# AI-assisted prediction of particle impact deformation simulated by Material Point Method

Saba Saifoori, Somayeh Hosseinhashemi, Mohammad Alasossi, Carsten

Schilde, Saeid Nezamabadi, Mojtaba Ghadiri

► To cite this version:

Saba Saifoori, Somayeh Hosseinhashemi, Mohammad Alasossi, Carsten Schilde, Saeid Nezamabadi, et al. AI-assisted prediction of particle impact deformation simulated by Material Point Method. Powder Technology, 2025, 459, pp.121018. 10.1016/j.powtec.2025.121018. hal-05050118

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Submitted on 28 Apr 2025

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# AI-assisted prediction of particle impact deformation simulated by Material Point Method

Saba Saifoori<sup>a, *</sup>, Somayeh Hosseinhashemi<sup>b</sup>, Mohammad Alasossi<sup>a</sup>, Carsten Schilde<sup>b</sup>, Saeid Nezamabadi<sup>a, c</sup>, Mojtaba Ghadiri<sup>a</sup>

## HIGHLIGHTS

* MPM is used to simulate impact of elastic-perfectly plastic particles.

* Hybrid AI framework is used to link input and output parameters of the simulations.

* Equations are derived predicting equivalent plastic strain and deformation extent.

* Equations are verified by MPM and validated by impact experiments.

* A correlation is suggested for estimation of the coefficient of restitution.

## GRAPHICAL ABSTRACT

<figure><img src="image_4.png" /></figure>

## ARTICLE INFO

**Keywords:**

Artificial intelligence

Machine learning

Material Point Method

Particle impact

Large plastic deformation

Coefficient of restitution

## ABSTRACT

A hybrid AI framework has been adopted to link the impact deformation of elastic-perfectly plastic particles with their material properties and impact velocity. Initially, Material Point Method (MPM) is employed to simulate the impact of an elastic-perfectly plastic particle with a rigid wall, covering an extensive range of material properties and impact velocities. The simulation results are then integrated into the AI framework to establish the relationship between the input and output parameters. Consequently, dimensionless equations are derived to predict the equivalent plastic strain and deformation extent based on the material properties and impact velocity of the particle, showing a strong agreement with the MPM results. The identified equations reveal that both the equivalent plastic strain and deformation extent depend on and can be determined from the yield strength of the material as well as the fraction of the incident kinetic energy that is spent on inducing plastic deformation. The validity of the equations is verified by comparing the MPM and predicted values of the equivalent plastic strain and deformation extent for cases with material properties and impact velocities beyond the initial dataset used for developing the equations. The equation identified by the framework for prediction of the deformation extent

<footnote><sup>a</sup> School of Chemical and Process Engineering, University of Leeds, Leeds LS2 9JT, UK</footnote>

<footnote><sup>b</sup> Institute for Particle Technology (iPAT), Technische Universität Braunschweig, Volkmaroder Str. 5, D-38104 Braunschweig, Germany</footnote>

<footnote><sup>c</sup> LMGC, CNRS, University of Montpellier, Montpellier, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: pm16ss@leeds.ac.uk (S. Saifoori).</footnote>
is further validated by impact experiments. Additionally, a correlation is suggested to estimate the coefficient of restitution based on the material properties, covering a wide range.

# 1. Introduction

The impact of particles with surfaces is of great interest across a wide range of technologies, including particle dispersion, deposition, coating, and surface contamination, as well as a number of natural phenomena, such as hailstorms. During the impact of two deformable bodies, kinetic energy of impact is partially retained within the bodies as recoverable elastic strain energy, while the remainder is dissipated through elastic wave propagation, as well as plastic deformation, given that yielding is initiated. Considering that the energy losses due to elastic wave propagation are normally less than 4 % of the initial kinetic energy [1–4], plastic deformation of impacting deformable bodies has conventionally been studied through investigating their rebound behaviour, experimentally [5–16] or analytically [17–24]. However, experimental investigation of the impact phenomenon is arduous due to its dynamic nature, while the nonlinearity of the problem leads to inadequacy of the analytical models that use simplified assumptions. Therefore, numerical methods have widely been adopted in the recent decades to study particle impact, namely Discrete Element Method (DEM) and Finite Element Method (FEM).

Considering the impact of an elastic-perfectly plastic particle with a rigid wall, DEM is only suitable for modelling small deformations, as the existing contact models [18,25–31] use simplifying assumptions. On the other hand, FEM is capable of modelling large deformations and several studies have been carried out to investigate contact between an elastic-perfectly plastic particle and a rigid surface in quasi-static [32–38] and dynamic [39] frameworks. Nevertheless, the accuracy of FEM calculations is easily compromised by the issue of element distortion during large deformations, giving rise to numerical instabilities. This is typically addressed by remeshing and remapping, which require high computational effort and if not handled correctly, can result in more errors [40,41]. In light of this, the so-called Material Point Method (MPM) [42] has drawn attention in recent years as it bypasses this drawback. MPM can be regarded as “FEM with moving integration points”, since it utilises Lagrangian material points (rather than a mesh) to discretise the material domain. These material points are tracked during deformation and hold the state variables (stress, strain, deformation gradient, etc.). Information is mapped from the material points to the nodes of an Eulerian background mesh, which is used to solve the momentum equations. The solution is then mapped back to the material points to update their position and state variables. Considering this, MPM can conveniently accommodate large deformations, making it an appealing numerical method for modelling various extents of deformation during particle impact.

Despite its potential, MPM has seldom been employed to study the impact of deformable particles [43–46]. In our recent work [46], a comprehensive study on impact deformation of elastic-perfectly plastic spherical particles was conducted using MPM. A wide range of material properties was covered, as opposed to the limited range considered in previous similar numerical work on particle impact, e.g. [39,44]. The focus of the study was on linking the coefficient of restitution, $e$, percentage of incident kinetic energy dissipated by plastic deformation, $W_p/W_i \times 100$, and equivalent plastic strain, $\epsilon_p$ (see Section 2.2.6), to the material properties and impact conditions. The results suggested that for large deformations, $W_p/W_i \times 100$ and $\epsilon_p$ merely depended on the yield strength and incident kinetic energy of the particle, whereas for small deformations, Young's modulus of the material also affected both $W_p/W_i \times 100$ and $\epsilon_p$. It was then demonstrated that the particle yield strength and incident kinetic energy could be used to predict the flattening extent. Moreover, the coefficient of restitution was shown to be insensitive to Young's modulus for an elastic-perfectly plastic particle.

