# Shell Surface Pressure Load Research Brief

## 1. Metadata

- feature_id: `shell-surface-pressure-load`
- source_requirement: `docs/shell-surface-pressure-load/requirements.md`
- requirement_range: `FESA-REQ-SHELL-PRESSURE-001`–`FESA-REQ-SHELL-PRESSURE-024`
- status: `draft-for-coordinator-gate-review`
- owner_agent: `research-agent`
- date: `2026-08-30`
- research_scope: `*SURFACE, TYPE=ELEMENT` + `*DSLOAD` uniform `P` on S4/S4R `SPOS|SNEG`, linear reference-configuration pressure, nonlinear current-configuration follower pressure와 external-load tangent
- source_policy: Project contract와 stored artifact가 범위/acceptance를 지배하며, Abaqus 문서는 observable input/behavior만, FEM 문헌은 formulation-neutral theory와 verification risk만 뒷받침한다.
- non_equivalence_statement: 이 문서는 Abaqus의 내부 shell formulation, pressure quadrature, stabilization, load-vector 또는 tangent 구현과 FESA의 동등성을 주장하지 않는다.

## 2. Research Questions

| id | 질문 | binding requirements |
| --- | --- | --- |
| `RQ-01` | 승인 syntax에서 element-based named surface와 `DSLOAD P`의 정확한 observable 의미는 무엇인가? | `001`–`004`, `014`–`015` |
| `RQ-02` | S4/S4R shell connectivity, `SPOS/SNEG`, selected surface normal과 pressure 부호는 어떻게 연결되는가? | `004`–`006`, `020` |
| `RQ-03` | Uniform pressure의 단위, 총 resultant와 work-equivalent nodal generalized force를 뒷받침하는 일반 FEM 원리는 무엇인가? | `002`, `007`, `011`–`012`, `020` |
| `RQ-04` | 선형 reference-configuration pressure와 기하비선형 current-configuration follower pressure를 어떤 evidence로 구분하는가? | `007`–`008`, `021` |
| `RQ-05` | Configuration-dependent pressure force가 Newton tangent에 기여하는 이유와 symmetry 위험은 무엇인가? | `009`, `016`, `021` |
| `RQ-06` | S4/S4R load verification에서 Abaqus element 특성과 generic surface-load theory의 적용 한계는 무엇인가? | `004`, `008`–`009`, `020`–`021` |
| `RQ-07` | 어떤 독립/저결합 benchmark 조합이 sign, resultant, objectivity, tangent, equilibrium, determinism과 stored reference를 각각 검증하는가? | `012`–`013`, `016`, `020`–`024` |
| `RQ-08` | Exact stored nonlinear case가 직접 증명하는 것과 증명하지 못하는 것은 무엇인가? | `022`–`024` |

## 3. Source Inventory

### 3.1 Project and stored evidence

| source_id | source_type | title | author_or_org | URL_or_path | access_date | reliability_tier | primary_or_secondary | notes / limits |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| `SRC-PRJ-REQ` | approved feature baseline | Shell Surface Pressure Load Feature Requirement Specification | FESA Requirement Agent / Coordinator | `docs/shell-surface-pressure-load/requirements.md` | 2026-08-30 | `Project Primary` | primary | 범위, acceptance, tolerance disposition의 binding source. 이 연구가 변경하지 않는다. |
| `SRC-PRJ-CONTRACT` | project contracts | `AGENTS.md`, `PRD.md`, `ARCHITECTURE.md`, `ADR.md`, `TOLERANCE.md`, `SOLVER_AGENT_DESIGN.md` | FESA project | repository paths | 2026-08-30 | `Project Primary` | primary | Independent solver, deterministic full-space load/reaction, unsymmetric nonlinear solver capability, HDF5 authority와 common external tolerance를 고정한다. |
| `SRC-LOCAL-REF` | stored reference bundle | `shellNL-meshfine-pressure` | FESA project / existing Abaqus export | `reference/shellNL-meshfine-pressure/` | 2026-08-30 | `Stored Reference` | primary artifact | `.inp`와 CSV가 target/input 값을 제공하지만 formulation derivation이나 Abaqus version/provenance는 제공하지 않는다. Read-only. |

### 3.2 Tier 1 authoritative manuals

모든 공식 문서 URL은 Dassault Systèmes SIMULIA Abaqus 2025 online documentation이며 access date는 `2026-08-30`이다.

