FESADev/docs/nonlinear-static-newton-raphson/numerical-review.md

Nonlinear Static Newton-Raphson 수치 검토

1. Metadata and source formulation

• feature_id: nonlinear-static-newton-raphson
• source_formulation: docs/nonlinear-static-newton-raphson/formulation.md
• source_formulation_status: ready-for-numerical-review
• source_formulation_sha256: DFA629888041D5762FD0DF4271353464A692B325DC0C821A8D8689688C1EAFD0
• source_requirements: docs/nonlinear-static-newton-raphson/requirements.md
• source_requirements_sha256: 910E351F3F90FFA42F07D15206B55E62C05EAE821E250424430148337506EBF2
• source_research: docs/nonlinear-static-newton-raphson/research.md
• source_research_sha256: C920A6B02381230D1DB419D73FE767D611B36720D27D6410507BA88C3CA28EEE
• repository_policy: AGENTS.md, docs/SOLVER_AGENT_DESIGN.md, docs/TOLERANCE.md
• related_linear_contracts: docs/linear-static-mitc4-shell/numerical-review.md, docs/linear-static-mitc4-shell/reference-model.md
• status: pass-for-io-definition
• owner_agent: numerical-review-agent
• date: 2026-08-23

이 검토는 정식화를 수학 및 수치 알고리즘 계약으로만 검토했다. C++ API, HDF5 schema, implementation plan, reference solver 실행, reference artifact 생성/수정은 수행하지 않았다.

2. Review Verdict

• verdict: pass-for-io-definition
• reason: 승인 정식화는 fixed-initial-reference Total Lagrangian MITC4, finite rotation state, nonlinear 24 -> 20 map, objective drilling, fixed-global load work, same-trial residual/tangent, Full Newton convergence, recovery와 state commit/rollback 의미를 구현 가능한 수치 계약으로 닫고 있다. 차원, 부호, DOF 순서, 적분, reaction 의미 또는 reference identity와 충돌하는 blocking defect는 확인되지 않았다.
• reference readiness: reference/shellNL 및 reference/shell의 선언 파일과 CSV는 읽을 수 있고 row identity/precheck/tolerance 계약을 정의할 수 있다.
• boundary: 이 verdict는 I/O Definition 진입 허가이며 implementation, reference comparison pass, physics evaluation 또는 release readiness 승인이 아니다.

reference/shellNL은 nlgeom=YES와 factor-one final frame을 갖는 저장 reference로 사용할 수 있지만, 주요 U3와 stress 값은 선형 reference/shell과 거의 같다. 관찰된 nonlinear 신호는 주로 약한 in-plane coupling이다(RF1/RF2 최대 약 1.9166, U1/U2 최대 약 1.29e-11). 따라서 이 CSV 세트만으로 finite-rotation objectivity, exact tangent, rollback 또는 큰 변위 benchmark를 통과했다고 판단하지 않는다. 해당 항목은 아래의 blocking internal tests로 분리한다.

3. Critical Findings

Confirmed defects

없음. Formulation Agent 또는 Research Agent로 즉시 되돌릴 confirmed numerical defect는 없다.

Numerical risks requiring downstream evidence

• shellNL의 stored reference는 geometric nonlinearity가 켜진 최종 상태 비교에는 충분하지만 response amplitude가 작아 finite-rotation/tangent/objectivity 검증을 대신하지 못한다.
• Abaqus stress CSV는 (instance, element label, element-corner node label, section point, component) identity로 완전하지만 FESA의 approved Gauss-location direct bottom/middle/top recovery와 location이 다르다. Current stored order는 element label 1..36 ascending block이며 이를 바꾸지 않는다. Nodal extrapolation/averaging을 만들지 않고 stress CSV는 comparison family에서 제외한다.
• External UR comparison은 기존 MITC4 정책에 따라 warning-only다. Rotation/director correctness는 Q, d, branch, objectivity invariant tests에서 blocking으로 확인해야 한다.

Open questions

현재 I/O Definition 진입을 막는 open question은 없다. Large-response public benchmark, MMS, broader locking/distortion characterization은 유용하지만 승인된 Stage 4 pass 조건은 아니다.