In the end, dimensionless equations were suggested to predict $e$, $W_p/W_i \times 100$ and $\epsilon_p$.

It should be noted that while our previous work [46] covered an extensive range of material properties, only one impact velocity was considered. Moreover, the empirical equations suggested for prediction of deformation behaviour were piecewise, separately covering two regions depending on whether the deformation was large or small. Therefore, the current study intends to expand on the work by including more impact velocities, in order to develop more comprehensive and universal equations that predict the particle deformation behaviour for an extensive range of material properties and impact velocities. The equations proposed previously were developed by examining the effect of different dimensionless combinations of material properties and impact velocity (input data) that suit the physics of the process. Subsequently, empirical formulations were suggested as a function of the dimensionless groups that unify the data. This is obviously a labour-intensive and time-consuming procedure, particularly when dealing with a relatively large dataset. Nevertheless, with the rapid growth of Artificial Intelligence (AI), particularly Machine Learning (ML), this constraint is overcome and large datasets can be intelligently analysed. Rather than being programmed to perform tasks, ML focuses on developing and training algorithms that can learn from the data and make predictions based on it. However, the efficiency of ML solutions/predictions heavily depends on the nature of the data and the performance of the ML algorithms, as different learning algorithms have different purposes and outcomes, depending on the data at hand [47]. Thus, it is important to select a learning algorithm that is adequate for the intended purpose [48]. In the case of data associated with nonlinear dynamic problems, such as particle impact, conventional regression methods are often inadequate. Therefore, building upon the methodologies described in prior research [49], this study employs Symbolic Regression (SR) [50], facilitated by Deep Neural Networks (DNN) [51], Genetic Algorithms (GA) [52], and Graph Attention Networks (GAT) [53] to refine and extend models that help with the prediction of particle impact behaviour. SR, which is a specialized form of ML, enables the identification of mathematical models that express the underlying physical laws from observed data in a concise form. The computational power of DNNs allows for identifying relevant mathematical expressions that capture the complex relationships within the data, by efficient navigation of the vast solution spaces. This process is complemented by Genetic Algorithms that optimise the expressions, ensuring that the models are not only accurate, but also generalizable across different conditions of material properties and impact velocities. The inclusion of GAT significantly improves the analytical capabilities of the model. Finally, integration of these different techniques to supplement the computational capabilities of MPM allows for convenient development of comprehensive models that predict and elucidate the deformation behaviour of impacting particles under varied operational scenarios. This advanced approach aims to bridge the gap between empirical data analysis and theoretical physics, offering deeper insights into the dynamics of particle impact phenomenon.

# 2. Methodology

## 2.1. Material Point Method simulation setup

For the current study, an in-house MPM code developed by Nezamabadi and Radjai [54] is implemented to investigate normal impact of an elastic-perfectly plastic particle with a rigid wall. An explicit total Lagrangian formulation of MPM (TLMPM) [55,56], coupled with an implicit contact algorithm is used to model the particle. In TLMPM, the
reference configuration is fixed and the material points are always associated with their initial positions. This overcomes the cell-crossing instability issue of the classic updated Lagrangian MPM (ULMPM). Details of the method can be found in [46,54]. The particle is modelled as a sphere with radius $R = 250$ µm and Poisson's ratio $v = 0.35$, normally impacting a rigid wall at five different impact velocities ($V_i$) of 10, 20, 30, 40 and 50 m/s. The wall is placed close to the particle (30 µm) to reduce the computational time. This is three times the distance between the material points to ensure that the material points are not interacting with/touching the wall before contact is made (Fig. 1). In order to include a wide range of material properties, four different densities ($ρ$) of 1000, 2000, 4000 and 8000 kg/m³ are considered for the particle. For each density, three different values of 1, 10 and 100 GPa for Young's modulus ($E$), and eight different values of 20, 40, 80, 160, 320, 640, 1280 and 2560 for the ratio of Young's modulus to yield strength ($E/Y$, where $Y$ denotes the yield strength) are taken into account, leading to a total of 480 cases being investigated. The range is selected based on the properties of common materials that exhibit both elastic and plastic properties (polymers and metals), with reference to [57]. After conducting a sensitivity analysis (verified using Johnson's [17] analytical model for impact of elastic particles [46]), the following discretisation settings are used: the particle is discretised into 74,227 material points. The mesh size is adjusted in a way that the ratio of the element size to the distance between the material points in each dimension is 1.5 for cases with a Young's modulus of 100 GPa, and 1.05 for cases with a Young's modulus of 1 GPa. For cases with a Young's modulus of 10 GPa, this is set as 1.5 when the density is 1000 and 2000 kg/m³, and as 1.05 when the density is 4000 and 8000 kg/m³. These settings affect the time step and are selected in this manner to reduce errors while optimising the computational cost. The ratios of 1.5 and 1.05 result in a total of 50,653 and 132,651 hexahedral elements for the mesh, respectively. A time step of 10 ps is selected following the approach suggested by de Vaucorbeil et al. [55], based on the element size, the dilatational wave speed and the impact velocity of the case with the largest Young's modulus, lowest density and the highest impact velocity (the case that requires the smallest time step for accuracy). Damping is not applied. An elastic-perfectly plastic constitutive law is implemented and the von Mises yield criterion is considered. Based on the simulation results for the considered material properties and impact velocities, the capability of the code in modelling deformation for an elastic-perfectly plastic material is limited to the equivalent plastic strain values (as defined by Eq.

<figure><img src="image_5.png" /><figcaption>Fig. 1. Visualization of the modelled particle, detailing the initial and boundary conditions (illustration is not to scale for presentation purposes).</figcaption></figure>

(7) in Section 2.2.6) ranging from $2.6 \times 10^{-6}$ (slight deformation) to 1.53 (extensive deformation).

