# Linear Static MITC4 Shell Research Brief

## Metadata

- feature_id: `linear-static-mitc4-shell`
- source_requirement: `docs/linear-static-mitc4-shell/requirements.md`
- status: `approved`
- owner_agent: `research-agent`
- date: `2026-08-13`
- 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: the sole acceptance comparison is the existing read-only full-integration S4 case at `reference/shell/`; S4R source support is verified without consuming an Abaqus reference artifact
- 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
Reference Model gates. It does not approve the existing formulation draft, finalize a C++
design, run a reference solver, create reference results, or claim that FESA MITC4 is
formulation-equivalent to Abaqus S4 or S4R.

## Research Questions

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. 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 element-level checks and the declared full-integration S4 source-solver case fit the approved implementation scope while retaining non-reference S4R mapping coverage?

## Source Reliability Tiers

- **Tier 1:** original peer-reviewed formulation or evaluation paper, authoritative standards benchmark, or official solver theory/user documentation.
- **Tier 2:** peer-reviewed implementation summary, graduate thesis, or technical review whose formulas and numerical results are useful but require confirmation against Tier 1 evidence before becoming a FESA numerical constant.
- **Tier 3:** derived wiki synthesis or informal study notes. These are navigation and terminology aids, not sole authority for a numerical decision.
- **Project contract:** approved FESA requirement or architecture decision. It defines product meaning but is not external validation evidence.

