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Scalable Stress-Constrained Topology Optimization: A Super-resolution and Simultaneous Approach Based on Machine Learning

Xiangyu Sun, Shirin Hosseinmardi, Ramin Bostanabad

arXiv:2610.28703v1cs.LG

TL;DR

Stress-constrained topology optimization is computationally demanding because local constraints and conventional analysis-and-design solves become costly at scale. SNTO jointly trains neural state, adjoint, and design fields and partitions additive loss gradients for distributed training. On 2D benchmarks, it produces sharper, less-gray designs with lower volume fractions than PolyStress while satisfying local stress constraints to the same degree.

  • Problem

    Local stress constraints grow with discretization resolution, and conventional nested finite-element solves become costly for large-scale problems.

    Source: §1 · pp. 1–2

  • Method

    SNTO jointly trains neural state, adjoint, and design fields on a composite loss and distributes additive loss gradients across subdomains and GPUs.

    Source: §1 · p. 3

  • Results

    Across 2D benchmarks, SNTO produces sharper, less-gray designs and lower volume fractions than PolyStress while satisfying local stress constraints to the same degree.

    Source: §5 · p. 31

  • Takeaways & Limitations

    The benchmarks include 3D stress-constrained designs with up to eight million nodal collocation points.

    Source: §5 · p. 31

    The benchmark setup uses an isotropic, linear-elastic aluminum-like material with E0 = 70 GPa, Poisson’s ratio 0.25, and yield strength 100 MPa.

    Source: §4 · p. 17

Abstract

from arXiv · show

Large-scale topology optimization (TO) remains computationally demanding, especially for stress-constrained problems. To address this challenge, we propose the simultaneous neural topology optimization (SNTO) framework where we represent state, adjoint, and design variables with custom neural networks (NNs) whose parameters are jointly optimized within a single optimization loop that leverages continuous adjoint analysis and automatic differentiation. This optimization minimizes a multi-component loss function that combines the design objective, an aggregation of the local stress constraints, and the potential energy functionals. Our NNs have a unique architecture that accelerates convergence while enabling the networks to capture sharp gradients and boundaries which are critical to stress-constrained TO. To achieve scalability to millions of degrees of freedom (DoF), we develop a hardware-aware training scheme in which the design domain is partitioned into subdomains that are distributed across multiple GPUs. To control peak GPU memory usage, we further slice each subdomain into smaller regions whose loss gradients are evaluated sequentially and accumulated to update the NN parameters. Unlike domain decomposition approaches, our partitioning naturally ensures field continuity across the subdomains and does not require the introduction of additional terms into the loss function. For validation, we first evaluate the proposed framework on various 2D stress-constrained benchmarks and compare our results to a conventional finite element-based approach, showing that SNTO produces super-resolution designs with sharper boundaries, smaller gray areas, and lower volume fractions. We then consider large-scale 3D problems with up to eight million collocation points and distribute the training over up to eight GPUs.

1 Introduction

Stress-constrained topology optimization is difficult because local constraints multiply with discretization resolution, while conventional nested solves become costly at scale. SNTO jointly trains neural state, adjoint, and design fields, with additive-gradient partitioning for multi-GPU training.