| source_id | source_type | exact title / section | author_or_org | URL | reliability_tier | primary_or_secondary | evidence supplied | limitation |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| `SRC-ABA-DSLOAD` | official keyword manual | `*DSLOAD` | Dassault Systèmes SIMULIA | https://docs.software.vt.edu/abaqusv2025/English/SIMACAEKEYRefMap/simakey-r-dsload.htm | `Tier 1` | primary | Step-level history data; pressure data line is surface name, `P`, reference magnitude. | Abaqus syntax 전체가 넓으며 FESA allowlist/fail-closed 세부는 `SRC-PRJ-REQ`가 지배한다. |
| `SRC-ABA-SURFACE-KEY` | official keyword manual | `*SURFACE` | Dassault Systèmes SIMULIA | https://docs.software.vt.edu/abaqusv2025/English/SIMACAEKEYRefMap/simakey-r-surface.htm | `Tier 1` | primary | `NAME`, `TYPE=ELEMENT`, element/ELSET와 face identifier data 형식. | 허용 가능한 전체 surface type을 설명하므로 FESA 승인 subset과 동일하지 않다. |
| `SRC-ABA-ELEM-SURFACE` | official modeling manual | Element-Based Surface Definition, especially “Creating Surfaces on Structural, Surface, and Rigid Elements” | Dassault Systèmes SIMULIA | https://docs.software.vt.edu/abaqusv2025/English/SIMACAEMODRefMap/simamod-c-deformablesurf.htm | `Tier 1` | primary | S4 계열의 `SPOS/SNEG`, single-sided surface, orientation consistency와 distributed-load orientation risk. | Contact 등 다른 surface consumer도 포함한다. FESA는 single-sided pressure subset만 승인한다. |
| `SRC-ABA-DISTLOAD` | official analysis manual | Distributed Loads, “Surface Tractions and Pressure Loads” and “Follower Surface Loads” | Dassault Systèmes SIMULIA | https://docs.software.vt.edu/abaqusv2025/English/SIMACAEPRCRefMap/simaprc-c-loaddistributed.htm | `Tier 1` | primary | Surface pressure 단위 `force/area`, `DSLOAD surface,P,magnitude`, 양의 pressure는 surface normal 반대, nonlinear follower/current-surface behavior, `DLOAD`와 `DSLOAD` 구분. | 많은 load type/procedure를 다루므로 feature allowlist의 근거가 아니라 observable convention 근거다. |
| `SRC-ABA-SHELL-NORMAL` | official element guide | Shell normals and shell surfaces | Dassault Systèmes SIMULIA | https://docs.software.vt.edu/abaqusv2025/English/SIMACAEGSARefMap/simagsa-c-shlnormal.htm | `Tier 1` | primary | 3-D shell positive normal은 connectivity node order의 right-hand rule, top=`SPOS`, bottom=`SNEG`; element-based와 surface-based pressure sign 차이 경고. | Abaqus element-based `DLOAD` sign은 이 feature의 `DSLOAD` sign이 아니다. |
| `SRC-ABA-SHELL-CHOICE` | official element guide | Choosing a Shell Element | Dassault Systèmes SIMULIA | https://docs.software.vt.edu/abaqusv2025/English/SIMACAEELMRefMap/simaelm-c-shellelem.htm | `Tier 1` | primary | S4/S4R은 4-node general-purpose finite-membrane-strain shell; large rotation capability; S4 full, S4R reduced integration 및 hourglass risk. | Abaqus element internal formulation 차이는 FESA pressure kernel 또는 quadrature 선택 근거가 아니다. |
| `SRC-ABA-PRESSURE-THEORY` | official theory manual | Pressure load stiffness | Dassault Systèmes SIMULIA | https://docs.software.vt.edu/abaqusv2025/English/SIMACAETHERefMap/simathe-c-pressloadsolids.htm | `Tier 1` | primary | Current surface virtual work, current area-vector representation, displacement-dependent equivalent nodal load와 Jacobian contribution. | 3-D solid surface derivation이다. Generic surface kinematics evidence로만 사용하며 S4/S4R 구현 동등성을 뜻하지 않는다. |
| `SRC-ABA-SHELL-PRESSURE` | official theory manual | Shear flexible small-strain shell elements, “Pressure load stiffness” | Dassault Systèmes SIMULIA | https://docs.software.vt.edu/abaqusv2025/English/SIMACAETHERefMap/simathe-c-shearflexshells.htm | `Tier 1` | primary | Shell surface virtual work와 load stiffness, uniform magnitude/free-edge 가정 아래 symmetric form을 설명한다. | 대상이 S4R5/S8R5/S9R5/S8R 등 small-strain shell이며 S4/S4R이 아니다. FESA tangent symmetry나 algorithm 선택 근거로 사용할 수 없다. |

