Real-time prediction of projectile penetration to laminates by training machine learning models with finite element solver as the trainer

Pushkar Wadagbalkar.G.R.Liu

Department of Aerospace Engineering and Engineering Mechanics.University of Cincinnati.Cincinnati.OH 45219.USA

Keywords: Finite element simulations Machine learning Neural networks Impact analysis Protective laminates Projectile Decision tree

ABSTRACT Studies on ballistic penetration to laminates is complicated,but important for design effective protection of structures.Experimental means of study is expensive and can often be dangerous.Numerical simulation has been an excellent supplement,but the computation is time-consuming.Main aim of this thesis was to develop and test an effective tool for real-time prediction of projectile penetrations to laminates by training a neural network and a decision tree regression model.A large number of finite element models were developed; the residual velocities of projectiles from finite element simulations were used as the target data and processed to produce sufficient number of training samples.Study focused on steel 4340+polyurea laminates with various configurations.Four different 3D shapes of the projectiles were modeled and used in the training.The trained neural network and decision tree model was tested using independently generated test samples using finite element models.The predicted projectile velocity values using the trained machine learning models are then compared with the finite element simulation to verify the effectiveness of the models.Additionally.both models were trained using a published experimental data of projectile impacts to predict residual velocity of projectiles for the unseen samples.Performance of both the models was evaluated and compared.Models trained with Finite element simulation data samples were found capable to give more accurate predication,compared to the models trained with experimental data.because finite element modeling can generate much larger training set,and thus finite element solvers can serve as an excellent teacher.This study also showed that neural network model performs better with small experimental dataset compared to decision tree regression model.

1.Introduction

Computational study of penetration resistance of materials is widely performed and used as a substitute for experimental tests,as it makes the examination affordable compared to experimental techniques.However,finite element simulations of ballistic impact problems require heavy computer memory and computational time to get intended results.A combination for FEA simulations and machine learning techniques would drastically reduce the study time of ballistic impact investigations.if predictions are fairly accurate.

The contribution of polyurea coating/laminate has been shown in the form of 2 basic mechanisms which are retarding the occurrence of fracture in the steel plate and then the absorption of remaining kinetic energy of the projectile [3].The numerical models in this study consisted of a total of 6 cases [3].An experimental investigation on penetration performance of polymeraluminum laminates has also been studied by researchers[11].

Research on implementation of deep neural networks to supplement other approximation methods like finite element method have been done recently.In an Energy Approach to the Solution of Partial Differential Equations [34]authors discuss the implementation and ability of deep neural networks in approximating the solutions of partial differential equations.In another study-‘Artificial Neural Network Methods for the Solution of Second Order Boundary Value Problems’ solutions to partial differential equations using artificial neural networks have been demonstrated[36].Use of neural networks in predicting the deflections in Kirchhoff’splate was discussed in a study-A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate [26].This study also shows a proper method to select hyper parameters for the neural networks by plotting the relative error against the number of hidden layers and hidden neurons.In a recent study-‘A deep energy method for finite deformation hyperelasticity’ [35]authors implemented a deep neural network and compared the solutions with finite element method and demonstrated how computational time can be saved with machine learning approaches.Studies have been conducted to test machine learning approaches in predicting mechanical response of objects.In Ref.[9]machine learning models were trained using data from finite element simulations of breast tissue compression using two displacement plates.These models were then evaluated based on their accuracy to predict the real time compression of the breast tissue models.In another study[10]it was investigated if a neural network could be able to predict the change in mean heart dose based of the position of heart during radiotherapy.A progressive learning neural network has also been implemented for material characterization[22].Similarly,decision tree models have also been implemented for mechanical applications like detecting unbalance parameters of multi discs rotor[23].A numerical study of penetration resistance of steel 4340/Polyurea laminates was done[12]considering different shapes of projectiles,the position of polyurea-front of the steel plate.back of the steel plate and in between plates.Further investigation on energy absorption by the laminates for each of the combination of projectile shape and laminae orientation was also present in the study.For the current study we have built models including 4 different projectile shapes and 3 different laminae orientation from Ref.[12]and additionally considering 3 different angles of impacts,a total of 36 simulation cases were identified.For every simulation 12 fixed values of initial velocities were considered for the projectile ranging from 200 m/s to 750 m/s.These resulted in 360 possible Finite element simulations.out of which half of the simulationswere performed and the residual velocity of projectile was recorded for 20 intervals of a time step of each simulation data giving a total of 3220 data points.Data points consisted of residual velocity value of projectile as the dependent variable for training the models which would predict the residual velocities of unseen cases.For our Finite element simulations.we followed the model proposed in Ref.[12]in ABAQUS as validation with experimental results was available in that literature.However.there was a scope for consideration of angle of impact of the projectiles on the laminae in this study and we incorporated that in our models by simulating all the cases for 0.30 and 45°of angle of impact.An experimental investigation on penetration performance of polymer-aluminumlaminates was performed and published by researchers [11].The data from this research was used to and was split up into 2 sets,one to train the models and the other to act as unseen data and validate it.

