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Field-level prediction of mid-plane stress tensor fields in concrete target penetration: a cross-velocity graph neural operator surrogate

Wenpu Du, Peng Zhou, Yunlong Xia, Sinuo Xin, Congcong Zhang, Boyang Zhang, Yi Zhang, Wenzheng Xu

arXiv:2609.10032v1cs.LG

TL;DR

The paper addresses the lack of a framework linking mesoscale concrete heterogeneity to full-field stress-tensor prediction. It generates a 400-case aggregate-resolved dataset and trains a field-level graph neural operator, while verifying terminal states and benchmarking by configuration similarity. The study reports bounded stress-field rollout, rapid inference, and explicit scope limits from terminal-state, numerical, and validation constraints.

  • Problem

    Existing approaches do not jointly provide full-scale, cross-velocity, per-seed coverage with complete mid-plane six-component stress tensors while retaining mesoscale heterogeneity.

  • Method

    The paper combines 400 aggregate-resolved LS-DYNA cases with a field-level graph neural operator and case-by-case terminal-state verification.

  • Results

    The surrogate rolls out from frame 11 to frame 39 in about 144 ms, reducing computational time by about 3.6 × 10^3 to 4.3 × 10^3 relative to single-core LS-DYNA.

  • Takeaways & Limitations

    Reliable penetration observables are rigid-body motion and field-level stress evolution, while the database provides a reproducible resource for simulation-trained decision support within the studied parameter space.

  • Takeaways & Limitations

    Validation is bounded by configuration-similarity benchmarking, a 1 ms terminal-state window, and reported surrogate amplitude collapse despite bounded rollout.

Abstract

from arXiv · show

Although the impact resistance of concrete has been studied extensively, a framework linking mesoscale heterogeneity to full-field stress-tensor prediction has been lacking. Data were generated with a full-scale aggregate-resolved LS-DYNA model (projectile diameter 45 mm, mass 2.13 kg, target diameter 500 mm x thickness 200 mm, mesh 10 mm), verified against published penetration experiments (Frew 2006, Hanchak 1992, Forrestal 1996) by configuration similarity. The dataset contains six-component stress-tensor fields on the X-Z mid-plane for 400 cases (4 impact velocities x 100 aggregate seeds). Three contributions are reported. First, case-by-case verification of the terminal penetration state delimited the rest-state validity of penetration depth and anchored reliable observables to rigid-body motion and field-level stress evolution. Second, a field-level graph neural operator surrogate learned the time-varying stress-field evolution and evaluated cross-velocity leave-one-out extrapolation. Third, the full-scale, aggregate-resolved, cross-velocity, per-seed database was established as a reproducible resource. Cases at 100, 135 and 200 m/s still moved at window end (negative velocity, i.e. rebound), and only one 165 m/s case arrested. Penetration depth is therefore not reported as a rest-state scalar except for the single arrested case (69.33 mm); nose-node depth differences were confirmed as numerical artifacts of displacement integration after erosion. The single-step relative L2 error was 0.6977, reported honestly; autoregressive rollout from frame 11 to 39 took about 144 ms, a speedup of about 3.6x10^3 to 4.3x10^3 relative to single-core LS-DYNA, reported as application value. Validation is bounded by configuration similarity and field-level self-consistency; the framework is a simulation-trained decision-support method within the studied parameter space.

1. Introduction

Concrete penetration assessment must represent heterogeneous, non-arrested responses and full-field stress evolution rather than relying on a single rest-state depth. The paper addresses this gap with aggregate-resolved data and a field-level graph neural operator evaluated across velocities.

