Abaqus validation report for representative Pareto-optimal designs
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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.

Representative Pareto-optimal cases
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The three Table 11 designs were analysed with the exact geometry convention used by the final notebook. All three are FG-X and use 11 through-thickness layers.

Case | Design type | a/b | a/h | h | d/b | alpha(deg) | Vg0 | FGM | Layers | Table11 area | Notebook area | Difference(%) | Jmin | Vertices
-----|-------------|-----|-----|---|-----|------------|-----|-----|--------|--------------|---------------|---------------|------|---------
BEST_OMEGA1 | Best omega_1 | 0.9 | 150.005 | 6.666463e-03 | 0.610551 | 72.5081 | 0.049992 | FG-X | 11 | 0.765785 | 0.765785 | 6.19714e-05 | 0.42046 | (0 0); (1 0); (1 1.11111); (0.203905 0.647021)
KNEE_POINT | Knee point | 0.37418 | 150.002 | 6.666567e-03 | 0.719005 | 82 | 0.05 | FG-X | 11 | 1.93033 | 1.93033 | 2.23075e-05 | 1.18814 | (0 0); (1 0); (1 2.67251); (0.267428 1.90285)
BEST_AREA | Best Area | 0.333333 | 150.944 | 6.624972e-03 | 1 | 82 | 0.043769 | FG-X | 11 | 2.35912 | 2.35912 | 7.976e-05 | 1.71825 | (0 0); (1 0); (1 3); (0.41752 2.97081)

Abaqus frequencies and optional ROM comparison
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Case | Mode | Abaqus omega(rad/s) | ROM omega(rad/s) | ROM-Abaqus error(%)
-----|------|---------------------|------------------|--------------------
BEST_OMEGA1 | 1 | 525.20518 |  | 
BEST_OMEGA1 | 2 | 1164.5256 |  | 
BEST_OMEGA1 | 3 | 1410.9521 |  | 
BEST_OMEGA1 | 4 | 1955.5158 |  | 
BEST_OMEGA1 | 5 | 2483.5547 |  | 
BEST_OMEGA1 | 6 | 2671.1077 |  | 
BEST_OMEGA1 | 7 | 2984.1989 |  | 
BEST_OMEGA1 | 8 | 3645.0015 |  | 
BEST_OMEGA1 | 9 | 4094.0607 |  | 
KNEE_POINT | 1 | 297.75387 |  | 
KNEE_POINT | 2 | 432.34598 |  | 
KNEE_POINT | 3 | 629.57517 |  | 
KNEE_POINT | 4 | 895.41674 |  | 
KNEE_POINT | 5 | 1009.7707 |  | 
KNEE_POINT | 6 | 1209.6388 |  | 
KNEE_POINT | 7 | 1247.0866 |  | 
KNEE_POINT | 8 | 1472.653 |  | 
KNEE_POINT | 9 | 1594.9238 |  | 
BEST_AREA | 1 | 275.86325 |  | 
BEST_AREA | 2 | 376.05492 |  | 
BEST_AREA | 3 | 486.00438 |  | 
BEST_AREA | 4 | 625.29004 |  | 
BEST_AREA | 5 | 804.87604 |  | 
BEST_AREA | 6 | 940.96983 |  | 
BEST_AREA | 7 | 1028.5574 |  | 
BEST_AREA | 8 | 1152.5247 |  | 
BEST_AREA | 9 | 1297.6034 |  | 

Surrogate versus Abaqus validation summary (Mode 1)
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Case | Design type | Surrogate omega_1(rad/s) | Abaqus omega_1(rad/s) | Surrogate-Abaqus error(%)
-----|-------------|--------------------------|-----------------------|--------------------------
BEST_OMEGA1 | Best omega_1 | 520.75141 | 525.20518 | -0.848
KNEE_POINT | Knee point | 299.82841 | 297.75387 | 0.6967
BEST_AREA | Best Area | 263.04346 | 275.86325 | -4.647

Important consistency note
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This script uses the final notebook geometry exactly: b=1/(a/b), h=1/(a/h), d=(d/b)b, and vertices [(0,0), (1,0), (1,b), (d cos(alpha), d sin(alpha))]. The reported Table 11 area is retained only for comparison. If the notebook-calculated area differs from the Table 11 area, resolve that inconsistency before using the Abaqus results as validation of the published Pareto solution.
