cs.LGOct 12, 2026

A physics-coupled graph neural operator for mid-plane six-component stress tensor field prediction in concrete penetration

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

Organizations: School of Environment and Safety Engineering, North University of China, Taiyuan, 030051, China

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Abstract

The penetration-resistant design of concrete structures requires the spatiotemporal evolution of the stress field within the target, whereas in design only the strength grade, the geometry and the projectile parameters are known, and the aggregate arrangement is not. Empirical formulae and scalar-output surrogates serve design screening, and mesoscale finite element models resolve the aggregate-scale response, but one solution cannot cover random aggregate samples at scale, and no framework links mesoscale heterogeneity to field-level prediction. A physics-coupled graph neural operator (PC-GNO) is proposed, which learns in function space the evolution of the mid-plane six-component stress tensor field. The training data are generated by full-scale, aggregate-resolved finite element models and comprise 359 cases at the impact velocities of 100, 135 and 165 m/s, corresponding to compressive strengths of 14, 23 and 48 MPa; the data source is verified by laboratory measurements. The coupling strength of message passing is constructed from the impedance contrast across the material interface, and the update is constrained by spectral normalization. Under the single-step primary setting, the mean accuracy of the three levels is 80.16% (91.45%, 77.80% and 71.22%), i.e. 27.81, 42.46 and 34.16 percentage points above the preceding-frame field of the same window; with 20 664 parameters it exceeds the 87.9% of the best Fourier neural operator configuration (about 812 times as many parameters) and the 71.0% of the closed-form linear regression. The output is the complete six-component tensor field rather than a fixed engineering quantity, so that strength criteria and energy indicators follow from one set of weights. For field-level response prediction in concrete penetration, this result provides a data-driven route, with conditions of applicability defined by the training and the evaluation.