### 3.3 Tier 2 theory sources and Tier 3 navigation

| source_id | source_type | title | author_or_org | URL_or_DOI_or_ISBN | access_date | reliability_tier | primary_or_secondary | notes / access limits |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| `SRC-FEM-HIBBITT79` | peer-reviewed short communication | Some Follower Forces and Load Stiffness, *International Journal for Numerical Methods in Engineering* 14(6), 937–941 (1979) | H. D. Hibbitt | https://doi.org/10.1002/nme.1620140613 | 2026-08-30 | `Tier 2` | primary | Follower forces depend on displacement and contribute load stiffness needed by Newton/eigenvalue methods. Publisher metadata/abstract accessible; full text may require subscription. |
| `SRC-FEM-SR84` | peer-reviewed paper | Displacement Dependent Pressure Loads in Nonlinear Finite Element Analyses, *Computers & Structures* 18(6), 1099–1114 (1984) | K. Schweizerhof, E. Ramm | https://doi.org/10.1016/0045-7949(84)90154-8 | 2026-08-30 | `Tier 2` | primary | Body-/space-attached classification, conservativeness conditions, curved-surface pressure load stiffness와 numerical examples. Publisher full text is access-limited; abstract/metadata used. |
| `SRC-FEM-LOGAN` | textbook, stored project reference | A First Course in the Finite Element Method, 4th ed., Sections 3.10, 6.2–6.3, Appendix E | Daryl L. Logan; Nelson/Thomson | ISBN `0-534-55298-6`; vault source `.raw/AFirstCourseInTheFiniteElementMethod/` | 2026-08-30 | `Tier 2` | secondary textbook | 동일 interpolation/work statement로 distributed traction을 consistent nodal force로 바꾸며 resultant/reaction 검증을 연결한다. Copyrighted local reference; 장문 인용하지 않는다. |
| `SRC-VAULT-SYNTH` | configured wiki synthesis | `[[Finite Element Load Vector Assembly]]`, `[[Nonlinear Finite Element Analysis]]`, `[[Abaqus Surface and Assembly Modeling]]`, `[[Basic Shell Mathematical Model]]`, `[[Abaqus Loads and Predefined Fields]]` | MultiPhysicsVault | `C:/git/MultiPhysicsVault/wiki/` | 2026-08-30 | `Tier 3` | secondary navigation | Source discovery와 terminology routing에만 사용했다. Core claims는 위 official/publisher/project sources로 재검증했다. Follower-pressure tangent의 S4/S4R-specific coverage는 vault gap이다. |

Source count: `Project Primary/Stored Reference 3`, public `Tier 1 8`, `Tier 2 3`, vault `Tier 3 1`. Tier 1은 모두 official manual, Tier 2는 peer-reviewed primary 2건과 textbook 1건이다.