## Source Inventory

| id | source | reliability | applicable evidence | limits |
| --- | --- | --- | --- | --- |
| S1 | E. N. Dvorkin and K.-J. Bathe, “A Continuum Mechanics Based Four-Node Shell Element for General Non-linear Analysis,” *Engineering Computations* 1 (1984) 77–88, DOI `10.1108/eb023562`; local transcription `docs/reference-papers/MITC4/AContinuumMechanicsBasedFourNodeShell/`; [author-hosted paper](https://web.mit.edu/kjb/www/Publications_Prior_to_1998/A_Continuum_Mechanics_Based_Four-Node_Shell_Element_for_General_Nonlinear_Analysis.pdf) | Tier 1 | original director kinematics, five physical DOFs, assumed transverse shear field, plane-stress degeneration, quadrature used by the authors, patch and shell benchmark results | nonlinear capability in the paper does not expand the approved FESA linear-static product scope |
| S2 | E. Dvořáková and B. Patzák, “Four-Node Quadrilateral Shell Element MITC4,” *Applied Mechanics and Materials* 825 (2016) 99–106, DOI `10.4028/www.scientific.net/AMM.825.99`; local transcription `docs/reference-papers/MITC4/FourNodeQuadrilateralShellElementMITC4/` | Tier 2 | compact 20-DOF ordering, tying equations and OOFEM implementation summary, patch-test classes, Scordelis–Lo convergence | secondary implementation paper; one OCR component label in the local transcription appears inconsistent and must not override the tensor equations |
| S3 | 이희준, *유한요소해석법을 이용한 쉘 구조물의 동적 좌굴 해석*, 인하대학교 석사학위논문 (2012); local transcription `docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/` | Tier 2 | five-to-six DOF transformation example, printed drilling diagonal rule, `5/6` shear factor, `2x2x2` integration, patch, pinched-cylinder, and hemispherical examples | thesis-specific implementation; the drilling rule mixes stiffness diagonal families unless a dimensional restriction is added; nonlinear/dynamic sections are out of current product scope |
| S4 | P.-S. Lee and H.-C. Noh, “On the Finite Element Analysis of Shell Structures”; local transcription `docs/reference-papers/MITC4/쉘구조물의유한요소해석에대하여/` | Tier 2 | locking classes, shell asymptotic behavior, need for a benchmark portfolio and field/error evidence | review evidence; it does not define FESA constants or I/O semantics |
| S5 | K.-J. Bathe, A. Iosilevich, and D. Chapelle, “An Evaluation of the MITC Shell Elements,” *Computers & Structures* 75 (2000) 1–30; [author-hosted paper](https://web.mit.edu/kjb/www/Principal_Publications/An_Evaluation_of_the_MITC_Shell_Elements.pdf) | Tier 1 | discriminating shell tests across different asymptotic behaviors; warning against judging a general shell element from one response value | evaluates a family and problem portfolio, not the exact future FESA implementation |
| S6 | Y. Ko, P.-S. Lee, and K.-J. Bathe, “The MITC4+ Shell Element and Its Performance,” *Computers & Structures* 169 (2016) 57–68, DOI `10.1016/j.compstruc.2016.03.002`; [publisher record](https://www.sciencedirect.com/science/article/pii/S0045794916300487) | Tier 1 | original MITC4 leaves membrane strain unmodified; distorted curved meshes can exhibit membrane locking | MITC4+ is a different formulation and is not approved for FESA in this feature |
| S7 | Y. Ko, K.-J. Bathe, and X. Zhang, “Continuum Mechanics-Based Shell Elements with Six Degrees of Freedom at Each Node—the MITC4/D and MITC4+/D Elements,” *Computers & Structures* 308 (2025) 107622, DOI `10.1016/j.compstruc.2024.107622`; [publisher record](https://www.sciencedirect.com/science/article/pii/S0045794924003511) | Tier 1 | a physical drilling-rotation extension can improve membrane behavior and shell/beam or intersecting-shell coupling without an artificial factor | MITC4/D is not the approved “five physical DOFs plus numerical drilling stabilization” FESA element |
| S8 | T. J. R. Hughes and F. Brezzi, “On Drilling Degrees of Freedom,” *CMAME* 72 (1989) 105–121, DOI `10.1016/0045-7825(89)90124-2`; [publisher record](https://www.sciencedirect.com/science/article/pii/0045782589901242) | Tier 1 | independent-rotation variational formulations exist for physical membrane drilling DOFs | not evidence that a diagonal numerical penalty is a physical strain or may carry drilling load |
| S9 | Abaqus, [Shell Section Behavior](https://docs.software.vt.edu/abaqusv2024/English/SIMACAEELMRefMap/simaelm-c-shellsectionbehavior.htm) and [Shear Flexible Small-Strain Shell Elements](https://docs.software.vt.edu/abaqusv2025/English/SIMACAETHERefMap/simathe-c-shearflexshells.htm) | Tier 1 | homogeneous-shell transverse shear stiffness, small drill penalty proportional to transverse shear stiffness, and a dimensionally consistent area/thickness scaling precedent | Abaqus does not publish a FESA coefficient and its element formulation must not be copied by implication |
| S10 | Abaqus, [Defining the Initial Geometry of Conventional Shell Elements](https://docs.software.vt.edu/abaqusv2024/English/SIMACAEELMRefMap/simaelm-c-shellgeometry.htm) | Tier 1 | connectivity/midsurface normals, order-independent grouping, 20-degree normal-averaging precedent, and coarse-mesh/fold cautions | 20 degrees is an Abaqus modeling heuristic, not a universal mathematical smoothness threshold; Abaqus permits multiple/user normals that FESA excludes |
| S11 | Abaqus, [Finite-Strain Shell Element Formulation](https://docs.software.vt.edu/abaqusv2025/English/SIMACAETHERefMap/simathe-c-finitestrainshells.htm) and [Choosing a Shell Element](https://docs.software.vt.edu/abaqusv2025/English/SIMACAEELMRefMap/simaelm-c-shellelem.htm) | Tier 1 | S4 is fully integrated, S4R is reduced-integrated, their membrane treatments differ, and both use drill control | Abaqus finite-strain and hourglass algorithms are outside the FESA MITC4 identity and current small-deformation scope |
| S12 | Abaqus, [LE3 Hemispherical Shell with Point Loads](https://docs.software.vt.edu/abaqusv2024/English/SIMACAEBMKRefMap/simabmk-c-le3.htm) and [The Pinched Cylinder Problem](https://docs.software.vt.edu/abaqusv2025/English/SIMACAEBMKRefMap/simabmk-c-pinchcyl.htm) | Tier 1 | authoritative point-load-compatible curved-shell benchmark definitions, target displacements, S4/S4R convergence, and distorted-mesh evidence | official input decks contain semantics such as explicit normals or symmetry shorthand that require an approved FESA-subset adaptation |
| S13 | Abaqus, [Shell Thickness and Section Points](https://docs.software.vt.edu/abaqusv2024/English/SIMACAEGSARefMap/simagsa-c-shlthick.htm) and [Whole and Partial Model Variables](https://docs.software.vt.edu/abaqusv2024/English/SIMACAEOUTRefMap/simaout-c-std-wholeandpartialmodelvariables.htm) | Tier 1 | bottom/middle/top linear-elastic stress recovery precedent and separate reporting of artificial energy that includes drill constraints | no drill-specific acceptable artificial-energy percentage is stated |
| S14 | configured FEM wiki pages `[[MITC4 Shell Element]]`, `[[MITC Shell Kinematics]]`, `[[Assumed Transverse Shear Strain Interpolation]]`, `[[Shell Locking Phenomenon]]`, `[[Shell Element Benchmark Testing]]`, and `[[Scordelis-Lo Shell Benchmark]]` | Tier 3 synthesis | navigation between local raw sources; locking, tying, and benchmark terminology | key claims are cited to S1–S13 rather than relying on the wiki alone |
| P1 | `docs/linear-static-mitc4-shell/requirements.md`, `docs/PRD.md`, `docs/ARCHITECTURE.md`, and `docs/ADR.md` | Project contract | exact FESA scope, six-global-DOF interface, source identity, output, verification, lifecycle, and reference immutability | does not prove numerical correctness by itself |

