FESADev/docs/linear-static-mitc4-shell/numerical-review.md

Linear Static MITC4 Shell Numerical Review

1. Metadata

• feature_id: linear-static-mitc4-shell
• source_formulation: docs/linear-static-mitc4-shell/formulation.md
• source_requirements: docs/linear-static-mitc4-shell/requirements.md
• source_research: docs/linear-static-mitc4-shell/research.md
• source_io_definition: docs/linear-static-mitc4-shell/io.md
• source_reference_inventory: docs/linear-static-mitc4-shell/reference-model.md
• repository_policy: AGENTS.md, docs/SOLVER_AGENT_DESIGN.md, .codex/skills/fesa-numerical-review/SKILL.md
• reviewed_head: cf769aa (mathematical implementation baseline)
• prior_pass_commit: 60b42f4 (context-only; verdict not inherited)
• status: pass-for-implementation-planning
• owner_agent: numerical-review-agent
• date: 2026-08-13
• implementation_planning_authorized: true
• implementation_complete: false
• build_test_complete: false
• reference_comparison_complete: false
• physics_evaluation_complete: false
• release_ready: false

이번 재검토는 현재 HEAD의 요구조건, 연구, 정식화, I/O 및 reference-case 계약을 처음부터 상호 대조했다. 기존 review의 판정과 artifact 관찰 결과는 결론의 전제로 사용하지 않았고, 이전 finding은 현 문서의 수식으로 다시 검산한 뒤 disposition만 기록했다. 원 MITC4 local paper는 tying 위치와 covariant shear 보간을 확인하는 데 read-only로 사용했다.

이 단계에서는 Abaqus, Harness, C++ build/test, FESA 실행 및 reference comparison을 수행하지 않았다. Reference artifact를 생성, 수정, 복원 또는 정규화하지 않았다.

2. Review Verdict

• verdict: pass-for-implementation-planning
• critical_blockers: none
• confirmed_defects: none in the approved current linear-static scope
• open_blocking_questions: none
• reason: 현재 정식화는 24 global DOF와 20 physical DOF의 관계, MITC4 shear tying, plane-stress section law, 공통 2 x 2 x 2 quadrature, residual/stiffness, 고정 drilling 안정화, 물리 recovery 및 검증 불변식을 구현계획으로 옮길 수 있을 만큼 명시한다. 요구조건, I/O 및 reference 계약과 모순되는 차원, 부호, 위치 또는 pass/fail 의미도 발견되지 않았다.
• downstream_boundary: 이 판정은 Implementation Planning 진입만 허용한다. 구현, MSVC build/CTest, reference comparison, physics sanity 또는 release를 승인하지 않는다.

정식화 Section 15의 geometrically nonlinear residual/tangent는 future-only다. 완전한 Phi: R24 -> R20, map Hessian, objective drilling potential 및 finite-rotation load work가 미정인 사실은 미래 nonlinear 구현을 막지만 현재 linear-static 판정은 막지 않는다.

3. Critical Findings

3.1 Confirmed defects

현재 승인된 선형 정적 범위에서 구현계획 전에 Formulation 또는 Research로 돌려보낼 confirmed mathematical defect는 없다.

K20의 exact-arithmetic 대칭/positive-semidefinite 구조와 20-to-24 congruence는 일관된다. 다만 실제 구현의 rank, rigid action, patch field와 reference error는 문서 검토만으로 통과했다고 볼 수 없으며 Section 6의 downstream test evidence가 필요하다.