## 4. Extracted Facts and Key-Claim Evidence

### 4.1 Key-claim evidence table

| claim_id | verified fact / inference | evidence | direct applicability | limitation | requirement trace |
| --- | --- | --- | --- | --- | --- |
| `CL-01` | **Verified fact:** `*SURFACE`는 named model region이고 `TYPE=ELEMENT` data line은 element/ELSET와 face identifier를 받는다. `*DSLOAD`는 step history data이고 pressure line은 surface name, `P`, magnitude다. | `SRC-ABA-SURFACE-KEY`, `SRC-ABA-DSLOAD` | 승인 syntax와 model/history 분리 | FESA가 지원할 optional parameters는 requirements allowlist가 제한한다. | `001`, `003`, `014`, `015` |
| `CL-02` | **Verified fact:** S4 계열 structural surface는 `SPOS`와 `SNEG` single-sided face를 가진다. Double-sided/edge surface는 oriented distributed surface load와 다른 범주다. | `SRC-ABA-ELEM-SURFACE` | S4/S4R selected-side target | Abaqus가 지원하는 다른 structural element/surface type은 FESA scope가 아니다. | `004`, `006`, `014` |
| `CL-03` | **Verified fact:** 3-D shell positive normal은 connectivity node order의 right-hand rule이며 `SPOS`는 positive side, `SNEG`는 negative side다. | `SRC-ABA-SHELL-NORMAL`, `SRC-ABA-ELEM-SURFACE` | Connectivity/side reversal tests | Adjacent normals가 inconsistent하면 oriented surface load가 부정확할 수 있다; diagnostic 정책은 후속 I/O/Numerical Review 소유다. | `005`, `006`, `015` |
| `CL-04` | **Verified fact:** Surface-based positive pressure는 local selected surface normal의 반대 방향이고 단위는 force/area다. **Binding inference:** scalar magnitude의 부호를 반전하면 force direction도 반전한다; magnitude absolute-value 처리는 허용되지 않는다. | `SRC-ABA-DISTLOAD`, `SRC-PRJ-REQ` | `DSLOAD P` sign/magnitude contract | Abaqus shell element-based `DLOAD P`는 다른 sign convention이므로 근거로 혼합하면 안 된다. | `002`, `005`, `020` |
| `CL-05` | **Verified fact:** `DLOAD`는 element/ELSET + element-face label 방식이고 `DSLOAD`는 named surface + `P` 방식이다. | `SRC-ABA-DISTLOAD`, `SRC-ABA-DSLOAD` | `*DLOAD` fail-closed exclusion | Abaqus 전체 capability가 아니라 FESA subset의 명시적 구분이다. | `001`, `014` |
| `CL-06` | **Verified fact:** Distributed surface traction은 interpolation/work-equivalence를 통해 compatible nodal terms로 변환된다. Consistent load는 원 distributed load와 같은 virtual work를 수행해야 한다. | `SRC-FEM-LOGAN`, `SRC-VAULT-SYNTH` | Pressure generalized force와 total resultant/reaction check의 이론적 근거 | 이 사실만으로 quadrature order, drilling moment 또는 FESA element API는 결정되지 않는다. | `007`, `011`, `012`, `020` |
| `CL-07` | **Binding fact:** Linear static pressure는 reference coordinates, reference area/normal에서 평가한다. **Theory consistency:** geometrically linear kinematics에서는 load geometry가 displacement로 갱신되지 않는다. | `SRC-PRJ-REQ`; nonlinear contrast `SRC-ABA-DISTLOAD`, `SRC-ABA-PRESSURE-THEORY` | Linear patch expectation | Abaqus와 동일 quadrature를 뜻하지 않는다. | `007`, `020` |
| `CL-08` | **Verified fact:** Geometrically nonlinear follower surface load는 surface와 함께 회전한다. Pressure normal과 area measure는 current surface에 의존한다. | `SRC-ABA-DISTLOAD`, `SRC-ABA-PRESSURE-THEORY`, `SRC-FEM-HIBBITT79` | Rigid-rotation objectivity/current-area checks | Material/element internal response와 독립적으로 load kernel을 먼저 검증해야 한다. | `008`, `021` |
| `CL-09` | **Verified fact:** Current coordinates로 정의된 pressure virtual work를 nodal displacement에 대해 선형화하면 external load/Jacobian contribution이 생긴다. | `SRC-ABA-PRESSURE-THEORY`, `SRC-FEM-HIBBITT79`, `SRC-FEM-SR84` | Directional finite-difference tangent check | FESA residual sign, matrix assembly sign과 exact discrete expression은 Formulation Agent가 정한다. | `009`, `021` |
| `CL-10` | **Verified fact with disagreement boundary:** Pressure follower load가 자동으로 nonconservative/nonsymmetric인 것은 아니다. Conservativeness와 symmetry는 load attachment, loaded boundary/free-edge treatment와 global assembly 조건에 의존한다. Abaqus small-strain shell page는 uniform pressure/free-edge effects neglect 조건의 symmetric form을 기술한다. | `SRC-FEM-SR84`, `SRC-ABA-SHELL-PRESSURE` | Unsymmetric capability를 유지하고 symmetry를 검증 대상으로 삼을 이유 | `SRC-ABA-SHELL-PRESSURE`는 S4/S4R 대상이 아니므로 FESA에서 tangent를 강제 symmetrize할 근거가 아니다. | `009`, `016`, `021` |
| `CL-11` | **Verified fact:** S4/S4R은 four-node general-purpose finite-membrane-strain shells이며 large rotation에 사용 가능하다. S4와 S4R의 Abaqus internal integration/hourglass 차이는 존재한다. | `SRC-ABA-SHELL-CHOICE` | Source-type coverage와 두 source type 모두의 test 필요성 | FESA는 두 source type을 existing MITC4로 매핑한다. Abaqus 내부 element equivalence를 주장하지 않는다. | `004`, `020`, `021` |
| `CL-12` | **Inference from surface measure:** Surface tangent cross product/area vector가 영 또는 비유한이면 unit normal과 pressure direction은 유효하지 않다. Scale-aware invalid-geometry 기준이 필요하다. | `SRC-ABA-PRESSURE-THEORY`; `SRC-PRJ-REQ` | Degenerate/current geometry fail-closed test | Exact threshold는 이 연구가 정하지 않으며 Numerical Review가 승인한다. | `016` |
| `CL-13` | **Verified local fact:** Stored input은 `S4` 676개, node 729개, instance `Part-1-1`, `Surf-1` element 1–676 `SPOS`, `NLGEOM=YES`, `DSLOAD Surf-1,P,1e+06`이다. Required displacement CSV는 factor-one 729 node rows와 six components를 가진다. | `SRC-LOCAL-REF`, `SRC-PRJ-REQ` | Exact end-to-end nonlinear reference candidate | S4R/SNEG/negative pressure/tangent를 직접 분리 검증하지 않는다. | `022`, `023` |
| `CL-14` | **Binding project fact:** Stored displacement `U1/U2/U3`는 blocking, `UR1/UR2/UR3`는 warning-only다. Reaction CSV는 review-only이고 stress CSV는 excluded다. | `SRC-PRJ-REQ`, `SRC-PRJ-CONTRACT`, `SRC-LOCAL-REF` | Reference acceptance boundary | Research가 새 family/tolerance를 만들지 않는다. | `023`, `024` |
| `CL-15` | **Binding project fact:** HDF5가 authoritative output이며 external comparison은 stable source identity/component one-to-one precheck 뒤 `docs/TOLERANCE.md`를 사용한다. | `SRC-PRJ-CONTRACT`, `SRC-PRJ-REQ` | Downstream reference-model/I/O handoff | Analytical/objectivity/tangent tolerance는 external common tolerance로 대체하지 않는다. | `018`, `019`, `023` |