Fig.1.Methodology flow chart.

Fig.2.Types of projectiles (1) Jacket (2) Pointed (3) Round (4) Flat.

Fig.3.Simulation cases for pointed projectile.

Fig.4.Visuals of finite element simulation-before.during and after impact (Jacket projectile impacting steel plate is shown).

Table 1 Split up between simulation cases for generating dataset.

2.Methodology

This work provides an account of data driven approach to estimate the penetration of target plates from projectile impacts.by monitoring the residual velocity of the projectile.First finite element simulation models were developed for projectile impact on steel 4340 and polyurea laminates.The residual velocity of the projectiles was monitored in FE simulations at fixed intervals of time step.Independent variables were identified and then dataset was generated to train machine learning models.The neural network and decision tree machine learning model then learned the relationship between type of projectiles.laminate orientation,time step in simulation and angle of impact with respect to the residual velocity of projectile.The same machine learning models were then used to train on a much smaller experimental dataset from the experiments performed in Ref.[11]to the check the effectiveness of the model when used with smaller experimental dataset.Fig.1 shows the basic flowchart of the methodology of this work.

2.1.Finite element models

Fig.2 illustrates all the projectile types as proposed in the FE Models in Ref.[12]which are recreated in this work for FE simulations.

Fig.3 illustrates all the simulation cases considered for pointed projectile.Similar 9 cases were considered for all the 4 projectiletypes making a total of 36 cases.

Table 2 Experimental cases for generation of dataset.

A mesh pattern similar to the one used in Ref.[12]is used in our models as the results from that work have been validated against experimental data.The projectile was considered to be a rigid body as the deformation and damage of plate is of most concern in the study.the projectile was thus meshed with rigid elements.The plate was meshed with hex elements from the periphery and wedge elements in the center where the impact is observed.

For simulation of the physical behavior of the steel plate the equation of state model with parameters proposed in Ref.[12]was used in the FE models along with Johnson cook plasticity and damage model considering parameters as used in Ref.[12].The behavior of polyurea is assumed to be hyper-elastic[12]and thus a Mooney-Rivlin hyper-elastic model along with a simple shear damage model with model parameters used in Refs.[12]was implemented.

The problem was solved by fixing the periphery of the laminates and constraining all the degrees of freedom for the projectile expect for the direction of velocity and impact.Dynamic explicit solver was used with a step time of 12 ms to simulate all the cases to keep the time step data to a similar scale.Fig.4 illustrates visuals of a simulation case before,during and after impact.

2.2.Data generation from finite element simulations

Including 4 different projectile shapes and 3 different laminate orientation from Ref.[12]and additionally considering 3 different angle of impacts.0,30 and 45°.a total of 36 simulation cases were identified.For every simulation 12 fixed values of initial velocities were considered for the projectile ranging from 200 m/s to 750 m/s with increment of 50 m/s between each velocity.This resulted in 360 simulations.out of which almost half simulations were performed and the residual velocity of projectile was recorded for 20 intervals for an entire time step of each simulation giving us approximately 3220 sample residual velocity values for training the neural network and decision tree model.

Only half of the cases were simulated because the finite element simulation dataset was used to find answers to the following questions:

Table 3 One-hot encoded values for projectile shape feature.

Table 4 One-hot encoded values for angle of impact feature.

Table 5 One-hot encoded values for laminate orientation feature.

Fig.5.Feature visualization before scaling (left) and after scaling (right).

Fig.6.Relative Error vs Number of neurons for finite element dataset.

Fig.7.Plot of Training loss vs Validation loss for finite element dataset.

1) How do the machine learning models perform on the unseen testing/validation dataset for configurations cases used for training?

