FESADev/docs/shell-surface-pressure-load/numerical-review.md

Shell Surface Pressure Load Numerical Review

1. Metadata

• feature_id: shell-surface-pressure-load
• source_formulation: docs/shell-surface-pressure-load/formulation.md
• source_requirements: docs/shell-surface-pressure-load/requirements.md
• source_research: docs/shell-surface-pressure-load/research.md
• status: pass-for-io-definition
• owner_agent: numerical-review-agent
• date: 2026-08-30
• reviewed_requirement_range: FESA-REQ-SHELL-PRESSURE-001–FESA-REQ-SHELL-PRESSURE-024
• external_tolerance_authority: docs/TOLERANCE.md

2. Review Verdict

Verdict: pass-for-io-definition.

Blocking formulation defect, research gap, reference-artifact gap 또는 critical user decision은 없다. Pressure virtual work, discrete force, exact current-area derivative, residual/tangent sign과 reference/current geometry 구분은 서로 일관된다. S4/S4R 공통 2x2 rule은 FESA-owned discrete pressure force와 tangent를 정확히 적분하고, 기존 full unsymmetric nonlinear tangent solver와 호환된다. 이 verdict는 Stage 5가 I/O contract를 정의할 수 있다는 뜻이며 구현, reference agreement 또는 release approval이 아니다.

3. Critical Findings

3.1 Confirmed defects

없음.

3.2 Independently confirmed findings

1. Connectivity-positive area vector a=g_xi x g_eta, s=+1 for SPOS, s=-1 for SNEG와 f_i=-lambda*p*s*integral(N_i*a)의 조합은 positive P가 selected normal의 반대 방향으로 작용하게 한다. p 또는 s 하나만 반전하면 force, moment와 nonlinear tangent가 반전하고 둘을 함께 반전하면 원래 부호다.
2. delta W_p=sum(delta u_i dot f_i)는 식 (4)의 surface virtual work와 동일하다. sum N_i=1로 nodal resultant 식 (23)이, x_h=sum N_i*x_i로 current/reference arm을 쓰는 moment 식 (22)가 직접 따른다. 승인된 midsurface/no-offset 범위에서는 pressure가 nodal rotation에 의존하지 않으므로 rotational force entries와 tangent rows/columns가 exact zero인 계약이 맞다.
3. Da[d]=sum_j(-N_j,xi*[g_eta]_x+N_j,eta*[g_xi]_x)d_j의 cross/skew 부호가 맞다. 따라서 external derivative 식 (14)는 df_ext/du이고, r=f_int-f_ext에 대한 pressure residual directional derivative는 -K_ext*d, Newton tangent는 K_int-K_ext다.
4. 식 (14)의 open-facet element operator는 일반적으로 nonsymmetric이다. Silent symmetrization, symmetric-triangle storage 또는 SPD 가정은 금지한다. Existing nonlinear path는 full CSR과 oneMKL PARDISO mtype=11을 사용하므로 이 계약을 수용한다.
5. Bilinear surface mapping에서 a는 xi,eta의 affine vector다. N_i*a와 식 (19)의 scalar component는 각 natural coordinate에서 차수가 최대 2이므로 tensor-product 2x2 Gauss rule은 force와 exact derivative를 적분한다. 이는 Abaqus S4/S4R 내부 integration과의 동등성 주장이 아니다.
6. Reference geometry는 analysis 시작 전, nonlinear current geometry는 매 trial에서 같은 chord/mean/corner/Gauss 규칙으로 검사한다. Connectivity reorder, normal flip, absolute-area repair 또는 reference-normal dot test가 없으므로 proper large rigid rotation을 허용하면서 fold, local inversion, collapse와 nonfinite geometry를 fail closed한다.

3.3 Open questions

Critical open question은 없다. 직접 적용 가능한 public S4/S4R follower-pressure target이 없다는 research.md의 OI-01은 analytical/tangent/internal-equilibrium suite와 exact stored case로 보완되는 disclosed nonblocking limitation이다.

