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Neural topology optimization of ship structures under propulsion machinery vibrations

Shengyu Yan, Muhammad Muztahidul Hakim Zareer, Jasmin Jelovica

arXiv:2609.38089v1cs.CEcs.AIcs.LGcs.NAmath.NAphysics.comp-ph

TL;DR

Ship machinery vibrations motivate designs that reduce energy transmitted through structural paths, while conventional topology optimization can target dynamic objectives. This study applies KATO to forced-vibration design using active input power; the deck layouts reduce AIP versus engineering references, and the frame comparison reports close AIP matching with GCMMA and lower binary compliance at 300 Hz.

  • Problem

    Ship machinery vibrations can contribute to fatigue, discomfort, noise, and underwater radiated noise, while designer-judgment-based stiffener layouts are rarely tuned for vibration transmission at specific excitation frequencies.

    Source: §1 · p. 1

  • Method

    The study applies KATO’s convolutional Kolmogorov–Arnold network generators to minimize active input power in forced-vibration topology optimization of a ship deck and thruster-supporting frame.

    Source: §1 · p. 2, §7 · p. 21

  • Results

    Both deck TO families reduce AIP versus engineering references; KATO matches GCMMA within 0.5 dB for three 18 Hz frame variants and achieves 59.4× lower binary compliance at 300 Hz.

    Source: §7 · p. 21

  • Takeaways & Limitations

    KATO’s evaluated binary layouts remain connected and are compatible with feature-size control and stress regularization.

    Source: §7 · p. 21

    The objective is structural AIP; direct radiated-sound-power optimization requires a coupled structural-acoustic model and is identified as future work.

    Source: §6.4 · p. 21

Abstract

from arXiv · show

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.

1. Introduction

Ship vibration can harm fatigue life and comfort and contribute to underwater noise, while conventional forced-vibration objectives can behave pathologically near resonance. This paper applies KATO neural-reparameterized topology optimization with active input power to ship structures, addressing a gap in engineering-scale AIP-based design.

1. Introduction

  • Stiffened panels and damping treatments depend on designer judgment and are rarely tuned for dynamic transmission at specific excitation frequencies.Topology optimization offers a systematic alternative under specified design domains, boundary conditions, loading, volume, and manufacturing constraints.Source: §1 · p. 1
  • Dynamic compliance can change sign and become singular or non-monotonic near resonance and antiresonance, destabilizing gradient-based forced-vibration optimization.Single-frequency optimization can exploit antiresonance sign flips without genuinely reducing vibration.Source: §1 · p. 2
  • AIP is non-negative for passive structures and avoids dynamic compliance’s sign ambiguity, but its topology-optimization use has focused mainly on academic benchmarks.Its application to large engineering problems such as ship structural components had not been explored.Source: §1 · p. 2
  • Ship-structure topology optimization faces computational costs from fine discretizations, numerous load cases, and coupled multidisciplinary analyses, while systematic surrogate integration remains underexplored.Prior ship applications include tank and bracket redesign, vibration-suppressing reinforcements, and other structural optimizations.Source: §1 · p. 2
  • Neural reparameterization uses an untrained network to parameterize density during optimization, with architecture imposing an implicit spatial prior toward smooth, connected topologies.This differs from trained layout predictors, which depend on datasets and may generalize poorly outside their distributions.Source: §1 · p. 2
  • The paper extends KATO’s convolutional Kolmogorov–Arnold network generators to AIP minimization for an engine-supporting deck and a thruster-supporting frame.Its stated contributions include deck comparisons with size-optimized panels, a KATO-versus-GCMMA frame comparison, and analysis of stress and feature-size control.Source: §1 · pp. 2–3

2. Problem Formulation

The formulation uses density-based material interpolation and frequency-domain response to optimize active input power, with static compliance and stress terms available to regularize structural integrity. Adjoint sensitivities are computed through the frequency-domain solve.

