Optimized artificial neural networks to predict the impact response of functionally graded composite materials

Abstract

Accurately predicting the Depth of Penetration (DOP) is essential for understanding the impact behavior of Functionally Graded Composite Materials (FGCMs) under high-velocity conditions. However, the nonlinear dynamic behavior governing FGCMs, coupled with the high cost and time-consuming nature of experimental tests, make precise DOP estimation a challenging task. This study proposes an efficient predictive framework that combines finite element (FE) simulations with artificial neural networks (ANNs) to overcome the constraints of conventional numerical and experimental approaches. This work focuses on applying the Whale Optimization Algorithm (WOA) and the Arithmetic Optimization Algorithm (AOA) as independent optimization strategies to improve the control parameters during the initiation phase, avoid convergence to local optima, and accelerate the learning process. Finite Element (FE) simulations were used to generate input-output datasets, which were used to train three ANN models: conventional back-propagation (BP), WOA-ANN, and AOA-ANN. The optimized ANNs accurately captured the nonlinear mapping between impact velocity ( $V_{i m p}$ ) and DOP. Results confirm that WOA-ANN model outperformed both AOA-ANN and BP models, achieving lower mean square error (MSE) and higher correlation coefficients. Moreover, the optimized ANNs provide reliable predictions with a limited number of FE simulations, considerably reducing computational effort while maintaining accuracy.

Keywords

functionally graded composite materials artificial neural networks back-propagation WOA-ANN AOA-ANN DOP prediction

Introduction

Functionally Graded Composite Materials (FGCMs) are increasingly adopted across a wide range of engineering applications, including aerospace, shipbuilding, nuclear installations, automotive engineering, geological structures, biomechanics, ballistic protection, as well as sensor and actuator technologies.^{1,2,3,4,5,6,7} The application of FGCMs in armor systems has attracted growing research interest; however, studies specifically addressing on their ballistic performance under high-velocity impact conditions remain scarce compared to conventional homogeneous or laminated composites. The ballistic efficiency (DOP) of FGCMs is traditionally investigated through experimental testing or numerical simulations. While experimental tests provide valuable data, they are often costly and time-consuming. To overcome these limitations, finite element (FE) simulations offer a more efficient alternative by reducing the need for extensive experimental testing. Nevertheless, FE simulations remain computationally expensive, especially when parametric studies are required, limiting their applicability for optimization. Beyond experimental and numerical approaches, Artificial Neural Networks (ANN) have recently gained attention for their effectiveness in capturing complex nonlinear relationships in materials and structural engineering problems. Owing to their self-learning capability, robustness, and architectural flexibility^{8,9,10,11,12,13,14,15,16,17,18,19} ANNs have been successfully applied to a wide range of applications, including mechanical fault diagnosis, additive manufacturing, machining processes, vibration analysis, nonlinear material behavior, and composite property prediction.^{16,17,20–38} These studies consistently demonstrate the ability of ANN-based models to provide accurate and efficient predictions when trained on numerical or experimental datasets.

Despite these advances, the application of ANN models to dynamic impact and ballistic problems remains comparatively underexplored, particularly for FGCMs subjected to high-velocity loading. Existing studies have mainly focused on conventional composite laminates, low-velocity impact damage, or ballistic limit prediction using experimental testing, finite element analysis, and ANN models^{9,10,11,12,14}. Additionally, Xu Long et al.³⁹ introduced a Long Short-Term Memory (LSTM) network model to forecast the dynamic compressive performance of concrete-like materials subjected to high strain rates. Their findings show that the LSTM network outperforms the conventional BP algorithm and can effectively replace computationally expensive and time-consuming FE modeling. Moreover, Jairo Peinado et al.¹⁵ and J.A. Artero Guerrero et al.⁹ proposed a methodology based on the paradigm of experiment-FE analysis-ANN to assess the impact of stacking sequence on the ballistic performance of composite laminate structures. While these studies highlight the potential of ANN-based approaches, they often rely on relatively large datasets and do not explicitly address the challenges associated with highly nonlinear impact behavior and limited training data, which are typical in ballistic simulations of FGCMs.

Despite their growing applications in recent years, Artificial Neural Networks still present certain challenges, such as slow convergence speed, susceptibility to local minima, and uncertainty in determining the optimal number of hidden layer neurons^40,41,42. To address these challenges, advanced ANN architectures coupled with optimization algorithms have been proposed, such as ANN-WOA^34,43,44,45, ANN-GA³⁹, ANN-AOA¹³, ANN-JAYA^46,47, and ANN-PSO^36,48,49,50. Overall, optimizing ANN parameters through these methods plays a crucial role in improving performance and reliability in structural analysis, fault diagnosis, and strategy evaluation^51,52,53,54. For instance, Khatir et al. demonstrated substantial improvements in ANN performance through: (i) the use of the Arithmetic Optimization Algorithm (AOA) for damage assessment¹³, (ii) Jaya search algorithms for identifying single and multiple cracks⁴⁷, and (iii) damage detection in steel beam structures and composite laminates using ANN-PSO-IGA^48,55. Their findings confirm that enhanced ANN algorithms offer high accuracy and robustness in damage identification, quantification, and prediction^47,55,56,57. However, the use of advanced architecture based on optimization algorithms for predicting the high-velocity impact response of FGCMs, particularly under data-scarce conditions, has not been sufficiently investigated.