4. Numerical Risk Assessment

risk label	assessment	required control / downstream evidence
rigid_body_modes	물리/internal/drill energy가 finite rigid motion에서 objective해야 하며 pure drill만 regularized 되어야 한다.	finite rigid translation/rotation superposition에서 strain, stress, physical energy, drill gamma, residual을 blocking invariant로 확인한다.
patch_test	Green-Lagrange direct strain과 MITC tied shear가 같은 trial geometry와 같은 first/second variation을 써야 한다.	membrane, bending, twist, transverse-shear patch를 linear limit와 finite valid trial perturbation에서 각각 확인한다.
symmetry	applied nodal moment가 없으면 physical/drill tangent는 exact arithmetic에서 symmetric이다. Fixed-spatial moment work tangent는 일반적으로 nonsymmetric이다.	no-moment case만 scaled symmetry check 대상으로 두고, moment case를 임의 symmetrization하지 않는다.
positive_definiteness	nonlinear tangent는 load state, compression, near-buckling에서 indefinite일 수 있다. Positive definiteness는 일반 gate가 아니다.	stress-free constrained linear limit의 nonsingular solve와 singular negative case를 분리하고, factorization failure를 숨기지 않는다.
hourglass	S4/S4R 모두 2 x 2 x 2 full integration이므로 one-point reduced hourglass path는 승인 범위 밖이다.	source type으로 reduced/selective integration 또는 hourglass stabilization을 선택하지 않는 unit/integration test가 필요하다.
shear_locking	MITC shear tying은 transverse-shear locking을 완화하지만 모든 thin-shell mesh에서 locking-free를 보장하지 않는다.	patch/reference evidence를 blocking으로 두고 broader thin/thick convergence는 nonblocking characterization으로 둔다.
volumetric_locking	approved plane-stress shell, St. Venant-Kirchhoff-type material restriction에서는 N/A다.	3-D incompressible material 또는 nu >= 0.5 의미로 확장하지 않는다.
membrane_locking	original-style MITC4는 curved/distorted nonlinear meshes에서 membrane locking 위험이 남는다.	known limitation으로 기록한다. MITC4+ 성능 또는 Abaqus S4/S4R internal equivalence를 주장하지 않는다.
distortion	positive Jacobian이라도 severe distortion/warpage는 conditioning과 stress accuracy를 악화할 수 있다.	모든 Gauss/tying/center/recovery location의 finite, positive predicate를 fail-closed로 적용한다. 추가 quality cutoff는 요구하지 않는다.
singular_jacobian	J0 <= 0, j <= 0, invalid reciprocal basis, invalid polar stretch는 residual/tangent를 무효화한다.	failed location을 omit/average/clamp/repair하지 않고 model/trial failure로 처리한다.
conditioning	thin shells, small fixed drill scale, near-mechanism constraints는 Kff conditioning을 악화시킬 수 있다.	scaled matrix diagnostics만 사용하고 solver regularization/fallback으로 singularity를 숨기지 않는다.
convergence	Full Newton은 same-trial consistent tangent에 의존한다. No line search/cutback이므로 difficult step은 deterministic nonconvergence가 될 수 있다.	NSNR-069/070 four-channel gates, iteration 1..100, nonfinite rejection, no hidden retry를 integration tests로 고정한다.

5. Consistency Checks

check	result	evidence
units and dimensions	pass	Translation/force, rotation/moment, stress/resultant dimensions are separated. Convergence channels keep force, force-length, length and dimensionless rotation separate.
signs and residual convention	pass	Global residual is f_int - f_ext; constrained full residual is reaction and free residual is equilibrium evidence. Fixed-global dead force has no configuration derivative.
DOF ordering and free/constrained assumptions	pass	Per-node global order is [UX, UY, UZ, URX, URY, URZ]; Newton solve uses stable free coordinates with Delta q_c = 0. Valid 0 x 0 free system is trivial success.
coordinate transforms	pass	Q=Exp(phi^)Q0, left-spatial update, J_l variation, physical swing map and spatial reaction moment conversion are explicitly defined.
kinematic operator / B matrix	pass	Green-Lagrange strain, MITC shear projection and both first/second variations use the same trial identity.
constitutive contract	pass	Constant symmetric C5 maps engineering Green-Lagrange strain to second Piola-Kirchhoff stress; no material history is introduced.
Jacobian rules	pass	Fixed reference J0 is integration measure; current j and det F_c are validity checks. Failed locations are not skipped.
integration policy	pass	S4 and S4R share four midsurface Gauss points times two thickness points; reduced/selective integration is prohibited.
residual/tangent consistency	pass	Complete tangent includes material, geometric, map-Hessian, objective drill Hessian and fixed-spatial moment work derivative terms.
tangent symmetry expectation	pass	Symmetric only for conservative no-applied-moment case; generally nonsymmetric moment tangent must remain unsymmetrized.
output locations	pass-with-handoff	Nodal U/UR/RF/RM and shell stress/resultant logical quantities are defined; exact HDF5 projection remains I/O-owned. Current nodal SPOS/SNEG stress CSV는 Gauss direct recovery와 일대일 대응하지 않으므로 external numerical comparison은 N/A다.

