docs: simplify MITC4 verification and drilling scope

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2026-08-12 03:39:57 +09:00
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- feature_id: `linear-static-mitc4-shell`
- source_requirement: `docs/requirements/linear-static-mitc4-shell.md`
- status: `research-complete-awaiting-approval`
- status: `approved`
- owner_agent: `research-agent`
- date: `2026-08-11`
- date: `2026-08-12`
- product_scope: small-displacement, small-rotation, single-step linear static analysis
- evidence_route: local papers in `docs/reference-papers/MITC4/`, the configured FEM wiki, original peer-reviewed papers, and official Abaqus documentation
- reference_inventory_state: `reference/shell/` was not observable during this research; no filename, schema, provenance, unit, or result value is asserted for that user-declared candidate
- reference_inventory_state: existing read-only S4 case at `reference/shell/` and S4R case at `reference/shellR/`; exact comparison paths are fixed by the approved requirements
- source_policy: each external claim below is assigned a reliability tier; FESA decisions are labeled `Project contract`, and derived recommendations are labeled `Inference` or `Research recommendation`
This brief supplies evidence to the Formulation, Numerical Review, I/O Definition, and
@@ -22,10 +22,10 @@ formulation-equivalent to Abaqus S4 or S4R.
1. What kinematics and degrees of freedom define the original four-node continuum-mechanics-based MITC4 element in the linear regime?
2. How are the transverse shear components tied, and what evidence supports the quadrature and homogeneous-isotropic section behavior?
3. How can the physical five-DOF kernel be exposed through six global rotational components without treating drilling rotation as a physical MITC4 strain?
4. Which drilling-stiffness scales are dimensionally defensible, and which coefficient, sensitivity, and artificial-energy decisions remain unsupported?
4. What evidence and dimensional restriction support the approved fixed drilling stabilization without turning it into a physical strain or load channel?
5. What evidence supports connectivity-derived thickness directions, nodal-normal smoothing, local tangent frames, and geometry rejection?
6. What may and may not be inferred when Abaqus S4 and S4R input types are both mapped to one FESA MITC4 formulation?
7. Which patch, locking, distortion, curved-shell, and source-solver benchmarks fit the approved nodal-load-only scope?
7. Which element-level checks and the declared S4/S4R source-solver cases fit the approved implementation scope?
## Source Reliability Tiers
@@ -87,20 +87,20 @@ navigation aid. No key numerical decision relies on them.
- **F-12 — Verified transformation, Tier 2 (S3):** The nodal rotation projection can be written `[alpha,beta,gamma]^T = [t1^T;t2^T;n^T] theta_global`, followed by `K_global=T^T K_local T`. An orthonormal right-handed frame preserves virtual work and strain energy.
- **F-12A — Kinematic inference requiring explicit review:** A physical rigid rotation of a five-DOF director shell is represented by rigid midsurface translations plus the tangent-plane change of each director; the director-parallel drilling coordinate is a gauge and can be zero. A rigid-mode test must construct those director changes explicitly. Blindly assigning the full spatial rotation vector, including its normal projection, to every six-DOF shell rotation would excite the numerical penalty and test a different quantity.
- **F-13 — Verified thesis implementation, Tier 2 (S3):** The thesis fills each otherwise zero local drilling diagonal with `d=10^-3 min(Kii)`. It does not provide a dimensional restriction on which diagonals enter the minimum.
- **F-14 — Dimensional inference:** Translational stiffness diagonals have units `force/length`, while rotational stiffness diagonals have units `force*length`. Taking a minimum across all of them is not unit invariant and can change meaning under a length-unit conversion. The unqualified S3 rule is therefore not acceptable as the FESA rule and is already prohibited by P1.
- **F-14 — Dimensional inference and approved restriction:** Translational stiffness diagonals have units `force/length`, while rotational stiffness diagonals have units `force*length`. Taking a minimum across all of them is not unit invariant. P1 therefore applies the thesis coefficient only to finite strictly positive physical director-tangent rotational diagonals, all of which have `force*length` units.
- **F-15 — Verified precedent, Tier 1 (S9):** Abaqus states that a small drill penalty is proportional to transverse shear stiffness. Its small-strain shell theory presents a rotational constraint scale of the family `G h A_node / (1 + q A_node/h^2)`, multiplied by a small dimensionless factor, with `q=2.5e-5`. The base quantity has units `force*length` and transitions toward a thickness-cubed scale for thin shells. Abaqus says the small factor was selected numerically but does not disclose a general FESA-ready value.