Motivated by these gaps, the present study focuses on predicting the ballistic efficiency (DOP) of FGCMs under high-velocity impact via optimized ANN approaches. The novelty of this study lies in the use of the Whale Optimization Algorithm (WOA) and the Arithmetic Optimization Algorithm (AOA) as two independent optimization strategies for ANN training, aiming to improving control parameter estimation during the early learning phase, avoid premature convergence, and improve prediction accuracy when only a limited dataset is available. The remainder of this paper is organized as follows. Section 2 describes the validated FE model employed to generate the training dataset, where the initial impact velocity ( $V_{i m p}$ ) is used as input and the corresponding depth of penetration (DOP) as output. Section 3 describes the implementation of BP, WOA-ANN, and AOA-ANN models. Finally, Section 4 presents a comparative analysis of different ANN models, evaluating their performance through quantitative comparisons of prediction errors.

Impact responses of the FGCMs plate

This section aims to construct a database comprising input-output pairs ( $V_{i m p}$ , DOP) for training the ANN models. To evaluate the impact response of FGCMs, a series of simulation tests were conducted using a 3D finite element model (FEM) previously developed by Zemani et al⁵⁸.

In this numerical study, the 3D model consists of two parts: a 0.30 caliber steel fragment-simulating projectile (FSP) and a FGCMs plate, as illustrated in Figure 1. The numerical model consisted of an impacted circular plate with a 90 mm diameter, composed of five functionally graded (FG) layers composed of silicon carbide ceramic (SiC) and aluminum metal (Al) constituents with a total thickness of 16 mm (each layer measuring 3.2 mm). The compositional gradient across the thickness was adjusted to transition from a pure ceramic top layer (100% SiC) to a pure metal bottom layer (100% Al). Additionally, the pure ceramic layer was positioned at the front face against the projectile.

Figure 1.

FE model for 5-layer Al/SiC FGCMs⁵⁸.

The mechanical properties of FGCMs layers were assumed to vary through thickness following a power-law distribution governing the volume fraction of the metal phase, as defined in equation (1)^59,60.

V_{m} = {(\frac{2 z + h}{2 h})}^{p}; \frac{- h}{2} \leq z \leq \frac{h}{2}

(1)

where

V_{m}

represents the metal volume fraction, while

h

denotes the thickness of the plate. The compositional gradient exponent, denoted as '

p

', characterizes the material variation profile across the thickness of the FGC structure.

ZEMANI et al⁵⁸ found that Al-SiC functionally graded composite plate with a linear composition ( $p$ = 1) exhibited the highest ballistic efficiency compared to metal-rich ( $p$ = 0.1) and ceramic-rich ( $p$ = 10) composition when subjected to FSP projectile penetration. They concluded that the linear FG plate can reduce mostly the projectile’s kinetic energy due to its high energy absorption capacity. Following this approach, the numerical specimen model in this study also features a linear composition gradient (Figure 2). DOP was evaluated through impact simulations by systematically increasing the projectile’s initial velocity ( $V_{i m p}$ ). Modeling material response under impact loading requires a constitutive law and a failure model. In this simulation, the material behavior of the functionally graded plate was modeled as the same approach in the previous study in⁶¹ To this end, the effective material properties of the FG layers were computed using a micromechanics-based formulation (Self-Consistent Method), while the Dynamic-Tamura-Tomota-Ozawa (DTTO) model provides the stress–strain flow of FGCM layers under dynamic loading based on the dynamic response of the metal phase and the volume fraction of constituents at a specific thickness coordinate. In addition to the constitutive relationships of the materials used in the numerical model, a failure strain of the FG layers was defined through a calibration study based on the aluminum failure strain. The contact used to model the interaction between the layers of the FG plate is mainly a SURFACE-TO-SURFACE CONTACT; that is, one surface impacts another. The contact algorithm utilized between the impactor and the impacted plate is defined as CONTACT ERODING SURFACE TO SURFACE. The numerical model takes advantage of the geometric and loading symmetry of the projectile–plate configuration. Therefore, only one quarter of the structure was modeled using symmetry boundary conditions through the thickness, which reduced the computational cost while maintaining accuracy. The plate was assumed to be clamped along its outer diameter. Mesh sensitivity analysis has been conducted by trial simulations to find an optimal density of the mesh for the FG plats which balances the computational accuracy and cost. It is important to note that the material configuration, mesh type, boundary conditions, contact formulation, and impact conditions established by Zemani et al⁵⁸ are consistently maintained as input parameters in the present study, the validated numerical model was directly employed to generate the dataset used for ANN training. Further details on the constitutive model, test conditions, and FE model validation can be found in^58,61,62

Figure 2.

Linear distribution of 5-layers Al/SiC FGCMs⁵⁸.

Proposed optimized ANN trained using FE predictions

Back-propagation learning algorithm

Machine learning techniques serve as effective regression tools in scenarios where examples of an unknown process are available, but no explicit analytical model exists to define their relationships. Among these techniques, Artificial Neural Networks stand out as powerful computational models. A specific type of ANN, the feed-forward neural network (FFNN), consists of neurons arranged in sequential layers, where the output of one layer serves as the input for the next. The FFNN architecture comprises three primary layers: the input layer, hidden layer(s), and output layer. There are no cycles or loops in the FFNN network, no feedback connections⁶³. The structural design of the FFNN architecture is illustrated in Figure 3.

Figure 3.

Architecture of a multi-layer perceptron.