6. Verification Readiness

Blocking internal numerical tests

Implementation Planning shall carry these as feature-required tests before any reference-comparison success claim:

Mixed translation/rotation 검사는 fixed-reference characteristic length L=max(||X_I-X_J||)와 node block Dq=diag(L I3,I3)를 사용한다. Tangent test direction은 ||p_hat||inf=1, perturbation은 q +/- h Dq p_hat으로 두고 h={1e-3,3e-4,1e-4,3e-5,1e-5,3e-6,1e-6,3e-7} 전부를 검사한다. Roundoff plateau 전 최소 두 interval에서 centered-difference observed order >=1.5이고 한 interior point가 scaled error <=1e-6여야 한다. Single-h 우연 통과는 허용하지 않는다.

1. Linear limit: at stress-free q=0, nonlinear element residual/tangent/recovery reduce to approved linear MITC4 signs, 2 x 2 x 2 integration and physical/drill split.
2. Rotation state invariants: every committed/trial Q satisfies ||Q^T Q-I||_F <= 1e-12, |det(Q)-1| <= 1e-12, and d=Q e3. Global/swing/twist pi branch angular margin은 sqrt(epsilon_double), 약 1.4901161193847656e-8 rad보다 커야 한다.
3. Tangent directional derivative: compare Dq^T K_T Dq p_hat with the central residual derivative under the schedule above in separated force and moment channels. Zero analytical directions require exact-zero construction and scaled absolute error <=1e-10; nonfinite metric은 허용하지 않는다.
4. No-moment tangent symmetry: after work-conjugate congruence Dq^T K_T Dq, no-applied-moment tangent scaled Frobenius symmetry error <=1e-12; applied fixed-spatial moment cases are checked for exact derivative consistency, not symmetry.
5. Finite rigid objectivity: superposed finite rigid motion leaves physical strain/stress, physical energy, drill gamma/energy and internal residual at scaled <=1e-10 levels.
6. Patch/sign recovery: membrane, bending, twist and transverse shear patch states preserve component order and bottom/top stress signs. Analytical strain/resultant/stress error is relative <=1e-9.
7. Load and boundary path: fixed-global force/moment source identity, rho_M <= 1e-12 rejection/projection, absolute factor-one boundary path and zero constrained Newton correction are tested independently.
8. Newton lifecycle: factor identities use ordinal j*h values strictly below one, followed by one exact assigned 1.0; cumulative lambda += h와 tolerance-based duplicate endpoint는 금지한다. For shellNL the expected intended factors are 0.1, 0.2, ..., 1.0. Iterations are 1..100, pass at 100 is allowed, iteration 101 is forbidden.
9. Convergence gates: NSNR-069/070 four-channel inclusive <= rules, typed zero-scale 1e-3 thresholds and nonfinite rejection are tested without adding energy or line-search gates.
10. Commit/rollback and determinism: failed trial, recovery, solve or output does not mutate the prior committed state or prior authoritative final result; row/diagnostic order is stable.

Downstream reference comparisons

The required stored-reference cases and logical quantities are defined in docs/nonlinear-static-newton-raphson/reference-model.md.

• shellnl-s4-final is the primary nonlinear external reference at accepted factor one.
• shell-s4-linear-limit is the read-only linear-limit/reference-identity regression for the same S4 mesh. Current stress element-label ordering is inventoried without becoming a comparison quantity.

External comparison uses only accepted-state quantities and docs/TOLERANCE.md. Missing, extra, duplicate or nonfinite required rows fail before tolerance. The comparison result itself is Implementation Agent ownership.

7. Required Revisions

Formulation Agent

None before I/O Definition. The current formulation is numerically sufficient for the I/O contract. Current source element/node SPOS/SNEG stress CSV에 맞추기 위한 extrapolation, averaging 또는 새 recovery 의미는 추가하지 않는다.

Research Agent

None before I/O Definition. Public large-response benchmarks and shell MMS remain useful future evidence, but the current Stage 4 gate does not require another research pass.

I/O Definition Agent

Define parser/semantic/HDF5/log projection for the logical quantities and row identities in reference-model.md. Preserve the split between blocking internal nonlinear tests and external stored reference comparison; do not claim Abaqus internal equivalence.

8. Downstream Handoff

For I/O Definition Agent through Coordinator:

• Use accepted factor-one frames only for external reference comparison; trial Newton states are not reference quantities.
• Preserve required source identity: case, accepted state, instance, node label and component. Stress artifact의 element/node/section identity와 stored element-label order는 read-only inventory로 보존하되 FESA comparison row로 합성하지 않는다.
• Keep U1/U2/U3 blocking and UR1/UR2/UR3 warning-only. For shellNL, compare constrained RF1/RF2, RF3, and spatial physical RM1/RM2 in the separate blocking families from reference-model.md. RM3와 stress는 comparison에서 제외한다.
• Apply docs/TOLERANCE.md exactly; do not add an independent absolute floor, row averaging, source-order matching, or calibration gate.
• Treat shellNL as weak but valid nonlinear final-state reference evidence; require the blocking internal tests above for tangent/objectivity/rollback/finite-rotation correctness.