## 2.2. AI hybrid approach

A custom-developed AI hybrid model is utilised to find the link between the MPM input data (material properties and impact velocity) and output data (parameters describing the deformation behaviour of the impacting particle). The AI model implements Symbolic Regression (SR), integrated with Deep Neural Networks (DNN) [51], Genetic Algorithms (GA) [52], and Graph Attention Networks (GAT) [53]. For a comprehensive description of the employed AI techniques (DNN, GA and GAT), the reader is referred to [49,58]. A brief description of the general workflow, shown by Fig. 2, is as follows: The first step in the workflow is typically Dimensional Analysis (DA), where the input and output data are analysed to determine dimensionless groups that represent the physics of the process. These groups are then used to update the original dataset, in order to refine the input for the subsequent stages. After DA, the process moves to the next stage, in which Symbolic Regression (SR) techniques are used. In this stage, several tools and techniques such as GPLearn [59], Graph Attention Networks (GAT), Fragment Selection Techniques (FST, developed in-house) and several other methods are utilised to identify the potential mathematical relationships within the data. If the first round of SR does not produce satisfactory solutions, the workflow calls for iteration. This includes revisiting the results to make any necessary recalibrations to refine the model outputs, after which SR is applied again. In case the modified results are still unsatisfactory, the process is extended to a deeper level of problem solving, involving Deep Neural Networks (DNN) that are improved through Hyperparameter Optimisation (HPO). This approach is used to tune the model parameters to best fit the data. During this stage, the DNN employs a user-provided equation library (based on prior knowledge of the system) to check for existing patterns and solutions. It should be noted that the aforementioned stages are iterative, continuously looping back to earlier steps as needed until a solution is found. After a successful solution, the model selects the configuration with the highest fitness from the tested SR outputs and completes the process.

In the current study, the best results are achieved with the GAT model combined with advanced pre-processing and data preparation strategies. Therefore, a detailed description of these stages of the framework is provided in the following sections. Other components, such as DNNs, are not directly applied to the final results of this study,

<figure><img src="image_6.png" /><figcaption>Fig. 2. Flowchart of the implemented hybrid AI framework.</figcaption></figure>
and therefore are not discussed here. The reader is referred to [49,58] for a detailed description of DNNs typically applied in the framework.

### 2.2.1. Pre-processing and augmentation techniques

Initially, the process begins by fetching the raw dataset, named “original_data”, which often includes a variety of redundant or irrelevant features. To refine this dataset, Recursive Feature Elimination (RFE) driven by a Random Forest Algorithm is applied to select only the most significant variables. This step is crucial for reducing dimensionality, thereby enhancing both the accuracy and interpretability of the models. Alongside RFE, Spearman's correlation analysis [60] is employed (preferred over that of Pearson for its ability to capture non-linear relationships), to further inspect the feature interactions through heat map visualization. The results are expressed as scores, assisting in identification and elimination of redundant or collinear features. Following RFE, the “original_data” undergoes an ‘Expand & Reorder’ process where transformations such as cropping and rotating are applied. This enhances data diversity and robustness, resulting in the augmented dataset “train_data”, which improves the generalizability of subsequent modelling phases.

### 2.2.2. Graph Attention Network (GATv2) integration

The Graph Attention Network (GATv2) [61] is defined with specific configurations to intelligently handle the complexities of the graph-structured data. GATv2 builds upon traditional Graph Convolutional Networks (GCNs) [62,63] by integrating dynamic attention mechanisms that adjust the influence of node interactions based on their relevance, refining the model adaptability and interpretability. At this stage, the “train_data” is split into training and validation subsets, with 80 % used for training and the remaining 20 % for validation. This ensures the model learns to generalise beyond the training data and accurately performs on new, unseen data.

### 2.2.3. Data augmentation phase

New data points, called “predict_data”, are generated from the existing “train_data” using the trained GATv2 model. These “predict_data” points are then combined with the “train_data”, and the combined dataset is saved as “predict_data”. This phase assesses the framework’s capability to handle unfamiliar or novel data scenarios. The “predict_data” undergoes further refinement, where outliers are removed and biases adjusted, ensuring the data remains representative and accurate for further analysis.

### 2.2.4. Symbolic Regression and mathematical modelling

In this study, Symbolic Regression (SR) using GPLearn techniques is systematically applied to three datasets: “original_data”, “train_data”, and “predict_data”. The details of how the GPLearn model is applied as an SR model are described in detail in [49,58]. Each application of SR serves a distinct purpose. Initially, SR interprets the basic input-output relationships in “original_data”. Subsequently, the generalisation ability of the model is examined with “train_data”. Finally, the prediction accuracy and practical utility of the model are evaluated using “predict_data”. This multi-level application helps validate the complex interactions and relationships captured by the model. The framework employs symbolic manipulation tools to identify and extract stable, recurring mathematical patterns across these datasets. These patterns represent the underlying physical laws and result in a unified symbolic equation or set of equations that concisely describe the data across all scenarios.

The performance of the framework and the fit of the predicted models to the MPM data are evaluated using the mean absolute error (MAE), mean squared error (MSE) and the coefficient of determination (R²), shown by the equations below:

$$ \text{MAE} = \frac{\sum_{i=1}^{n} |y_{\text{pred},i} - y_{\text{MPM},i}|}{n} \qquad (1) $$

$$ \text{MSE} = \frac{\sum_{i=1}^{n} (y_{\text{pred},i} - y_{\text{MPM},i})^2}{n} \qquad (2) $$

$$ R^2 = 1 - \frac{\sum_{i=1}^{n} (y_{\text{pred},i} - y_{\text{MPM},i})^2}{\sum_{i=1}^{n} \left( y_{\text{pred},i} - \frac{1}{n} \sum_{i=1}^{n} y_{\text{MPM},i} \right)^2} \qquad (3) $$

where $y_{pred}$ and $y_{MPM}$ are respectively the predicted and MPM values of an output, and $n$ is the number of the data points.

