Abaqus lower/upper-bound validation report for ROM comparison
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Abaqus finite-element model
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Element formulation: S4R reduced-integration shell elements.
Mesh: structured/mapped quadrilateral mesh with 40 elements seeded on each boundary edge.
Modal extraction: Lanczos eigensolver; first 9 natural frequencies extracted.
FGM representation: per-case through-thickness composite shell layers; UD and FG-X use 11 layers, FG-O uses 21 layers; Simpson integration; 3 integration points per ply.
Material model: layer-wise rule-of-mixtures E and density; constant Poisson ratio nu=0.340.
Boundary conditions: U3=0 on all edges, U1=0 on bottom/top edges, U2=0 on left/right edges, V0 pinned in U1/U2/U3, and V1 constrained in U2. Rotations are left free.
Interpretation: Abaqus provides an independent shell-reference benchmark for the Kirchhoff-based ROM.

Design-space boundary cases
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Only two marginal designs were analyzed, and each was repeated for UD, FG-O, and FG-X.

Case | a/b | a/h | h | d/b | alpha(deg) | Vg0 | FGM | Layers | Area | Jmin | Vertices
-----|-----|-----|---|-----|------------|-----|-----|--------|------|------|---------
LOWER_BOUND_UD | 0.333333 | 150 | 2.222222e-03 | 0.6 | 72 | 0.01 | UD | 11 | 0.266667 | 0.2 | (0 0); (1 0); (0.708307 0.333333); (0.108307 0.333333)
LOWER_BOUND_FG_O | 0.333333 | 150 | 2.222222e-03 | 0.6 | 72 | 0.01 | FG-O | 21 | 0.266667 | 0.2 | (0 0); (1 0); (0.708307 0.333333); (0.108307 0.333333)
LOWER_BOUND_FG_X | 0.333333 | 150 | 2.222222e-03 | 0.6 | 72 | 0.01 | FG-X | 11 | 0.266667 | 0.2 | (0 0); (1 0); (0.708307 0.333333); (0.108307 0.333333)
UPPER_BOUND_UD | 0.9 | 300 | 3.000000e-03 | 1 | 82 | 0.05 | UD | 11 | 0.9 | 0.9 | (0 0); (1 0); (1.12649 0.9); (0.126487 0.9)
UPPER_BOUND_FG_O | 0.9 | 300 | 3.000000e-03 | 1 | 82 | 0.05 | FG-O | 21 | 0.9 | 0.9 | (0 0); (1 0); (1.12649 0.9); (0.126487 0.9)
UPPER_BOUND_FG_X | 0.9 | 300 | 3.000000e-03 | 1 | 82 | 0.05 | FG-X | 11 | 0.9 | 0.9 | (0 0); (1 0); (1.12649 0.9); (0.126487 0.9)