  • Motivation: A single scalar penetration depth cannot represent near-critical responses spanning arrest, rebound, and continued advance among heterogeneous concrete cases.The paper argues that such responses should instead be treated as a non-arrested interval or field-level observation.
  • Research gap: Existing approaches separately trade off computational cost, heterogeneity resolution, or complete tensor-field prediction.Mesoscale simulation resolves aggregate-scale randomness but is expensive; homogenized and empirical methods smooth heterogeneity; scalar surrogates omit the complete mid-plane stress tensor.
  • Approach: The dataset comprises 400 full-scale aggregate-resolved cases covering four impact velocities and 100 aggregate random seeds.Each case provides mid-plane six-component stress-tensor fields generated with LS-DYNA.
  • Approach: A field-level graph neural operator predicts time-varying six-component stress tensors in function space and tests unseen velocities through leave-one-velocity-out evaluation.The model aggregates geometric features, material properties, and stress states at nodes before iterative message passing updates the full field.
  • Contributions: Terminal-state verification anchors reliable observables to rigid-body motion and field-level stress evolution because only one 165 m/s case arrested within the solution window.Nose-node depth after erosion is treated as a numerical artifact rather than a rest-state scalar.

2. Full-scale mesoscale modelling of concrete and benchmarking against published penetration experiments

The study builds a full-scale mesoscale LS-DYNA model with randomly generated aggregate geometries, explicit dynamics, and a 10 mm mesh, then benchmarks the numerical framework against published penetration experiments by configuration similarity.

  • Full-scale geometry and material models: The model represents a 45 mm ogive projectile impacting a 500 mm diameter by 200 mm thick, three-phase mesoscale concrete target.The target contains mortar, coarse aggregates, and an interfacial transition zone; the projectile is modeled as a fourth phase.
  • Random generation of aggregates: 400 geometrically independent aggregate arrangements are generated by random seeds using a Fuller gradation over 5–25 mm particle sizes.The target sphere volume fraction is 42%, while 10 mm meshing yields about 33% element-level aggregate volume fraction.
  • Numerical solution settings: The explicit LS-DYNA solution uses central-difference integration with a Courant-limited stable timestep and mass scaling below the specified timestep threshold.The study declares that inertial error from mass scaling has not been independently quantified.
  • Numerical solution settings: The 10 mm single-point-integration mesh is a computational compromise, but systematic mesh convergence was not studied.Aggregates about 10–25 mm are resolved by roughly one to three elements relative to the mesh scale.
  • Benchmarking against published penetration experiments: Benchmarking uses Frew 2006, Hanchak 1992, and Forrestal 1996 as configuration-similarity anchors rather than exact per-velocity experimental reproductions.The 100 and 135 m/s cases fall below the Frew experimental range, so no direct experimental benchmarking is available there.

3. Cross-velocity random penetration dataset and mid-plane six-component stress tensor

The study constructs a 400-case cross-velocity dataset of aggregate-resolved mid-plane six-component stress fields and defines its valid observables through terminal-state verification. The dataset spans four impact velocities and 100 aggregate arrangements per velocity, while documenting numerical failures, unresolved terminal states, and stress-field representation choices.

  • 3.1. Parameterization: 400 cases combine four impact velocities—100, 135, 165 and 200 m/s—with 100 independent aggregate arrangements per velocity.The velocity and seed dimensions support cross-velocity evaluation and per-seed variability analysis.
  • 3.1. Parameterization: Benchmarking uses configuration similarity with published penetration experiments, while the 100 and 135 m/s levels lie below Frew 2006’s 160 m/s lower velocity bound.The low-velocity comparison therefore depends on extrapolating the velocity trend rather than direct experimental benchmarking.
  • 3.2. Field quantities: The workflow generates aggregate-resolved LS-DYNA solutions, resamples them onto a 64 × 64 X-Z grid, and stores normalized six-component stress fields over 40 frames.Each case is represented as a [6, 64, 64, 40] array, with stress values divided by 30 MPa.
  • 3.2. Field quantities: The complete tensor field is retained because aggregate–mortar interfaces produce evolving stress concentrations and a scalar equivalent stress would discard component-specific information.The six components comprise three normal and three shear stresses on the X-Z mid-plane.
  • 3.5. Data quality statement: Terminal-state verification found only one arrested case among the valid 165 m/s cases, with the remaining cases unresolved, rebounding, or still advancing at the 1 ms window end.Only the arrested case has a determined penetration depth of 69.33 mm; depth values for non-rest states are not reported as rest-state scalars.
  • 3.5. Data quality statement: Numerical failures were grouped into negative volume, time-step collapse and numerical divergence, while only four valid cases were available at 200 m/s and no representativeness assumption was made.These boundaries are reported as data-quality limitations without physical inference.