### 2.2.5. Input data

In the current work, rather than using DA, the number of input data ($V_i$, $\rho$, $E$ and $Y$) is reduced by replacing the individual variables with dimensionless groups that are of significance to the physics of the studied problem, previously identified in [46]. Group $E/Y$, which represents the mechanical properties of the material, is considered as an input, as it repeatedly emerges in formulae connected to plastic deformation in fundamental studies, e.g. [17,64–66]. To include the effect of particle impact velocity and density (and consequently, the kinetic energy of impact), group $\rho V_i^2/Y$ is decided as another input. This group is used by Johnson [17] to distinguish different regimes of deformation behaviour for impact of metal particles. Additionally, group $\rho(V_i^2-V_y^2)/Y$ is also considered as an input, in which $V_y$ is the “yield velocity” (the velocity required for onset of plastic deformation), as shown by Eq. (4) below [17]:

$$ V_y = \left( \frac{26Y}{\rho} \right)^{1/2} \left( \frac{Y}{E} \right)^2 \qquad (4) $$

where $E^*$ is the effective Young’s modulus of the material defined as $E^* = E/(1-v^2)$, noting that the wall is rigid. Group $\rho(V_i^2-V_y^2)/Y$ not only considers the effect of all the input variables, it also accounts for the portion of the incident kinetic energy that is spent on plastic deformation, i.e. by deducting the incident kinetic energy required for onset of yielding (represented by $\rho V_y^2$) from the total kinetic energy (represented by $\rho V_i^2$).

### 2.2.6. Output data

The output data considered for the study of deformation behaviour include the coefficient of restitution, $e$, the equivalent plastic strain, $\epsilon_p$, deformation extent, $1-(H_d/D)$ and the ratio of plastic work/energy to the incident kinetic energy, expressed by $1-e^2$. The coefficient of restitution is calculated using Eq. (5):

$$ e = \frac{V_r}{V_i} \qquad (5) $$

where $V_r$ is the magnitude of the rebound velocity of the particle in the impact direction. In order to find $V_r$, the velocity of the particle in the impact direction is first calculated for each time step using the following equation:

$$ \sum_{p=1}^{N_p} m_p v_p^2 / \sum_{p=1}^{N_p} m_p \qquad (6) $$

where $m_p$ and $v_p^2$ are respectively the mass of material point $p$ and its velocity in the impact direction (z axis here), and $N_p$ is the number of the material points. Subsequently, the first instant at which the calculated velocity reaches a constant value after contact is lost is considered the rebound instant and the corresponding velocity is taken as the rebound
velocity.

The equivalent plastic strain is calculated from Eq. (7) at the instant of rebound, where $\epsilon_{ij}^p$ is the deviatoric plastic strain.

$$ \epsilon_p = \sqrt{\frac{2}{3} \epsilon_{ij}^p \epsilon_{ij}^p} \qquad (7) $$

The deformation extent, $1-(H_d/D)$, is calculated using the flattening extent of the particle, $H_d/D$, where D is the particle diameter before impact, and $H_d$ is the length of the imaginary centre line connecting the top and bottom of the particle, parallel to the impact direction after rebound, as shown in Fig. 3. It should be mentioned that $H_d$ is measured after rebound as an average value, due to the fluctuations caused by the elastic waves within the particle, as damping is not considered.

# 3. Results and discussion

## 3.1. Equivalent plastic strain

The distribution of plastic strain inside the particle is displayed in Fig. 4 as contours of plastic strain in the impact direction ($\epsilon_{22}^p$) shortly after rebound for the case $\rho = 2000 \text{ kg/m}^3$, $E = 1 \text{ GPa}$ and $E/Y = 320$, as an example. It should be noted that the contours are on the surface of a plane slicing the particle in half, with the normal of the plane pointing towards the observer (along the x axis). Accordingly, there is extensive plastic strain (up to -3.3) in the contact area, accompanied by jetting at the bottom edges of the particle, though the top part of the particle remains mostly undeformed. Moreover, there is a slight positive strain close to the top edge of the particle, attributed to the reflection of the compressive elastic waves within the particle as tensile waves.

The equivalent plastic strain, $\epsilon_p$ (described in Section 2.2.6), is analysed by the Random Forest Variable Importance method, and the RFE and correlation scores of all the input variables are presented in Fig. 5. Considering both RFE and correlation scores, the yield strength of the material is the most important variable in influencing the equivalent plastic strain, as intuitively expected. This is followed by the particle impact velocity, density, yield velocity and Young's modulus, respectively.

The equation found by the framework for the equivalent plastic strain, along with the corresponding MAE, MSE and $R^2$ values with respect to the MPM data are displayed in Table 1. The identified equation is an excellent fit to the MPM data, while containing all the influential variables detected by the Random Forest Variable Importance method (Fig. 5). This is further shown by Fig. 6, where the $\epsilon_p$ values obtained from the MPM simulations fall perfectly on the line of equality when plotted as a function of the correlation identified by the framework. Moreover, the predicted values of $\epsilon_p$ are very close to the MPM values for all the data points, as shown in Fig. 1 of the Supplementary Data file.

According to Saifoori et al. [46], when an elastic-perfectly plastic particle undergoes large deformation, $\epsilon_p$ is only affected by the group $\rho V_i^2/Y$, which accounts for the incident kinetic energy and yield strength

<figure><img src="image_7.png" /><figcaption>Fig. 3. Illustration of parameters D and $H_d$ used in calculation of the extent of deformation, $1-(H_d/D)$.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 4. Contours of plastic strain in the impact direction ($\epsilon_{22}^p$) shortly after rebound for case $\rho = 2000 \text{ kg/m}^3$, $E = 1 \text{ GPa}$ and $E/Y = 320$. The contours are on a plane slicing the particle in half (normal of the plane is along the x axis).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 5. Random Forest Variable Importance scores for the equivalent plastic strain, $\epsilon_p$.</figcaption></figure>

**Table 1**

The equation identified by the framework for the equivalent plastic strain, $\epsilon_p$, and the corresponding MAE, MSE and $R^2$ values with respect to the MPM data.

