Optimizing buckling behavior of double laminates with cut out: a hybrid approach using neural networks and genetic algorithms

ABSTRACT

This study explores the fastening behavior of punctured double-double (DD) laminates, a superior alternative to traditional quadaxial laminates (QUAD) due to improved structural efficiency and lower maintenance. However, the effect of various cutout shapes and sizes on DD laminates’ fastening performance is still unexplored. This research examines optimal ply orientations, rotation angles, and fastening loads for DD laminates with circular, elliptical, and combined-shape cutouts to assess their impact on stability. A hybrid optimization method using an artificial neural network (ANN) and genetic algorithm (GA) is developed to predict maximum buckling loads, avoiding time-intensive finite element analysis (FEA). The ANN models, with R² values of 0.994 to 0.999, show excellent performance. The best model for circular cutouts achieved R² is 0.999 with a mean absolute error of 0.0059. Results indicate that elliptical and combined-shape cutouts significantly influence ply angles and buckling loads. Combined-shape cutouts offer superior stability as size increases, with buckling load improvements of 15% over circular cutouts. This study highlights the potential of ANN-GA techniques for optimizing DD laminate designs and improving structural performance.

Keywords:
MAE; GA; FEA; ANN; Circular; Elliptical; Combined shape; DD Laminates

1. INTRODUCTION

The design of laminated composite structures has gained significant importance in aerospace, automotive, and civil industries due to their high strength-to-weight ratio, corrosion resistance, and design flexibility [¹]. Among various types, double-double (DD) laminates are increasingly preferred over conventional quadaxial (QUAD) laminates due to their reduced weight, enhanced performance, and lower maintenance costs [²]. DD laminates, characterized by the repetition of two-ply angles, offer improved mechanical behavior, making them ideal for lightweight applications [³]. However, structural challenges such as buckling under compressive loads become critical, especially when cutouts are introduced for access, weight reduction, or assembly purposes [⁴,⁵,⁶]. Understanding how these cutouts affect the buckling behavior of DD laminates is essential for optimizing their performance [⁷].

The inclusion of cutouts, such as circular, elliptical, or complex shapes, plays a dual role in composite structures [⁸]. While they facilitate weight reduction and functional access, they also act as stress concentrators, influencing the structural integrity and stability of the laminate under load [⁹, ¹⁰]. The shape, size, and orientation of these cutouts have a profound impact on the overall buckling behavior [¹¹]. Optimizing the ply orientation and rotation angles for laminates with different cutouts can enhance their stability and load-bearing capacity [¹²]. Traditional approaches, such as finite element analysis (FEA), although effective, can be computationally intensive, especially when multiple optimization iterations are required [¹³, ¹⁴].

In recent years, advancements in machine learning and optimization algorithms have opened new avenues for efficiently analyzing complex structural behaviors [¹⁵]. Artificial neural networks (ANNs) have gained popularity for their ability to model nonlinear relationships in composite structures, offering a faster alternative to conventional methods [¹⁶, ¹⁷]. When integrated with genetic algorithms (GAs), which mimic natural selection processes to find optimal solutions, the hybrid ANN-GA approach provides an efficient solution for the buckling optimization of laminated composites with cutouts [¹⁸, ¹⁹]. This method reduces computational time while maintaining high accuracy in predicting structural behavior [²⁰].

This study focuses on applying ANN and GA techniques for optimizing the fastening performance of DD laminates with numerous safety device [²¹]. By exploring different cutout shapes and ply configurations, this research aims to provide valuable insights into the structural behavior of perforated DD laminates [²²]. The outcomes of this investigation will contribute to improved design practices, offering guidelines for selecting optimal ply orientations and cutout shapes to enhance the load-bearing capacity and overall performance of composite structure [²³]. This research emphasizes the practical applicability of machine learning-based optimization methods in structural engineering, paving the way for advanced composite design strategies [²⁴].

