Graph Neural Networks Predict Supersonic Metal Bonding in Cold Spray

The significance of the cold spray process lies in its ability to deposit coatings and build up material without the melting that characterizes conventional thermal spray techniques. Because metallic particles remain solid throughout their flight and only deform plastically upon striking the substrate, the process avoids oxidation, phase transformations, and the porosity that often degrades thermally sprayed layers. This makes cold spray particularly attractive for aluminum alloys, titanium, and other materials whose microstructures are sensitive to heat. The trade-off, however, is that successful bonding depends entirely on the mechanics of a microsecond-long impact event, in which a particle traveling at supersonic speed must undergo sufficient plastic deformation to bring fresh, oxide-free surfaces into intimate contact with the substrate.

The physical mechanism most widely invoked to explain particle adhesion is adiabatic shear instability. During impact, the periphery of the particle experiences extreme strain rates, often exceeding ten to the seventh per second. When thermal softening locally outpaces strain hardening, a narrow shear band forms, concentrating deformation and heat into a thin region at the particle-substrate interface. This localized softening promotes the formation of material jets and enables metallurgical bonding between surfaces that would otherwise rebound elastically. The Johnson-Cook constitutive model, which couples strain, strain-rate, and temperature effects into a single flow stress description, is the standard framework for capturing this behavior in finite element simulations, and its parameters directly shape how the response surface varies across the process parameter space.

Single-particle finite element models, such as the spherical particle on a cylindrical substrate configuration used in this study, have become the canonical tool for interrogating these mechanisms. The geometry chosen here, a forty-micrometer-radius aluminum particle striking an aluminum substrate of two hundred fifty micrometer radius and depth, reflects the typical scale of cold spray powder feedstock and allows the impact event to be resolved with sufficient mesh refinement to capture the deformation gradients near the contact zone. Normal incidence is assumed, which is a reasonable first approximation since most particles in a cold spray jet strike the substrate at or near perpendicular orientation, although oblique impacts and particle-particle interactions in real deposits introduce additional complexity that single-particle studies deliberately set aside.

The five output targets selected for prediction capture complementary aspects of the impact response. Maximum equivalent plastic strain quantifies the severity of deformation, which correlates with the extent of interfacial contact area development and hence with bonding likelihood. Average contact plastic strain provides a more spatially averaged measure of deformation at the interface. Maximum temperature indicates whether adiabatic heating approaches the softening regime necessary for jetting. Maximum von Mises stress characterizes the mechanical loading experienced by the material, relevant to residual stress development and potential substrate damage. The deformation ratio, finally, describes the geometric flattening of the particle, a quantity experimenters can measure directly in cross-sectioned deposits, making it a useful bridge between simulation and experimental validation.

The choice of particle velocity, particle temperature, and friction coefficient as the three input parameters reflects their dominant roles in governing impact outcomes. Velocity controls the kinetic energy available for plastic work and is widely regarded as the single most influential cold spray parameter, with critical velocities below which particles rebound and above which they bond. Particle temperature, set by the gas temperature in the spray nozzle, pre-softens the material and lowers the energy barrier for deformation. The friction coefficient at the interface, though harder to control experimentally, governs tangential restraint and energy dissipation during sliding contact, influencing jet formation and the distribution of plastic strain around the contact periphery.

The surrogate modeling strategy adopted in this work addresses a persistent bottleneck in process simulation. A single resolved finite element impact simulation can require substantial computational resources, and exploring a three-dimensional parameter space at useful resolution demands hundreds or thousands of such runs. Once trained, a surrogate model evaluates new process conditions in milliseconds, enabling optimization studies, sensitivity analyses, and inverse design tasks that would be intractable with direct simulation. The practical value of the surrogate, however, depends entirely on its accuracy across the operating envelope, which is precisely where the choice of machine learning architecture becomes consequential.

The central representational innovation of the study is the treatment of each simulation sample as a node in a k-nearest-neighbour graph constructed in feature space, rather than as an isolated feature vector. In conventional feedforward networks, each training example is processed independently, and any information about the similarity between neighboring process conditions is implicit only in the aggregate statistics of gradient descent. By contrast, graph neural networks explicitly pass messages along edges connecting similar samples, allowing each node’s prediction to be conditioned on the responses of its parametric neighbors. For a physical system like cold spray impact, where the response surface is smooth and continuous in the vicinity of any given operating point, this inductive bias aligns naturally with the structure of the underlying data-generating process.