<table><thead><tr><th>Identified Equation</th><th>MAE</th><th>MSE</th><th>R<sup>2</sup></th></tr></thead><tbody><tr><td>$\epsilon_p = 0.475 \left[ \frac{\rho (V_i^2 - V_y^2)}{Y} \right]$</td><td>0.003</td><td>1.556 &times; 10<sup>-5</sup></td><td>0.999</td></tr></tbody></table>

of the material. Alternatively, when the extent of deformation is small, Young's modulus of the material also has an effect on $\epsilon_p$, which is reflected through the insertion of the dimensionless group $E/Y$ into the equation for small deformation. However, as opposed to the piecewise equation suggested previously [46], the current correlation appears to be adequate for both large and small deformation, though it seemingly overlooks the influence of $E/Y$ (Fig. 6). This is due to the fact that the effect of $E/Y$ on $\epsilon_p$ is already accounted for in the current correlation by replacing $\rho V_i^2$ with $\rho(V_i^2-V_y^2)$, as $E/Y$ is a component of $V_y$. Thus, by considering the portion of the incident kinetic energy that is spent on plastic deformation (which is dependent on both the mechanical properties and impact velocity) rather than the entirety of the incident kinetic energy, the correlation can predict the equivalent plastic strain, regardless of the deformation extent. Consequently, considering the identified correlation and Fig. 6, $\epsilon_p$ intuitively increases with increase in $\rho(V_i^2-V_y^2)/Y$, as the increase in the latter is due to a decrease in the yield strength of the material, or an increase in the energy that is consumed by plastic deformation.
<figure><img src="image_10.png" /><figcaption>Fig. 6. Equivalent plastic strain, $\epsilon_p$, obtained from the MPM simulations as a function of the equation identified by the framework, 0.475[ρ(Vᵢ²-Vⱼ²)/Y].</figcaption></figure>

## 3.2. Deformation extent

The deformation extent, as described in Section 2.2.6, is analysed by the Random Forest Variable Importance method. The scores of all the input variables for the prediction of the deformation extent, 1-(H₉ /D), are displayed in Fig. 7. Accordingly, groups ρ(Vᵢ²-Vⱼ²)/Y and ρVᵢ²/Y strongly influence the deformation extent. This is respectively followed by the yield velocity, impact velocity, yield strength, Young's modulus, group E/Y, and particle density.

The identified equation for the deformation extent, along with the corresponding MAE, MSE and R² with respect to the MPM values are shown in Table 2. Moreover, 1-(H₉ /D) is plotted as a function of the identified equation, shown in Fig. 8. The predicted values for all the data points are compared to the MPM values, given in Fig. 2 of the Supplementary Data file. Considering MAE, MSE and R² values in Table 2, the identified equation is a good fit to the MPM data, though Fig. 8 shows that the predicted values are slightly underestimated with respect to the MPM values, especially for large deformation. Nevertheless, the equation identified by the framework provides a fair prediction of the MPM values, as well as including the most influential group identified by the Random Forest Variable Importance technique (Fig. 7).

Based on the identified equation (Table 2) and consistent with the observations for $\epsilon_p$, the extent of deformation for an elastic-perfectly plastic particle impacting a rigid wall depends on the yield strength of

<figure><img src="image_11.png" /><figcaption>Fig. 7. Random Forest Variable Importance scores for the deformation extent, 1-(H₉ /D).</figcaption></figure>

<caption>Table 2</caption>

The equation identified by the framework for the deformation extent, 1-(H₉ /D), and the corresponding MAE, MSE and R² values with respect to the MPM data.

<table><thead><tr><th>Identified Equation</th><th>MAE</th><th>MSE</th><th>R<sup>2</sup></th></tr></thead><tbody><tr><td>1 - H<sub>d</sub> / D = 0.33 [ρ(V<sub>i</sub><sup>2</sup> - V<sub>j</sub><sup>2</sup>) / Y]<sup>0.66</sup></td><td>0.006</td><td>4.855 × 10<sup>-5</sup></td><td>0.997</td></tr></tbody></table>

<figure><img src="image_12.png" /><figcaption>Fig. 8. The deformation extent, 1-(H₉ /D), obtained from the MPM simulations as a function of the equation identified by the framework, 0.33[ρ(Vᵢ²-Vⱼ²)/Y]<sup>0.66</sup>.</figcaption></figure>

the material and the fraction of the incident kinetic energy expended on plastic deformation, as intuitively expected. Additionally, Fig. 8 suggests that 1-(H₉ /D) increases with a decrease in the yield strength of the material, or an increase in the energy spent on inducing plastic flow. It is important to note that the identified equation allows for prediction of the deformation extent based on the material properties and impact velocity. This is valuable for coating techniques like aerosol deposition and cold spraying, where the deformation extent of the particles affects the quality of the final coating, i.e. the impact velocity required to induce a desired extent of deformation can be determined based on the material properties of the feed particles.

## 3.3. Numerical verification

In order to verify the adequacy of the identified equations, MPM simulations are carried out for two further cases with material properties and impact velocities outside the original input data range. Subsequently, the MPM values of the equivalent plastic strain and deformation extent are compared to the values obtained using the equations in Table 1 and Table 2. For the first case, a lead particle with an impact velocity of 5 m/s is considered. The second case addresses the impact of a nickel particle at 80 m/s. The material properties for both of the particles are taken from [57], as shown in Table 3. For both cases, the

<caption>Table 3</caption>

Material properties of lead (Pb) and nickel (Ni) used in the MPM simulations for verification, taken from [57].

<table><thead><tr><th>Material</th><th>ρ (kg/m³)</th><th>E (MPa)</th><th>Y (MPa)</th><th>ν (–)</th></tr></thead><tbody><tr><td>Pb</td><td>11,400</td><td>15,800</td><td>5.90</td><td>0.44</td></tr><tr><td>Ni</td><td>8900</td><td>204,000</td><td>148</td><td>0.31</td></tr></tbody></table>
particle is modelled as a sphere with the same settings discussed in Section 2.1. However, the mesh size is adjusted here in a way that the ratio of the element size to the distance between the material points in each dimension is 1.5 for the nickel particle and 1.05 for the lead particle.