First, the input layer consists of a vector of input parameters, which directly serve as the outputs of this initial layer. Subsequently, the outputs $O_{i - 1, k}$ from the ${i - 1}^{t h}$ layer are multiplied by a set of constants $w_{i, j, k}$ , known as synaptic weights. These weighted sums are then used as inputs $u_{i, j}$ for the $j^{t h}$ neuron in the following $i^{t h}$ layer. In both the hidden and output layers, neurons are governed by an activation function $f_{a} (u_{i, j})$ , which processes the input and generates the output value for the $j^{t h}$ neuron in the $i^{t h}$ layer, as expressed by the following equation

a_{i, j} = f_{a} (u_{i, j}); u_{i, j} = \sum_{k = 0}^{k} (O_{i - 1, k} w_{i, j, k})

(2)

The synaptic weights $w_{i, j, k}$ are parameters of an ANN to be determined during the training process. $k$ is the number of neurons in the $i - 1$ layer. The choice of activation function typically depends on the nature of the function being approximated. For continuous problems, the sigmoid activation function, as defined in equation (3), is commonly used in hidden layers due to its smooth and differentiable properties. It maps the weighted sum of inputs at a neuron to a value in the range [0,1].

O_{i, j} = f_{a} (u_{i, j}) = \frac{1}{1 + e^{- u_{i, j}}}

(3)

The optimal network architecture is typically determined through a trial-and-error process. Among feed-forward neural networks, the multilayer perceptron is the most commonly used, employing a backpropagation learning algorithm. The primary objective of this algorithm is to approximate the unknown function governing the process. The training process consists of two main steps: (i) feed-forward propagation of the input training pattern and (ii) weight adjustment based on the computed error, enabling the network to learn and improve its predictive accuracy.

During the training process, the synaptic weights are iteratively adjusted to minimize the error between the predicted and actual outputs using the backpropagation algorithm. The learning process concludes once the error falls within the specified accuracy threshold, yielding a trained neural network. If the accuracy requirement is not satisfied, the process continues iterating until the error is reduced to an acceptable level. This iterative cycle ensures optimal weight adjustments for improved predictive performance. Further details on the learning parameters are discussed in the next section.

Optimized-ANN models

The training process of ANN remains one of the most challenging aspects of machine learning, drawing significant research interest in recent years. The primary difficulty lies in the nonlinear nature of neural networks and the uncertainty in determining the optimal set of controlling parameters, particularly the synaptic weights. Traditional training algorithms are often prone to issues such as local optima stagnation and slow convergence. However, optimization algorithms, such as Whale Optimization Algorithm (WOA) and the Arithmetic Optimization Algorithm (AOA), have proven to be highly effective in addressing complex and diverse optimization problems across various fields. Their robustness, flexibility, and adaptability make them suitable for handling continuous, discrete, and combinatorial optimization tasks while efficiently exploring the search space and avoiding local optima. These make the optimization algorithm a reliable alternative to alleviate these drawbacks and motivated our attempts to employ it as a trainer for ANN particularly in generation phase of controlling parameters (i.e., weights) during the initiation phase.

The WOA and the AOA utilize unique strategies to balance exploration and exploitation, making them effective for tackling complex optimization problems. WOA, proposed by Seyedali Mirjalili and Andrew Lewis⁶⁴, balances exploration and exploitation, making it effective for complex optimization problems. Inspired by the hunting behavior of humpback whales, WOA mimics their bubble-net hunting strategy. The algorithm begins with a randomly generated population of solutions and iterates through three key phases: (i) Encircling Prey – involves the whales adjusting their positions relative to the best-known solution, (ii) Bubble-Net Attacking method which combining shrinking encircling (converging toward the optimal solution) and spiral position updating (simulating helical movements around their prey), and (iii) searching for prey. By effectively combining local refinement with global search, WOA demonstrates strong optimization capabilities, making it a powerful tool for solving challenging problems.

In contrast, the AOA, introduced by Abualigah et al⁶⁵ mimics arithmetic operations to dynamically adjusts its exploration and exploitation phases, balancing between global and local searches to avoid premature convergence and enhance solution quality. It initializes a population of solutions and iteratively refines them through: (i) addition and subtraction focus on exploration by altering solutions to cover a broad search area, while multiplication and division concentrate on exploitation by fine-tuning solutions for better accuracy. AOA dynamically adjusts these operations to prevent premature convergence, continuing until an optimal solution or iteration limit is reached. Here is a simplified overview of both algorithms in a single tale, for a detailed description of WOA and AOA, please refer to the works documented in studies^{64,65,66,67,68}.

As illustrated in Figure 4, optimization algorithms are integrated in the initiation phase of network to optimize the weight values during the training process. Once the structure of the ANN model is established, each connection weight is represented by a real number. The optimization algorithms iterate until the maximum number of iterations is reached. The goal is to determine suitable weight values using WOA and AOA, where the Mean Square Error ( $M S E$ ) function, defined in equation (4), is used as the objective function for estimating the initial ANN training parameters.

M S E = \frac{1}{n} \sum_{i = 1}^{n} {(y_{i}^{p r e d} - y_{i})}^{2}

(4)

where

y_{i}^{p r e d}

is the output corresponding to

i

th data point in the training set by the initial network,

y_{i}

denotes actual output as consider in the target set and n is the data size. Since the

M S E

is defined by the average squared difference between the estimated and actual values, a smaller

M S E

value indicates better consistency between the two sets of data. After the optimization, the optimal weights are determined and set as the initial parameters for the final back-propagation learning algorithm. The quality of the trained ANN is then assessed by identifying the minimum

M S E

value obtained from the training dataset.

Figure 4.

General step of the optimized ANN models (WOA-ANN and AOA-ANN).

