in the approved mixed-tolerance form.
5. Reference Model and Numerical Review shall define the nonblocking UR large-error threshold and reporting severity.
6. Formulation shall fix quadrature, tying-point interpolation, shear correction, local axes, sign conventions and recovery locations consistent with this baseline.
7. `reference/shell/` shall be re-inventoried when visible. Until then its exact filenames, units, provenance and CSV schema are not established evidence.
8. Separate `S4` end-to-end evidence remains required even if the declared `S4R` candidate is valid.
9. The existing formulation draft's 5-DOF-only and no-drilling statements conflict with this approved baseline and shall be revised during the Formulation gate, not patched silently during implementation.

## Downstream Handoff

### Research Agent

- Establish source-backed MITC4 linear kinematics, tying, quadrature, shear correction and benchmark applicability.
- Research dimensionally consistent 6-DOF drilling stabilization alternatives and sensitivity/energy criteria; distinguish the thesis's printed `10^-3 min(Kii)` rule from a FESA-approved scale-aware rule.
- Research nodal-director generation, smooth-patch angle handling and distorted/warped geometry limits.
- Identify nodal-load-compatible flat, locking, distorted and curved-shell benchmark families.

### Formulation Agent

- Revise `docs/formulations/mitc4-shell-formulation.md` to align with global 6-DOF input/output and a physical 5-DOF MITC4 kernel plus numerical drilling embedding.
- Keep current-product equations strictly linear static; retain geometric-nonlinear residual/tangent only as clearly separated future formulation.
- Define local frames, transformations, generalized component order, quadrature/tying, stress/resultant recovery, energy split and consistent units/signs.
- Do not introduce distributed-load product support or make `S4R` select reduced integration.

### Numerical Review Agent

- Independently review the revised formulation for rigid modes, rank, symmetry, invariance, locking, distortion, Jacobian/director validation and drilling contamination.
- Approve every downstream numerical threshold listed above before Implementation Planning.
- Require coefficient sensitivity and artificial-energy evidence rather than accepting a small coefficient by assertion.

### I/O Definition Agent

- Define the exact Abaqus keyword/data subset for `S4`, `S4R`, single-layer `*SHELL SECTION`, material, BC and CLOAD semantics.
- Define source identity, auto-director metadata, unsupported-drilling-load projection, diagnostics and exact HDF5 dataset/row schemas.
- Preserve source element type separately from FESA formulation and define bottom/middle/top stress location identity.

### Reference Model Agent

- Re-inventory the user-declared `reference/shell/` candidate read-only and report exact files, S4R identity, provenance, units, step/frame and CSV row schema or `needs-reference-artifacts`.
- Do not rename or repair noncanonical files; propose an explicit legacy alias only if retaining them is justified.
- Prepare coverage for both S4 and S4R, flat/thin/thick/distorted/curved response and the approved displacement-only blocking comparison.
- Propose evidence-backed U tolerance coefficients/floors and UR warning thresholds for Numerical Review approval.

### Implementation Planning Agent

- Do not start until Research, revised Formulation, Numerical Review, I/O and Reference Model contracts have resolved every named downstream decision.
- Trace every `must` requirement to RED/GREEN/VERIFY tests and preserve current solver ownership, deterministic assembly and failure-atomic HDF5 boundaries.
- Include independent tests for source-type mapping, auto directors, drilling rank/energy/sensitivity, all recovery quantities, schema failures and U-versus-UR comparison behavior.