2) Could the machine learning models predict the results for the configuration cases not considered for training?

Table 1 shows the split up between simulation cases for generating dataset.

Field variables can be monitored in FE simulation for desired intervals of the time step.The total time step for all the simulations was fixed at 12 ms and the field variable V3 which is the residual velocity of the projectile was monitored for 20 equidistant increments in the entire time step.The independent and depend variable for the models are discussed in section 2.4.

2.3.Data generation from published experimental results

After training the models on FE simulation data and getting predictions for unseen cases.the models were also tested with an available experimental data from Ref.[11].The intent was to check the effectiveness of the models to predict penetration from experimental data which have fewer input parameters as we cannot monitor projectile velocity during intervals of experimental time as we can do it in FE simulations.

Fig.8.Relative error vs minimum sample splits for finite element dataset.

In recent study on the ballistic resistance of polymer-aluminum laminated plates by Xianglin Huanga.Wei Zhanga.Yunfei Dengb,Xiongwen Jianga [11]an experimental investigation of projectile impacts on polycarbonate (PC) and the polymethyl methacrylate(PMMA) has been presented.The experiments involved 2 types of projectiles namely Blunt and Ogival and 7 types of laminate (AL2,AL3,AL4,PCF,PCB,PMF,PMB)orientations,details of which can be found in the paper referenced[11].Thus forming 2×7=14 overall impact cases which were studied experimentally and have been used in this present study.

Cases used for training the neural network and decision tree model can be seen in Table 2 below.Nomenclature is ‘Projectile -Laminate orientation’for example:BLUNT-AL2 means Blunt type projectile impacting AL2 configuration of laminate.

The experimental dataset was used to find answers to the following question:

1) How do the machine learning models perform on the smaller unseen Testing dataset from experimental observations?

Fig.9.Training vs testing accuracy for corresponding number of minimum sample splits for finite element dataset.

Fig.10.Training vs testing accuracy for corresponding number of minimum sample splits for experimental dataset.

2.4.Neural network model

Neural network models have been built for various applications in engineering.including for solving complex inverse problems[20]and automatic detection of structural damage [21].In this study it was built using Keras with TensorFlow as backend.Use of TensorFlow for machine learning applications has been studied and discussed by researchers[1,2].Keras is a high-level neural networks python API which was developed with a focus on enabling fast experimentation [5].TensorFlow is an open source library for developing machine learning models.like neural networks and decision trees in this work [4].A sequential model [13]from Keras library was used in the study with an input layer.3 hidden layers and an output layer.The network used dense layers from Keras library[14]with ReLU activation function[15]which is often used in simple neural network applications.The independent variables used in the training were 1) Time step increment of simulation 2)initial velocity of projectile 3) type of projectile 4) orientation of laminate 5) angle of impact and the dependent variable was the instantaneous residual projectile velocity during impact simulation.The five input variables were converted into a 12-node input as one hot encoding was implemented on categorical features which is discussed in detail in section 2.4.1.Type of projectile,laminate orientation and angle of impact were identified as categorical features.The number of hidden layers and number of hidden neurons were optimized considering the relative error involved in various configurations which is discussed in detail in section 2.4.3.

Fig.11.Plot of Actual vs Predicted projectile velocity values for testing set for finite element dataset by neural network.

Fig.12.Plot of Actual vs Predicted projectile velocity values for testing set for experimental dataset by neural network.

2.4.1.One hot encoding

Since the dataset consisted of categorical features like projectile shape,laminate orientation and angle of impact,one hot encoding method was used in the neural network code.One hot encoding converts categorical integer features as a one-hot numeric array[6].It converts categorical features into arrays of binary(0 s and 1 s).For achieving this.first Label encoder function was used from Scikit learn library [19]which encodes the target labels with value between 0 and number of classes-1.Over this Label encoder,One hot encoder command was used to convert categorical integer features,created with Label encoder,as one-hot numeric arrays.Tables 3 and 5 show categorical feature and the one-hot encoded array for the corresponding features (see Table 4).