- **F-16 — Research recommendation:** Carry at least two dimensionally compatible candidates into Numerical Review: (A) the S9 transverse-shear/area transition family and (B) the isotropic bending rigidity `D_iso=E t^3/[12(1-nu^2)]`. A statistic formed only from the physical rotational block is a third implementation-dependent comparator. Do not silently choose among them in code.
- **F-17 — Research recommendation with explicit evidence limit:** `10^-3` may be used only as the center of a logarithmic sensitivity experiment because S3 supplies that order of magnitude; it is not an approved nominal coefficient. Numerical Review must examine lower and higher orders, equation rank/conditioning, physical `U/N/M/Q` sensitivity, and separate drilling energy before fixing a nominal value and range.
- **F-18 — Verified evidence gap, Tier 1 (S13):** Abaqus exposes artificial strain energy associated with singular-mode and drill constraints but gives no drill-specific acceptable percentage. Guidance for kinetic energy or dynamic hourglass energy is not transferable to this static drilling penalty. A numerical `E_drill/E_physical` warning limit remains a blocking Numerical Review decision.
- **F-16 — Historical alternatives, not current gates:** A transverse-shear/area transition scale and `D_iso=E t^3/[12(1-nu^2)]` are dimensionally compatible alternatives. The project has instead approved the implementation-local statistic `k_ref=min(R+)`, where `R+` contains only positive physical rotational diagonals; no comparison among these alternatives is required in this feature.
- **F-17 — Approved project decision with evidence limit:** P1 fixes `k_d=10^-3 k_ref` and `K_drill_local=k_d I4`. The `10^-3` value is a project choice informed by S3, not a claim of universal optimality or Abaqus equivalence. Coefficient sweeps, plateau/conditioning calibration, and response-sensitivity studies are outside the approved implementation gate.
- **F-18 — Scope consequence:** S13 supplies no drill-specific acceptable artificial-energy percentage. P1 therefore defines no `E_drill/E_physical` threshold and requires no drilling stiffness, ratio, or energy result dataset. This absence is an explicit scope decision, not missing numerical evidence.
- **F-19 — Verified boundary, Tier 1 (S7/S8):** MITC4/D and independent-rotation membrane formulations give drilling rotation physical/variational content. FESA's approved diagonal regularization is not MITC4/D, must not carry a director-parallel applied moment, and does not justify intersecting-shell, sharp-fold, hinge, or shellbeam drilling transfer.
### Initial director, tangent frame, and geometry evidence
- **F-20 — Verified precedent, Tier 1 (S10):** Abaqus computes normals from adjacent shell midsurfaces and uses order-independent grouping; its default averaging heuristic requires all normals in a smooth group to remain within 20 degrees. The manual warns that a coarse mesh can create a false fold or smooth a real fold.
- **F-21 — Project contract, informed by F-20:** FESA uses one deterministic area-weighted unit director at a smooth shared node and fails discontinuous/opposed incident directions, requiring duplicated source nodes at a physical fold. It does not adopt Abaqus's ability to retain multiple normals at one source node.
- **F-22 — Research recommendation:** Use 20 degrees as the first Numerical Review candidate for the maximum incident-normal deviation in a smooth patch, not as a proven theorem. Because it becomes a hard FESA rejection rather than an Abaqus grouping choice, it must be tested against curved benchmark mesh densities and documented as a mesh-resolution limit before approval.
- **F-22 — Evidence limit and project decision:** The 20-degree value is an Abaqus modeling heuristic, not a universal MITC4 constant. P1 does not adopt or calibrate a smooth-patch angle in this feature; `NR-O03` is removed. Supported inputs still require finite nonzero, consistently oriented incident normals.
- **F-23 — Research recommendation:** Build each tangent frame by selecting the global basis least aligned with the unit director, projecting or crossing it into the tangent plane, normalizing, and forming the second tangent by a cross product. This avoids the near-parallel fixed-axis singularity seen in simpler source examples and is deterministic, but the exact sign/axis rule belongs in Formulation.
