# Linear Static MITC4 Shell Research Brief

## Metadata

- feature_id: `linear-static-mitc4-shell`
- source_requirement: `docs/requirements/linear-static-mitc4-shell.md`
- status: `research-complete-awaiting-approval`
- owner_agent: `research-agent`
- date: `2026-08-11`
- 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
- 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. Which drilling-stiffness scales are dimensionally defensible, and which coefficient, sensitivity, and artificial-energy decisions remain unsupported?
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?

## 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/requirements/linear-static-mitc4-shell.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:** 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-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-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 — 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-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.

### 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

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.

| 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, 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-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 optional Scordelis–Lo must be run as mesh/thickness sequences. Underpredicted displacement on a single thin mesh is evidence of possible locking, not a complete diagnosis.
- **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.
- **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. 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 Scordelis–Lo only as a documented equivalent-nodal-load auxiliary case.

### Blocking numerical decisions before Implementation Planning

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.

These issues do not require a new user scope choice, but Items 1–5 must be closed by
Numerical Review before implementation planning can truthfully claim a complete numerical
contract.

## 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` | 20-degree and scale-aware geometry candidates; constants remain blocking Numerical Review decisions |
| `031-038` 5-to-6 DOF/drilling | F-12–F-19, `MITC4-DRILL-SENSITIVITY` | dimensional rejection of mixed diagonals; scale/coefficient/energy decisions carried forward |
| `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/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-25–F-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 |

## 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.
- 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.

### 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.
- 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.

### 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.
- 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.

### 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.

### 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.
- Reopen Requirements only if physical drilling loads, fold/intersection coupling, explicit normals, distributed loads, MITC4+, or nonlinear execution is proposed.