1 Introduction

  • Local stress constraints scale with discretization resolution, and p-norm or Kreisselmeier–Steinhauser aggregation can approximate maximum stress loosely.Stress concentrations at corners, crack tips, and loading regions require fine meshes, further increasing constraint counts.Source: §1 · pp. 1–2
  • Each conventional nested analysis-and-design iteration requires two mesh-sized finite-element solves and stress evaluation at every element.For large 3D problems, the cost becomes prohibitive on a single machine.Source: §1 · p. 2
  • Prior physics-informed neural topology optimization studies gave little attention to stress-constrained problems and mostly covered 2D cases with modest constraint counts.Data-driven generative approaches also depend on training databases built by conventional solvers.Source: §1 · p. 2
  • PGCAN-based networks are designed to capture sharp gradients, while finite-element shape functions and reduced Gauss quadrature approximate loss derivatives and integrals.The numerical approximation is for loss evaluation; design and state representations remain mesh-free.Source: §1 · p. 3
  • Additive loss gradients allow subdomain and sequential sub-slice training across GPUs without interface loss terms.A shared network parameterization ensures continuity across partitions; sequential evaluation reduces peak GPU memory.Source: §1 · p. 3
  • SNTO simultaneously trains separate neural networks for state, adjoint, and design fields using a composite physics-informed loss.The discrete adjoint sensitivity for the local stress penalty is solved through minimization of the adjoint potential-energy functional.Source: §1 · p. 3
  • SNTO produces sharper, lighter designs than PolyStress at comparable stress levels and is demonstrated on 3D problems with up to 8M local stress constraints.The paper also reports training distributed over up to eight GPUs.Source: §1 · p. 3

2 Governing Physics

The governing-physics passages describe linear-elastic stress and equilibrium through a variational energy formulation. They also connect that formulation to the network pipeline used in SNTO.

2 Governing Physics

  • The framework figure combines outputs from three independent PGCANs with stress-penalty and volume-objective terms in one jointly minimized loss.The PGCAN decoder uses interpolated convolutional features passed through a shallow gated decoder.Source: Figure 1 · p. 4, §2 · p. 4
  • Stress follows the linear-elastic constitutive relation, with stiffness varying by position because Young’s modulus is interpolated from density.The strain tensor is symmetric under the infinitesimal-strain assumption.Source: §2 · p. 4
  • Minimizing total potential energy over kinematically admissible displacements gives equilibrium; the functional combines strain energy and external-traction work.Stationarity recovers equilibrium and traction boundary conditions, while free surfaces arise as natural boundary conditions.Source: §2 · p. 5
  • The deep energy formulation uses only first displacement derivatives, is quadratic for linear elasticity, and provides an equilibrium energy-balance diagnostic.At equilibrium, twice the strain energy equals the external work.Source: §2 · p. 5

3 Proposed Method

The proposed method presents a density-based stress-constrained formulation using neural representations of displacement, adjoint, and design fields, with finite-element approximations and memory-efficient training for large problems. It details discretized loss evaluation, recovery of implicit stress sensitivities through an adjoint mechanism, and staged training with continuation for simultaneous optimization.

3 Proposed Method

  • Stress constraints use von Mises stress evaluated with solid stiffness, then relax it by the density interpolation function so their contribution vanishes in void regions.This avoids satisfying the stress limit merely by lowering density and allows gradient-based optimizers to remove material and evolve the design through voids.Source: §3 · p. 6
  • A normalized quadratic penalty weights positive stress violations by squared magnitude, while smooth Heaviside projection pushes density toward binary values.The stress-violation form grows more steeply for severe violations; the penalty is normalized by the number of constraints, and the projection uses threshold density η = 0.5.Source: §3 · pp. 6–7

3.1 Neural Parameterization

  • PGCAN uses trainable spatial features and a shallow decoder to capture sharp gradients; separate displacement, adjoint, and density networks use analytical output transformations to enforce prescribed conditions.The displacement transformation satisfies homogeneous boundary conditions exactly, without adding penalty terms to the loss.Source: §3.1 · pp. 7–8

3.2 Memory-Efficient Multi-GPU Training

  • The training scheme partitions the domain across GPUs and processes local sub-slices sequentially, accumulating gradients while retaining only one sub-slice’s autograd graph at a time.A shared global neural representation keeps fields continuous across subdomain boundaries, requiring no interface-loss terms; GPU gradients are synchronized before parameter updates.Source: §3.2 · pp. 8–10, Figure 2 · p. 9