- **F-24 — Research recommendation:** Geometry acceptance should use dimensionless or geometry-scaled measures, such as positive Jacobian at every Gauss and tying location, `J_min/J_max`, and `J/L_char^2`; it must not use `max(1,geometry_scale)`. No reviewed source establishes a universal numerical cutoff for all valid warped shells, so the constants remain a Numerical Review decision supported by distortion sweeps.
- **F-24 — Evidence limit and project decision:** No reviewed source establishes a universal distortion/warp cutoff. P1 therefore requires only the formulation/I/O finite, nonzero-area, topology, and positive-Jacobian predicates and removes `NR-O04`; a distortion/warp threshold sweep is not an implementation-readiness gate.
### Abaqus S4/S4R mapping and comparison meaning
@@ -115,9 +115,9 @@ navigation aid. No key numerical decision relies on them.
## Candidate Benchmarks
Exact acceptance tolerances are intentionally not assigned here. Published values are
source targets whose geometry, units, boundary conditions, and loading must be reproduced
and independently checked before becoming a Reference Model contract.
This catalog records useful future evidence, not the minimum implementation-completion
portfolio. The approved blocking source-solver cases are only the existing S4 and S4R
input/displacement pairs named in P1. Published values below do not create additional gates.
| benchmark_id | source/evidence | configuration and target quantities | verifies | does not verify / adaptation limit |
| --- | --- | --- | --- | --- |
@@ -131,8 +131,8 @@ and independently checked before becoming a Reference Model contract.
| `MITC4-PINCHED-CYLINDER` | S1/S3/S12 | thin cylinder with end diaphragms and concentrated pinching load; radial displacement and mesh convergence. S12 cites `1.825e-5`; S3 uses `L=600`, `R=300`, `t=3`, `E=3e6`, `nu=0.3`, `P=1` and reports `1.8248e-5` | nodal-load-compatible inextensional bending, complex membrane response, curvature, convergence, regular/irregular mesh sensitivity | one response point cannot certify stresses or drilling; diaphragm semantics must fit approved BCs without rigid elements |
| `MITC4-NAFEMS-LE3` | S12 | radius-10 hemispherical shell, `t=0.04`, `E=68.25 GPa`, `nu=0.3`, opposite radial `2 kN` point loads; target `Ux(A)=185 mm`; S4 and S4R official cases exist | positive Gaussian curvature, point load, symmetry, automatic directors, S4/S4R source-label coverage candidate | official decks use explicit nodal normals and shorthand symmetry/perturbation semantics; FESA adaptation and mesh refinement are required, and the target is not a tolerance |
| `MITC4-SCORDELIS-LO` | S1/S2/S4/S5 | quarter cylindrical roof, mesh convergence of free-edge displacement and preferably field/resultant evidence | mixed-dominated shell behavior and classical convergence comparison | original dead-weight loading is outside P1; only a documented deterministic equivalent nodal CLOAD version may enter FESA product tests |
| `MITC4-DIRECTOR-GEOMETRY` | S10 plus P1 | connectivity reversal, smooth curved patches around the candidate angle, opposed normals, duplicated-node fold, valid warp, bow-tie, inversion, degeneracy, and Gauss/tying Jacobian sweeps | deterministic director generation and fail-closed geometry policy | physical shell accuracy |
| `MITC4-DRILL-SENSITIVITY` | S3/S9/S13 plus inference | dimensionally compatible scale candidates; logarithmic coefficient sweep around the S3 order; unconstrained rank, constrained solve, `U/N/M/Q`, condition evidence, `E_physical`, and `E_drill` | regularization sufficiency and physical contamination risk | no source supplies an automatic passing coefficient or energy ratio |
| `MITC4-DIRECTOR-GEOMETRY` | S10 plus P1 | connectivity reversal, opposed normals, bow-tie, inversion, degeneracy, and Gauss/tying Jacobian checks | deterministic director generation and fail-closed basic geometry policy | `NR-O03`/`NR-O04` calibration and physical shell accuracy |
| `MITC4-DRILL-FIXED` | S3 plus P1 | exact positive physical-rotational-diagonal selection, fixed `10^-3` factor, free-element rank, symmetry, and physical-recovery exclusion | implementation of the approved numerical regularization | coefficient optimality, sensitivity plateau, or energy ratio |
| `MITC4-S4-S4R-SAME-PATH` | S11 plus P1 | identical supported model written once as S4 and once as S4R; FESA HDF5 numeric rows equal while source metadata differs | approved semantic mapping and deterministic internal path | Abaqus S4/S4R equivalence; their reference displacements are expected to differ on finite meshes |
The local S3 hemispherical example with target displacement `0.0924` and its reported
@@ -143,10 +143,10 @@ problems must not be mixed.