Results and discussions

The finite element (FE) results of the FGCMs specimen studied by ZEMANI et al⁵⁸ provide the input-output sequences for the network (i.e. the initial velocity of the training sample that used as a set of input to predict the corresponding depth of penetration). All finite element simulations were performed under fixed and controlled conditions (projectile geometry, impact angle, material properties, and plate configuration), ensuring that impact velocity is the dominant parameter governing the depth of penetration. Consequently, the ANN was formulated as a single-input model to capture its nonlinear relationship with DOP for this study. This nonlinear mapping was then established using the back-propagation (BP) learning algorithm and the proposed optimized ANN models.

The definition of a neural network’s architecture involves selecting the number of parameters that define the input data, the number of hidden layers, and the number of neurons in each layer. At the beginning of this section, input-output sequences of the network are used to determined ANN parameters. However, the node number $l$ in the hidden layer is determined using the empirical formulas given in Equations 5 and 6,³⁹ based on the number of nodes $n$ in the input layer and the number of nodes $n^{'}$ in the output layer. Additionally, the constant $c$ is chosen within the range of 1.0 to 10.

l = \sqrt{n + n^{'}} + c

(5)

l \geq \log 2 n

(6)

The neural network model selected has a single hidden layer with 20 nodes, according to Equations (5) and (6). Using more hidden layers would result in a more complex ANN structure, making optimization techniques more challenging. To provide a fair assessment of the predictive capability of these ANN models, the dataset was randomly partitioned into 54 training sets and 6 validation sets, ensuring a training-to-verification ratio of 0.9:0.1, which aligns with standard ANN modeling practices³⁹. This partitioning strategy was applied consistently across all ANN models (BP, WOA-ANN, and AOA-ANN) to enable a reliable comparative evaluation.

Various training functions can be employed, as these functions update the weight values using different optimization algorithms. Some commonly used training functions include Levenberg-Marquardt optimizations, Bayesian Regularization, Scaled Conjugate Gradient method, and One Step Secant method. Among these, Levenberg-Marquardt (LM) and Bayesian Regularization (BR) are more efficient compared to the Scaled Conjugate Gradient and One Step Secant methods, despite requiring more memory and computational time. In this study, the Levenberg-Marquardt training function is selected for both the BP training algorithm and the optimized ANN models.

To enhance the statistical significance of the results, the regression parameter ( $R$ -value) is adopted alongside Mean Squared Error ( $M S E$ ) as suggested by Botchkarev⁶⁹. The $R$ -value is used to objectively measure the correlation between the predicted values from FE simulations and the outputs of the proposed ANN models during the projectile penetration process. Since the $R$ -value represents the linear correlation between two datasets, it is generally accepted that an $R$ -value between 0.8 and 1.0 indicates a strong correlation, ensuring a high degree of reliability between the predicted and actual values in the ballistic performance evaluation of FGCMs.

As outlined in Section 3.2, the key parameters used in the optimized ANN are detailed in Table 1. To evaluate the impact of varying the main controlling parameters of AOA on its performance, multiple tests were conducted by adjusting the parameters µ and α of AOA. The average fitness values (Ave) and standard deviation (STD) of the obtained results across 30 independent runs are reported in Table 2. The results indicate that test 9 (i.e., µ = 0.9 and α = 9) yields the best performance, followed by test 8 and test 2. The consistency of average and STD values across all tests confirms the stability of AOA, demonstrating its ability to generate consistent and reliable solutions across multiple executions.

Table 1.

Parameters of the database with three ANN models.

	BP	AOA	WOA
Total data sets	60	60	60
Max epochs	500	500	500
Train function	Levenberg-marquardt	Levenberg-marquardt	Levenberg-marquardt
Neurons numbers in layer	20	20	20
Population size	-	100	100
Maximum of iterations	-	500	500
Target goal (MSE)	0.0001
Evaluation metrics	$R$ , MSE
µ	-	0.1, 0.5, 0.9	-
α	-	1, 5, 9	-

Table 2.

The influence of changing µ and α parameters value of AOA on its performance.

	Test 1	Test 2	Test 3	Test 4	Test 5	Test 6	Test 7	Test 8	Test 9
Mu	0.1	0.1	0.1	0.5	0.5	0.5	0.9	0.9	0.9
Alpha	1	5	9	1	5	9	1	5	9
Worst (max)	0.966769033	0.476680723	0.468480738	1.810542777	2.145419047	1.782517761	0.629609344	0.477872924	0.3839007
Best (min)	0.190923936	0.162998901	0.157745291	0.159588519	0.179713584	0.189972169	0.173640082	0.16092524	0.1538929
Aver	0.502791692	0.334784685	0.286130455	0.498814937	0.737746856	0.617685957	0.352557592	0.252781072	0.2324285
STD	0.182218617	0.079441544	0.088482988	0.33158936	0.582497326	0.444711878	0.089106452	0.075240515	0.0715009

In order to analyze the convergence performance of the proposed WOA compared to AOA algorithms, Figure 5 presents exemplary convergence curves based on averages of the fitness function over 500 iterations. Please note that average fitness function indicates the average of the solutions obtained in each iteration across 30 independent runs. For a fair comparison, all ANN model inputs and settings were kept identical during the optimization process of WOA and AOA algorithms in the optimized ANN process.

Figure 5.

Evolution of the Fitness Function value during iterations for the DOP prediction using WOA-ANN and AOA-ANN.

It is evident from Figure 5 that the WOA and AOA exhibit a steady convergence and a fast convergence acceleration. However, the steady convergence of both algorithms means they progressively approach the global optimum solution without major fluctuations. These results demonstrate their ability to provide consistent solutions over several runs. This also implies that both algorithms do not behave erratically and follow a smooth trajectory towards the global optimum. The fast convergence acceleration may indicate that the algorithm takes less time to significantly reduce the value of the fitness function, which may be more efficient in terms of execution time. These findings confirm that both WOA and AOA are robust optimization techniques capable of efficiently training ANN models while ensuring convergence to optimal solutions.