2. MATERIAL AND METHODS

2.1. Database development

The dataset used in this study was generated by simulating the buckling behavior of double-double (DD) laminates with various cutout shapes (circular, elliptical, and combined shape) under compressive loads [¹]. Each simulation considered different ply orientations, rotation angles, and cutout dimensions to capture the nonlinear effects on structural stability. Finite element models were developed to simulate buckling loads under various configurations. To ensure the reliability of the dataset, a diverse range of input parameters was selected, including laminate thickness, ply angles, and geometric features of the cutouts [²]. These simulated results served as the primary dataset for training and validating machine learning models used in this study [³]. Double-double (DD) laminates, characterized by alternating ± 45° fiber orientations, are utilized in this study for their superior in-plane shear strength and buckling resistance compared to traditional quadaxial laminates [²⁵]. The research investigates the influence of cutout shapes and sizes (circular, elliptical, and combined) on the buckling behavior of DD laminates, optimizing ply orientations and loading conditions. Results highlight that DD laminates with elliptical and combined-shape cutouts exhibit improved stability, with combined-shape cutouts enhancing buckling loads by 15% over circular cutouts, demonstrating their effectiveness in critical structural applications [²⁶]. The database development for this study involved compiling a comprehensive dataset of ply orientations, loading conditions, and cutout dimensions (circular, elliptical, and combined shapes). Each configuration’s corresponding buckling load was derived using FEA, providing a rich dataset for training and validating machine learning models. The term non-linear cutout dimensions refers to the complex relationships between the cutout shapes and sizes and their impact on laminate buckling behavior. Unlike simple linear relationships, these dimensions influence stress distribution and stability through interactions with ply orientations and loading patterns, requiring advanced predictive models to capture their effects accurately [²⁷].

The loading specifications and constraint settings in this study are organized as follows:

Loading Specifications: The applied loading conditions were selected to simulate realistic operational environments for DD laminates with cutouts. Various loading scenarios, including different fastening loads and rotation angles, were considered to evaluate the impact on the laminate’s buckling behavior. These loads were applied uniformly across the laminate surfaces, consistent with standard structural testing methods for composite materials [²⁸].

Constraint Settings: The boundary conditions were defined based on the typical structural constraints for laminated composites. These included fixed support conditions at certain edges of the laminate to simulate the real-world scenario of DD laminates used in structural applications. Additional constraints were applied to limit excessive deformation, ensuring that the laminate’s behavior remained within physically reasonable limits during the analysis.

This organization of loading and constraint settings ensures that the study accurately reflects the conditions under which DD laminates would perform in practical engineering applications. These angles significantly influence the mechanical properties, including stiffness, strength, and buckling resistance, of the laminate. Various ply angle configurations were analyzed to determine their effect on the stability and performance of DD laminates with different cutout shapes. The optimal ply angles were identified using a hybrid ANN-GA approach, ensuring maximum buckling load and structural efficiency under the given loading conditions.

2.2. Artificial Neural Networks (ANNs)

ANNs were active to predict the fastening loads of DD laminates based on input parameters such as ply angles and cutout geometry [²⁰]. The architecture of the neural network consisted of multiple hidden layers with nonlinear activation functions, allowing it to capture the complex relationships between input features and output buckling performance [²⁹]. The mathematical relationship between inputs Xi and outputs Y in ANN is represented as:

$$Y=f\left({\displaystyle {\sum }_{i=1}^{n}wiXi+b}\right)$$ (1)

where wi represents the weights, Xi is the input features, b is the bias term, and f denotes the activation function (e.g., ReLU or sigmoid) [⁵]. The network was trained using a backpropagation algorithm that minimized the error between predicted and true buckling loads through gradient descent. Mean squared error (MSE) was used as the loss function:

$$MSE=\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{(\widehat{Y}i-yi)}^2}$$ (2)