4. Numerical Risk Assessment

risk label	assessment	blocking evidence / disposition
rigid_body_modes	Pressure는 support 또는 artificial stiffness가 아니다. Unconstrained patch의 six rigid modes를 regularization으로 숨기면 안 된다.	supported patch solution과 별도로 unconstrained singular behavior 및 pressure rigid-translation action을 검사한다.
patch_test	Stored case만으로 sign, S4R 또는 exact resultant를 분리할 수 없다.	Flat/distorted S4/S4R의 resultant, moment, side/sign, reaction과 mesh subdivision test가 blocking이다.
symmetry	K_ext는 일반적으로 nonsymmetric이고 global closure 조건이 우연히 symmetry를 만들 수 있다.	독립 식 (14) matrix oracle, nonsymmetric open-patch precondition과 full-matrix assembly가 blocking이다. Symmetrization 금지.
positive_definiteness	K_int-K_ext는 follower loading에서 positive definite일 필요가 없다.	Existing unsymmetric tangent solver를 사용하며 pivot perturbation, SPD fallback 또는 hidden regularization을 허용하지 않는다.
hourglass	Pressure force/tangent는 S4/S4R 모두 2x2라 pressure under-integration mode를 만들지 않는다.	Existing MITC4 drilling/hourglass policy는 변경하지 않는다. Source S4R은 reduced pressure integration을 뜻하지 않는다.
shear_locking	Existing MITC4 internal kinematics의 위험이며 external pressure interpolation의 결함이 아니다.	Existing shell evidence를 보존하고 pressure kernel을 locking correction으로 바꾸지 않는다.
volumetric_locking	Pressure kernel에는 volumetric constitutive mode가 없다.	N/A; material/section scope를 확장하지 않는다.
distortion	Warped/distorted bilinear patch는 pointwise vector area와 consistent tangent가 필요하다.	Mean, four corners와 four Gauss points의 dimensionless validity 및 distorted analytical/FD tests가 blocking이다. 추가 calibrated distortion portfolio는 요구하지 않는다.
singular_jacobian	Collapsed, folded, inverted 또는 nonfinite surface에서는 normal/area와 force가 유효하지 않다.	§6.1의 strict tau_J gate를 force/tangent 평가 전에 적용하고 failed candidate를 commit하지 않는다.
conditioning	Near-collapse와 extreme aspect ratio는 normalization 및 tangent conditioning을 악화한다.	Chord-relative gate와 finite-first checks를 사용한다. Absolute length floor, max(1,...) 또는 clamp는 금지한다.
convergence	Force와 tangent의 common sign error는 FD만 통과할 수 있고, tangent omission은 full Newton 수렴을 악화한다.	Independent sign/resultant oracle, external-force 및 residual FD, accepted free residual과 global balance를 모두 통과해야 한다.

5. Consistency Checks

check	result	independent evidence
units and dimensions	pass	p:[F/L^2], a:[L^2], f:[F], translational K_ext:[F/L]; generalized rotational force의 단위는 [F L]이고 값은 exact zero다.
matrix/vector dimensions	pass	q_e,f_e는 24x1, E_i는 3x24, K_ij는 3x3, K_e는 24x24; four translation blocks와 exact-zero rotational blocks가 일치한다.
signs	pass	SPOS/SNEG, positive/negative p, external force, reaction, r=f_int-f_ext, K=K_int-K_ext가 하나의 sign chain을 이룬다.
DOF ordering and constraints	pass	Global nodal order [UX,UY,UZ,URX,URY,URZ]; full-space stable assembly 후 free/constrained projection; constrained residual은 reaction, free residual은 equilibrium evidence다.
coordinates and transforms	pass	Geometry/force/reaction/balance는 global Cartesian; pressure mapping에는 physical in-plane inverse Jacobian이 필요 없다. 기존 MITC4 local frame, B와 recovery transform은 변경하지 않는다.
kinematic / B operator	pass	New load kinematics는 midsurface positional mapping과 식 (9)–(10)뿐이다. B_MITC4는 internal strain용이며 pressure force/tangent에 혼합되지 않는다.
constitutive contract	pass	Pressure는 D, stress update, thickness와 material state에 독립이다. Existing linear-isotropic MITC4 internal force/tangent/recovery를 변경하지 않는다.
Jacobian / geometry	pass with fixed decision	ell_e, mean area, corner/Gauss signed projection과 finite checks가 충분하며 final tau_J=1.0e-12를 §6.1에서 고정한다.
integration	pass	S4/S4R force와 tangent 모두 fixed-order four-point 2x2, w_q=1; polynomial exactness와 same-rule consistency 확인.
tangent symmetry	pass	General nonsymmetry가 예상되며 full unsymmetric solver path와 일치한다. Symmetric storage 또는 averaging은 계약 위반이다.
output locations	pass	Nodal global displacement/rotation, full residual/reaction, factor-one global balance와 existing MITC4 result locations만 사용한다. Pressure-only recovery row는 없다.
deterministic identity	pass	Stable (load source order, facet order, instance, element label, source type, side, local row[,column]), exact-once local completion, fixed reduction, full-space-before-projection 계약이 완전하다.