### 4.2 Observable Abaqus contract versus FESA-owned formulation

| observable contract backed by Abaqus sources | FESA-owned decision left downstream |
| --- | --- |
| `SURFACE, TYPE=ELEMENT, NAME=...`와 element/ELSET + `SPOS/SNEG`; `DSLOAD` data의 surface + `P` + magnitude | Semantic record layout, source-order retention, exact duplicate/ambiguity diagnostics |
| Shell connectivity-derived positive normal, `SPOS/SNEG` side | FESA MITC4 surface interpolation, normal/area evaluation and geometry validity scale |
| Positive surface pressure acts opposite selected surface normal; force/area units | Discrete equivalent nodal generalized-force expression, numerical integration rule |
| NLGEOM pressure is current-surface follower loading | Trial/committed state integration and exact residual sign convention |
| Configuration dependence contributes load stiffness/Jacobian | Exact external tangent expression, assembly sign, symmetry treatment |

The second column cannot be inferred from Abaqus result agreement and must be finalized independently by Formulation and Numerical Review.

## 5. Theory Summary for Formulation Handoff

### 5.1 Pressure virtual work and equivalent nodal force

Generic surface-loading theory represents external virtual work as a surface integral of virtual displacement against traction. For uniform pressure, traction direction is tied to the oriented surface normal and magnitude has force/area units. After inserting the same surface displacement interpolation used by the element, the work-equivalent nodal generalized force is obtained. This establishes three formulation invariants without selecting a final formula:

1. A rigid virtual translation must recover the total pressure resultant.
2. A rigid virtual rotation must recover the total moment about the declared origin.
3. Element-local contributions must assemble exactly once into the full generalized load before free/constrained projection.

`SRC-ABA-PRESSURE-THEORY` gives a current-surface area-vector representation and `SRC-FEM-LOGAN` gives the work-equivalence principle. The Formulation Agent still owns interpolation details, quadrature and DOF embedding.

### 5.2 Sign map

Let the connectivity-derived positive shell normal be the orientation reference. The binding observable sign map is:

| selected side | selected surface normal | positive `P` resultant direction | negative `P` direction |
| --- | --- | --- | --- |
| `SPOS` | positive shell normal | opposite positive normal | reversed |
| `SNEG` | negative shell normal | opposite negative normal | reversed |

This table is an input/behavior contract, not a finalized residual equation. Residual and tangent signs must be stated together in `formulation.md`.

### 5.3 Linear configuration

For the approved linear procedure the binding baseline freezes surface geometry at reference coordinates. Thus pressure direction and surface measure do not vary with displacement and there is no displacement derivative of the external pressure force in that procedure. Flat and distorted reference patches should recover the analytical area-vector resultant independent of any test-only candidate displacement.

### 5.4 Nonlinear current configuration and objectivity

For `NLGEOM=YES`, the load must remain normal to and integrated over the current surface. A superposed rigid rotation must rotate the global force resultant by the same rotation without changing scalar pressure or source identity. A deformation that changes surface area may change total resultant magnitude even when the scalar pressure remains constant. These two effects distinguish follower pressure from a fixed-global `CLOAD` approximation.