The comparative results carry a clear message about which architectural assumptions suit this problem. GraphSAGE-style spatial aggregation and the geometric attention network both achieved coefficients of determination above 0.93 for most targets, with the attention model reaching 0.97 for maximum plastic strain. These architectures share a common principle: they aggregate information from spatially proximate neighbors in feature space, weighting that information either uniformly or through learned attention coefficients. The attention mechanism’s slight edge is physically sensible, since the velocity-dominated nature of the response means that neighbors at different velocities carry unequal informational value, and attention weights can adaptively emphasize the most relevant ones.

The comparatively poor performance of the Chebyshev spectral graph convolution network and the topologically augmented multilayer perceptron, including negative R-squared values on several targets, is instructive rather than merely negative. Spectral methods operate through polynomial approximations of the graph Laplacian, which excel at multi-scale feature extraction when the graph structure itself carries meaningful community or frequency information. For a k-nearest-neighbour graph built from a relatively smooth parametric dataset, the spectral structure may be too weak or too sensitive to the choice of graph construction to provide a useful signal. Similarly, persistent homology descriptors encode global topological features of the point cloud, such as connected components and loops across scales, which may be largely uninformative for a response surface governed by local parametric gradients rather than by global shape features.

Negative R-squared values deserve particular emphasis for readers less familiar with regression diagnostics. A negative coefficient of determination indicates that the model’s predictions are worse than simply predicting the mean of the training data for every input. This is not a marginal failure but a categorical one, signaling that the architecture has failed to extract any generalizable input-output relationship from the training set. In surrogate modeling applications, such failures typically arise from an inductive bias mismatched to the data structure, or from architectures whose capacity is poorly matched to the available training set size, rather than from noise in the underlying simulation data, which is deterministic in this case.

The velocity-dominated character of the input-output relationships, confirmed by the three-dimensional feature space visualizations and two-dimensional contour projections, is consistent with decades of experimental cold spray research. Deposition efficiency, critical velocity, and coating quality all vary steeply with particle velocity, while temperature and friction act as secondary modulators. A surrogate model that respects this hierarchy, as the attention-based spatial aggregation evidently does, can allocate its representational capacity where the response varies most sharply. This interpretability of model performance in terms of constitutive physics is one of the study’s more valuable contributions, since it transforms an empirical architecture comparison into a statement about the physics of the process itself.

Several limitations frame the scope of these findings. The dataset derives from a single material system, aluminum on aluminum, with a single particle size and normal impact geometry, so extrapolation to dissimilar material pairs, oblique impacts, or multi-particle interactions remains untested. The friction coefficient is treated as a fixed input parameter, whereas in reality interfacial friction evolves with temperature, pressure, and surface state during the impact itself. Furthermore, the surrogate learns from simulation data and inherits any idealizations embedded in the finite element model, including the constitutive parameters of the Johnson-Cook model and the assumed contact behavior. Experimental validation against measured deformation ratios or deposition efficiencies would strengthen confidence in the surrogate’s predictions beyond the simulation domain.

Nevertheless, the framework points toward practical applications in cold spray process development. A validated surrogate could accelerate the identification of operating windows that maximize interfacial plastic strain while keeping substrate stresses within acceptable limits, or support real-time process control where nozzle gas conditions are adjusted in response to measured particle velocities and temperatures. The graph-based representation could also be extended to incorporate additional parameters, such as particle size distributions, substrate preheating, or nozzle standoff distance, as further node features, provided the training dataset is expanded accordingly through continued automated simulation campaigns.

More broadly, the study contributes to a growing recognition in computational materials science that the structure imposed on training data can matter as much as the choice of model family. Where physical responses vary smoothly across a parametric space, encoding that continuity directly into the learning architecture, as graph-based neighborhood aggregation does, provides a form of physics-informed bias that improves both accuracy and data efficiency. For solid-state deposition processes, and potentially for other impact-dominated manufacturing problems with expensive simulations and smooth response surfaces, this representational insight may prove as consequential as the specific performance numbers reported.

Subject of Research: Application of geometric and topological deep learning to predict thermo-mechanical performance in cold spray deposition process modeling

Article Title: Geometric and topological deep learning for predicting thermo-mechanical performance in cold spray deposition process modeling

Article References: Mishra, A. (2026). Geometric and topological deep learning for predicting thermo-mechanical performance in cold spray deposition process modeling. Discover Informatics, 1(1), Article 13. https://doi.org/10.1007/s44564-026-00012-3

Image Credits: AI Generated

DOI: 10.1007/s44564-026-00012-3

Keywords: cold spray deposition, geometric deep learning, graph neural network, GraphSAGE, finite element simulation, Johnson-Cook model, topological data analysis, graph attention network, thermal spray coating, additive manufacturing, surrogate modeling, supersonic particle impact