Table 4 provides the MPM values of $\epsilon_p$ and 1-(H$_d$ / D) for lead and nickel particles, along with those calculated using the equations identified by the Hybrid AI Framework (HAIF). As expected, the nickel particle undergoes larger deformation (17 % of its original diameter) compared to the lead particle (5 % of its original diameter), as it has a significantly larger impact velocity. Considering Table 4, the predicted values by the framework for both $\epsilon_p$ and 1-(H$_d$ / D) are very close to the MPM values, despite the fact that the material properties and impact velocities of the tested cases are outside the range of the original input data that have been used to develop the equations. This verifies the adequacy of the equations and confirms the capability of the framework for finding appropriate equations that describe the relationship between the input and output data.

### 3.4. Experimental validation

For the purpose of validating the identified equation for 1-(H$_d$ / D), impact experiments are carried out using spherical copper particles (Goodfellow Cambridge Limited, Cambridge, UK) with the properties shown in Table 5 (provided by the supplier). With reference to the work of Mao et al. [67], the copper particles used in the current work are expected to have a very small strain-rate sensitivity (regardless of the strain rate they experience), as their grain size does not exceed 10 µm. Thus, they are assumed to be elastic-perfectly plastic. It should be noted that the reported values of the yield strength vary widely in the range 54 to 270 MPa for soft and hard particles, respectively. Thus, an average value is additionally used, based on the arithmetic mean value of the aforementioned. The particles are subjected to impact on a rigid target using an in-house device called the “single particle impact tester” (Fig. 9). The device comprises an air eductor which is connected to a collection chamber via an acceleration tube. The collection chamber, in which the target assembly is located, is connected to a vacuum pump with adjustable pressure. Once the vacuum pump is adjusted, the pressure difference between the collection chamber and the inlet of the rig (where the particles are fed) provides different air velocities in the acceleration tube and consequently, different particle impact velocities.

The particle impact velocity is first established against the applied vacuum for large particles (350 to 1000 µm) of different materials by the use of high-speed motion analysis (velocities ranging from 1.5 to 53 m/s). As high-speed imaging cannot track the small particles used in this work, the impact velocity is estimated using a non-dimensional empirical correlation (specifically developed for the device based on the aforementioned analysis of large particles), as shown by Eq. (8) below:

$$N_v^{\frac{1}{3}} = 240.527 \left[ \frac{\left( \frac{p}{\rho_{air}gh} \right)^{0.472}}{(S-1)^{0.624} (N_d)^{0.221}} \right] \quad (8)$$

where $N_v$ and $N_d$ are dimensionless particle impact velocity and size, respectively. P is the vacuum gauge pressure (positive values), $\rho_{air}$ is the air density, g is the gravitational acceleration and h is the full length of

<table><caption>**Table 4**
The values of $\epsilon_p$ and 1-(H$_d$ / D) for lead and nickel particles calculated by MPM and the equations identified by the Hybrid AI framework (HAIF).</caption><thead><tr><th rowspan="2">Method/Equation</th><th colspan="2">ε<sub>p</sub> (-)</th><th colspan="2">1-(H<sub>d</sub> / D) (-)</th></tr><tr><th>Pb</th><th>Ni</th><th>Pb</th><th>Ni</th></tr></thead><tbody><tr><td>MPM</td><td>0.02</td><td>0.17</td><td>0.05</td><td>0.17</td></tr><tr><td>HAIF</td><td>0.02</td><td>0.18</td><td>0.04</td><td>0.17</td></tr></tbody></table>

<table><caption>**Table 5**
Material properties of copper particles (Cu) used in the impact experiments for validation, taken from the supplier.</caption><thead><tr><th>Material</th><th>D (µm)</th><th>ρ<sub>p</sub> (kg/m<sup>3</sup>)</th><th>E (MPa)</th><th>Y (MPa)</th><th>ν (-)</th></tr></thead><tbody><tr><td>Cu</td><td>&lt;150</td><td>8960</td><td>129,808</td><td>54-162-270</td><td>0.343</td></tr></tbody></table>

<figure><img src="image_13.png" /><figcaption>**Fig. 9.** Schematic drawing of the single particle impact tester device.</figcaption></figure>

the acceleration tube (0.401 m). $S = \rho_p/\rho_{air}$, where $\rho_p$ is the particle density. $N_v$ and $N_d$ are calculated from Eqs. (9) and (10), respectively:

$$N_v^{\frac{1}{3}} = V_i \left[ \frac{3}{4g\nu(S-1)} \right]^{\frac{1}{3}} \quad (9)$$

$$N_d^{\frac{1}{3}} = D \left[ \frac{4g(S-1)}{3\nu^2} \right]^{\frac{1}{3}} \quad (10)$$

where $\nu = \mu_{air}/\rho_{air}$ with $\mu_{air}$ being the viscosity of air. To demonstrate the reliability of the correlation in predicting the impact velocity, the experimental values of the dimensionless impact velocity calculated by Eq. (9), $N_{v,e}^{1/3}$, are plotted as a function of those predicted by Eq. (8), $N_{v,p}^{1/3}$, for the large particles mentioned before, as shown in Fig. 10. The graph suggests that the developed correlation provides acceptable predictions of the impact velocity for all the large particles.

For the current experiment, a vacuum gauge pressure of 19 kPa is selected and a small amount of copper particles is fed to the device using a vibratory trough. After the impact, the particles are collected from the collection chamber in order to image their deformed area using Scanning Electron Microscopy (SEM). For this purpose, the collected particles are poured onto a glass slide, which is gently shaken so that they fall on their plane of maximum stability, i.e. their deformed area. Subsequently, an SEM stub with a carbon adhesive tape on top is gently pressed on the glass slide to pick up the particles. A Hitachi TM3030 table top microscope (Hitachi Ltd., Tokyo, Japan) is used to capture images of the particles with their deformed area presented for viewing by the microscope. An example of such images is shown in Fig. 11 (a). Fiji ImageJ software [68] is then used to measure the projected area of the particle along with the deformed area, from which the particle diameter, D, and diameter of the deformed area, $d_d$, are obtained, respectively. $H_d$ is estimated from D and $d_d$, following a geometrical assumption, as illustrated in Fig. 11 (b). Moreover, using the measured
<figure><img src="image_14.png" /><figcaption>**Fig. 10.** The experimental values of the dimensionless impact velocity calculated by Eq. (9), $N_{v,e}^{1/3}$, as a function of those predicted by Eq. (8), $N_{v,p}^{1/3}$, for all the large particles used in developing the correlation: high density polyethylene (HDPE), polypropylene homopolymer (PP Homo), polypropylene random copolymer (PP RaCo), glass beads, and SAC305 solder balls.</figcaption></figure>

$D$, $N_d$ is calculated from Eq. (10) and inserted in Eq. (8) to find $N_v$. Finally, $V_i$ is obtained from Eq. (9).