### 5.5 External-load linearization and symmetry risk

Because current surface position, tangent directions and oriented area depend on displacement, the external force derivative is part of a consistent Newton tangent. A directional finite-difference derivative of external force is the formulation-independent verification oracle for the assembled external tangent action.

Pressure load stiffness must not be presumed symmetric or nonsymmetric solely from the word “follower.” Literature distinguishes conservative and nonconservative cases, and the official Abaqus small-strain shell theory states extra assumptions for its symmetric form. FESA already permits an unsymmetric nonlinear tangent path; Formulation must derive the discrete operator, and Numerical Review must inspect its symmetry properties and boundary assumptions rather than imposing Abaqus's internal choice.

## 6. Candidate Benchmarks

No new artifact is created in this stage. `candidate artifact` marked `to-be-authored` means a later approved implementation plan may add a test input; it is not an existing reference target.

| benchmark_id | candidate / artifact | verifies | target quantities | applicability | cannot verify / limits | requirements |
| --- | --- | --- | --- | --- | --- | --- |
| `BM-01` | Flat single-element S4 and S4R pressure patch; `to-be-authored` | uniform magnitude, unit/sign, `SPOS/SNEG`, positive/negative/zero reversal | total applied force, constrained reaction, moment about patch centroid, source side identity | Planar nondegenerate quad; linear reference configuration; fully constrained or statically determinate support | Nonlinear follower tangent, curved geometry, displacement accuracy | `002`, `004`–`007`, `012`, `020` |
| `BM-02` | Distorted coplanar S4/S4R quad; `to-be-authored` | surface mapping/integration resultant and connectivity reversal | analytical area-vector force/moment, reaction | Convex nondegenerate distorted four-node patch, uniform pressure | Warped/nonplanar exact solution and material response | `006`–`007`, `016`, `020` |
| `BM-03` | Same flat region as one/multiple faces with shuffled declaration/member order; `to-be-authored` | mesh resultant invariance, exact-once membership, deterministic semantic/reduction order | global force/moment, reaction, stable source order, repeated-run diagnostic/order evidence | Linear uniform pressure; equivalent geometry and BC | Floating value bitwise equality is implementation/order contract, not continuum accuracy | `003`, `010`–`013`, `020` |
| `BM-04` | Planar prescribed trial geometry under rigid rotation and independent homogeneous in-plane area change; `to-be-authored` | current normal objectivity and current-area response | rotated force vector, force norm versus current planar area, source identity | Initially/currently planar nondegenerate S4/S4R patch; nonlinear load evaluation isolated from internal element force | Warped-surface vector-area behavior, full equilibrium convergence and tangent | `008`, `016`, `021` |
| `BM-05` | Directional centered finite-difference external-force check on distorted current patch; `to-be-authored` | external tangent consistency | `(f_ext(u+εq)-f_ext(u-εq))/(2ε)` versus assembled external-tangent action | Nondegenerate distorted S4 and S4R; both sides and pressure signs; perturbation remains valid | Cannot detect a common sign error if force and tangent share it; pair with `BM-01/04`. `ε` and norm tolerance remain Numerical Review decisions. | `009`, `016`, `021` |
| `BM-06` | Supported nonlinear pressure patch solved to factor one; `to-be-authored` | integration into Newton residual, convergence and global physics | accepted free residual, applied+reaction force and moment balance, load-factor scaling | Existing full-Newton procedure, linear elastic MITC4, valid supports | Does not isolate load tangent from internal tangent; pair with `BM-05`. | `010`–`012`, `019`, `021` |
| `BM-07` | Exact stored case: `reference/shellNL-meshfine-pressure/shellNL-meshfine-pressure.inp` and `... displacements.csv` | end-to-end parsing, surface resolution, nonlinear pressure response and accepted HDF5 displacement projection | `U1/U2/U3` blocking, `UR1/UR2/UR3` warning-only; exact row identity | 729-node/676-S4 model, `SPOS`, positive `P=1e6`, `NLGEOM=YES`, linear elastic shell | No S4R, SNEG, negative/zero pressure, isolated tangent, independent analytical truth or stress acceptance. Abaqus version/provenance absence is not a readiness failure. | `022`–`024` |
| `BM-08` | Focused unsupported/invalid input matrix; `to-be-authored` | fail-closed capability boundary | failure category, source location, zero contribution, output atomicity | `*DLOAD`, unsupported DSLOAD label/parameter, node/double/edge surface, wrong element, ambiguous/empty surface, nonfinite magnitude | Numerical accuracy or formulation | `014`–`017` |