Figures & tables

Figure 1: Geometry and mesh of the full-scale model. (a) The geometrical relation between the projectile and the cylindrical target plate and the coarse aggregates randomly distributed within the target; (b) the local mesh refinement in the impact region. The two panels are taken from one and the same rendering, the projectile, the mortar matrix and the coarse aggregates being distinguished by different colours, and the dimension lines give the projectile diameter, the projectile length and the initial projectile-target spacing.
Figure 2: Aggregate gradation curve and geometric scatter. (a) The cumulative passing curve of the Fuller continuous gradation, with a particle size range of 5 to 25 mm and a target sphere volume fraction of 42%; (b) the aggregate distribution generated by random placement on the mid-plane section at the impact face, the aggregate and mortar phases being coloured separately; the case is taken from the 100 m/s level with the number 1001, and the section is the layer of elements containing the target axis, with a lateral window of 150 mm and a depth of 200 mm; the aggregate proportion shown in the figure is an element-level value after meshing and is lower than the target sphere volume fraction.
Figure 3: Six-component stress snapshots of a representative case. The six panels are, in order, σxx\sigma_{xx} , σyy\sigma_{yy} , σzz\sigma_{zz} , σxy\sigma_{xy} , σyz\sigma_{yz} and σxz\sigma_{xz} , and are taken from the mid-plane field of one and the same case and frame. Each component is presented as a rectangular block, the range of each panel being given separately with an independent colour bar, and the colour bars are displayed with a symmetric power-law spacing; the stress values are the normalized values.
PhaseConstitutive modelParameterValue
ProjectileMAT_RIGIDDensity / elastic modulus / Poisson ratio / mass7.85×10−37.85\times 10^{-3} g/mm 3 / 210000 MPa / 0.30 / 2.2414 kg
Mortar matrixMAT_PLASTIC_KINEMATIC (MAT_003)Density / elastic modulus / Poisson ratio2.2×10−32.2\times 10^{-3} g/mm 3 / 10000 MPa / 0.20
Target—Concrete compressive strength (three levels)14 / 23 / 48 MPa
Coarse aggregateMAT_JOHNSON_HOLMQUIST_CERAMICS (MAT_110)Density2.7×10−32.7\times 10^{-3} g/mm 3
Aggregate geometryFuller gradationParticle size range / target sphere volume fraction / placement domain5 to 25 mm / 42% / ϕ\phi 500 ×\times 500 mm cylinder
MeshELFORM=1 single-point integration hexahedronElement size / target element count / projectile element count7.0 mm / 271859 / 1840
Table 1: Material and solution parameters
Figure 4: Projectile assembly and post-penetration morphology. (a) The assembly section of the projectile; (b) the acceleration sensor; (c) the front target face after penetration; (d) the projectile after penetration.
Figure 5: Comparison between the simulated and the measured deceleration time histories. The simulation is the case with a total projectile length of 340 mm, an impact velocity of 134 m/s and a target plate compressive strength of 23 MPa; the measurement is the deceleration curve recorded in the second group of tests on the same apparatus and at the same target strength. The horizontal axis is time and the vertical axis is the projectile deceleration.
Figure 6: Crater morphology and section. (a) The target face after penetration, with the crater and the annular spalling zone marked; (b) the projectile embedded in the crater; (c) the three-dimensional laser scanning reconstruction of the crater; (d) the section profile through the crater centre, with a crater depth of 133.0 ±\pm 1.2 mm.
Figure 7: Dataset generation workflow. Four steps in total, which are aggregate generation (parameter setting and random placement), the explicit solution, the resampling to a regular grid and normalization and storage, corresponding in order to the numbered panels 1 to 4 of the figure. The stress tensor field is stored as a four-dimensional array of [6,64,64,82][6,64,64,82] , and the field quantities are linearly normalized in the export layer.
LevelConcrete compressive strength / MPaImpact velocity / (m ⋅\cdot s -1 )CasesTraining casesValidation casesTest casesEvaluation windows
v10014100149831719608
v13523135107601214476
v16548165103581214476
Total——35920141471560
Table 2: Composition of the dataset
Figure 8: Terminal-state verification of the three velocity levels. (a) The distribution over cases of the projectile axial velocity at the end of the window, a positive value meaning that the projectile is still advancing into the target; (b) the velocity and displacement time histories of one case at the 135 m/s level; (c) the scatter over cases of the peak penetration depth within the window and the penetration depth at the end of the window. The numbers of cases of the three levels are 149, 107 and 103 respectively.
Figure 9: Overall structure of the field-level neural operator. The input is the normalized six-component stress tensor field of consecutive frames within the look-back window together with the impact velocity scalar, and the encoder concatenates the nodal stress components, the nodal coordinates and the impact velocity into node features and compresses them into a latent representation; the operator layers perform 8 rounds of message passing on the cross-section node graph with a latent width of 32, the aggregation being weighted by edge weights constructed from physical quantities, and the propagation operator applying a graph Laplacian spectral normalization (20 power iterations); the decoder outputs the six stress components of the next frame through two linear mappings. The number of nodes shown in the figure is 64×6464\times 64 of the storage grid; the grid on which the operator of the present work actually operates is 32×3232\times 32 , that is, 1024 nodes, and the relation between the two is given in Section 5.2 .
Velocity levelConcrete compressive strength / MPaEvaluation windowsAccuracy %Preceding- frame field %Gain (per- centage points)
100 m/s1460891.4563.64+27.81
135 m/s2347677.8035.34+42.46
165 m/s4847671.2237.06+34.16
Average of the three levels——80.16——
Table 3: Field-level accuracy at the three velocity levels
ModelParameter countAccuracy %Preceding-frame field %Gain (percentage points)
PC-GNO20 66491.4563.64+27.81
Fourier neural operator (configuration m32w64l2)16 788 03887.963.64+24.27
Fourier neural operator (configuration m16w64l4)8 399 68682.963.64+19.22
Closed-form linear regression13271.063.64+7.39
Table 4: Comparison of field-level accuracy under the same configuration (100 m/s level, 608 windows)
Figure 10: Peak stress profile, decay exponent and decay depth. (a) Peak stress profile along the depth direction, with a logarithmic vertical axis; (b) decay exponent of the profile; (c) decay depth difference defined by the 1/e1/e drop of the peak. The reference values of the three velocity levels are all taken as the solver ground-truth field of the same window, the solid line being the reference solution and the dashed line being the operator prediction. Single-step prediction with ground-truth history and a frame interval of 16 are taken for all three levels; the evaluation windows are frames 48 to 79 (100 m/s level) and frames 48 to 81 (135 and 165 m/s levels) of the test split.
Figure 11: Evolution of the Frobenius norm of the stress tensor with time. The reference curves and the prediction curves of the three velocity levels share one and the same normalization basis, which is taken as the norm of the ground-truth field of each level at frame 10. The reference curves cover frames 10 to 81 and the prediction curves cover frames 48 to 81; the colors distinguish the impact velocity levels, the solid line being the reference solution and the dashed line being the operator prediction.
Figure 12: Comparison of the predicted field with the solver reference field. Taking one test case of the 135 m/s level as an example, the four rows give the solver reference field, the single-step prediction field of the operator and the absolute difference between the two at four advancing frames. The reference field and the prediction field of the same row share one color scale centered at zero, and the error column uses a separate color scale; the stress values are normalized values.
Figure 13: Distribution of the node-level residuals. (a) Node-wise σzz\sigma_{zz} absolute residual at four advancing frames, the horizontal axis being the node index ordered by depth, the error being displayed on a logarithmic color scale, and the nodes outside the mask being marked in gray; (b) node-wise comparison of the predicted values with the reference values, the diagonal being the line of equality. The node set and the conditions of evaluation of the node-level coefficient of determination differ from those of the field-level relative L2L_{2} error of Table 3 , so that the two are not interchangeable.
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