3.2 Previous finding disposition

previous item	rerun disposition	current independent basis
NR-C01 Jacobian/geometry inventory	resolved	Formulation 9.2-9.3은 center, eight stiffness points, four tying points 및 committed recovery points를 공통 fail-closed inventory로 두고 finite bases, nonzero surface measure와 J>0를 요구한다. 승인 범위는 calibrated distortion/warp cutoff를 요구하지 않는다.
NR-C02 drilling normalization	resolved	Formulation 12.2는 R+를 오직 8 physical tangent-rotation diagonals의 finite positive 값으로 제한하므로 모든 후보의 단위가 force*length로 같다.
NR-C03 mixed-DOF spectrum scaling	resolved	Formulation 12.5의 (L_e I3,I2) 및 (L_e I3,I3) congruence는 rank/condition evidence에서 translation/rotation 단위 혼합을 제거한다. Raw mixed-unit spectrum은 금지된다.
NR-C04 20/24 weak-form mismatch	resolved	Formulation 5.2-5.3과 7.1-7.2는 physical, drilling, external work를 모두 V24에서 T_p^T와 T_d^T로 결합한다.
NR-C05 nonlinear 20-to-24 closure	resolved for current scope	Section 15는 physical chart tangent와 conditional global pullback을 분리하고 map-curvature 항을 보존하며, 미정인 global map/objective drill을 future-only blocker로 명시한다.
NR-D01 drilling-direction moment	retained and consistent	Exact-zero moment는 별도 처리하고 nonzero moment에 rho_M=abs(d dot M)/norm(M)<=1e-12를 적용한다. Numerical drilling은 거부된 moment를 운반하지 않는다.
NR-D02 normalized algebraic checks	retained and consistent	1e-12 symmetry/frame/energy와 1e-10 rigid/residual/equilibrium 기준은 scaled matrices와 unclamped denominators에 적용된다.
NR-O01 coefficient sweep/plateau	closed by product decision	k_d=1e-3 min(R+)가 고정 계약이다. Sweep, plateau 및 coefficient optimality는 구현 gate가 아니다.
NR-O02 drilling-energy ratio	removed from scope	Drilling energy는 내부 quadratic identity일 뿐 physical energy나 mandatory output이 아니며 ratio/warning threshold도 요구하지 않는다.
NR-O03 smooth-director calibration	removed from scope	Pairwise positive incident-normal orientation, finite/nonzero averaging 및 duplicate-node fold modeling이 승인된 exact predicate다. 별도 angle calibration은 gate가 아니다.
NR-O04 distortion/warp calibration	removed from scope	Basic topology, finite/nonzero surface measure 및 required-point J>0가 승인된 predicate다. Quality sweep이나 cutoff는 gate가 아니다.
NR-O05 U/UR tolerance	resolved	모든 관련 문서가 sole S4 reference에 공통 family-scale row/RMS 규칙, U blocking, UR warning-only를 동일하게 정의한다. S4R은 reference gate가 아닌 common-path evidence다.

이전의 needs-reference-model 판정에 포함됐던 canonical naming, README, metadata.json, provenance, expanded portfolio 및 아직 없는 comparison result는 현재 프로젝트 정책상 formulation verdict의 blocker가 아니다. 현 Reference Model 문서는 정확한 기존 input/displacement path와 row/tolerance 계약을 제공한다.

3.3 Open questions

• current_linear_scope: none blocking
• future_geometric_nonlinearity: finite global rotation coordinate, Phi와 그 1/2차 미분, chart recentering, objective drilling, nodal-moment work 및 nonlinear output/state 계약이 미정이다. 이는 별도 future formulation/review가 소유한다.
• optional_characterization: near-singular positive-J geometry의 conditioning과 original MITC4의 distorted-curved membrane locking을 더 넓게 정량화할 수 있으나 현재 승인된 planning/completion gate는 아니다.
• downstream_results: implementation rank/patch evidence, S4 reference comparison과 S4R common-path test 결과는 해당 후속 Agent가 판정한다. 부재 자체는 pre-implementation review의 결함이 아니다.