## Verification Relevance
- **Element code verification:** rigid modes, stiffness symmetry, tangent-frame orthonormality, transformation-energy invariance, quadrature cross-checks, individual tying values, and patch fields isolate algebraic mistakes before a source-solver comparison.
- **Locking and convergence:** thin/thick cantilevers, pinched cylinder, LE3, and optional ScordelisLo must be run as mesh/thickness sequences. Underpredicted displacement on a single thin mesh is evidence of possible locking, not a complete diagnosis.
- **Locking and convergence:** thin/thick cantilevers, pinched cylinder, LE3, and ScordelisLo remain useful future studies. They are not additional completion gates for the approved two-case implementation scope.
- **Geometry verification:** tests must evaluate every formulation-required Gauss and tying location, not only the element center. Director smoothing and Jacobian quality are separate checks; a smooth director cannot rescue a self-intersecting or inverted mapping.
- **Drilling verification:** rank success alone is insufficient. Numerical Review must compare physical outputs across the coefficient sweep and examine separate artificial energy. Drilling stabilization must remain absent from `E/N/M/Q/stress` recovery.
- **Reference comparison:** a valid S4 and S4R artifact portfolio can test source mapping and global displacement. It cannot prove formulation identity. Schema, node/component identity, units, and finite values fail before P1's mixed displacement tolerance is evaluated.
- **Drilling verification:** verify the fixed formula, dimensional family, symmetry, positivity, four-mode regularization, and absence from physical `E/N/M/Q/stress` recovery. Sensitivity and artificial-energy evidence are excluded.
- **Reference comparison:** the declared S4 and S4R cases test source mapping and global displacement. They cannot prove formulation identity. Missing, extra, duplicate, nonfinite, or source-node/component-mismatched required rows fail before P1's mixed displacement tolerance is evaluated.
- **Physics sanity:** force and global moment balance, symmetry, displacement direction, reaction sign, positive physical energy, free residual, and consistency of recovered resultants remain mandatory even when all reference displacement rows pass.
- **Validation boundary:** the identified sources provide analytical, benchmark, and source-solver verification. No experimental dataset was established for the approved homogeneous linear-static feature; physical validation remains N/A unless separately added.
@@ -172,72 +172,65 @@ problems must not be mixed.
2. Use a single documented `2x2` midsurface integration path for source S4 and S4R, subject to independent Formulation derivation and Numerical Review rank/patch checks.
3. Use homogeneous-isotropic plane-stress resultants with `5/6` transverse-shear correction as the formulation candidate, and recover linear in-plane stress at `-t/2,0,+t/2`.
4. Project global rotations with deterministic right-handed nodal frames and keep drilling stiffness and energy algebraically separate from all physical shell results.
5. Reject the unqualified `10^-3 min(all Kii)` rule. Compare the dimensionally compatible F-16 candidates before selecting a scale.
6. Treat 20 degrees as a documented smooth-normal candidate requiring mesh/benchmark calibration, not as a self-justifying universal threshold.
7. Make pinched cylinder and NAFEMS LE3 the preferred nodal-load-compatible curved tests; retain ScordelisLo only as a documented equivalent-nodal-load auxiliary case.
5. Apply `10^-3` only to the minimum finite positive physical director-tangent rotational diagonal and use the resulting scalar uniformly for the four local drilling coordinates.
6. Do not adopt a calibrated smooth-normal angle or distortion/warp threshold in this feature; retain finite, orientation, topology and positive-Jacobian validation.
7. Keep pinched cylinder, NAFEMS LE3 and ScordelisLo as optional future evidence rather than implementation-completion requirements.
### Blocking numerical decisions before Implementation Planning
### Closed decisions and nonblocking evidence limits
1. **Drilling reference scale:** choose between or formally combine the F-16 candidates after unit-rescaling, thickness, mesh-size, rank, and conditioning studies.
2. **Drilling coefficient:** define the tested logarithmic range and nominal dimensionless value. S3 supports only considering `10^-3` as an experiment center, not approving it.