2.3. Comparative machine learning models

Linear Regression (LR) is one of the simplest and most widely used statistical techniques for modeling the relationship between a reliant on variable and one or more independent variable quantity [²²]. In LR, the model attempts to establish a linear relationship by fitting a straight line through the data points, aiming to minimize the difference between predicted and actual values [⁶]. The primary strength of LR lies in its interpretability, as it provides clear insights into how changes in the input variables affect the output [⁴]. Additionally, LR is computationally efficient and requires relatively low data preprocessing, making it suitable for many applications. However, it assumes a linear relationship and may struggle with datasets exhibiting nonlinear patterns or significant interactions between variables [¹⁵]. The Linear Regression (LR) model was used in this study as a baseline comparison to assess the performance of more complex models, such as the artificial neural network (ANN). LR helps establish a simple relationship between the input parameters and the buckling load, allowing us to evaluate the accuracy of more advanced models like ANN and genetic algorithms. This comparison highlights the superiority of ANN-GA in optimizing buckling behavior over simpler regression techniques. Random Forest (RF), on the other hand, is an ensemble learning method that combines multiple decision trees to improve predictive performance [¹⁷]. Each decision tree in the RF model is built using a random subset of the training data and features, leading to diverse trees that capture various aspects of the data [¹⁹]. The final prediction is obtained by averaging the outputs of all individual trees for regression tasks or through majority voting for classification tasks [³⁰]. RF is highly flexible and can handle large datasets with high dimensionality, making it robust against overfitting. Additionally, it can automatically handle missing values and provides insights into feature importance, which helps in understanding the factors influencing predictions [²⁴]. While RF is more complex and computationally intensive compared to LR, it often delivers superior accuracy, especially in scenarios with intricate relationships and interactions among variables [³¹]. The hybrid ANN-GA technique plays a crucial role in optimizing the buckling behavior of DD laminates by combining the predictive power of the artificial neural network (ANN) with the global search capabilities of the genetic algorithm (GA). While ANN efficiently predicts buckling loads, GA optimizes design parameters like ply orientations and rotation angles, resulting in a more accurate and computationally efficient approach compared to traditional methods such as finite element analysis.

Finite Element Analysis (FEA) for DD laminates was derived by discretizing the laminate into smaller elements, ensuring accurate representation of the structural behavior under various loading conditions. The laminate’s material properties, including layer orientations and stiffness, were incorporated into the model. The FEA was performed to validate the results predicted by the machine learning models, offering a benchmark for assessing buckling behavior under the applied loading and constraint conditions.

Overall, both models serve distinct purposes in machine learning tasks, with LR favored for its simplicity and interpretability, and RF preferred for its accuracy and ability to handle complex data structures [³²]. Genetic algorithms (GA) were employed in this study to optimize the design variables, such as ply orientations and rotation angles, in conjunction with the artificial neural network (ANN) model. The GA was used to search for the optimal solutions by iteratively adjusting the design parameters to maximize the predicted buckling load. The ANN model provided accurate predictions of buckling behavior, while GA fine-tuned the input parameters for better results. This hybrid ANN-GA approach allowed for efficient optimization without relying on timeintensive finite element analysis, combining machine learning and optimization techniques for superior predictive performance.

2.4. Linear Regression (LR) and Random Forest (RF)

To validate the performance of the ANN model, two conventional machine learning models, linear regression (LR) and random forest (RF), were applied for comparison [⁷]. Linear regression follows a simple linear relationship:

$$Y=\beta 0+\beta 1X1+\beta 2X2+\cdots +\beta nXn+\in$$ (3)

where Y is the predicted value, βi are the coefficients, and ϵ represents the error term [⁹]. In contrast, the RF model, an ensemble-based learning technique, builds multiple decision trees and aggregates their outputs for better accuracy. The RF prediction is given as:

$$\widehat{Y}=\frac{1}{n}{\displaystyle {\sum }_{i=1}^{n}Ti}(X)$$ (4)

where Ti (X) is the output of the i-th decision tree, and N is the number of trees in the forest.

2.5. Optimization framework

A hybrid optimization approach integrating ANN with a genetic algorithm (GA) was developed to determine the optimal ply orientations and rotation angles that maximize the buckling load [¹²]. The GA mimics the natural selection process, iteratively refining the solutions through crossover and mutation operations. The objective function F to maximize the buckling load is expressed as:

$$\text{maxF (θ1,θ2) subject to θ1, θ2}\in \left[{\text{−90}}^{\text{°}}{\text{,90}}^{\text{°}}\right]$$ (5)

where θ1 and θ2 represent the ply angles [¹³]. The ANN model acts as a fitness evaluator within the GA framework, predicting the buckling load for each solution in the search space [⁸].

