Neural topology optimization of ship structures under propulsion machinery vibrations

Shengyu Yan; Muhammad Muztahidul Hakim Zareer; Jasmin Jelovica

cs.CESep 29, 2026

Neural topology optimization of ship structures under propulsion machinery vibrations

Authors: Shengyu Yan, Muhammad Muztahidul Hakim Zareer, Jasmin Jelovica

Organizations: Department of Mechanical Engineering, The University of British Columbia, 6250 Applied Science Ln, Vancouver, V6T 1Z4, Canada · Department of Civil Engineering, The University of British Columbia, 6250 Applied Science Ln, Vancouver, V6T 1Z4, Canada

Abstract

Ship structural vibrations contribute to noise, fatigue, and equipment damage, while dynamic-compliance topology optimization can produce pathological designs near resonance. This study extends neural-reparameterized topology optimization using a convolutional Kolmogorov-Arnold network (KATO) to forced-vibration design with active input power (AIP) as the objective. Applications include a 100 Hz engine-supporting deck panel and an 18 Hz thruster foundation frame. Helmholtz PDE filtering and Heaviside projection control feature sizes and manufacturing tolerance. Across both deck families, all eight optimized layouts reduce AIP relative to size-optimized references and, after finite-depth extrusion, also achieve lower static compliance. For unrestricted, manufacturing-aware, and stress-aware frame variants, KATO matches GCMMA in AIP within 0.5 dB while yielding 22-36x lower static compliance after matched-volume binary re-analysis. In a near-resonant 300 Hz case, both methods reduce initial AIP by more than 32 dB; KATO maintains a connected design, achieves 59x lower binary static compliance, and reduces maximum AIP over 1-500 Hz by 2.7 dB. KATO runs 6.4-10.4x faster than GCMMA for the implemented stress-aware formulations. The results demonstrate neural AIP-driven topology optimization as an efficient approach for designing connected, feature-size-controlled ship structures with improved forced-vibration performance.

Figures & tables

Benchmark	Variant	$w_{\mathrm{AIP}}$	$w_{\sigma}$	$\sigma_{\mathrm{ref}}$ (MPa)
Deck (double-skin)	Unrestricted / Mfg.-aware	0.9	0	–
	Stress-aware	0.9	0.05	35
	Stress + Mfg.	0.9	0.08	35
Deck (single-skin)	Unrestricted / Mfg.-aware	0.9	0	–
	Stress-aware	0.9	0.20	25
	Stress + Mfg.	0.9	0.25	25

Table 1 : Objective weights and stress treatment used by KATO and GCMMA in the deck-panel and frame cases. The stress-aware KATO variants use the weighted stress term in Eq. ( 12 ), whereas the stress-aware GCMMA variants impose an explicit 26 MPa stress constraint.

Figure 1 : Two-dimensional deck-panel design domains and loading: (a) double-skin and (b) single-skin configurations. The gray interiors are designable, the black edge strips are passive skins, and the hatched end symbols denote clamped supports. The orange arrows indicate the two 15 kN harmonic forces, and the blue arrows indicate the uniformly distributed 12 kN static load.

Figure 2 : Three-dimensional re-analysis model formed by extruding the optimized deck cross sections by 2.0 m. The two 15 kN harmonic loads

F_{1}

and

F_{2}

are distributed along the extrusion direction at

x=0.25

m and

x=1.75

m, respectively. The 12 kN static load

q

is distributed over the full top surface, and the hatched lateral faces denote clamped boundaries.

Figure 3 : Cross-sections of the size-optimized engineering references: (a) X-core double-skin and (b) stiffened single-skin panels.

	Parameter	Range (cells)	Range (mm)	Selected
X-core	Top-face thickness $t_{t}$	2–6	12.5–37.5	$37.5$ mm
	Bottom-face thickness $t_{b}$	2–6	12.5–37.5	$18.75$ mm
	Diagonal-web thickness $t_{d}$	2–6	12.5–37.5	$18.75$ mm
	Core height $h_{c}$	16–60	100–375	$343.75$ mm
	Number of X-units $n_{u}$	3–16	–	$8$
Stiffened	Top-plate thickness $t_{p}$	2–8	12.5–50.0	$50.0$ mm

Table 2 : Parameter ranges for the size-optimized cross-sections of the X-core double-skin and stiffened single-skin panels.

Figure 4 : Design domain, boundary conditions, passive-solid regions, and loading for the thruster foundation frame. The gray regions denote passive-solid connection and load-introduction zones. The orange arrows mark the 30 kN side and top load components, applied over

y=0.08

–

0.52

m and

x=1.00

–

1.25

m, respectively. The blue arrows indicate the uniformly distributed 12 kN static uplift

q_{\mathrm{b}}

over the full bottom edge.