2.4.2.Feature scaling

The numeric features involved in this study have a considerable difference in their scales.Time is measured in the order of milliseconds and velocity in meters per second.This creates a lot of difference in the numeric values of the two features.Feature scaling was performed to reduce the scale difference between the numeric values of features.Fig.5 illustrates feature visualization based on KDE plots before and after scaling.KDE which stands for kerneldensity estimation is a technique of data smoothing of a variable in a finite dataset [24]where it shows probability density on the vertical axis and scaled variable on the horizontal axis.It can be seen that before scaling the time data spikes vertically,whereas the velocity data expands horizontally.This difference in scales is managed my minmax scaler [6]and the scaled is plotted on the right side which corresponds to a better visual distribution of data.Average prediction error before and after scaling the data was calculated to show the importance of data scaling which is described in detail in Appendix A.

Table 6 Actual and predicted residual velocity values for experimental testing samples.

Fig.13.Plot of Actual vs Predicted projectile velocity values for testing set for finite element dataset by decision tree model.

2.4.3.Hyperparameters

A study to select the optimal hyperparameters (number of Hidden layers and neurons)was done in Ref.[26]Similar technique was implemented in this study.relative error between the actual residual velocity values and the values predicted by the neural network on the unseen testing dataset was monitored by increasing the number of hidden layers and hidden neurons.Fig.6 shows the plot of relative error against the number of neurons in particular hidden layers.

It was observed that a configuration with 30 hidden neurons gave better predictions for the unseen testing dataset for corresponding number of hidden layers.The relative error was high for 12 neurons with 1 and 2 hidden layer configuration,which showed convergence at 30 neurons for both the layers.When the number of hidden neurons was increased for 30 to 40.the relative error increased slightly irrespective of the number of hidden layers.This can be attributed to slight overfitting of training dataset,and thus a less generalized performance on the testing dataset.The least amount of relative error was observed for the configuration of 3 Hidden layers with 30 neurons each.and thus this scheme was selected for the FE simulation dataset.Similar study was performed to finalize the parameters for experimental dataset,details of which are shown in Appendix B.

Fig.14.Plot of Actual vs Predicted projectile velocity values for testing set for experimental dataset by decision tree model.

Table 7 Actual vs predicted residual velocity values for experimental testing samples.

For selecting the optimal activation function.an automated selection tool GridSearchCV [27]from Scikit learn was used.Relu,softplus,linear and sigmoid activation functions were tested for all the above-mentioned configurations of neurons and hidden layers.The estimator used by the selection tool for selecting the activation function and layers was model accuracy.Models for all the combinations of activations and layer configurations were build and a neural network with Relu activation and 3 hidden layers with 30 neurons in each hidden layer (which was also found to perform with the least relative error on testing data)was determined to be the optimum model for the dataset and hence finalized.

2.4.4.Early stopping

While training the neural network it was expected to train the network with optimum generalization performance.Generalization performance is termed as small error on testing dataset which is unseen by the network[25].Early stopping is a technique used to stop training the network when a monitored quantity stops improving.In this case,monitored quantity was validation loss.The minimum change in the monitored quantity to qualify as an improvement was set to be 1e-3 and the number of epochs that produced the validation loss with no minimum improvement,before the network stops training was set to 5.The maximum number of epochs was set to 1000.Early stopping was introduced to counter the overfitting of training data and thus improve the generalization performance.

2.4.5.Training and validation

Training of the model was done using Adam optimizer [8,31]and mean squared error technique.Adam optimizer was selected as it has been found to provide good results with small to moderate datasets [10].Fig.7 shows that training and validation loss converge over time for a run of the model for different patience values of the early stopping algorithm.If values of training and validation loss end up being roughly the same and also if the values are converging.then the model is said be well trained [16].

Table 8 Average error in predicting residual velocity values(Finite element dataset).

Fig.15.Velocity-Time curve for Round projectile impacting Steel plate at 0 Degree angle and 580 m/s velocity.

2.5.Decision tree model

Result from a preliminary study performed by researchers showed that decision tree-based models perform better when trained with finite element solver data [9].thus a decision tree model was trained using Scikit learn [19]library as a second machine learning model to evaluate the same dataset.The same dataset generated from finite element simulations and the published experimental results as discussed in section 2.2 and section 2.3 respectively,was used to train a Decision tree regressor model.The decision tree model was implemented by using the Scikit learn[19]library for python.Mean squared error or‘mse’was used as the function to measure the quality of decision splits.Minimum sample split which is known as the minimum number of nodes required to split an internal node[17]was set to 30,The max depth of the tree was set to ‘none’which is default from the library and means all the nodes of trees were expanded until they had less than minimum number of samples for split which was set to 30 [17].