3.3 Numerical Approximation

  • The fixed background mesh supports finite-element shape-function differentiation and Gauss-quadrature integration, while the unknowns remain the global neural-network parameters.Cached shape-function gradient matrices reduce the cost of computing spatial derivatives at large numbers of collocation points.Source: §3.3 · p. 10
  • Single-point center Gauss quadrature evaluates density and stress at element centers, yielding one stress constraint per element and enabling direct comparison with PolyStress.Full Gauss quadrature increases computational cost without noticeably affecting optimized designs.Source: §3.3 · p. 12

3.4 Adjoint Sensitivity Analysis

  • The discrete adjoint recovers the stress penalty’s omitted implicit density sensitivity, with adjoint loads assembled only from active, violated constraints.Rather than separately solving the adjoint linear system, the method minimizes an adjoint energy functional as another loss term, reusing the same cached matrices.Source: §3.4 · pp. 12–14

3.5 Stabilization and Curriculum Training

  • Training uses two 5%-epoch equilibrium warm-ups with the design frozen, followed by full-loss optimization of all three networks.The initial projected volume fraction is approximately 0.60 and remains fixed through the first 10% of epochs.Source: §3.5 · p. 14
  • After a 10% warm-up, SIMP exponent, projection sharpness, and stress-constraint weight ramp to p=5, β=8, and ω_C=10^3 by 60%; the stress limit tightens to 0.9 times nominal yield strength.This leaves a 10% final margin below nominal yield strength to buffer stress-discretization errors.Source: §3.5 · p. 15

4 Results and Discussion

The evaluation covers four 2D stress-constrained benchmarks compared with PolyStress and two 3D benchmarks used to assess scalability. The 3D studies extend to eight million nodal control points, while training times across resolutions are not directly comparable because hardware allocation differs.

3D scalability evaluation

  • The 3D studies assess scalability on L-beam and portal-frame benchmarks, reaching 8M nominal nodal control points and up to four GPUs.The benchmarks are formed by extruding the 2D geometries; the 8M case uses four GPUs with ten sub-slices per GPU.Source: §4 · pp. 15–17, Figure 4 · p. 16

2D benchmark evaluation

  • SNTO is evaluated on four 2D stress-constrained benchmarks against PolyStress, an FE-based code using augmented Lagrangian local stress constraints.PolyStress runs use approximately 250,000 elements; SNTO is also tested across meshes from 125K to 1M nodal control points and multiple random seeds.Source: §4 · pp. 15–16
  • The 2D cases include an L-beam re-entrant-corner singularity, a crack-tip stress concentration, a loaded curved hook, and a half-domain portal frame.The examples assume plane stress; the hook includes a prescribed solid ring around its inner arc to maintain a feasible load path.Source: §4 · pp. 15–16

3D scalability evaluation

  • Training times across 3D mesh resolutions are not directly comparable because GPU and sub-slice allocations change with problem size.The 500K cases use one GPU, while the 4M and 8M cases use four GPUs.Source: §4 · pp. 16–17

4.1 2D Benchmarks with Comparison Against an FE-based Method

Across the 2D benchmarks, SNTO produces designs with smaller gray fractions and lower volume fractions than PolyStress while satisfying stress constraints almost everywhere. Training evolution and mesh-refinement analyses show how these designs form and how their topology and optimization behavior vary with resolution.

4.1 2D Benchmarks with Comparison Against an FE-based Method

  • Both methods form load-bearing frames, but SNTO changes member layouts across benchmarks and removes idle material retained in some PolyStress designs.The hook’s nearly unstressed region around the upper clamped hole is filled by PolyStress but supported by two separate SNTO clamps.Source: §4.1 · p. 17, Figure 5 · p. 18, §4.1.3 · p. 25
  • SNTO’s gray fractions are two to seven times smaller than PolyStress’s, and its designs have lower volume fractions while satisfying stress constraints almost everywhere.SNTO gray fractions range from 0.01 to 0.06, compared with 0.05 to 0.13 for PolyStress.Source: §4.1 · p. 17, Table 3 · p. 19, §4.1.3 · p. 25