Abaqus frequencies and optional ROM comparison
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Case | Mode | Abaqus omega(rad/s) | ROM omega(rad/s) | ROM-Abaqus error(%)
-----|------|---------------------|------------------|--------------------
LOWER_BOUND_UD | 1 | 261.9083 |  | 
LOWER_BOUND_UD | 2 | 388.06209 |  | 
LOWER_BOUND_UD | 3 | 590.84561 |  | 
LOWER_BOUND_UD | 4 | 848.85833 |  | 
LOWER_BOUND_UD | 5 | 925.38753 |  | 
LOWER_BOUND_UD | 6 | 1073.1681 |  | 
LOWER_BOUND_UD | 7 | 1139.5185 |  | 
LOWER_BOUND_UD | 8 | 1347.6804 |  | 
LOWER_BOUND_UD | 9 | 1466.307 |  | 
LOWER_BOUND_FG_O | 1 | 177.21724 |  | 
LOWER_BOUND_FG_O | 2 | 262.61201 |  | 
LOWER_BOUND_FG_O | 3 | 399.80536 |  | 
LOWER_BOUND_FG_O | 4 | 574.34597 |  | 
LOWER_BOUND_FG_O | 5 | 626.088 |  | 
LOWER_BOUND_FG_O | 6 | 726.21056 |  | 
LOWER_BOUND_FG_O | 7 | 771.0725 |  | 
LOWER_BOUND_FG_O | 8 | 912.00435 |  | 
LOWER_BOUND_FG_O | 9 | 992.24062 |  | 
LOWER_BOUND_FG_X | 1 | 327.5236 |  | 
LOWER_BOUND_FG_X | 2 | 485.2504 |  | 
LOWER_BOUND_FG_X | 3 | 738.83976 |  | 
LOWER_BOUND_FG_X | 4 | 1061.4813 |  | 
LOWER_BOUND_FG_X | 5 | 1157.2371 |  | 
LOWER_BOUND_FG_X | 6 | 1341.9627 |  | 
LOWER_BOUND_FG_X | 7 | 1424.9636 |  | 
LOWER_BOUND_FG_X | 8 | 1685.2131 |  | 
LOWER_BOUND_FG_X | 9 | 1833.5591 |  | 
UPPER_BOUND_UD | 1 | 143.55822 |  | 
UPPER_BOUND_UD | 2 | 334.29059 |  | 
UPPER_BOUND_UD | 3 | 384.71315 |  | 
UPPER_BOUND_UD | 4 | 564.33685 |  | 
UPPER_BOUND_UD | 5 | 672.92915 |  | 
UPPER_BOUND_UD | 6 | 780.81144 |  | 
UPPER_BOUND_UD | 7 | 872.04329 |  | 
UPPER_BOUND_UD | 8 | 987.33974 |  | 
UPPER_BOUND_UD | 9 | 1136.4397 |  | 
UPPER_BOUND_FG_O | 1 | 75.794064 |  | 
UPPER_BOUND_FG_O | 2 | 176.48211 |  | 
UPPER_BOUND_FG_O | 3 | 203.10397 |  | 
UPPER_BOUND_FG_O | 4 | 297.95493 |  | 
UPPER_BOUND_FG_O | 5 | 355.2513 |  | 
UPPER_BOUND_FG_O | 6 | 412.21466 |  | 
UPPER_BOUND_FG_O | 7 | 460.43182 |  | 
UPPER_BOUND_FG_O | 8 | 521.3096 |  | 
UPPER_BOUND_FG_O | 9 | 599.96252 |  | 
UPPER_BOUND_FG_X | 1 | 190.1229 |  | 
UPPER_BOUND_FG_X | 2 | 442.73837 |  | 
UPPER_BOUND_FG_X | 3 | 509.52235 |  | 
UPPER_BOUND_FG_X | 4 | 747.38489 |  | 
UPPER_BOUND_FG_X | 5 | 891.207 |  | 
UPPER_BOUND_FG_X | 6 | 1034.1495 |  | 
UPPER_BOUND_FG_X | 7 | 1154.9123 |  | 
UPPER_BOUND_FG_X | 8 | 1307.5937 |  | 
UPPER_BOUND_FG_X | 9 | 1505.0742 |  | 

Suggested SI wording
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Additional Abaqus checks were performed at the lower and upper margins of the final notebook design space. The lower-bound case used a/b=1/3, a/h=150, d/b=0.60, alpha=72 deg, and Vg0=0.01. The upper-bound case used a/b=0.90, a/h=300, d/b=1.0, alpha=82 deg, and Vg0=0.05. Each case was analyzed for UD, FG-O, and FG-X distributions using the same S4R shell model, mapped mesh, layered FGM representation, and boundary-condition treatment as the manuscript Abaqus validation. Based on the through-thickness layer convergence check, UD and FG-X were retained with 11 layers while FG-O was evaluated with 21 layers. The resulting frequencies were compared with the ROM predictions to assess whether the ROM remains suitable at the final sampled design-space margins. If the current final notebook CONFIG uses a/b upper = 1.00, replace only the reported upper-bound a/b value accordingly.