Structure	AIP@100 (dB)	Static comp. (N m)	Dyn. $p$ -norm (MPa)	Dyn. max (MPa)	Solid frac.
Double-skin designs:
Unrestricted	131.14	6.12	41.8	97.6	0.300
Manufacturing-aware	131.22	6.65	35.4	89.1	0.300
Stress-aware	130.02	46.5	29.2	67.8	0.300
Stress + Manufacturing	130.09	22.8	29.7	73.0	0.300
X-core double-skin (SO)	136.30	11.3	69.0	183.2	0.291

Table 3 : Performance at 100 Hz for the eight KATO panel designs and two size-optimized engineering references. Stress values use the combined dynamic von Mises measures, and the solid fraction includes passive material. Bold values identify the best KATO result within each structural family.

Figure 5 : KATO-optimized cross-sections and corresponding 100 Hz dynamic von Mises stress fields for the double-skin and single-skin designs. Each family is grouped with its corresponding size-optimized reference. The common stress scale is capped at the 98th percentile of the eight KATO layouts; values above this limit are saturated.

Figure 6 : Extruded three-dimensional models of the eight KATO designs and two size-optimized references. Columns show the optimized structure, element-wise static-compliance contribution, and 100 Hz combined dynamic von Mises stress field. The compliance and stress scales are capped at the 98th percentile of the eight KATO layouts; values above these limits are saturated.

Figure 7 : AIP frequency responses for extruded models over 5–500 Hz for the four double-skin KATO designs and the size-optimized X-core double-skin reference. The vertical dashed line marks the 100 Hz design frequency.

Figure 8 : AIP frequency responses for extruded models over 5–500 Hz for the four single-skin KATO designs and the size-optimized stiffened single-skin reference. The vertical dashed line marks the 100 Hz design frequency.

Structure	AIP (dB)	$C_{s}$ (N m)	$\sigma_{p}$ (MPa)	$f_{1}$ (Hz)	$f_{\max}$ (Hz)
Double-skin designs:
Unrestricted	105.38	$5.43\times 10^{-3}$	0.148	233	404
Manufacturing-aware	105.49	$5.58\times 10^{-3}$	0.091	242	428
Stress-aware	104.36	$4.32\times 10^{-3}$	0.069	183	341
Stress + Manufacturing	104.61	$4.58\times 10^{-3}$	0.070	226	405
X-core double-skin (SO)	110.06	$1.59\times 10^{-2}$	0.197	563	500 †

Table 4 : Performance of the eight extruded KATO deck models and two size-optimized references. AIP and the dynamic stress

p

-norm are evaluated at 100 Hz;

C_{s}

is static compliance,

f_{1}

is the first elastic modal frequency, and

f_{\max}

is the frequency of maximum AIP within the 5–500 Hz sweep.

Method	Variant	AIP (dB)	Static comp. (N m)	Dyn. $p$ -norm (MPa)	Max (MPa)	$M_{nd}$ (%)	$N_{c}$	Run (s)	Eff. evals	Peak RSS (MB)
KATO	Unrestricted	126.19	3.60	38.22	105.0	6.2	1	49.2	160	1651
	Manufacturing-aware	126.24	3.62	36.05	99.3	6.4	1	50.7	160	1660
	Stress-aware	126.93	3.78	29.49	78.0	5.5	1	56.0	160	1739
	Stress + Manufacturing	126.50	3.72	27.53	74.9	6.4	1	57.2	160	1730
GCMMA	Unrestricted	125.84	95.62	36.45	100.4	19.3	3	52.6	382	598
	Manufacturing-aware	125.83	80.38	36.32	100.1	25.9	2	51.6	362	612

Table 5 : Performance and optimization cost for the four KATO and four GCMMA frame variants at 18 Hz. The non-discreteness measure

M_{nd}

is evaluated before thresholding; AIP, compliance, stress, and the connected-component count

N_{c}

use the post-processed designs. Timing and peak resident set size use six-thread cached PARDISO.

Figure 9 : Optimized designs and 18 Hz combined dynamic von Mises stress fields for the four KATO variants and the corresponding GCMMA variants. The upper and lower pairs of rows show KATO and GCMMA, respectively; the rightmost column shows the uniform baseline with a top flange. The stress scale is clipped at the 98th percentile of the optimized cases.

Figure 10 : Continuous final density fields before volume-preserving thresholding for the four KATO variants (top) and four GCMMA variants (bottom). The reported $M_{nd}$ values quantify pre-threshold non-discreteness.

Figure 11 : AIP frequency responses over 1–500 Hz for the optimized 18 Hz frame designs obtained with the four KATO and four GCMMA variants. The vertical dashed line marks the design frequency.