3. **Drilling acceptance:** define physical displacement/resultant sensitivity and an `E_drill/E_physical` warning rule. If physical energy is zero or near zero, the ratio must not be hidden by an arbitrary denominator clamp; the two energies need explicit classification.
4. **Director angle:** approve the hard smooth-patch criterion after checking representative curved mesh densities. The consequence of rejection versus mesh refinement must be documented.
5. **Geometry scale:** approve dimensionless Jacobian/warp measures and constants at Gauss and tying locations. No universal source value was identified.
6. **Reference tolerances:** P1 assigns U floors/coefficients and UR warning thresholds to Reference Model and Numerical Review; this Research brief supplies no invented values.
7. **Artifact inventory:** re-inventory `reference/shell/` read-only when it becomes visible. Until then the user-declared S4R candidate provides no auditable benchmark evidence, and separate S4 evidence is still required.
1. **Drilling:** P1 fixes the positive physical-rotational-diagonal scale and `10^-3` factor. Alternative-family comparison, coefficient sweep, conditioning plateau and artificial-energy threshold are not required.
2. **Director/geometry:** `NR-O03` and `NR-O04` are removed. The absence of a calibrated smooth angle or distortion/warp cutoff is not missing evidence.
3. **Reference tolerance:** P1 reuses the B33 formula `1e-9 + 1e-6*reference_scale_c`; U exceedance fails and UR exceedance only warns.
4. **Reference cases:** the existing `reference/shell/` S4 and `reference/shellR/` S4R input/displacement pairs are the complete required inventory. Administrative bundle metadata and an expanded portfolio are not gates.
These issues do not require a new user scope choice, but Items 15 must be closed by
Numerical Review before implementation planning can truthfully claim a complete numerical
contract.
No research-owned numerical decision remains blocking for Implementation Planning.
## Requirement Traceability
| requirement area | research evidence | downstream result |
| --- | --- | --- |
| `006-010` material/section | F-09F-11 | homogeneous isotropic plane-stress and section recovery candidate; exact I/O remains downstream |
| `011-016` director/geometry | F-20F-24, `MITC4-DIRECTOR-GEOMETRY` | 20-degree and scale-aware geometry candidates; constants remain blocking Numerical Review decisions |
| `031-038` 5-to-6 DOF/drilling | F-12F-19, `MITC4-DRILL-SENSITIVITY` | dimensional rejection of mixed diagonals; scale/coefficient/energy decisions carried forward |
| `011-016` director/geometry | F-20F-24, `MITC4-DIRECTOR-GEOMETRY` | basic deterministic validity rules; `NR-O03`/`NR-O04` calibration removed |
| `031-038` 5-to-6 DOF/drilling | F-12F-19, `MITC4-DRILL-FIXED` | fixed dimensionally compatible rotational-diagonal rule; calibration and energy output excluded |
| `037` S4/S4R common path | F-25F-27, `MITC4-S4-S4R-SAME-PATH` | source mapping supported only as a FESA product decision, never an Abaqus formulation claim |
| `039-048` shell outputs | F-09F-11/F-18 | physical/drilling energy split and bottom/mid/top stress-location evidence |
| `049-057` element verification | F-28/F-29 and Candidate Benchmarks | invariant, patch, locking, distortion, curved, drilling, equilibrium, and energy portfolio |
| `058-064` U/UR tolerance | F-25F-27 | displacement comparison is meaningful but numerical tolerance/UR warning values remain with Reference Model/Numerical Review |
| `065-072` reference artifacts | S11/S12 and current inventory state | official public S4/S4R candidates identified; local user artifact remains `needs-reference-artifacts` and immutable |
| `039-048` shell outputs | F-09F-11/F-18 | physical shell output and bottom/mid/top stress locations; no drilling-specific dataset |
| `049-057` element verification | F-28/F-29 and Candidate Benchmarks | required invariant/patch/fixed-drill checks; broader portfolio remains optional |
| `058-064` U/UR tolerance | F-25F-27 plus P1 | exact B33 mixed tolerance; U blocking and UR warning-only |
| `065-072` reference artifacts | S11/S12 and current inventory state | exact existing S4/S4R paths, source-row/component mapping, and immutability |
## Downstream Handoff
### Formulation Agent
- Re-derive the exact bilinear geometry, physical `20x20` kernel, local component order, edge-midpoint shear tying, `B` matrices, plane-stress section matrices, `2x2` quadrature, and bottom/middle/top recovery from S1/S2 rather than copying OCR text blindly.