2.1. Density-based topology optimization

  • SIMP assigns each element a density and penalizes stiffness by a power law, while mass density is interpolated linearly to avoid artificially high local frequencies.The stiffness floor is 10^-9E0, and the penalization exponent p>1 is ramped during continuation; linear mass interpolation can instead introduce localized low-frequency modes.Source: §2.1 · p. 3

2.2. Frequency-domain dynamic equation

  • The frequency-domain equation uses a density-dependent dynamic stiffness operator; broadband calculations use stiffness-proportional structural damping with constant loss factor η_s=0.02.This corresponds to ζ≃1% for light damping and avoids the frequency-dependent damping increase produced by a fixed Rayleigh coefficient.Source: §2.2 · p. 3

2.3. Active input power objective

  • Harmonic force and velocity define time-averaged input power; minimizing its magnitude safeguards against roundoff-scale negative values near zero crossings.For the passive-system solutions considered, this safeguard does not alter positive AIP away from numerical zero crossings.Source: §2.3 · p. 4
  • AIP is the non-negative, time-averaged real power injected by harmonic forces into a passive structure, avoiding dynamic compliance’s sign ambiguity.It measures structure-borne power, not radiated acoustic power, and comparisons are meaningful only under the same finite-element model and load case.Source: §2.3 · p. 4

2.4. Mixed objective and stress-aware regularization

  • The mixed objective adds static-compliance regularization to AIP minimization, addressing weakly connected or hinge-like designs that can arise without it.The compliance term is normalized by a reference value during optimization; reported unnormalized compliance values are compared only within a case study.Source: §2.4 · p. 4
  • Stress-aware variants aggregate element-center von Mises stress over solid-like material with a weighted p-norm, using p=8 in optimization and binary post-evaluation.The combined dynamic stress measure conservatively bounds cycle-maximum von Mises stress and is exact when real and imaginary stress tensors are proportional.Source: §2.4 · pp. 4–5

2.5. Adjoint sensitivity analysis

  • Adjoint sensitivities differentiate the objective through the real-block frequency-domain solve, reusing stored LU factors for back-substitution without another factorization.The resulting sensitivities are passed to the selected optimizer.Source: §2.5 · p. 5

3. Methodology

The methodology compares KATO’s neural density parameterization with GCMMA’s direct density updates under shared finite-element responses and stress aggregation. Filtering, projection, physical-unit analysis, and modal reduction support feature control and frequency-sweep evaluation.

3.1. Optimization strategies

  • KATO updates a cKAN generator’s parameters with Adam, while GCMMA updates filtered element densities and enforces volume through a constrained optimization update.Both use the same responses and normalization references; stress-aware KATO adds a weighted stress term, whereas GCMMA imposes a 26 MPa stress constraint.Source: §3.1 · pp. 5–6, Table 1 · p. 6

3.2. KATO generator parameterization

  • The cKAN generator maps a trainable latent code to a 2D logit field, then a constrained sigmoid, filtering, and projection produce the physical density field.Generator weights and latent code are updated by Adam, with sensitivities propagated through the pipeline by automatic differentiation.Source: §3.2 · p. 6

3.3. Density pipeline and volume constraint

  • Helmholtz filtering smooths densities, and Heaviside projection drives them toward binary values; three deck thresholds define nominal, eroded, and dilated fields for feature-size and tolerance assessment.Deck AIP and active stress are evaluated on the intermediate field; frame Helmholtz variants use a single volume-adjusted projection threshold.Source: §3.3 · pp. 6–7

3.4. Physical-unit FEA and differentiable implementation

  • Physical-unit finite elements model 2D designs and extruded 3D structures, while one frequency-domain factorization supplies AIP and dynamic stress responses.The same forward displacement serves both measures; LU factors are reused for adjoint back-substitution, with sparsity cached across iterations.Source: §3.4 · p. 7

3.5. Modal reduction and frequency sweeps

  • For 3D models with approximately 640,000 free degrees of freedom, frequency sweeps use a truncated modal ROM and directly solve an m×m system at each frequency; unified deck sweeps retain 200 modes.Below the first elastic resonance (f1 ≈180–560 Hz), the extruded 3D deck response is stiffness-controlled and unresolved residual flexibility biases active input power low; a static residual-flexibility (mode-acceleration) correction is added.Source: §3.5 · pp. 7–8

4. Case Definitions and Numerical Experiments

The numerical experiments compare topology-optimized deck panels and a thruster foundation frame under harmonic and static loading, then evaluate binary designs. The deck designs are also re-analyzed after finite-depth extrusion.