The informal `docs/reference-papers/MITC4/MITC공부/` notes were used only as a
navigation aid. No key numerical decision relies on them.

## Theory Summary

### Continuum-degenerated kinematics and physical DOFs

- **F-01 — Verified, Tier 1 (S1):** MITC4 represents a point through the shell thickness with bilinear midsurface interpolation plus a thickness coordinate multiplying nodal thickness and nodal director vectors. In compact form,
  `X(xi,eta,zeta) = sum(N_i X_i) + zeta/2 sum(t_i N_i n_i)`.
- **F-02 — Verified, Tier 1 (S1), plus Project contract (P1):** The original director need not be exactly normal to an individual element midsurface. FESA deliberately narrows this freedom: its initial unit thickness director is generated from consistently oriented connectivity normals and thickness remains a separate positive scalar.
- **F-03 — Verified, Tier 1/2 (S1/S2):** The physical nodal variables are three midsurface translations and two rotations in the director tangent plane, giving 20 physical element DOFs. For a right-handed nodal frame `(t1,t2,n)`, the linear director increment has the form `delta_n = -alpha t2 + beta t1`.
- **F-04 — Project-contract consequence:** Global `[URX,URY,URZ]` is an interface vector. Only its projections onto `t1` and `t2` enter the physical MITC4 strains. The projection along `n` is a numerical drilling coordinate and must not appear in membrane, bending, transverse-shear, resultant, or section-point stress output.

### Assumed transverse shear interpolation

- **F-05 — Verified, Tier 1/2 (S1/S2):** Direct bilinear displacement/director interpolation cannot make transverse shear vanish throughout a thin element under constant bending, producing parasitic shear energy and shear locking.
- **F-06 — Verified, Tier 1/2 (S1/S2):** MITC4 evaluates the covariant shear components at the four edge-midpoint tying locations and interpolates them as
  `e13_hat(xi,eta) = 0.5(1+eta)e13_A + 0.5(1-eta)e13_C` and
  `e23_hat(xi,eta) = 0.5(1+xi)e23_D + 0.5(1-xi)e23_B`, with the tied covariant components transformed consistently to the local Cartesian shell basis.
- **F-07 — Verified, Tier 1 (S1):** The original elastic examples used `2x2` Gauss integration on the midsurface and two Gauss points through the thickness. The authors reported the expected six rigid-body modes and no spurious zero-energy modes in the undistorted and distorted element checks they performed.
- **F-08 — Research recommendation from F-06/F-07:** Formulation should evaluate standard membrane/bending terms and MITC tied shear terms on one common, documented `2x2` midsurface quadrature path for both source S4 and S4R. This is evidence-backed as a candidate, but the Formulation and Numerical Review gates must still re-derive the exact `B` rows, Jacobian use, and rank.