6. Verification Readiness

6.1 Geometry validity decision

Final single threshold is

[ \boxed{\tau_J=1.0\times10^{-12}}. ]

Formulation 식 (20)–(21)의 모든 strict inequality에 동일 값을 사용한다. 즉 ||bar(a)||/ell_e^2 > tau_J와 four corners plus four Gauss points의 a(z) dot bar(n)/ell_e^2 > tau_J가 모두 필요하다. 값이 threshold와 같으면 fail이다. 모든 coordinate, chord, tangent, area, normalized mean, integrand, accumulated vector/matrix는 finite여야 하고 ell_e=0은 즉시 fail이다. 이 값은 double roundoff보다 약 4,500배 큰 dimensionless guard이며 별도 aspect-ratio calibration gate를 만들지 않는 최소 fail-closed 선택이다. Stored Abaqus value에 맞춰 조정하지 않았고 absolute length floor도 없다.

6.2 Analytical arithmetic decisions

Configuration c에서 다음 unclamped scales를 쓴다.

[ A_\Sigma=\sum_qw_q|a_c(q)|2,\qquad F\Sigma=|\lambda p|A_\Sigma, ]

[ M_\Sigma=|\lambda p|\sum_qw_q |x_h(q)-O|2|a_c(q)|2,\qquad K\Sigma=F\Sigma/\ell_e. ]

• Natural nodes와 four Gauss points에서 |sum_i N_i-1|<=1e-14 및 |sum_i N_i,xi|, |sum_i N_i,eta|<=1e-14; nodal Kronecker values는 exact 0/1이다.
• 식 (23) resultant error는 ||error||_2/F_Sigma<=1e-12, 식 (22) moment error는 ||error||_2/M_Sigma<=1e-12다.
• Side/sign reversal, double reversal, lambda={0,0.5,1} proportionality, flat analytical force/reaction과 mesh/order invariance도 해당 F_Sigma 또는 M_Sigma로 <=1e-12다.
• Scale이 exact zero이면 expected와 computed vector가 모두 exact zero여야 한다. p=0은 force/tangent exact zero지만 geometry/model validation을 우회하지 않는다.
• Rotation entries와 tangent rotation rows/columns은 exact zero다. Rigid-translation tangent action은 direction의 maximum nodal translation을 ell_e로 scaling한 뒤 ||K_ext*d||_2/F_Sigma<=1e-12; F_Sigma=0이면 exact zero다.

6.3 Centered directional tangent decision

Nonzero deterministic translational direction d에 대해 D=max_i||d_i||_2와 dhat_i=ell_e*d_i/D를 사용한다. Dimensionless perturbation is

[ \boxed{h=1.0\times10^{-5}},\qquad u^\pm=u\pm h,\hat d. ]

Both perturbed surfaces must independently pass §6.1. Define

[ d_{FD}=\frac{f^{ext}(u^+)-f^{ext}(u^-)}{2h},\quad d_A=K^{ext}(u)\hat d,\quad S_D=\max(|d_{FD}|_2,|d_A|_2). ]

If S_D>0, require ||d_FD-d_A||_2/S_D<=1e-8; if S_D=0, both actions must be exact zero. The pressure-residual check independently compares centered D(-f_ext) with -K_ext*dhat by the same rule. A nonzero-pressure non-rigid direction must have S_D>0 so an accidentally zero oracle cannot pass. S4/S4R, SPOS/SNEG와 positive/negative p를 각각 다룬다. p=0, rigid translation and rotation-only directions use their stated zero rules rather than this relative branch. This fixed centered rule reflects the quadratic current-area mapping and is not selected from stored reference results.