4.1. Case I: Ship deck panel

  • The 100 Hz deck uses a 2.0 m × 0.4 m domain, 320 × 64 mesh, 30% total solid fraction, and two 15 kN harmonic loads plus a 12 kN static load.The double-skin and single-skin panels have different passive-solid skin arrangements; both are steel.Source: §4.1 · pp. 8–9, §4 · p. 8
  • Four KATO variants are compared with size-optimized X-core double-skin and stiffened single-skin references selected by a volume-screened grid search for lowest AIP.The two reference types are matched to their corresponding topology-optimized panel families.Source: §4.1 · p. 9

4.2. Case II: Thruster foundation frame

  • The thruster-frame case uses an 18 Hz, 128 × 64 model with 30 kN side and top harmonic loads, 12 kN static uplift, and 45% total solid fraction.Passive-solid regions include the left connection column, load-introduction zones, and bottom load rows.Source: §4.2 · pp. 9–10
  • Thruster-frame KATO and GCMMA runs compare unrestricted, manufacturing-aware, stress-aware, and combined variants under common settings, with manufacturing control using a Helmholtz filter.An additional unrestricted optimization targets 300 Hz, near the 306 Hz response peak of the uniform starting field.Source: §4.2 · p. 11

4.3. Binary post-evaluation protocol

  • Binary re-analysis thresholds each converged density field to match volume, then recomputes AIP, compliance, stress, and connectivity; one connected solid component denotes a single topology.The 2D analysis assigns void elements an ersatz density of 10^-3.Source: §4.3 · pp. 11–12

5. Results

Across the deck and thruster-frame tests, KATO designs reduce forced-vibration AIP while retaining favorable static stiffness after post-processing. Near resonance, KATO also yields a connected binary design with lower worst-case AIP over the frequency sweep.

5.1. Ship deck panel

  • At 100 Hz, double-skin KATO panels reach 130.02–131.22 dB AIP versus 136.30 dB for the X-core reference, while single-skin panels reach 135.03–135.99 dB versus 147.99 dB for the stiffened reference.All eight topology-optimized layouts reduce AIP relative to their corresponding engineering references.Source: §5.1 · p. 12, Table 3 · p. 13
  • Stress-aware KATO reduces the dynamic stress p-norm from 41.8 to 29.2 MPa for double-skin panels and from 54.7 to 49.1 MPa for single-skin panels, with higher static compliance.AIP changes by less than 1 dB in both families; Manufacturing-aware control also reduces double-skin stress from 41.8 to 35.4 MPa.Source: §5.1 · pp. 12–13
  • After 3D extrusion, double-skin KATO designs remain 4.6–5.7 dB below the X-core reference and single-skin designs remain 9.0–9.5 dB below the stiffened reference at 100 Hz.The extruded topology-optimized layouts also remain below both engineering baselines in AIP and static compliance; sweep responses vary by layout.Source: §5.1 · pp. 15–16, Table 4 · p. 16

5.2. Thruster foundation frame: KATO versus GCMMA

  • At 18 Hz, KATO reaches 126.2–126.9 dB AIP and 3.6–3.8 N m binary static compliance, while comparable GCMMA variants reach 125.8–126.7 dB and 80–135 N m, respectively.The GCMMA compliance is 22–36× larger for the Unrestricted, Manufacturing-aware, and Stress-aware variants.Source: §5.2 · p. 16, Table 5 · p. 17
  • For stress-constrained frame variants, KATO completes in 49–57 s versus 366–581 s for GCMMA, using 160 effective evaluations versus 1112–1227.KATO’s peak memory is about 1.65–1.74 GB, compared with 0.60–0.61 GB for GCMMA.Source: §5.2 · p. 16, Table 5 · p. 17
  • At 18 Hz, KATO fields have 5.5–6.4% non-discreteness versus 19.3–60.4% for GCMMA, and all four KATO variants retain one connected component after thresholding.Stress-aware KATO reduces the stress p-norm from 38.2 to 27.5 MPa; thresholding GCMMA gray fields is associated with degraded binary stiffness.Source: §5.2 · pp. 16–17

5.3. Near-resonant frame test at 300 Hz

  • At 300 Hz, KATO and GCMMA reduce binary AIP from 172.02 dB to 139.76 and 139.30 dB, respectively; KATO has 59.4× lower binary static compliance and one connected component versus GCMMA’s four.Across 1–500 Hz, KATO’s peak AIP is 2.73 dB below GCMMA’s, although KATO has slightly higher AIP at the selected frequency.Source: §5.3 · pp. 17–18, Table 6 · p. 19, Figure 12 · p. 19

6. Discussion

The discussion connects KATO’s connected binary load paths and feature-size controls to its stress and AIP trade-offs, then reviews computational performance and limits on the study’s scope.