2.5.1.Parameter selection for finite element dataset

Fig.16.Velocity-Time curve for a flat projectile impacting Steel plate- Polyurea laminate at 0 Degree angle and 720 m/s velocity.

Fig.17.Velocity-Time curve for a round projectile impacting Steel plate at 0 Degree angle and 620 m/s velocity.

To select the optimum value of the minimum sample split for the decision tree model.relative error on actual and predicted values was calculated on the testing dataset.The plot of relative error vs minimum number of sample splits is shown in Fig.8.The value of sample splits was varied and corresponding relative error for each decision tree model was calculated.

The relative error decreased as the value of minimum number of sample splits was reduced and converged at a value of approximately 30 sample splits,with the exception when the value was set to 2 where the relative error was higher compared to when the value was set to 10.This can be attributed to overfitting of the decision tree model.which was also studied by monitoring the training and testing accuracy of the model to avoid overfitting and select the optimal value for minimum sample splits.The plot of training vs testing accuracy for corresponding values of minimum samples split is show in Fig.9.

The training and testing accuracy converge as the value of minimum sample splits is reduced to 30.At 30 sample splits.the accuracy of 0.994 was obtained on training dataset an accuracy of 0.990 was obtained on testing dataset.The testing accuracy dropped considerably when the minimum samples splits was set to 2 which shows signs of overfitting model as the testing accuracy diverges from training accuracy.Thus.decision tree model with minimum sample splits of 30 was selected.

2.5.2.Parameter selection for experimental dataset

The plot of training vs testing accuracy for corresponding values of minimum samples split for experimental dataset is shown in Fig.10.

The training and testing accuracy showed considerable divergence when the maximum number of sample splits was set to 2which clearly showed overfitting of data.The difference between training and testing accuracy was monitored.details of which are mentioned in Appendix C and the least difference was found at minimum sample split value of 20.With training accuracy of 81.9%and testing accuracy of 76.23%.Thus.a decision tree model with minimum sample split of 20 was selected for the experimental dataset.

Fig.18.Velocity-Time curve for a pointed projectile impacting Steel plate at 30 Degree angle and 220 m/s velocity.

Fig.19.Velocity-Time curve for a pointed projectile impacting Steel plate at 30 Degree angle and 425 m/s velocity.

3.Results and discussion

3.1.Results from neural network model

3.1.1.Results for finite element data set

There was a total of 3222 data points extracted from FE simulation of projectile impacts.Out of these 2576 data samples were used to train the network while 322 samples were used as a validation set and 323 samples as pure unseen testing set.This was achieved by using train test split functionality from Scikit learn[6].Fig.11 shows plot of actual residual velocity values of projectile against the predicted values of the validation set of the network.The general trend of the plot shows a linear fit with a R squared value of 0.987 which signifies a good result from the neural network as the predicted values are close to the actual values and at same time the model was not over trained.

3.1.2.Results for published experimental dataset

The dataset from the published experimental work as discussed earlier in section 2.3 consisted of 108 training samples.Out of these 22 samples were used for testing and 86 were used for training of the neural network model.Fig.12 shows the plot of actual vs predicted values for the experimental dataset by the neural network model.

Fig.20.Velocity-Time curve for Jacket projectile impacting Steel plate at 0 Degree angle and 450 m/s velocity.

Fig.21.Velocity-time curve for a flat projectile impacting Steel plate at 0 Degree angle and 600 m/s velocity.

It can be inferred from the plot that the neural network model is able to predict the residual velocities for the experimental dataset with a decent accuracy as the plot shows a trend with a linear fit of R squared value of 0.9687.Table 6 shows the actual vs predicted values by the network.

The neural network model did not output exact zero values for the cases in which the projectile did not penetrate the target.Instead the model gave a slightly negative or close to zero positive output.The lowest value of residual velocity from the experimental dataset for which the projectile penetrated the plates was 13.43 m/s,thus all the predicted output values below that were considered to be zero for simplification of results.

3.2.Results from decision tree regression model

3.2.1.Results for finite element data set

The dataset discussed in section 2.2 was used to train the decision tree model.The dataset was split up into training and testing sets similar to the technique discussed in section 2.4.1.thus 2576 data samples were used to train the network while 645 samples were used as a testing set.Fig.13 shows plot of actual residual velocity values of projectile against the predicted values of the testing set of the decision tree regressor model.The general trend of the plot shows a linear fit with a R squared value of 0.99 which signifies an excellent result from the decision tree regressor model as the predicted values are close to the actual values.The relativeerror calculated for actual and predicted values was 0.2.