Furthermore, the results indicate that WOA outperforms AOA technique in terms of accuracy, despite AOA’s superior initial estimates in the earliest iterations. The superior accuracy of WOA algorithm originates from the saving of the best prey and adaptive search around it. The search agents in WOA tend to search more locally around the prey proportional to the number of iterations. The WOA-ANN trainer inherits this feature from the WOA and managed to outperform the AOA-ANN algorithm.

The detailed training and validation results for the three examined ANN models are presented in Table 3 and Figure 6. The optimization techniques notably enhance the artificial neural network’s performance by optimizing the weight parameters during the initiation phase. For the BP learning algorithm, the training

M S E

values decrease from 693.6453 to 0.165885. While, the optimization of weights reduces the training

M S E

values respectively from 2.874391 to 0.028923 and 0.786948 to 0.145844 due to the WOA and AOA optimization of the initial network parameters. According to these results, Table 3 shows that the error at the beginning of training is very high for the BP algorithm due to the poor estimation of the weight parameters during initiation training parameters. It can be clearly seen that the optimized ANN models show good estimation at the network initiation phase due to the optimization of the initial network population, they have a much faster decreasing loss function and a smaller number of epochs.

Table 3.

Accuracy results for BP learning algorithm and optimized ANN models in terms of $M S E$ values.

Epochs	BP training	BP validation	WOA training	WOA validation	AOA training	AOA validation
0	693.6453	964.6835	2.874391	3.279305	0.786948	0.675819
1	26.30081	37.02165	0.746255	1.558282	0.283397	0.028361
2	23.2946	41.35068	0.176039	0.393834	0.197686	0.011499
3	14.62698	65.67435	0.094548	0.200216	0.18994	0.008872
4	7.345743	54.63913	0.073715	0.135987	0.187325	0.008593
5	6.42192	54.05758	0.065691	0.106406	0.184515	0.008332
6	2.785578	36.34774	0.061295	0.092422	0.181412	0.008183
7	0.889294	23.31729	0.057929	0.083962	0.177894	0.008174
8	0.77046	20.59151	0.055028	0.077738	0.173863	0.008357
9	0.609638	17.11989	0.052437	0.072692	0.169257	0.008812
10	0.580727	17.29548	0.046976	0.083405	0.164072	0.009633
11	0.447001	14.93449	0.032601	0.037235	0.158363	0.010909
12	0.397747	14.82173	0.030302	0.037056	0.152238	0.012692
13	0.332534	13.43975	0.028923	0.039891	0.145844	0.014985
14	0.306396	13.31314
15	0.245701	13.83974
16	0.234705	14.17138
17	0.165885	13.33196

Figure 6.

Optimized-ANN models performance.

Figure 6 presents the training and validation curves for the WOA-ANN and AOA-ANN models, highlighting the performance of the optimized ANN. The training curves illustrate how the model’s error (or performance) evolves over epochs, while the validation curves depict the error on the validation set, which is a subset of the data that the model did not used during training.

As shown in Figure 6, the optimized ANN initially experiences a rapid decline in error as it learns to adjust its parameters. The steady decrease in both training and validation errors continues until the validation error stabilizes, demonstrating effective training and the generalization capability of the optimized ANN model. Furthermore, the WOA-ANN model achieves a lower MSE compared to AOA-ANN, indicating superior prediction accuracy.

To evaluate the predictive performance of the three ANN architectures, the Depth of Penetration estimated from FE simulations are compared with those from the BP training algorithm and the optimized ANN models (i.e. WOA-ANN and AOA-ANN) in Figure 7(a). The results indicate that all three models achieve $R$ values above 0.98. After WOA and AOA optimizations, the $R$ values are improved to be 0.999 and 0.994, respectively compared to the original BP network with $R$ value of 0.98132. Although a systematic sensitivity analysis of the training/validation split was not performed, the strong convergence behavior, low validation MSE, and high R-values achieved on the independent validation set collectively indicate that the optimized ANN models generalize well within the studied impact velocity range. This is consistent with recent studies in mechanics employing similar data partitioning strategies under data-limited conditions³⁹. It is not difficult to see that optimization algorithms can enhance the neural network prediction performance, bringing the $R$ value closer to 1.0. Based on these comparisons, it can be found that the optimized ANN models provide more precise predictions than the BP training algorithm. Therefore, the comparisons of $R$ values indicate that the predictions by the optimized-ANN models accurately reproduce the FE predictions, and the WOA-ANN with the best $R$ value network can be accepted to be equivalent to the FE simulations.

Figure 7.

Comparisons of impact responses predicted using BP and optimized ANN models. (a) Predicting results of DOP using different models (b) Error probability distribution; (c) Percentage error (PE) Analysis.

To further assess the predictive accuracy of the improved ANN models, Figure 7(b) compares FE data with DOP predictions from BP, WOA-ANN, and AOA-ANN methods. The results indicate that WOA-ANN provides the most accurate predictions for initial velocities of 200, 400, 500, 600, 700, 900, and 1000 m/s. Meanwhile, AOA-ANN performs best at 100, 300, and 800 m/s, with predictions closest to the FE values. In all cases, the BP learning algorithm shows the highest gap from the FE results, highlighting the superior accuracy of the optimized ANN models.

To objectively compare the DOP prediction performance of different models, percentage error ( $P E$ ) and $M S E$ are used as evaluation metrics, with results presented in Figure 7(c). Generally, a $P E$ range within [-3%, 3%], highlighted in gray area, is considered acceptable for accurate predictions [79]. As shown, all $P E$ values from the optimized ANN models fall within this range, except at an initial velocity of 100 m/s.