The analysis is carried out for 18 different particles. The measured particle diameter, $D$, estimated particle impact velocity, $V_i$, along with the measured and predicted (using the identified equation in Table 2) values of the flattening extent are shown in Table 6. The measured flattening extent is denoted by $(H_d/D)_m$, while the values predicted by the framework, considering yield strengths of $Y = 54, 162$ and $270$ MPa, are denoted by $(H_d/D)_{Y54}$, $(H_d/D)_{Y162}$ and $(H_d/D)_{Y270}$, respectively. The variations in the input variables $D$ and $V_i$ are shown in Table 6, while the values for the flattening extent are reported as the arithmetic mean

<figure><img src="image_15.png" /><figcaption>**Fig. 12.** Random Forest Variable Importance scores for the coefficient of restitution, $e$.</figcaption></figure>

for the 18 particles. Evidently, considering a yield strength of 162 MPa, the experimental results are in perfect agreement with the predicted values obtained from the equation identified by the framework (Table 2). This is further depicted in Table 7 of the Appendix, where $(H_d/D)_m$ and $(H_d/D)_{Y162}$ values for each individual particle are provided. This highlights the fact that the precision of the predictions is highly dependent on the accuracy of the mechanical properties used in the calculations.

### 3.5. Coefficient of restitution

Random Forest Variable Importance scores of all the input variables for prediction of the coefficient of restitution, $e$, are displayed in Fig. 12. Accordingly, $e$ is largely affected by the yield velocity, followed by group $\rho(V_i^2-V_y^2)/Y$, group $\rho V_i^2/Y$, impact velocity, yield strength, group $E/Y$, Young's modulus and density, respectively.

<figure><img src="image_16.png" /><figcaption>**Fig. 11.** Measuring the particle diameter, $D$, and the diameter of the deformed area, $d_d$, after impact for copper particles: (a) an example of the SEM images used for analysis, with the deformed area presented for viewing by the microscope, and (b) schematic diagram of the geometrical approach taken to estimate $H_d$ using $D$ and $d_d$.</figcaption></figure>

**Table 6**

The measured particle diameter, $D$, estimated particle impact velocity, $V_i$, along with the measured values of the flattening extent, $(H_d/D)_m$, as well as the values predicted by the framework, considering yield strengths of $Y = 54, 162$ and $270$ MPa, denoted by $(H_d/D)_{Y54}$, $(H_d/D)_{Y162}$ and $(H_d/D)_{Y270}$, respectively, for 18 different copper particles after impact.

<table><thead><tr><th>D (μm)</th><th>V<sub>i</sub> (m/s)</th><th>(H<sub>d</sub>/D)<sub>m</sub> (–)</th><th>(H<sub>d</sub>/D)<sub>Y54</sub> (–)</th><th>(H<sub>d</sub>/D)<sub>Y162</sub> (–)</th><th>(H<sub>d</sub>/D)<sub>Y270</sub> (–)</th></tr></thead><tbody><tr><td>81 ± 5</td><td>35 ± 1</td><td>0.94</td><td>0.88</td><td>0.94</td><td>0.96</td></tr></tbody></table>
<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /><figcaption>**Fig. 13.** Identifying a correlation for the coefficient of restitution, *e*: (a) the relationship between the dimensionless groups $1-e^2/\epsilon_p$ and $2.42[Y/\rho(V_i^2-V_y^2)]$, as identified by the framework and (b) the link between the equivalent plastic strain, $\epsilon_p$, and the deformation extent, $1-(H_d/D)$.</figcaption></figure>

The framework identifies a linear relationship between the dimensionless groups $1-e^2/\epsilon_p$ and $Y/\rho(V_i^2-V_y^2)$, as shown in Fig. 13 (a). The group $1-e^2$ represents the ratio of the plastic work to the incident kinetic energy, by deducting the rebound energy ($1/2mV_i^2$) from the incident kinetic energy ($1/2mV_y^2$) to find the plastic work. Thus, for cases where the fraction of the incident kinetic energy spent on plastic work is large, the rebound velocity of the particle decreases, leading to an increase in $1-e^2$. On the other hand, higher values of $1-e^2/\epsilon_p$ correspond to cases of small deformation, even though both $1-e^2$ and $\epsilon_p$ decrease in such cases. The reason for this is that in instances of smaller deformation, the value of $\epsilon_p$ is significantly smaller than $1-e^2$ (by at least two orders of magnitude), resulting in an overall increase in the ratio $1-e^2/\epsilon_p$. This is well demonstrated in Fig. 13 (a) by the trend of increase in $1-e^2/\epsilon_p$ with increase in $Y/\rho(V_i^2-V_y^2)$, i.e. an increase in the yield strength of the material or a decrease in the energy that is consumed by plastic deformation. Moreover, Fig. 13 (a) shows that even though $1-e^2/\epsilon_p$ correlates well with $2.42[Y/\rho(V_i^2-V_y^2)]$ for the majority of the studied cases (up to 2.42

$[Y/\rho(V_i^2-V_y^2)] = 1.2 \times 10^4$), there is scatter in the data for the cases corresponding to very small deformation. This can be attributed to either the sensitivity of the unloading dynamics to the selected time step (especially for very small deformation), or the energy dissipation caused by the use of a lumped mass matrix in the MPM approach [55,69].