Fig.22.Velocity-Time curve for a pointed projectile impacting polyurea-Steel plate at 0 Degree angle and 275 m/s velocity.

Table 9 Average error in predicting residual velocity values (Experimental dataset).

3.2.2.Results for published experimental dataset

The dataset from the published experimental work as discussed earlier in section 2.3 consisted of 108 training samples.Out of these 22 samples were used for testing and 86 were used for training of the decision tree regressor model.The minimum samples split parameter for the experimental dataset was set to 20 as discussed earlier in section 2.5.2.Fig.14 lists the plot of actual and predicted values for the validation set of from decision tree regressor model trained with experimental dataset.It can be inferred from the plot that the results from decision tree regressor for the experimental dataset are not as reliable as a near perfect linear fit trendline cannot be fit to this plot.In contrast,a near perfect linear fit in case of finite elements dataset observed in section 3.2.1 which indicates that actual and predicted values were almost equal with relatively small error.The R squared value on the linear fit for actual vs predicted plot of experimental dataset is 0.7758 which is much lower compared to FE simulation data plot.

Table 7 shows the actual vs predicted values by decision tree model.

It was observed from the experimental testing dataset that the decision tree model was not able to predict the residual velocity of the projectiles correctly and involved a significant amount of error in estimation.The decision tree was only able to predict one critical case correctly for which the projectile did not penetrate the target.Hence it can be agreed with[18]that decision tree is not a perfect model to use in this case against a small experimental dataset and we need more data to get accurate results.

3.3.Comparison between neural network model and decision tree regressor model

As mentioned earlier in section 2.2 and section 2.3 the main aim of this research was to answer the following questions:

1) How do the machine learning models perform on the unseen testing/validation dataset for configurations cases used for training?

2) Could the machine learning models predict the results for the configuration cases not considered for training?

3) How do the machine learning models perform on the smaller unseen Testing dataset from experimental observations?

This section makes a comparison between neural network and decision tree model on their generalization and real time prediction ability on finite element dataset for both configurations used for training and not used for training.and the experimental dataset.

3.3.1.Finite element dataset (configuration cases used for training)

The average error in predicting the residual velocity values for the Finite element dataset for the neural network and the decision tree model is showed below in Table 8.

Both the models performed better with the larger finite element dataset compared to the much smaller experimental dataset.The decision tree model performed slightly better than the neural network model.As the neural network models make use of randomness,the decision tree model is found to be more stable as it can reproduce the same results on each run of the model unlike neural network model which can tend to give slightly different results with different runs owing to the randomness of its parameters.

Results for 5 random configuration cases used in training but with unknown initial velocity 1)A round projectile impacting Steel plate at 0 Degree angle and 580 m/s velocity 2) A flat projectile impacting Steel plate- Polyurea laminate at 0 Degree angle and 720 m/s velocity 3) A round projectile impacting Steel plate at 0 Degree angle and 620 m/s velocity 4) A pointed projectile impacting Steel plate at 30 Degree angle and 220 m/s velocity 5)A pointed projectile impacting Steel plate at 30 Degree angle and 425 m/s velocity predicted by the neural network and decision tree regressor are shown here to discuss the ability of the models to predict outcomes for unseen initial velocities for configuration cases used in training.Figs.15-19 show the actual and predicted velocity curves for the 5 cases by neural network and decision tree model.The actual velocity curves are obtained from FE simulations of the mentioned cases.

For all the cases shown above in this section.the predicted velocity-time curve by the neural network and the decision tree model are close to the actual curve obtained from finite element simulations of the cases.A total of 5 cases are shows above with 20 data points in each,making a total of 100 data samples and most of these lie within the ±5% error bar range of the actual data points.

3.3.2.Finite element dataset (configuration cases not used for training)

A configuration case not used in training is referred to one of the cases identified in section 2.2 which was not used for training the machine learning models.