4.1.1 Topology Evolution During Optimization

  • As design updates begin, stress penalties add material to reduce widespread violations; the volume objective then removes material where stress constraints are inactive.The volume fraction peaks at 0.65–0.75, while stress-relieving features emerge before the design becomes crisp and binary.Source: §4.1.1 · p. 19, Figure 6 · p. 20
  • Primal and adjoint networks maintain mechanical equilibrium during training, as checked without FE solves by comparing twice the strain energy with the corresponding work.The adjoint load’s spatial distribution changes as local constraints switch state.Source: §4.1.1 · p. 21, Figure 7 · p. 21

4.1.2 Mesh Sensitivity Analysis

  • Across 125K–1M nodal CPs, principal load-carrying features persist, with observed local variations attributed to optimization non-convexity rather than mesh dependence.Small variations occur in secondary members; the study reports design metrics varying by at most 0.02 in volume fraction and 0.03 in gray area fraction.Source: §4.1 · p. 18, §4.1.2 · pp. 22–23, Figure 8 · p. 22
  • Refinement improves optimization stability rather than changing designs: at 500K CPs and above, late stress-violation drift and adjoint-scaling fluctuations subside, while training time grows proportionally with CP count.At coarser resolutions, steep stress gradients at fine-member boundaries are less accurately resolved.Source: §4.1.2 · p. 23

4.1.3 Stress Constraint Verification

  • Independent FE analysis finds both methods satisfy stress limits almost everywhere; their 99.9th-percentile stress-ratio differences are small and do not exceed about 0.05.PolyStress is slightly lower for the L beam, crack, and hook; SNTO is lower for the portal frame.Source: §4.1.3 · pp. 24–25, Figure 10 · p. 24

4.2 3D Benchmarks with Scalability to Millions of Local Stress Constraints

The 3D benchmarks retain consistent designs as resolution rises to 8M nodal CPs, while FE verification finds few stress violations and refinement reduces violations in the portal frame. Sub-slicing trades longer training for lower peak memory, and adding GPUs accelerates training.

4.2 3D Benchmarks with Scalability to Millions of Local Stress Constraints

  • The 3D benchmarks use a reduced density-network learning rate and adjoint-force target to maintain equilibrium across random runs.The learning rate changes from 2×10^-5 to 10^-5, and the scaled adjoint-force target changes from 10 to 1.Source: §4.2 · p. 25

4.2.1 Scalability

  • The 8M designs have the highest volume fractions, and the portal frame varies by 0.034 across resolutions as finer CPs resolve sharper stress concentrations.L-beam volume fraction varies by less than 0.01; the authors suggest added material near concentrated stresses as a plausible explanation.Source: §4.2.1 · pp. 25–26, Table 4 · p. 26
  • Portal-frame training is slower than L-beam training by 5% through 2M CPs, 13% at 4M, and 23% at 8M, attributed to uneven active-element distribution across GPU slices.Because gradients synchronize each epoch, the GPU with the most active elements sets epoch time.Source: §4.2.1 · p. 26
  • From 500K to 8M nodal CPs, both designs retain their primary load paths, with differences mainly in member thickness and fillet shape, without explicit filtering.The same PGCAN feature-grid resolution is used at all five mesh resolutions, so refinement affects loss evaluation rather than design parameterization.Source: §4.2.1 · p. 26, Figure 12 · p. 27

4.2.2 Stress Constraint Verification

  • At 8M nodal CPs, portal-frame FE violations fall to 0.06%, while its volume fraction rises from 0.228 at 2M to 0.258; L-beam violations remain approximately 0.4%.The portal-frame 99.9th-percentile stress ratio drops from 1.07 at 500K to 0.97 at 8M; the L-beam percentile ranges from 1.07 to 1.14.Source: §4.2.2 · p. 28, Figure 14 · p. 29