4. Numerical Risk Assessment

risk label	assessment	required in-scope control
rigid_body_modes	Physical K20은 six rigid modes와 expected rank 14를 가져야 한다. 24-DOF embedding은 네 drill null coordinates를 더하고 fixed drill block 뒤 expected rank 18/nullity 6이다.	세 translation과 세 rotation을 명시적으로 구성한다. Rotation mode는 u_I=omega x X_I, theta_I=omega-(omega dot d_I)d_I, gamma_I=0를 사용한다.
patch_test	Bilinear membrane/bending field와 MITC tied shear는 required patch states를 표현할 계약을 갖는다.	E11/E22/G12, K11/K22/K12, G13/G23를 독립 시험하고 signs/component order/resultants/stress를 함께 확인한다.
symmetry	B^T C B, T_p^T K20 T_p, T_d^T(k_d I)T_d는 exact arithmetic에서 symmetric이다.	Scaled Frobenius check <=1e-12; deterministic assembly가 대칭을 깨지 않는지 확인한다.
positive_definiteness	Free element는 six-mode semidefinite이고, 충분히 구속된 nonsingular Kff는 positive definite가 기대된다. Geometry 또는 supports가 부적절하면 singularity가 정당하다.	Scaled spectrum/rank, non-rigid positive physical energy, constrained solve 및 singular negative cases를 분리한다.
hourglass	1 x 1 reduced integration을 쓰지 않으므로 Abaqus-style hourglass path는 N/A다.	Full 2 x 2 midsurface rank test는 유지한다. S4R source label로 reduced rule을 선택하지 않는다.
shear_locking	Edge-midpoint MITC projection이 transverse-shear locking을 다루지만 모든 mesh/thickness에서 완전 제거를 주장할 수 없다.	Required shear/bending patch와 declared S4 reference를 통과한다. Broader thin/thick convergence는 nonblocking characterization이다.
membrane_locking	Original MITC4는 membrane strain을 수정하지 않아 distorted curved meshes에서 알려진 locking 위험이 남는다.	Known limitation을 유지하고 MITC4+ 성능을 주장하지 않는다. Expanded curved/distorted portfolio는 optional이다.
volumetric_locking	승인된 homogeneous plane-stress shell에는 N/A다.	C5를 3D nearly-incompressible law로 확장하지 않는다.
distortion	양의 J를 유지하는 심한 distortion/warpage는 정확도와 rank/conditioning을 악화할 수 있다.	모든 required location의 exact predicates와 rank/finite-result checks를 시행한다. 승인되지 않은 quality cutoff를 추가하지 않는다.
singular_jacobian	Nonfinite/nonpositive J, zero surface measure, invalid reciprocal basis는 mapping을 무효화한다.	Center, stiffness, tying 및 recovery inventory 전체를 omission/clamp 없이 fail closed한다.
conditioning	Thin shells, near-degenerate positive-J geometry 및 작은 fixed drill scale에서 Kff conditioning이 나빠질 수 있다.	Spectrum/condition evidence는 오직 length-scaled matrix를 사용한다. Threshold calibration은 gate가 아니지만 factorization failure는 결정적으로 진단한다.
convergence	Current solve는 direct linear solve라 Newton convergence는 N/A; spatial convergence와 locking trend는 모델 의존이다.	Free residual/global equilibrium <=1e-10과 declared S4 reference case를 확인한다. Broader mesh sequences는 optional이다.
drilling_contamination	Numerical drill이 physical strain/recovery에 들어가면 비물리 결과가 생긴다.	T_p/T_d를 분리하고 pure drill에서 physical strain/resultant/stress/energy가 zero임을 시험한다. Full residual은 의도대로 stabilized system 전체를 포함한다.
future_nonlinear_misuse	Section 15만으로 global nonlinear element를 만들면 nonobjective drill 또는 inconsistent tangent가 된다.	Current plan에서 완전히 제외하고 별도 승인 전 실행하지 않는다.