The proposed WOA-ANN model demonstrates the best DOP prediction accuracy based on the $P E$ standard. In contrast, the BP learning algorithm exhibits a much wider $P E$ range of [-18%, 8%], indicating lower accuracy. Furthermore, the $M S E$ values for WOA-ANN (0.0525) and AOA-ANN (0.0527) are considerably lower than that of the BP learning algorithm (0.6034), confirming the superior predictive performance of the optimized ANN models. This implies that the WOA and AOA algorithms have a greater capacity for optimizing ANN in DOP prediction. Apparently, the WOA-ANN model has the best accuracy and the superior DOP prediction performance, despite initial weight estimation challenges compared to the AOA-ANN model.

Finally, Table 4 summarizes the prediction accuracy and computational cost for DOP predictions. All BP and optimized ANN analyses were conducted on a standard computer (Intel Core i7-6600U CPU @ 2.60 GHz 2.81 GHz, 16 GB RAM). Training time refers to the duration required to complete the learning process. While the BP model benefits from a shorter training time, it suffers from higher prediction errors. In contrast, the optimized ANN models substantially enhance accuracy but require longer computation times, primarily due to the population-based search mechanisms, iterative fitness evaluations for parameter optimization, and the larger number of weight updates involved. Among them, the WOA-ANN model demonstrates the best predictive performance despite its higher time cost. Comparisons in Figure 7 and Table 4 confirm that WOA-ANN outperforms other ANN models in predicting impact responses.

Table 4.

Comparison of time cost and accuracy with three ANN models.

	BP	AOA-ANN	WOA-ANN
$R$	0.981	0.994	0.999
MSE	0.6034	0.0527	0.0525
Training time (s)	2.090229 s	68.295805 s	67.591870 s

Conclusion

This study proposes three ANN models, including the back-propagation training algorithm and two optimized Artificial Neural Networks (WOA-ANN and AOA-ANN) models to predict the impact response of Al-SiC functionally graded composite materials. The key novelty lies in the independent application of WOA and AOA to optimize the initial weight parameters, effectively overcoming the inherent limitations of conventional back-propagation (BP) algorithms, including slow convergence and local optima stagnation.

Various statistical measures were used to assess prediction performance. The high average accuracy and low standard deviation achieved by the optimized models demonstrate their effectiveness in preventing premature convergence and identifying optimal initial parameters for ANN training. Furthermore, the results confirm that the effective local optima avoidance does not compromise convergence speed. Comparative results demonstrate that the proposed WOA-ANN model achieves superior predictive accuracy even with limited data availability, with the lowest Mean Square Error (MSE = 0.0525), the highest correlation coefficient (R = 0.999), and a narrow percentage error band, outperforming both the AOA-ANN and standard BP models. Moreover, the optimized ANNs reliably capture the complex nonlinear relationship between impact velocity and depth of penetration (DOP) using a limited dataset derived from computationally intensive finite element simulations, thereby offering a practical and efficient alternative to extensive numerical or experimental campaigns.

It is important to emphasize that the present findings are specific to the studied linear Al/SiC FGCM configuration, no external validation using alternative gradation profiles or experimental data has been performed. This constraint is explicitly acknowledged, and claims regarding generalizability have been carefully moderated accordingly. Future work will extend the framework by incorporating additional governing parameters, such as the influence of constituent distribution through the thickness of FGCMs and the influence of manufacturing-induced porosity, to further enhance its predictive scope and applicability.

Footnotes

ORCID iDs

Kada Zemani

Samir Khatir

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

Data will be made available on request.

References

Natarajan

Annabattula

Martínez-Pañeda

. Phase field modelling of crack propagation in functionally graded materials. Compos B Eng 2019; 169: 239–248.

Miyamoto

Kaysser

Rabin

, et al. Functionally graded materials: design, processing and applications. Springer Science & Business Media, 2013.

Paulino

Jin

Dodds

. Failure of functionally graded materials. Comprehensive structural integrity. ISBN Elsevier, 2023.

Cheng

Liu

Zhao

, et al. Numerical simulation of crack propagation and branching in functionally graded materials using peridynamic modeling. Eng Fract Mech 2018; 191: 13–32.

Zheng

Sladek

, et al. Fracture analysis of functionally graded material by hybrid meshless displacement discontinuity method. Eng Fract Mech 2021; 247: 107591.

Gayen

Tiwari

Chakraborty

. Static and dynamic analyses of cracked functionally graded structural components: a review. Compos B Eng 2019; 173: 106982.

Ozdemir

Kefal

Imachi

, et al. Dynamic fracture analysis of functionally graded materials using ordinary state-based peridynamics. Compos Struct 2020; 244: 112296.

Haykin

. Neural networks and learning machines, 3/E. Pearson Education India, 2009.

Artero-Guerrero

Pernas-Sánchez

Martín-Montal

, et al. The influence of laminate stacking sequence on ballistic limit using a combined Experimental/FEM/Artificial neural networks (ANN) methodology. Compos Struct 2018; 183: 299–308.

10.

Carriere

Cherniaev

. Honeycomb parameter-sensitive predictive models for ballistic limit of spacecraft sandwich panels subjected to hypervelocity impact at normal incidence. J Aero Eng 2022; 35: 04022039.

11.

D'Addona

Teti

Caprino

. Residual strength prediction of artificially damaged composite laminates based on neural networks. J Intell Fuzzy Syst 2012; 23: 217–223.

12.