### Homogeneous isotropic section behavior and recovery

- **F-09 — Verified, Tier 1/2 (S1/S3/S9):** The shell constitutive law is formed in a local orthonormal frame under a through-thickness normal-stress condition (`sigma33=0`). For the approved homogeneous isotropic layer, `G=E/[2(1+nu)]`; a `5/6` transverse-shear correction is a documented homogeneous rectangular-section precedent.
- **F-10 — Inference from linear section kinematics:** With midsurface membrane strain `epsilon0`, curvature `kappa`, and thickness coordinate `z`, in-plane strain and stress are linear through thickness. Under one candidate curvature convention this is `epsilon(z)=epsilon0+z kappa`, `sigma(z)=Cps epsilon(z)`; Formulation must fix the final sign against its director and positive-face convention. Bottom, middle, and top in-plane stresses are then direct evaluations at `z=-t/2,0,+t/2`, while section resultants follow the familiar `A=Cps t` and `D=Cps t^3/12` integrals for a centered homogeneous layer.
- **F-11 — Verified precedent, Tier 1 (S13):** Official Abaqus guidance likewise treats bottom/middle/top as the default linear-elastic shell stress locations and states that three section points are exact for a linear through-thickness problem. This supports location choice, not Abaqus formulation equivalence.

### Six-global-DOF embedding and drilling stabilization

- **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 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 — 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 shell–beam 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 — 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 — 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

- **F-25 — Verified, Tier 1 (S11):** Abaqus S4 is fully integrated, while S4R uses reduced integration and associated control; the membrane treatment differs. S4 and S4R therefore do not identify the original Dvorkin–Bathe MITC4 formulation.
- **F-26 — Project-contract conclusion:** Both source labels may select the single FESA MITC4 path only because P1 explicitly defines that input mapping. Source type remains metadata. The mapping is not evidence that the Abaqus elements or their recovered rotations, forces, stresses, integration points, or stabilization energies are equal to FESA's.
- **F-27 — Verification consequence:** Abaqus displacement rows are useful source-solver evidence for the same physical model, especially across mesh refinement. Only global `U1/U2/U3` is blocking under P1; large `UR1/UR2/UR3` differences are warning-only because the drilling and rotation representations are not equivalent.

### Known accuracy limits of original MITC4

- **F-28 — Verified, Tier 1 (S6):** Original MITC4 specifically treats transverse shear locking but leaves membrane strains unmodified. Distorted elements on curved geometries can therefore suffer membrane locking; MITC4+ was introduced to address that limitation.
- **F-29 — Verified, Tier 1/2 (S1/S4/S5):** Shell behavior depends on geometry, boundary conditions, thickness, asymptotic class, and mesh. A patch test or one point displacement cannot establish general robustness. Convergence sequences, distortion, curvature, field/resultant behavior, equilibrium, and energy must be examined together.

## Candidate Benchmarks

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 |
| --- | --- | --- | --- | --- |
| `MITC4-RIGID-RANK` | S1 plus P1 | free valid planar, rotated, and smoothly curved elements; six physical rigid motions, `K r`, symmetry, rank, and positive non-rigid energy | physical null modes, transform, drilling regularization rank, energy invariance | global supports, curved-shell accuracy, Abaqus agreement |
| `MITC4-MEMBRANE-PATCH` | S1/S2/S5 | independent constant `E11`, `E22`, and `G12` states on regular and distorted multi-element patches; generalized strain, `N`, stress, residual | membrane completeness, signs, component order, stress/resultant recovery | bending, shear locking, curved membrane locking |
| `MITC4-BENDING-PATCH` | S1/S2/S3 | constant curvature in both principal directions for thick and very thin thicknesses; rotations, `K11/K22`, `M`, bottom/top stress | bending consistency, thickness-cubed scaling, stress sign | complex curved membrane response |
| `MITC4-SHEAR-PATCH` | S1/S2/S3 | zero-rotation constant transverse shear states in each local direction; `G13/G23` and `Q13/Q23` | MITC shear tying, `5/6` section factor, component signs | thin bending convergence by itself |
| `MITC4-TWIST-PATCH` | S1/S2/S3 | constant twist on thin and thick patches; `K12`, `M12`, displacement/rotation symmetry | mixed bending terms and tying consistency | S3 reports thick-case sensitivity; it cannot set a universal accuracy tolerance |
| `MITC4-THIN-THICK-CANTILEVER` | S1 | nodal tip force or moment on regular and intentionally distorted meshes over documented `t/L` sequence; tip displacement/rotation, reaction, resultants, energy | shear-locking trend, thick response, distortion, equilibrium | curved membrane locking and general shell behavior |
| `MITC4-QUADRATURE-CROSSCHECK` | S1 plus inference | one constant-property planar element; `2x2` stiffness versus independently integrated high-order/analytical section result; tying-point shear values | quadrature, Jacobian, `B` rows, stiffness symmetry | benchmark validation or curved geometry |
| `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, 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
mesh convergence is useful corroborating evidence, but NAFEMS LE3 has stronger benchmark
provenance and a directly accessible official S4/S4R definition. The two hemispherical
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 Scordelis–Lo 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:** 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 case tests full-integration source mapping and global displacement. It cannot prove formulation identity. Missing, extra, duplicate, nonfinite, or source-node/component-mismatched required rows fail before P1's mixed displacement tolerance is evaluated. S4R source support is established separately by FESA mapping/kernel/metadata tests.
- **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.