5. Consistency Checks

5.1 Units, dimensions, DOF order, and constrained/free system — pass

• Per-node global order is exactly [UX,UY,UZ,URX,URY,URZ]; element order is 24 global coordinates and 20 physical coordinates [uX,uY,uZ,alpha,beta] per node plus four separately selected gamma coordinates.
• T_p is 20 x 24, T_d is 4 x 24, K20 is 20 x 20, and both global stiffness contributions are 24 x 24.
• Translation-translation, translation-rotation, and rotation-rotation stiffness blocks have units force/length, force, and force*length; R+ therefore excludes every translational diagonal.
• The constrained/free equation is Kff*df=Ff-Kfc*dc. Stiffness partition and factorization precede load assembly, and an all-constrained valid 0 x 0 Kff is not reclassified as singular.
• r=K*d-F fixes the internal-minus-external sign. Constrained entries are the required reaction rows and free entries remain residual evidence.

5.2 Local/global transforms, congruence, and energy — pass

• (a_I,b_I,d_I) and (e1,e2,e3) are deterministic right-handed orthonormal frames. The least-aligned-axis nodal rule avoids a fixed-axis parallel singularity.
• [alpha,beta,gamma]^T=R_I^T theta_I^g gives delta d=beta*a-alpha*b=theta x d with the required sign.
• q20=T_p qg and gamma=T_d qg preserve virtual work. Congruence gives Kphys24=T_p^T K20 T_p and Kdrill24=T_d^T(k_d I4)T_d; the corresponding local and global quadratic energies are identical.
• A physical rigid rotation uses only the tangent projection of omega, so gamma=0 and drilling does not destroy the six physical rigid modes.

5.3 Kinematic operators and MITC tying — pass

• Bilinear N_I satisfies partition, Kronecker and derivative-sum identities.
• Membrane and bending content comes from the direct covariant small-strain operator. Only epsilon_xi-zeta and epsilon_eta-zeta are replaced.
• epsilon_xi-zeta is tied at (0,-1,0) and (0,+1,0) and interpolated in eta; epsilon_eta-zeta is tied at (-1,0,0) and (+1,0,0) and interpolated in xi. Each interpolation reproduces its own edge value and is constant along the edge direction, matching the original MITC4 construction.
• The assumed covariant tensor is reconstructed through reciprocal bases, projected into the stored local Cartesian frame, and converted once to engineering shear gamma_ij=2 epsilon_ij. The same projected B_bar drives strain, internal force, stiffness and recovery.

5.4 Constitutive and section matrices — pass

• Cps=E/(1-nu^2)[[1,nu,0],[nu,1,0],[0,0,(1-nu)/2]] uses engineering G12; its shear coefficient is exactly G=E/[2(1+nu)].
• C5=diag(Cps,(5/6)G I2) is symmetric positive definite for E>0 and -1<nu<0.5. sigma33=0 and absent thickness stretch remain assumptions.
• A=t Cps, B=0, D=t^3 Cps/12, and As=(5/6)Gt I2 have consistent dimensions. Membrane/shear strains are dimensionless, curvature is 1/length, N/Q is force/length, M is force, stress is force/length^2, and energy is force*length.

5.5 Jacobian, derivative transform, and integration — pass

• The three-dimensional degenerated mapping uses X=sum(N X_I)+(t*zeta/2)sum(N d_I) with a unit nodal director and separate thickness, preventing nodal thickness double counting.
• J=det[G_xi,G_eta,G_zeta] and finite covariant/reciprocal bases are checked at all eight stiffness points, all four midsurface tying points, center, and every committed bottom/middle/top recovery evaluation. Failed points are not skipped, averaged, clamped or repaired.
• Direct natural derivatives are converted covariantly and then to the local Cartesian tensor through contravariant bases; no flat-element derivative shortcut is substituted for curved/warped accepted geometry.
• Both S4 and S4R use the common in-plane 2 x 2 points +-1/sqrt(3) with unit weights and two identical thickness points, for eight volume evaluations. Tied shear is evaluated at zeta=0 and reused at both thickness points while the remaining mapping and direct components use the actual thickness point.