4.2.3 Computational Cost and Memory

  • SNTO partitions the domain across GPUs and processes smaller sub-slices sequentially, accumulating gradients to reduce memory while preserving globally continuous fields.This approach requires no interface conditions or additional interface residual terms.Source: §4.2.3 · pp. 28–30
  • Increasing sub-slices from 2 to 16 per GPU cuts peak memory from 68% to 14% of device capacity, while training time rises from 2.5 to 4.2 hours.The tested 1M L beam uses two GPUs; all configurations produce nearly identical topologies.Source: §4.2.3 · pp. 29–30, Figure 15 · p. 30
  • For the 2M portal frame, eight GPUs reduce training time from 9.4 to 1.5 hours, a 6.3× speedup and 79% parallel efficiency.Peak memory remains approximately 68% for configurations using two or more GPUs; all four configurations yield consistent topologies.Source: §4.2.3 · p. 30, Figure 15 · p. 30

5 Conclusion

The paper presents SNTO for stress-constrained topology optimization, jointly training neural representations of displacement, adjoint, and density fields. It reports competitive 2D results and scalable 3D benchmarks, while the current framework assumes linear elasticity.

5 Conclusion

  • SNTO jointly trains custom neural networks for displacement, adjoint, and density fields using a composite loss for volume, stress constraints, and state and adjoint potential energies.Its PGCAN architecture is intended to capture stress concentrations and fine-scale design features; distance-based functions impose boundary conditions and design constraints exactly.Source: §5 · p. 30
  • Gradient accumulation over sequential sub-slices and across GPUs reduces peak memory while global neural representations preserve field continuity without interface residual terms.The loss gradient is additive over collocation points, enabling partitioning across devices.Source: §5 · p. 31
  • Across the reported benchmarks, SNTO yields sharper, lower-gray designs and lower volume fractions than PolyStress in all four 2D examples, while scaling to 8M nodal CPs in 3D.The conclusion reports similar stress-constraint satisfaction to PolyStress, largely resolution-insensitive designs without density filtering, and reduced training time with more GPUs.Source: §5 · p. 31
  • The current framework assumes linear elasticity; nonlinear material behavior and transient stress-constrained problems are proposed as extensions.The authors also identify adaptive sampling near high-stress regions as a possible way to reduce remaining violations near geometric singularities.Source: §5 · p. 31

CRediT authorship contribution statement

The contribution statement assigns conceptualization and methodology to all three authors, with distinct roles in analysis, software, supervision, funding, and writing.

CRediT authorship contribution statement

  • Sun led formal analysis, software, data curation, and figures; Hosseinmardi contributed to conceptualization and methodology; Bostanabad provided supervision and funding acquisition.All three authors contributed to conceptualization and writing or editing.Source: CRediT authorship contribution statement · p. 31

A Sensitivity of the training histories to the number of nodal CPs

Across the reported 2D and 3D mesh-resolution studies, design and stress histories generally follow similar trajectories as nodal CP counts increase. The 3D portal-frame adjoint body work, however, remains resolution-dependent.

A Sensitivity of the training histories to the number of nodal CPs

  • Across 125K–1M CPs, the 2D L beam, hook, and portal frame show similar active-element, volume, and gray-area histories, with stress violation approaching 5% for the L beam and portal frame and 4% for the hook.Warm-up, peak volume, material removal, and design sharpening occur at nearly the same training stages across resolutions.Source: A Sensitivity of the training… · pp. 31–32, Figure 16 · p. 32, Figure 17 · p. 33
  • For the 3D portal frame, active-element, volume, and gray-area histories remain similar across 500K–8M CPs, while adjoint body work is consistently higher at 8M.Stress violation and adjoint scaling factor remain within narrow bands after the initial transient.Source: A Sensitivity of the training… · p. 32, Figure 19 · p. 34
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