Results for three random configuration cases not used in training 1) A Jacket projectile impacting Steel plate at 0 Degree angle and 450 m/s velocity 2)A flat projectile impacting Steel plate at 0 Degree angle and 600 m/s velocity 3) A pointed projectile impacting polyurea-Steel plate at 0 Degree angle and 275 m/s velocity predicted by the neural network and decision tree regressor are shown here to discuss the ability of the models to predict outcomes for configuration cases not used in training.Figs.20-22 show the actual and predicted velocity curves for the three cases by neural network and decision tree model.The actual velocity curves are obtained from FE simulations of the mentioned cases.

For both the neural network and decision tree models.most of the data points on the predicted velocity curve lied outside of the 10% error bars and thus the accuracy in predicting the velocity curves for configuration cases not used in training can beconsidered equally weak for these models with the training scheme finalized in this study.Out of the 3 cases.the machine learning models performance is worst for the pointed projectile impacting polyurea-Steel plate at 0 Degree angle and 275 m/s velocity.this can be attributed to the fact that only one configuration case consisting laminate orientation of Polyurea first and then steel plate was used for training.Thus.making it clear from this result that training with all the configuration cases is key to get good predictions on new unknown samples.

3.3.3.Experimental dataset

The average error in predicting the residual velocity values for the experimental dataset for the neural network and the decision tree model is showed below in Table 9.

The neural network performed better with less amount of data in the experimental dataset.As seen in Table 6 from section 3.1.2 and Table 7 from section 3.2.2 the neural network was able to predict 5 out of 8 critical residual velocities i.e.the cases where the projectile did not penetrate the target and was recorded as zero in the experiment.On the other hand.the decision tree model was able to predict only one critical velocity for the same testing samples.

4.Conclusion

A real-time approach is developed in this paper to predict residual velocity of various projectile penetrating/impacting on steel 4340 and polyurea laminates,by training of a neural network and a decision tree regression model.The predictions were based on input parameters which were available from the simulations,such as the shape of projectile.laminate orientation.initial velocity of impact and angle of impact.Experimental data are also used for comparison study.It is found that.

1) The lacking limited experimental data can be well supplemented using finite element simulations.which ensures the reliability of the trained real-time models.

2) The trained neural network is able to predict projectile residual velocities for unseen testing cases of finite element data as well as experimental data with good accuracy

3) The decision tree model works better with large finite element dataset compared to the smaller experimental dataset.

4) Both neural network model and the decision tree model fail to predict residual velocity values for unseen case configurations which are not used in training.with good accuracy.Thus,including all the cases in training the models is necessary for better predictions.

5) The decision tree regression model is found stable compared to the neural network model when used on finite element dataset and is capable to reproduce results with less deviation on each run compared to the neural network results as neural network makes use of randomness.

6) The combined approach of finite element analysis and machine learning models can be a practical predictive approach and can significantly save the time on the time-consuming FE simulations.

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.

Appendix A.Effect of data scaling

The advantage of data scaling was demonstrated by calculating the average error involved in prediction of residual velocities for testing dataset by the neural network model with the finalized architecture of 3 hidden layers and 30 neurons in each layer.Table 10 shows the average error monitored for different number of neurons while selecting the neural network architecture scheme.

Table 10 Average error for scaled and original data for corresponding number of neurons and hidden layers.

The error involved in prediction was significantly reduced by scaling the data.The average error was reduced by 61.18%by scaling the time and initial velocity inputs to the model.

Appendix B.Neural network hyperparameters selection for experimental dataset

Average error between the actual residual velocity values and the values predicted by the neural network on the unseen testing experimental dataset was monitored by increasing the number of hidden layers and hidden neurons.Fig.23 shows the plot of average error against the number of neurons in particular hidden layers.

Fig.23.Average Error vs Number of neurons for experimental dataset.

The lowest error was observed for a configuration of 30 hidden neurons with 3 hidden layers,which is similar to the architecture of neural network for finite element dataset.The only difference in the two models was the input layer.which had 12 units for FE dataset and 10 units for experimental dataset.

Appendix C.Decision tree parameter selection for experimental dataset

The difference between training and testing accuracy was monitored by varying the minimum number of samples split of the decision tree regressor to select the optimal parameter value.Table 11 shows the training and testing accuracy for increasing number of minimum sample splits and the difference between those value.

Table 11 Training and testing accuracy for increasing number of minimum sample splits.

The maximum difference was observed at minimum sample split 2.which shows overfitting of training data and poor generalization.The minimum difference between training and testing accuracy was found at minimum sample split value of 20,which indicated an optimal model.