6.1. Topology regularization, stress, and binary performance

  • Thresholded KATO fields retain coherent load paths where thresholding removes gray bridges from several GCMMA fields.The lower non-discreteness and connected KATO binaries support the generator’s role as an architectural prior and implicit regularizer.Source: §6.1 · p. 20
  • Helmholtz filtering and three-field projection control solid and void feature sizes while preserving principal load paths under erosion and dilation.The resulting deck layouts contain fewer, thicker internal members.Source: §6.1 · p. 20
  • Helmholtz filtering controls feature size, while stress regularization produces the main reduction in the frame’s stress p-norm.Across the deck families and frame, the aggregated stress treatment lowers dynamic stress p-norm with small AIP changes.Source: §6.1 · p. 20

6.2. Relation to prior AIP topology optimization

  • Compared with the earlier 2D cantilever demonstration, this study applies AIP-based optimization to ship decks and foundations with physical units, feature-size control, stress regularization, and binary re-analysis.The KATO generator adds an architectural prior and preserves connected load paths in the reported designs.Source: §6.2 · p. 20

6.3. Computational performance

  • 6.4–10.4× faster stress-aware KATO runs, while cached PARDISO reaches 26.8× over SuperLU and corrected ROM spot checks are 990–2125× faster.The 200-mode basis takes approximately 400–2900 s per deck structure, with its cost amortized over the frequency sweep.Source: §6.3 · p. 20
  • Below 0.02% corrected active-power error across five 50–400 Hz full-order spot checks, versus up to 11.8% uncorrected error.The correction accounts for a 1.7–16.7% residual-flexibility fraction across the ten layouts.Source: §6.3 · p. 20, Table 8 · p. 21

6.4. Limitations

  • The study is limited to isotropic steel and structural AIP; fabrication-specific constraints and full 3D deck optimization remain extensions, while X-core can compete at selected higher-frequency peaks.Deck topologies are optimized in 2D and evaluated in finite-depth 3D models; direct radiated-sound-power optimization requires coupled structural-acoustic modeling.Source: §6.4 · pp. 20–21

7. Conclusions

The paper applies KATO-based AIP minimization to ship decks and a foundation frame, with connected binary layouts and feature-size control. The reported comparisons show reduced AIP against deck references and competitive frame AIP with lower binary compliance; future work includes direct acoustic and broadband-random-excitation optimization.

Main findings

  • KATO’s binary layouts remain structurally connected in the final evaluations and are compatible with feature-size control and stress regularization.Stress regularization lowers the dynamic stress p-norm with small AIP changes.Source: §7 · p. 21
  • Both deck topology-optimization families reduce AIP relative to stiffened-panel and X-core references at 100 Hz, with the advantage retained in extruded 3D models.Helmholtz filtering and three-field projection control solid and void feature sizes and preserve main load paths after binarization.Source: §7 · p. 21
  • At 18 Hz, KATO matches GCMMA in AIP within 0.5 dB for three frame variants while achieving 22–36× lower binary compliance.At 300 Hz, KATO achieves 59.4× lower binary compliance and 2.73 dB lower maximum AIP over 1–500 Hz; its binary layouts remain connected in both comparisons.Source: §7 · pp. 21–22

Future work

  • Future work could integrate structural-acoustic coupling for direct underwater-radiated-noise optimization and extend the framework to broadband random excitation.Source: §7 · p. 22

Statements and Declarations

The declarations acknowledge research funding, describe contributions, and state how to request or access supporting data and code.

Funding

  • The research received NSERC Discovery and Alliance Advantage grants, with additional support from Seaspan Shipyards.Source: Funding · p. 22
  • The listed contributions include formal analysis, investigation, original-draft writing, and review and editing.Source: Funding · p. 22

Data availability

  • Study data and code are available from the corresponding author on reasonable request and are slated for release after journal acceptance.Source: Data availability · p. 22

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