5.6 Internal force, residual, stiffness, and future tangent — pass

• Current K20=integral(B_bar^T C5 B_bar dV) and f_int20=K20 q20 are mutually consistent and symmetric positive semidefinite in exact arithmetic.
• The complete current weak form is in V24 and adds the numerical drilling gradient before subtracting the global nodal CLOAD vector.
• No geometric stiffness or nonlinear state enters the current product path. Future Section 15 correctly separates Kmat and the stress-dependent Kgeo and, conditionally on a future Phi, includes both A^T K20 A and the residual-weighted map-Hessian term.

5.7 Fixed drilling contract — pass

• R+ contains only finite strictly positive diagonals of the eight physical tangent-rotation coordinates. k_ref=min(R+), k_d=1e-3 k_ref, and Kd_local=k_d I4 are dimensionally consistent and deterministic; empty R+ fails validation.
• T_d^T(k_d I4)T_d is symmetric and positive on the four pure drilling coordinates. It must remove those four nonphysical null modes without changing the physical rank/null modes.
• Drilling is excluded from generalized strain/resultant/stress and reported physical strain energy. A director-parallel applied nodal moment is rejected as unsupported-drilling-load; no numerical drill load channel exists.

5.8 Recovery, signs, locations, units, and external comparison — pass

• Nodal [U1,U2,U3,UR1,UR2,UR3] and full-residual [RF1,RF2,RF3,RM1,RM2,RM3] are global and source-node ordered.
• Four midsurface Gauss rows recover [E11,E22,G12,K11,K22,K12,G13,G23] and [N11,N22,N12,M11,M22,M12,Q13,Q23] in the stored local frame. The definition e_m(z)=epsilon0+z*kappa fixes curvature, moment, and bottom/top stress signs.
• [S11,S22,S12] is evaluated directly at zeta=-1,0,+1; S33=0 is documented but not emitted, and S13/S23 point stress is not synthesized. Different natural or section locations are never averaged.
• Reference comparison first rejects missing, extra, duplicate, nonfinite or identity-mismatched rows. Every U/UR family then uses the common reference-only scale, near-zero ratio 0.01, row relative tolerance 0.05 and scale-relative RMS tolerance 0.01. S=0 requires exact-zero FESA values; there is no zero clamp or independent absolute-error gate. U1/U2/U3 is blocking and UR1/UR2/UR3 is warning-only.
• Source S4 and S4R select the same FESA MITC4 kernel/quadrature/recovery path while preserving source type. This is an input mapping, not an Abaqus formulation, integration, stabilization or recovery equivalence claim.

6. Verification Readiness

6.1 Downstream unit and invariant tests

Implementation Planning shall convert the following to RED -> GREEN -> VERIFY:

1. Shape identities; nodal/integration frame orthonormality, handedness and axis tie-break determinism.
2. T_p/T_d dimensions, orthogonal channel selection, virtual-work equality and nonzero transformation-energy equality.
3. Hand-calculated direct membrane/bending columns, all four covariant tying values, interpolation weights and engineering-shear factors.
4. Cps/C5/A/D/As coefficients, symmetry, positive definiteness, dimensions and force/length unit-rescaling invariance.
5. Common 2 x 2 x 2 point/weight order and an independent analytical or higher-order flat-element stiffness/recovery cross-check.
6. Required-location geometry validation: valid planar/rotated/warped cases and duplicate, bow-tie/self-intersecting, zero-area, reversed, nonfinite, nonpositive-J and opposed-normal negative cases.
7. Scaled symmetry <=1e-12, physical rigid action <=1e-10, expected physical rank 14, stabilized rank 18/nullity six, and positive non-rigid physical energy.
8. Exact R+ membership, exclusion of translations, fixed coefficient, empty-R+ failure, pure drill action and zero physical recovery/energy.
9. Stable COO/reduction, source/result/diagnostic order and thread-count repeatability.
10. Kff/Kfc effective RHS, nonzero prescribed values, full-residual reaction, singular-support negative case, and valid all-constrained 0 x 0 Kff case.
11. Exact-zero and accepted/rejected rho_M moment projections, including proof that rejected drilling moments never reach stabilization.
12. Mandatory HDF5 locations/components/units, physical-only energy, nonfinite recovery failure and atomic finalization.