### Stored artifact inventory

| role | exact artifact | observed research-stage evidence | disposition |
| --- | --- | --- | --- |
| input | `reference/shellNL-meshfine-pressure/shellNL-meshfine-pressure.inp` | SHA-256 `53718ccc9b41c23fcc1b509a51d66f1347e87aca43a74f1d6cec4d2085f4b90e`; S4, `Surf-1/SPOS`, `NLGEOM=YES`, `DSLOAD P=1e+06` spot-checked | required read-only |
| displacement | `reference/shellNL-meshfine-pressure/shellNL-meshfine-pressure displacements.csv` | SHA-256 `2fe53f98eac9f04779100721c3edf8049eb1dbfe733ebcbd819aeeed03b81198`; header and factor-one node rows spot-checked | required external comparison |
| reaction | `reference/shellNL-meshfine-pressure/shellNL-meshfine-pressure reactions.csv` | Header contains `RF1/RF2/RF3/RM1/RM2/RM3` | optional review-only |
| stress | `reference/shellNL-meshfine-pressure/shellNL-meshfine-pressure stresses.csv` | Header contains element/node/section-point `S11/S22/S33/S12` | excluded from feature numerical comparison |

## 7. Verification Relevance

| evidence class | relevant candidates | purpose | acceptance boundary |
| --- | --- | --- | --- |
| code verification | `BM-01`–`BM-05`, `BM-08` | Sign, geometry mapping, exact-once scale, current-surface force and tangent derivative를 작은 독립 oracle로 검증 | Numerical Review가 analytical/geometry/finite-difference scale-aware tolerance를 확정한다. |
| solution verification | `BM-03`, `BM-06` | Mesh/order invariance, nonlinear convergence, accepted free residual과 global balance | Requirement의 internal blocking contract; external common tolerance로 대체하지 않는다. |
| external reference comparison | `BM-07` | FESA authoritative HDF5 displacement를 existing Abaqus CSV observable과 비교 | `U` blocking, `UR` warning-only; `docs/TOLERANCE.md`; row precheck exact |
| validation | 없음 | 실제 실험/물리 시스템과의 validation은 이 numerical load feature 범위가 아니다. | Correctness를 validation으로 과장하지 않는다. |
| physics evaluation preparation | `BM-01`, `BM-04`, `BM-06` | 방향, reaction, rigid rotation, force/moment equilibrium 공통-mode 오류 탐지 | Gate 7 owner가 Implementation evidence 이후 판단한다. |

Reference displacement 단독 일치는 sign/tangent/common-mode error를 배제하지 못하므로 `BM-01`, `BM-04`, `BM-05`, `BM-06`을 대체할 수 없다.

## 8. Applicability Limits

| dimension | applicable evidence | excluded / caution |
| --- | --- | --- |
| deformation | Linear: reference geometry. Nonlinear: current geometry, large rotation and valid current surface. | Material nonlinearity, dynamics, contact, buckling perturbation은 이 feature 범위 밖이다. |
| element | FESA source types S4/S4R mapped to existing MITC4; four-node shell surface. | S3/S8, continuum shell, membrane, solid, beam/pipe, Abaqus internal S4/S4R formulation equivalence 없음. |
| material/section | Existing linear isotropic elastic shell material/section. | Hyperelastic, plastic, composite, thickness/offset pressure correction 없음. |
| load | Uniform finite scalar `DSLOAD P`; positive/zero/negative; single-sided `SPOS/SNEG`. | `DLOAD`, `PNU`, `HP`, `VP`, `SP`, `TRVEC`, `TRSHR`, amplitude/user distributions, double/edge/node surfaces 제외. |
| geometry | Nondegenerate oriented four-node surface; flat/distorted patches and stored curved/deformed response. | Self-intersection, zero/near-zero current area, inconsistent normals require fail-closed criteria; threshold downstream. |
| coordinates | Surface orientation from source connectivity; assembled quantities and balance in global Cartesian; nonlinear moments use current `X+u` arms. | Local shell stress/result conventions are not new pressure acceptance quantities. |
| units | Pressure `force/length²`, nodal force `force`, physical moment `force·length`, displacement `length`, rotation dimensionless. | No implicit SI assumption; model-consistent unit system required. |
| solver behavior | Linear static and approved full-Newton nonlinear static; unsymmetric tangent capability retained. | This research does not choose symmetric tangent, quadrature or solver backend. |
| reference | Exact stored S4 positive-SPOS nonlinear case; current filenames/paths. | Reaction distribution review-only, stress excluded, no S4R or sign reversal evidence. |