4. Field-level graph neural operator

The paper uses a graph neural operator to predict time-varying mid-plane six-component stress fields as an autoregressive function-space mapping. Its single-step error is slightly worse than the Fourier baseline, while the reported value is tied to full-field autoregression and physical verification.

  • Operator formulation: The graph neural operator maps historical stress fields, node coordinates and impact velocity to the next six-component stress field, then advances frame by frame autoregressively.The model predicts a stress increment that is added to the current field, enabling multi-frame rollout independent of LS-DYNA at inference time.
  • Graph architecture: The surrogate performs message passing on a 64 × 64 cross-section graph whose node features combine three stress frames, coordinates and impact velocity.Eight message-passing rounds use physically constructed edge weights based on acoustic impedance, traction amplitude and learnable correction factors.
  • Training configuration: Training uses three-step teacher forcing with decreasing loss weights of 0.5, 0.3 and 0.2 for successive predictions.A three-seed ensemble reports prediction dispersion of ±0.0034, making the single-step result an ensemble mean.
  • Interpretation: The method’s claimed advantage lies in consistent structural-level physical quantities and bounded autoregression rather than superior single-step absolute accuracy.The paper explicitly states that the graph architecture and physical edge weights become apparent in field-level verification and bounded rollout.

5. Results I: terminal-state verification of penetration and selection of observables

Terminal-state verification shows that most low-velocity cases had not arrested within the 1 ms window, making nose-node penetration depth unreliable after erosion. Reliable interpretation therefore uses rigid-body motion and energy, while numerical failures vary with velocity and aggregate configuration.

  • Terminal-state verification: Only one 165 m/s case arrested within 1 ms; the 100, 135 and 200 m/s cases remained in motion, including rebound at the lower and higher levels.The arrested case had approximately −0.7 m/s final velocity and a determined penetration depth of 69.33 mm; the other 165 m/s cases provide only lower bounds.
  • Observable selection: Rigid-body velocity and kinetic energy provide the reliable observables used to classify arrest, rebound and continued advance.At 100, 135 and 200 m/s, final velocities were negative and residual kinetic energy remained approximately 5.5%–10.5% of the initial value.
  • Observable selection: Nose-node penetration depth is invalid after erosion because displacement continued integrating at the velocity frozen when the node failed.For 100 m/s cases, erosion occurred at approximately 0.30–0.475 ms, producing spuriously increased or stagnated depth values.
  • Perforation state: Perforation below 200 m/s was geometrically impossible within the 1 ms window, whereas the 300–400 m/s probe cases were not assigned a perforation state.High-velocity erosion contaminated rigid-body readings, and no retained full-field projectile geometry was available for confirmation.
  • Numerical failures: Fifteen numerical failures were concentrated at 200 m/s, while 135 m/s failures were linked more plausibly to local aggregate-induced stress concentration and element distortion.Failure mechanisms included negative volume, time-step collapse and numerical divergence; no failures occurred at 165 or 400 m/s.

6. Results II: field-level prediction and physical verification

The graph neural operator reproduces time-varying stress-field structure and cross-velocity behavior, while physical verification reveals bounded rollouts, systematic amplitude smoothing, and substantial computational acceleration.