## Applicability Limits

- **analysis:** one small-displacement, small-rotation linear-static step. Nonlinear tangent, buckling, dynamics, finite rotation, and follower-load results present in S1/S3 are research background only.
- **element:** original four-node MITC4 transverse-shear treatment with unmodified membrane strain, not MITC4+, MITC4/D, Abaqus S4, or Abaqus S4R.
- **degrees of freedom:** five physical director-shell DOFs embedded in six global DOFs. Drilling is numerical regularization, not a physical load/result channel.
- **geometry:** smooth shell patches with one auto-generated director per source node. Explicit normals, discontinuous shared-node normals, physical folds without duplicate nodes, hinges, intersections, and shell–beam joints are excluded.
- **material/section:** one centered homogeneous isotropic linear-elastic layer with constant positive thickness. No composites, offsets, orthotropy, plasticity, field dependence, or thickness stretch.
- **loads:** nodal global forces and non-drilling moments only. Pressure, gravity, body force, edge traction, distributed load, and follower load are outside the product path.
- **locking:** assumed transverse shear addresses shear locking; original MITC4 does not guarantee immunity to membrane locking for distorted curved meshes.
- **quadrature:** `2x2` midsurface integration is a source-backed candidate for this feature, not permission to reinterpret S4R as a reduced-integration FESA element.
- **stress:** local in-plane stress is recovered at bottom/middle/top. `S33=0` is an assumption; pointwise `S13/S23` is not emitted, and S1 notes that transverse shear stress may be inaccurate in distorted cantilever tests.
- **reference:** Abaqus U comparison is model-specific. Rotations are warning-only; reaction, stress, shell force/moment, integration layout, and stabilization energy are not equality gates under P1.
- **units:** user-consistent units. All drilling formulas and geometry tolerances must remain dimensionally invariant under a consistent change of length/force units.

## Research Recommendations and Open Issues

### Recommendations supported for Formulation

1. Retain the original MITC4 five-DOF kinematics and the S1/S2 edge-midpoint covariant shear interpolation.
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. 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 Scordelis–Lo as optional future evidence rather than implementation-completion requirements.

### Closed decisions and nonblocking evidence limits

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:** The approved downstream contract uses fixed absolute `1.0e-5` for every U/UR row without a reference-scale decision term; U exceedance fails and UR exceedance only warns. The B33 mixed tolerance remains unchanged and separate.
4. **Reference case:** the existing `reference/shell/` S4 input/displacement pair is the complete required acceptance inventory because FESA-MITC4 uses full integration. `reference/shellR/` is not consumed by acceptance comparison. Administrative bundle metadata and an expanded portfolio are not gates.

No research-owned numerical decision remains blocking for Implementation Planning.

## Requirement Traceability

| requirement area | research evidence | downstream result |
| --- | --- | --- |
| `006-010` material/section | F-09–F-11 | homogeneous isotropic plane-stress and section recovery candidate; exact I/O remains downstream |
| `011-016` director/geometry | F-20–F-24, `MITC4-DIRECTOR-GEOMETRY` | basic deterministic validity rules; `NR-O03`/`NR-O04` calibration removed |
| `031-038` 5-to-6 DOF/drilling | F-12–F-19, `MITC4-DRILL-FIXED` | fixed dimensionally compatible rotational-diagonal rule; calibration and energy output excluded |
| `037` S4/S4R common path | F-25–F-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-09–F-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-25–F-27 plus approved user decision | fixed absolute `1.0e-5`; U blocking and UR warning-only |
| `065-072` reference artifacts | S11/S12 and current inventory state | exact existing S4 paths, source-row/component mapping, S4R non-consumption, 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, 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.
- 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.
- 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.
- 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 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

- 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 absolute MITC4 tolerance `1.0e-5`; do not require provenance, naming policy, README/metadata, duplicated model descriptions, or an expanded portfolio.

### Implementation Planning Agent

- 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 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.