Fernández-Fdz

López-Puente

Zaera

. Prediction of the behaviour of CFRPs against high-velocity impact of solids employing an artificial neural network methodology. Compos Appl Sci Manuf 2008; 39: 989–996.

13.

Khatir

Tiachacht

Le Thanh

, et al. An improved artificial neural network using arithmetic optimization algorithm for damage assessment in FGM composite plates. Compos Struct 2021; 273: 114287.

14.

Laban

Gowid

Mahdi

, et al. Experimental investigation and artificial intelligence-based modeling of the residual impact damage effect on the crashworthiness of braided carbon/kevlar tubes. Compos Struct 2020; 243: 112247.

15.

Peinado

Jiao-Wang

Olmedo

, et al. Use of artificial neural networks to optimize stacking sequence in UHMWPE protections. Polymers 2021; 13, 1012. s Note: MDPI stays neutral with regard to jurisdictional claims in published.

16.

Pham

Q-H

Nguyen

P-C

Tran

. Free vibration response of auxetic honeycomb sandwich plates using an improved higher-order ES-MITC3 element and artificial neural network. Thin-Walled Struct 2022; 175: 109203.

17.

Tao

Liu

, et al. Learning composite constitutive laws via coupling abaqus and deep neural network. Compos Struct 2021; 272: 114137.

18.

Bishop

. Neural networks for pattern recognition. Oxford University Press, 1995.

19.

Mobarak

Mimona

Islam

, et al. Scope of machine learning in materials research—A review. Appl Surf Sci Adv 2023; 18: 100523.

20.

Zhang

Huang

, et al. Application of multi-scale convolutional neural networks and extreme learning machines in mechanical fault diagnosis. Machines 2023; 11: 515.

21.

Bessaoudi

Habbouche

Benkedjouh

, et al. A hybrid approach for gearbox fault diagnosis based on deep learning techniques. Int J Adv Manuf Technol 2024; 133: 1–14.

22.

Xie

Zhan

, et al. Gearbox fault diagnosis based on RGT-MFFIN and multi-sensor fusion image generation. Eng Res Express 2024; 6: 035577.

23.

Zhang

Zhou

, et al. Intelligent bearing fault diagnosis based on open set convolutional neural network. Mathematics 2022; 10: 3953.

24.

Liu

Zhang

, et al. A novel intelligent fault diagnosis method for gearbox based on multi-dimensional attention denoising convolution. Sci Rep 2024; 14: 24688.

25.

Xiong

Zhang

, et al. Bead geometry prediction for robotic GMAW-Based rapid manufacturing through a neural network and a second-order regression analysis. J Intell Manuf 2014; 25: 157–163.

26.

Upadhyay

Jain

Mehta

. Artificial neural network modeling of cutting force in turning of Ti-6Al-4V alloy and its comparison with response surface methodology. Proceedings of the International Conference on Soft Computing for Problem Solving SocProS 2011; 2012, pp. 761–768.

27.

Çelik

Gündoğdu

Özdilek

, et al. Artificial neural network (ANN) validation research: free vibration analysis of functionally graded beam via higher-order shear deformation theory and artificial neural network method. Appl Sci 2023; 14: 217.

28.

Kallannavar

Kattimani

Soudagar

MEM

, et al. Neural network-based prediction model to investigate the influence of temperature and moisture on vibration characteristics of skew laminated composite sandwich plates. Materials 2021; 14: 3170.

29.

Tiboni

Remino

Bussola

, et al. A review on vibration-based condition monitoring of rotating machinery. Appl Sci 2022; 12: 972.

30.

Yang

Guo

Tang

, et al. Derivation of heterogeneous material laws via data-driven principal component expansions. Comput Mech 2019; 64: 365–379.

31.

Yang

Qiu

Xiang

, et al. Exploring elastoplastic constitutive law of microstructured materials through artificial neural Network—A mechanistic-based data-driven approach. J Appl Mech 2020; 87: 091005.

32.

Zobeiry

Reiner

Vaziri

. Theory-guided machine learning for damage characterization of composites. Compos Struct 2020; 246: 112407.

33.

Ouladbrahim

Belaidi

Khatir

, et al. Sensitivity analysis of the GTN damage parameters at different temperature for dynamic fracture propagation in X70 pipeline steel using neural network. FRATTURA ED INTEGRITA STRUTTURALE-FRACTURE STRUCTURAL INTEGRITY 2021; 58: 442–452.

34.

Ouladbrahim

Belaidi

Khatir

, et al. Experimental crack identification of API X70 steel pipeline using improved artificial neural networks based on whale optimization algorithm. Mech Mater 2022; 166: 104200.

35.

Zhao

Yang

, et al. Synergistic effect of ZnO microspherical varistors and carbon fibers on nonlinear conductivity and mechanical properties of the silicone rubber-based material. Compos Sci Technol 2017; 150: 187–193.

36.

Fadlallah

Anderson

Nates

. Artificial neural network–particle swarm optimization (ANN-PSO) approach for behaviour prediction and structural optimization of lightweight sandwich composite heliostats. Arabian J Sci Eng 2021; 46: 12721–12742.

37.

Liu

Yang

Sun

, et al. Review of deep learning-based aerodynamic shape surrogate models and optimization for airfoils and blade profiles. J Physics of Fluids 2025; 37: 041304.

38.

Hao

Shi

Yang

, et al. Multi-mode fatigue life prediction using machine learning inspired by damage physics. Int J Mech Sci 2025; 304: 110723.

39.

Long

Mao

, et al. Machine learning method to predict dynamic compressive response of concrete-like material at high strain rates. Def Technol 2023; 23: 100–111.