- Revise the existing formulation draft to expose global six-DOF input/output while keeping only two tangent rotations in physical strains. Define `T`, signs, frame construction, physical/drilling energy split, and all units explicitly.
- Revise the existing formulation draft to expose global six-DOF input/output while keeping only two tangent rotations in physical strains. Define `T`, signs, frame construction, the fixed drilling embedding, and all units explicitly.
- Keep any geometric-nonlinear residual/tangent material in a clearly marked future-only section; it is not part of the approved executable analysis.
- Present drilling scales as candidates pending Numerical Review, not as an implementation default.
- State the approved fixed drilling rule exactly and do not reintroduce candidate sweeps or drilling-specific recovery.
### Numerical Review Agent
- Independently check six physical rigid modes, non-rigid rank, symmetry, transform energy, patch consistency, and every Gauss/tying Jacobian.
- Resolve all seven blocking decisions above. Require consistent-unit rescaling and thickness/mesh sweeps for drilling, not only a successful factorization.
- Confirm the fixed drilling rule's dimensions, symmetry, positivity and physical-recovery separation without reopening coefficient calibration.
- Audit the original MITC4 distortion/membrane-locking limitation and set convergence expectations that do not imply MITC4+ behavior.
- Approve benchmark mesh densities compatible with the hard director-angle policy.
- Treat `NR-O03`/`NR-O04` and expanded benchmark portfolios as removed/nonblocking scope.
### I/O Definition Agent
- Preserve source S4/S4R identity separately from `FESA-MITC4`, while mapping both to one quadrature/kernel path.
- Define automatic director metadata, the exact unsupported-drilling-load projection rule, and fail-closed diagnostics for folds, normals, Jacobians, section/material data, and excluded loads.
- Define automatic director data, the exact unsupported-drilling-load projection rule, and fail-closed diagnostics for normals, topology, Jacobians, section/material data, and excluded loads without calibrated `theta_smooth`.
- Distinguish Gauss, tying, and section positions in HDF5 identities; do not average mismatched result locations.
### Reference Model Agent
- Prefer pinched-cylinder and NAFEMS LE3 adaptations for curved nodal-load coverage, and explain every change from the published model needed to fit the approved FESA keyword subset.
- Provide at least one S4 and one S4R model, but never use their different Abaqus algorithms to select different FESA kernels.
- Re-inventory `reference/shell/` without creating, renaming, repairing, or normalizing artifacts. Establish provenance, units, node/component identity, and supported semantics before using any row.
- Propose U mixed-tolerance values and nonblocking UR warning thresholds from actual artifact scales and mesh/convergence evidence for Numerical Review approval.
- Record the exact existing S4 and S4R input/displacement paths without creating, renaming, repairing, or normalizing artifacts.
- Define deterministic HDF5-to-CSV source-node/component mapping and the fixed B33 tolerance; do not require provenance, naming policy, README/metadata, duplicated model descriptions, or an expanded portfolio.
### Implementation Planning Agent
- Do not start until revised Formulation, Numerical Review, I/O, and Reference Model contracts close the blocking decisions.
- Translate every selected benchmark and negative geometry case into TDD `RED -> GREEN -> VERIFY` work while preserving the project linear-static lifecycle and deterministic assembly/output rules.
- Start after revised Formulation, Numerical Review, I/O, and lightweight Reference Model inventory are mutually consistent.
- Use the project Harness skill to draft self-contained RED/GREEN/VERIFY Steps, obtain user approval before writing `phases/` planning files, and never run the executor without a separate explicit request.
### Coordinator Agent
- Treat Research as evidence-complete once this brief is approved, but track drilling scale/coefficient/energy, director/Jacobian constants, U/UR tolerance, and missing reference artifacts as downstream blockers.
- Treat Research as approved and do not track removed drilling calibration, `NR-O03`/`NR-O04`, tolerance calibration, bundle administration, or portfolio expansion as downstream blockers.
- Reopen Requirements only if physical drilling loads, fold/intersection coupling, explicit normals, distributed loads, MITC4+, or nonlinear execution is proposed.