Method	AIP at 300 Hz (dB)	Compliance (N m)	Stress $p$ -norm (MPa)	$M_{nd}$ (%)	$N_{c}$	FRF peak (Hz)	Peak AIP (dB)
KATO	139.76	3.74	40.95	8.0	1	425	146.67
GCMMA	139.30	222.12	43.67	32.9	4	382	149.40

Table 6 : Performance of the Unrestricted KATO and GCMMA frame designs optimized at 300 Hz. The non-discreteness measure

M_{nd}

uses the continuous final fields; AIP, compliance, stress, and

N_{c}

use the post-processed designs, and the FRF peaks are obtained from the 1–500 Hz sweeps.

Figure 12 : Optimized designs obtained from the Unrestricted KATO and GCMMA runs at 300 Hz. The labels report the connected-component count $N_{c}$ .

Figure 13 : AIP frequency responses over 1–500 Hz for the uniform initial field and the optimized KATO and GCMMA designs at 300 Hz. The vertical dashed line marks the design frequency.

Case	Backend	$t_{\mathrm{step}}$ (ms)	Speedup vs. SuperLU	Gradient rel. err.
Frame 18 Hz	SuperLU	919	$1.0\times$	0
	PARDISO stock	347	$2.7\times$	$8.5\times 10^{-13}$
	PARDISO cached	84	$11.0\times$	$7.7\times 10^{-13}$
Deck 100 Hz	SuperLU	$4\,902$	$1.0\times$	0
	PARDISO stock	929	$5.3\times$	$2.8\times 10^{-11}$
	PARDISO cached	183	$26.8\times$	$2.9\times 10^{-11}$

Table 7 : Sparse-solver timing for one forward-and-adjoint evaluation of the 18 Hz frame and 100 Hz deck cases. All backends use the same objective and adjoint path; speedups and gradient errors are measured relative to SuperLU.

$f$ (Hz)	AIP FOM (dB)	$\epsilon_{\Pi}$ uncorr. (%)	$\epsilon_{\Pi}$ corr. (%)	$t_{\mathrm{FOM}}$ (s)	$t_{\mathrm{ROM}}$ (s)	Speedup
50	$102.33$	11.75	0.004	742	0.65	$1140\times$
100	$105.38$	11.65	0.001	661	0.67	$990\times$
150	$107.21$	11.47	0.003	662	0.63	$1050\times$
250	$109.85$	10.43	0.015	1011	0.69	$1470\times$
400	$117.07$	3.17	0.014	1316	0.62	$2125\times$

Table 8 : Accuracy and per-frequency cost of the uncorrected and residual-flexibility-corrected 200-mode ROM for the double-skin Unrestricted design at five full-order spot-check frequencies. The AIP error

\epsilon_{\Pi}

follows Eq. ( 24 ); FOM and ROM times are from the dedicated PARDISO benchmark.

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Sep 23, 2026, cs.CEJ/K move · Enter open · S save

KATOsuper: Surrogate-accelerated neural topology optimization with sensitivity-consistent Fourier neural operators

Shengyu Yan, Jasmin Jelovica

Department of Mechanical Engineering, The University of British Columbia, 6250 Applied Science Ln, Vancouver, V6T 1Z4, Canada · Department of Civil Engineering, The University of British Columbia, 6250 Applied Science Ln, Vancouver, V6T 1Z4, Canada

Topology Optimization Fourier Neural Operators Neural Network Optimization Shape Optimization Finite Element Method 3D Generation

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Topology optimization (TO) remains computationally intensive due to repeated finite element analysis (FEA) evaluations required at each iteration. While neural network-based surrogates offer potential acceleration, existing approaches often suffer from gradient inconsistency between predicted objectives and sensitivities, leading to optimization instability. This work presents KATOsuper, an objective-agnostic framework that couples neural-reparameterized topology optimization with a Sensitivity-Consistent Fourier Neural Operator (SC-FNO). The framework employs the forward_split architecture, which derives deployed sensitivities via automatic differentiation through the predicted objective field and thereby preserves consistency between the predicted objective and the gradient used for optimization. The case studies include three 2D benchmark problems and three 3D structures considering compliance or stress minimization. A physics-informed multi-channel input encoding with Fourier position embedding enables resolution-invariant learning, supporting zero-shot extrapolation beyond the training resolution, with useful performance at moderate scaling factors and topology-preserving exploration at up to 64x without retraining. The framework extends to 3D through KATO3D, featuring novel KANConv3D blocks with learnable B-spline activations. KATOsuper demonstrates 15--110x deployment-time speedup over MATLAB baselines while maintaining competitive optimality, with the clearest gains observed in complex 3D and stress-optimization cases. The insight that sensitivity direction matters more than magnitude enables robust optimization even with approximate physics evaluation, extensible to other differentiable physics-driven design objectives.

Abstract

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