  • 6.1. Field-level accuracy: 0.6977 ± 0.0034 single-step relative L2 error, slightly above the Fourier neural operator baseline of 0.6907.The comparison used a three-level mixture excluding 200 m/s and 1072 samples.
  • 6.1. Field-level accuracy: 0.75 extrapolation relative L2 error at the held-out 200 m/s level versus 0.70 for interpolation, indicating retained but degraded cross-velocity prediction.The 200 m/s level was excluded from training; the reported extrapolation used leave-one-velocity-out evaluation.
  • 6.2. Field-level physical verification: The model reproduces peak-region position and general shape, but peak-profile relative errors of 0.5506 ± 0.179, 0.4526 ± 0.217 and 0.5762 ± 0.143 indicate systematically low amplitudes.These values correspond to 100, 135 and 165 m/s, respectively.
  • 6.2. Field-level physical verification: The decay exponent is lower than reference by +0.74, +0.46 and +0.46, consistent with conditional-mean smoothing of sharp aggregate-induced stress concentrations.Reference/model values are 2.76/2.02, 2.75/2.29 and 2.81/2.35, respectively.
  • 6.2. Field-level physical verification: The predicted peak zone is deeper than reference by +2.29±4.61, +0.92±3.99 and +0.90±4.36 mm, while the active-zone area is 2.81, 2.30 and 2.11 times reference.The deeper peak zone and enlarged active area form part of the reported amplitude-side smoothing pattern.
  • 6.3. Energy level: The autoregression remains bounded but systematically collapses amplitude by about 50%, so stability does not imply amplitude fidelity.The reported mechanism is smoothing of concentrated stress energy over a wider area.
  • 6.4. Computational cost comparison: The rollout from frame 11 to frame 39 takes about 144 ms, corresponding to about 3.6×10^3 to 4.3×10^3 lower computational time than single-core LS-DYNA.The comparison is reported as application value, not as a methodological advantage, because hardware and dimensional settings differ and data-generation and training costs are excluded.

7. Conclusions

The study establishes a field-level surrogate and aggregate-resolved database for concrete penetration while redefining reliable observables around rigid-body motion and stress-field evolution. Results are bounded by the studied numerical parameter space, configuration-similarity validation, and limitations in field accuracy and discretization.

  • Work completed and core results: The study establishes a 400-case, four-velocity, per-seed library of aggregate-resolved mid-plane six-component stress fields for surrogate training and evaluation.The cases cover 100, 135, 165 and 200 m/s with 100 aggregate seeds per velocity.
  • Key findings: The graph neural operator learned time-varying six-component stress-field evolution with bounded autoregression and cross-velocity extrapolation to an unseen velocity.The reported single-step relative L2 error was 0.6977; the cross-velocity extrapolation error was 0.75 against an interpolated 0.70.
  • Work completed and core results: 144 ms reduced rollout time by about 3.6 × 10^3 to 4.3 × 10^3 relative to single-core LS-DYNA, reported as application value rather than a methodological claim.The comparison used frames 11–39, with frames 0–10 supplied as a solver warm start.
  • Key findings: Only one 165 m/s case reached rest within 1 ms, so penetration observables are anchored to projectile rigid-body motion and field-level stress evolution rather than scalar depth.Nose-node depth differences after erosion were identified as numerical artifacts.
  • Key findings: Field-level observables and cross-velocity prediction address near-critical variation that a single deterministic simulation or arrest depth cannot characterize.The supported assessment centers on rigid-body motion and full-field six-component stress evolution.
  • Boundaries: The framework is a simulation-trained decision-support methodology within the studied numerical parameter space, not an immediately general design tool.Limitations include conditional-mean prediction, no mesh-convergence study, about 17.7% hourglass energy, and configuration-closeness benchmarking.

CRediT authorship contribution statement

The authors contributed across conceptualization, methodology, software, validation, analysis, investigation, data curation, visualization, writing, resources, and project administration.

  • Wenpu Du handled conceptualization, methodology, software, validation, formal analysis, investigation, data curation, visualization, original drafting, and project administration.
  • Peng Zhou, Yunlong Xia, Sinuo Xin, Congcong Zhang, Boyang Zhang, and Yi Zhang contributed writing review and editing and resources.
  • Wenzheng Xu contributed conceptualization and writing review and editing.

Data availability

The paper’s penetration data were generated through a documented LS-DYNA workflow and can be independently reproduced, with the dataset available from the authors upon reasonable request.

  • The reproducible dataset contains stress time series for 400 cases and aggregate volume-fraction fields generated by the described LS-DYNA workflow.Generation scripts accompany the code, and the data can be made available by the authors upon reasonable request.

Code availability

The graph neural operator and data-generation workflow are implemented with documented settings and code available from the authors upon reasonable request.

  • The PyTorch implementation documents data splitting, ensemble strategy, evaluation convention, training hyperparameters, and entry points for generation, training, evaluation, and ablations.The repository can be made available by the authors upon reasonable request.
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