6.2 Required patch and sign tests

• independent constant E11, E22, and G12 membrane fields with N and middle-stress signs;
• pure K11 and K22 bending with M order and bottom/top stress reversal;
• pure K12 twist with M12 sign;
• constant G13 and G23 transverse shear with Q13/Q23 order;
• six physical rigid states and four pure drilling states;
• source-type-only S4/S4R variants producing identical FESA numeric rows and different preserved source metadata.

6.3 Reference and physics handoff readiness

The sole declared read-only acceptance pair is:

• reference/shell/shell.inp with reference/shell/shell displacements.csv for full-integration S4.

S4R remains required input coverage through source mapping, common-kernel, deterministic-assembly and HDF5 metadata tests, but no S4R Abaqus artifact is consumed by reference acceptance.

The Reference Model and I/O documents define deterministic HDF5-to-CSV identity, precheck and tolerance sufficiently for later comparison. This review did not assert that results.h5 exists or that any row passes. Reference Verification owns numeric U/UR outcome; Physics Evaluation owns force/moment balance, displacement direction, symmetry, result signs, recovered-resultant consistency and physical plausibility.

6.4 Missing evidence classification

• blocking_for_current_formulation: none
• required_after_implementation: invariant, patch, MSVC build/CTest, declared reference comparison and physics evidence above
• nonblocking_optional: drilling coefficient sweep/energy ratio, NR-O03, NR-O04, canonical naming, README/metadata/provenance, expanded benchmark portfolio and broader mesh convergence studies
• future_only: nonlinear directional-derivative/objectivity/Newton evidence after its missing formulation decisions are separately approved

7. Required Revisions

Formulation Agent

• None for the approved current linear-static implementation scope.
• Do not promote Section 15 to executable status until a separate formulation closes the nonlinear global coordinate map, objective drilling and load-work contracts.

Research Agent

• None before current Implementation Planning.
• Optional locking/convergence characterization must remain clearly outside the approved implementation gate and must not imply MITC4+ or Abaqus equivalence.

I/O Definition Agent

• None for the current numerical verdict. Preserve exact physical/full-residual distinction, source identity, locations, units and U-versus-UR decision rule.

Reference Model Agent

• None for the current numerical verdict. Preserve the two declared S4 files and every optional existing reference artifact read-only, exclude S4R artifacts from acceptance comparison, and do not add administrative or portfolio gates.

8. Downstream Handoff

Implementation Planning Agent

Implementation Planning is authorized and shall:

• trace the required tests in Section 6 to the approved requirement IDs before production work;
• keep 24 global -> 20 physical + 4 drilling transforms, covariant MITC tying, common 2 x 2 x 2 integration, fixed drilling and physical recovery as explicit independent test seams;
• preserve stiffness assembly/partition/factorization-before-load, stable reduction, full-residual reaction and failure-atomic HDF5 lifecycle;
• keep future nonlinear execution, coefficient calibration, drilling output, NR-O03/NR-O04, reference-artifact mutation and Abaqus-equivalence claims outside the plan.

This handoff authorizes planning only. It does not authorize Harness execution, production implementation, reference artifact changes, or completion claims.

Reference Verification Agent

• Compare authoritative FESA HDF5 rows directly with the matching declared Abaqus displacement CSV after exact row-set precheck.
• Let only U1/U2/U3 affect pass/fail; report every UR1/UR2/UR3 warning without changing the verdict.

Physics Evaluation Agent

• After reference verification, independently evaluate force and global moment balance, free residual, reaction sign, displacement direction, symmetry, positive physical energy and consistency of local resultants/stresses.

Coordinator and Release Agents

• Record the Numerical Review gate as passed for planning at HEAD a058ef7.
• Do not infer implementation or release completion. Build/test, reference, physics-sanity and release-readiness gates remain pending.