40.

Zhao

. Prediction of critical properties of biodiesel fuels from FAMEs compositions using intelligent genetic algorithm-based back propagation neural network. Energy Sources, Part A Recovery, Util Environ Eff 2021; 43: 2063–2076.

41.

Zhang

Wei

Xie

, et al. Direct interval forecasting of wind speed using radial basis function neural networks in a multi-objective optimization framework. Neurocomputing 2016; 205: 53–63.

42.

Hong

W-K

. Artificial intelligence-based design of reinforced concrete structures: artificial neural networks for engineering applications. Elsevier, 2023.

43.

Aljarah

Faris

Mirjalili

. Optimizing connection weights in neural networks using the whale optimization algorithm. Soft Comput 2018; 22: 1–15.

44.

Zhao

Guo

Zhao

. Energy-related CO2 emissions forecasting using an improved LSSVM model optimized by whale optimization algorithm. Energies 2017; 10: 874.

45.

Brodzicki

Piekarski

Jaworek-Korjakowska

. The whale optimization algorithm approach for deep neural networks. Sensors 2021; 21: 8003.

46.

Fahem

Belaidi

Brahim

, et al. Prediction of resisting force and tensile load reduction in GFRP composite materials using artificial neural network-enhanced jaya algorithm. Compos Struct 2023; 304: 116326.

47.

Khatir

Boutchicha

Le Thanh

, et al. Improved ANN technique combined with jaya algorithm for crack identification in plates using XIGA and experimental analysis. Theor Appl Fract Mech 2020; 107: 102554.

48.

Khatir

Boutchicha

, et al. An efficient hybrid TLBO-PSO-ANN for fast damage identification in steel beam structures using IGA. Mart Structures Systems 2020; 25: 605–617.

49.

Seguini

Khatir

Boutchicha

, et al. Crack prediction in pipeline using ANN-PSO based on numerical and experimental modal analysis. Smart Struct Syst 2021; 27: 507–523.

50.

Gambhir

Malik

Kumar

. PSO-ANN based diagnostic model for the early detection of dengue disease. New Horizons in Translational Medicine 2017; 4: 1–8.

51.

Liu

Dai

Zhao

, et al. A bidirectional reflectance distribution function model of space targets in visible spectrum based on GA-BP network. Appl Phys B 2020; 126: 1–9.

52.

Wang

. Risk assessment of knowledge fusion in an innovation ecosystem based on a GA-BP neural network. Cogn Syst Res 2021; 66: 201–210.

53.

Dong

Mohamed

, et al. Prediction of tubular T/Y-joint SIF by GA-BP neural network. KSCE J Civ Eng 2020; 24: 2706–2715.

54.

Zhang

. Mold breakout prediction in slab continuous casting based on combined method of GA-BP neural network and logic rules. Int J Adv Manuf Technol 2018; 95: 4081–4089.

55.

Khatir

Tiachacht

Thanh

C-L

, et al. Damage assessment in composite laminates using ANN-PSO-IGA and cornwell indicator. Compos Struct 2019; 230: 111509.

56.

Samir

Brahim

Capozucca

, et al. Damage detection in CFRP composite beams based on vibration analysis using proper orthogonal decomposition method with radial basis functions and cuckoo search algorithm. Compos Struct 2018; 187: 344–353.

57.

Zenzen

Khatir

Belaidi

, et al. A modified transmissibility indicator and artificial neural network for damage identification and quantification in laminated composite structures. Compos Struct 2020; 248: 112497.

58.

Zemani

May

Khatir

, et al. Numerical analysis of an experimental ballistic test of Al/SiC functionally graded materials. Compos Struct 2024; 333: 117909.

59.

Moita

Araújo

Correia

, et al. Material distribution and sizing optimization of functionally graded plate-shell structures. Compos B Eng 2018; 142: 263–272.

60.

Smahat

Mankour

Slimane

, et al. Numerical investigation of debris impact on spacecraft structure at hyper-high velocity. J Braz Soc Mech Sci Eng 2020; 42: 117.

61.

Zemani

May

Gilson

, et al. Numerical investigation of the dynamic behavior of a Ti/TiB functionally graded material using split hopkinson pressure bar test. J Compos Mater 2022; 56: 1053–1070.

62.

Aydin

Apalak

Gül Apalak

. Experimental study on structure optimization of functionally graded sandwich plates under ballistic impact. J Compos Mater 2020; 54: 3967–3980.

63.

Basheer

Hajmeer

. Artificial neural networks: fundamentals, computing, design, and application. J Microbiol Methods 2000; 43: 3–31.

64.

Mirjalili

Lewis

. The whale optimization algorithm. Adv Eng Software 2016; 95: 51–67.

65.

Abualigah

Diabat

Mirjalili

, et al. The arithmetic optimization algorithm. Comput Methods Appl Mech Eng 2021; 376: 113609.

66.

Rana

Latiff

MSA

SiM

, et al. Whale optimization algorithm: a systematic review of contemporary applications, modifications and developments. Neural Comput Appl 2020; 32: 16245–16277.

67.

Gharehchopogh

Gholizadeh

. A comprehensive survey: Whale optimization algorithm and its applications. Swarm Evol Comput 2019; 48: 1–24.

68.

Dhal

Sasmal

Das

, et al. A comprehensive survey on arithmetic optimization algorithm. Arch Comput Methods Eng 2023; 30: 3379–3404.

69.

Botchkarev

. A new typology design of performance metrics to measure errors in machine learning regression algorithms. Interdiscipl J Inf Knowl Manag 2019; 14: 045–076.