Considering Fig. 13 (a), the coefficient of restitution can be found using the material properties and $\epsilon_p$. However, unlike $1-(H_d/D)$, $\epsilon_p$ is merely a numerical parameter in MPM and cannot be measured realistically. Therefore, it is beneficial to find the link between $\epsilon_p$ and $1-(H_d/D)$, in order to use them interchangeably as required. This is done by a simple regression analysis for the MPM values of $1-(H_d/D)$ and $\epsilon_p$, leading to the correlation shown in Fig. 13 (b).

## 4. Conclusions

A hybrid AI framework has been implemented to assist with the generalisation of the trends obtained from numerical simulations of impact deformation for elastic-perfectly plastic particles by MPM. First, MPM is used to simulate the impact of an elastic-perfectly plastic particle with a rigid wall, considering a wide range of material properties and impact velocities. The results are then fed to the AI framework in order to identify the link between the input parameters (material properties and impact velocity) and the resulting output, i.e. the equivalent plastic strain, $\epsilon_p$, the deformation extent, $1-(H_d/D)$, and the ratio of plastic work to the incident kinetic energy of the impacting particle, expressed by $1-\epsilon^2$. As a result, dimensionless equations are obtained for the prediction of the equivalent plastic strain and deformation extent, showing a strong correlation with the MPM results. The adequacy of the identified equations is then verified by comparing the MPM and predicted values of $\epsilon_p$ and $1-(H_d/D)$ for test cases with material properties and impact velocities which are outside the original studied data range. Furthermore, the equation identified by the framework for the prediction of $1-(H_d/D)$ is validated by impact experiments using copper particles. Moreover, the framework detects a linear equation for relating the dimensionless group $1-e^2/\epsilon_p$ to the material properties and impact velocity. This is a good fit for the majority of the studied cases, though there is scatter in the data for cases of very small deformation, due to numerical errors caused by either the selected time step, or the use of a lumped mass matrix. By replacing $\epsilon_p$ with $1-(H_d/D)$ through their link, the correlation for $1-e^2/\epsilon_p$ can be used to give an estimate of the coefficient of restitution. In this first attempt, only materials that are not strain-rate sensitive have been studied and the results are validated using experiments. The foundation is therefore laid for the work to be expanded by including different constitutive laws that account for thermo-mechanical effects in the future. Ultimately, the implementation of the current hybrid AI approach not only enhances our understanding of the dynamics inherent in the MPM data, but also improves the predictive accuracy and practical value of the identified models.

### CRediT authorship contribution statement

**Saba Saifoori:** Writing – original draft, Visualization, Validation, Methodology, Investigation, Formal analysis, Data curation. **Somayeh Hosseinhashemi:** Writing – original draft, Visualization, Formal analysis. **Mohammad Alasossi:** Data curation. **Carsten Schilde:** Supervision. **Saeid Nezamabadi:** Writing – review & editing, Supervision, Software, Conceptualization. **Mojtaba Ghadiri:** Writing – review & editing, Supervision, Resources, Project administration, Funding acquisition, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
## Acknowledgements

The support of Engineering and Physical Sciences Research Council (EPSRC) under grant EP/S029036/1 and its Principal Investigator, Professor Steven Milne is gratefully acknowledged. This work was undertaken on ARC4, part of the High Performance Computing facilities at the University of Leeds, UK. The authors also acknowledge the support

of the French Agence Nationale de la Recherche (ANR), under grant ANR-20-CE08-0011 (PaMaCo project), and that of the Bundesministerium für Bildung und Forschung (BMBF) for funding within the “Kompetenzcluster Intelligente Batteriezellproduktion” (InZePro) program and the Deutsche Forschungsgemeinschaft (DFG) within the Heisenberg program.

## Appendix A. Appendix

<caption>Table 7</caption>

The measured particle diameter, *D*, estimated particle impact velocity, $V_i$, along with the measured values of the flattening extent, $(H_d/D)_m$, as well as the values predicted by the framework, considering a yield strength of $Y = 162$ MPa, denoted by $(H_d/D)_{Y162}$, for 18 different copper particles after impact.

<table><thead><tr><th>D (μm)</th><th>V<sub>i</sub> (m/s)</th><th>(H<sub>d</sub>/D)<sub>m</sub> (-)</th><th>(H<sub>d</sub>/D)<sub>Y162</sub> (-)</th></tr></thead><tbody><tr><td>88.90</td><td>34.94</td><td>0.94</td><td>0.94</td></tr><tr><td>88.44</td><td>34.98</td><td>0.94</td><td>0.94</td></tr><tr><td>84.62</td><td>35.32</td><td>0.94</td><td>0.94</td></tr><tr><td>84.33</td><td>35.35</td><td>0.94</td><td>0.94</td></tr><tr><td>82.96</td><td>35.48</td><td>0.95</td><td>0.94</td></tr><tr><td>81.09</td><td>35.66</td><td>0.94</td><td>0.94</td></tr><tr><td>81.04</td><td>35.66</td><td>0.95</td><td>0.94</td></tr><tr><td>79.99</td><td>35.76</td><td>0.93</td><td>0.94</td></tr><tr><td>76.84</td><td>36.08</td><td>0.93</td><td>0.94</td></tr><tr><td>76.34</td><td>36.13</td><td>0.94</td><td>0.94</td></tr><tr><td>75.55</td><td>36.22</td><td>0.93</td><td>0.94</td></tr><tr><td>75.39</td><td>36.23</td><td>0.94</td><td>0.94</td></tr><tr><td>72.70</td><td>36.53</td><td>0.94</td><td>0.94</td></tr><tr><td>93.39</td><td>34.90</td><td>0.94</td><td>0.94</td></tr><tr><td>85.85</td><td>35.55</td><td>0.94</td><td>0.94</td></tr><tr><td>76.30</td><td>36.49</td><td>0.94</td><td>0.94</td></tr><tr><td>75.50</td><td>36.58</td><td>0.94</td><td>0.94</td></tr><tr><td>78.12</td><td>36.39</td><td>0.93</td><td>0.94</td></tr></tbody></table>

## Appendix B. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.powtec.2025.121018.

## Data availability

the data will be made available on request but the code will not be shared

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