## 9. Requirement Traceability

| requirement IDs | research disposition | evidence / gap |
| --- | --- | --- |
| `001`–`002` | covered | `CL-01`, `CL-04`; official keyword/load units plus binding subset |
| `003` | project-contract only; benchmark supplied | `BM-03`; external literature does not define FESA deterministic identity order |
| `004`–`006` | covered | `CL-02`–`CL-04`, `CL-11`; S4/S4R and side/sign evidence |
| `007`–`009` | covered for formulation handoff | `CL-06`–`CL-10`; exact discrete formulation remains Gate 3 |
| `010`–`013` | project-contract plus verification candidates | `BM-03`, `BM-06`; no external tolerance invented |
| `014`–`017` | boundary covered; diagnostics downstream | official distinction plus `BM-08`; exact failure categories/I/O representation remain Gate 5 |
| `018`–`019` | project-contract only | `CL-15`; research found no need for new authoritative output |
| `020`–`021` | covered with bounded benchmark portfolio | `BM-01`–`BM-06`; perturbation/geometry/objectivity tolerance remains Numerical Review |
| `022`–`024` | covered | `CL-13`–`CL-15`, `BM-07`, exact artifact inventory |

Traceability gaps: normative requirement IDs `0`. Downstream-owned numerical detail gaps are listed below and are not silent research gaps.

## 10. Open Issues and Source Disagreements

| issue_id | classification | evidence / search performed | owner / disposition |
| --- | --- | --- | --- |
| `OI-01` | missing public benchmark evidence, nonblocking | Abaqus 2025 Verification/Benchmark Guide searches for S4/S4R + `DSLOAD`/follower/pressure stiffness did not locate a directly applicable case with downloadable target values. Available shell verification hits concerned curvature/contact/submodeling, not this load contract. | Numerical Review uses analytical patches plus exact stored case; no invented public target. |
| `OI-02` | formulation choice, nonblocking | Generic/current-surface linearization can yield nonsymmetric contributions; conservative/global boundary conditions may restore symmetry; Abaqus small-strain shell manual documents a symmetric form under uniform/free-edge assumptions. | Formulation derives FESA operator. Numerical Review examines symmetry/free-edge implications. No user decision unless approved physics/scope changes. |
| `OI-03` | numerical acceptance detail, nonblocking | Sources establish area-vector degeneracy and tangent derivative but not a FESA scale-aware invalid-geometry threshold, finite-difference perturbation or norm tolerance. | Numerical Review fixes these before implementation. |
| `OI-04` | implementation detail, nonblocking | Sources support work equivalence but do not choose FESA quadrature or S4/S4R-specific load integration. | Formulation owns quadrature and discrete consistency; no Abaqus internal equivalence claim. |
| `OI-05` | reference limitation, accepted | Stored bundle contains no Abaqus version/provenance. Requirements/project contracts explicitly state these are not readiness conditions. | Preserve exact artifacts read-only; do not escalate or create metadata. |

Critical user question: 없음. 승인 scope, physics, tolerance, authoritative output 또는 reference acceptance를 변경해야 하는 source conflict는 발견되지 않았다.

## 11. Downstream Handoff

### 11.1 To Formulation Agent through Coordinator

- Use `CL-03`–`CL-10` as the evidence boundary for connectivity/side sign, work-equivalent surface force, reference/current configuration and external-load linearization.
- State pressure force and residual signs together; do not import element-based `DLOAD` shell sign.
- Derive FESA's own interpolation, numerical integration and tangent. Do not cite Abaqus result agreement as internal equivalence.
- Resolve `OI-02` and `OI-04`: discrete tangent symmetry/free-edge assumptions, quadrature and generalized DOF embedding.
- Preserve full-space assembly, exact-once load-factor scaling and existing reaction/free-residual meaning from the binding requirements.

### 11.2 To Numerical Review Agent through Coordinator

- Review `BM-01`–`BM-06` as the minimum independent suite before relying on `BM-07`.
- Fix scale-aware tolerances for analytical area/resultant, geometry validity, rigid-rotation objectivity and directional finite-difference tangent; do not substitute `docs/TOLERANCE.md` for these internal checks.
- Keep exact stored paths and disposition: displacement `U` blocking, `UR` warning-only, reaction CSV review-only, stress CSV excluded.
- Confirm one-to-one source identity/component/frame precheck and `docs/TOLERANCE.md` only for the external displacement families.
- Carry `OI-01` as a disclosed public-benchmark gap, not as permission to invent targets or modify reference artifacts.

Research stops here. No formulation, C++ API, implementation, reference solver execution, reference artifact modification or Gate 3 decision was performed.