6.4 Objectivity and current-area decisions

• Translation test uses finite ||c||_2<=10*ell_e. Force/area invariance and tangent invariance must pass normalized <=1e-12 using F_Sigma, A_Sigma, K_Sigma; zero scales require exact zero.
• Proper rigid rotation uses an independently constructed orthogonal Q, det(Q)=+1, including a 120 deg non-coordinate-axis rotation. Require force covariance, vector-area covariance, scalar area invariance and T K T^T tangent covariance <=1e-12 with the same unclamped scales. The rotated geometry must remain valid; reference-global-normal comparison is forbidden.
• A planar current patch with homogeneous in-plane scale gamma=1.25 must produce area and total pressure resultant magnitude factors gamma^2 within <=1e-12; pressure scalar and source identity remain unchanged.
• The open distorted-patch tangent oracle must use a case whose independent 식 (14) has nonzero antisymmetric part, compare every full-matrix entry by K_Sigma within <=1e-12, and demonstrate that no symmetrization occurred.

6.5 Assembly, reaction, residual and global equilibrium

Blocking assembly tests cover unique facet exact-once application, distinct load-record addition, fixed Gauss/contribution order, serial/parallel stable reduction, shuffled declaration/member order, full-space-before-projection, nonfinite rejection and failed-candidate atomicity.

Accepted factor-one global equilibrium preserves the existing nonlinear contract. With current x=X+u and global origin 0, form applied/reaction nodal force contributions a_F,I, c_F,I and current-arm plus spatial physical moment contributions a_M,I, c_M,I. Pressure has no separate nodal moment. Define

[ e_F=\sum_I(a_{F,I}+c_{F,I}),\quad S_F=\max(\sum_I|a_{F,I}|2,\sum_I|c{F,I}|_2), ]

[ e_M=\sum_I(a_{M,I}+c_{M,I}),\quad S_M=\max(\sum_I|a_{M,I}|2,\sum_I|c{M,I}|_2). ]

Both ||e_F||_2/S_F<=1e-8 and ||e_M||_2/S_M<=1e-8 are blocking. Exact-zero scale requires exact-zero balance; there is no absolute floor, max(1,S) or clamp.

Accepted free translation-force and rotation-moment residual channels remain separate and finite-first. For a nonzero matching external scale, the existing infinity-norm relative gate is <=1e-2. For an exact-zero external scale, the existing typed absolute infinity-norm gate is <=1e-3 force or force-length. Both channels must pass; neither global balance nor local reaction distribution substitutes for this evidence.

6.6 Evidence classes and requirement traceability

evidence class	classification	readiness / requirement trace
input/surface/source identity	blocking-internal	Exact grammar/semantic projection remains Stage 5-owned; stable scope/failure contracts cover 001–004, 013–017.
sign/resultant/moment/reaction patches	blocking-internal	§6.2 and flat/distorted S4/S4R portfolio cover 005–007, 010–012, 020.
current geometry/objectivity/tangent	blocking-internal	§§6.1, 6.3–6.4 cover 008–009, 016, 021.
deterministic assembly/output preservation	blocking-internal	Exact identity/order/atomicity and unchanged existing result meanings cover 011, 013, 017–019.
factor-one free residual/global balance	blocking-internal	§6.5 covers 012, 019, 021.
stored U1/U2/U3	blocking external	Exact required input/CSV, source rows and common tolerance are ready for 022–023; success remains Implementation-owned.
stored UR1/UR2/UR3	warning-only external	Same identity/precheck and common tolerance, deterministic warning disposition for 023.
reaction CSV	review-only	Optional inventory only; internal reaction/equilibrium remains blocking under 012, 021, 024.
stress CSV	excluded	No comparison family; existing stress recovery remains unchanged under 019, 024.

Coverage: normative requirements 24, reviewed requirements 24, numerical/reference readiness gaps 0.

7. Required Revisions

Formulation Agent

없음.

Research Agent

없음. Public follower-pressure benchmark absence remains a disclosed nonblocking gap and does not justify an invented target or extra portfolio.

8. Downstream Handoff

Coordinator를 통해 I/O Definition Agent에 전달한다.

• Preserve logical source facet (instance, element label, source element type, side), distinct load source order and exact-once semantics; final diagnostic representation is I/O-owned.
• Preserve global nodal component order, pressure rotational zero contract, source node identity, r=f_int-f_ext, constrained reaction/free residual distinction and current-arm factor-one balance.
• Preserve the exact external logical families and row contract in reference-model.md; I/O owns the final authoritative dataset projection.
• Do not add a pressure-only result quantity unless Stage 5 finds binding evidence; none was found in this review.
• Internal thresholds in §6 are test gates only. They do not modify docs/TOLERANCE.md and must not be calibrated to Abaqus output.