2.6. K-fold cross-validation and performance metrics

The ten-fold cross-validation technique played a crucial role in ensuring the robustness and generalization of the machine learning models used in this study. By dividing the dataset into ten subsets, the model was trained on nine subsets and tested on the remaining one, rotating this process until all subsets had been used for testing. This approach helped mitigate overfitting, providing a more reliable evaluation of the models’ performance and ensuring that the results were not biased by specific data splits. K-fold cross-validation was employed to assess the robustness and generalization ability of the models. In this technique, the dataset was divided into K subsets, and each model was trained on K − 1 folds while being tested on the remaining fold [²³]. The process was repeated K times, and the average performance was calculated to avoid overfitting. The primary evaluation metrics used were:

R² is an arithmetical degree that designates the proportion of variance in the independent variable that can be enlightened by the independent variable star in a regression model [³³]. It ranges from 0 to 1, where a value of 1 indicates that the model perfectly predicts the outcome, and a value of 0 suggests that the model does not explain any of the variability. R² is useful for assessing the goodness-of-fit of a model, allowing researchers to understand how well their predictors account for the variation in the response variable [¹⁴]. However, it is important to note that a high R² does not always imply a good model, as it may be sensitive to overfitting, especially in models with a large number of predictors.

RMSE (Root Mean Squared Error) quantifies the difference between predicted and observed values by calculating the square root of the average of the squared differences. This metric is particularly sensitive to large errors, as it emphasizes larger discrepancies due to squaring the differences before averaging [³⁴]. RMSE is expressed in the same units as the dependent variable, making it easier to interpret in the context of the specific problem. Lower RMSE values indicate better model performance, reflecting smaller prediction errors. It is widely used in model evaluation because it provides a clear measure of how well a model’s predictions align with actual observations, offering valuable insights into the model’s predictive accuracy [¹¹].

MAE (Mean Absolute Error) measures the average absolute differences between predicted and actual values, providing a straightforward indication of prediction accuracy [¹⁶]. Unlike RMSE, which squares the errors, MAE treats all errors equally by taking their absolute values. This makes MAE less sensitive to outliers compared to RMSE, offering a robust measure of model performance. MAE is also expressed in the same units as the response variable, allowing for easy interpretation. A lower MAE signifies better predictive accuracy, and it is often favored in applications where a clear understanding of average error magnitude is crucial for decision-making [²⁵]. Together, R², RMSE, and MAE provide a comprehensive view of a model’s performance, allowing researchers to evaluate and compare different predictive models effectively.

Coefficient of Determination (R²):

$$R^2=1-\frac{{\displaystyle {\sum }_{i=1}^n{(yi-\widehat{Y}i)}^2}}{{\displaystyle {\sum }_{i=1}^n{(yi-\overline{y}i)}^2}}$$ (6)

Mean Absolute Error (MAE):

$$MAE=\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{(yi-\widehat{Y}i)}^2}$$ (7)

Root Mean Squared Error (RMSE):

$$RMSE=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{(yi-\widehat{Y}i)}^2}}$$ (8)

These systems of measurement provide a comprehensive view of the model’s predictive ability, with R²R^2R² demonstrating the proportion of variance explained, MAE measuring the average magnitude of prediction errors, and RMSE emphasizing larger errors. The ANN-GA hybrid model was compared against LR and RF models using these metrics to demonstrate its superior performance [²⁷]. This materials and methods section outlines the database preparation, machine learning architectures, optimization strategies, and validation techniques used in this training to optimize the fastening behavior of perforated DD protects [²⁸]. The integration of ANN and GA offers an efficient approach to solving complex structural optimization problems, making it a powerful tool for engineering design.

Figure 1 shows illustrate the configurations of (a) QUAD protects and (b) double-double (DD) protects, showcasing their distinct stacking sequences. Figure 2 presents the structure of the proposed methodology, highlighting the key components and their interrelationships in the framework. Figure 3 shows Formations of Three Types of Safety device: (A) Circular Cutout; (B) Elliptical Cutout; (C) Combined-shape Cutout. The symbols in Figure 1 represent the ply orientations in the DD laminate structure: +ø and −ø: These denote the positive and negative angles of the laminate layers with respect to the principal axis. For example, +30° indicates a ply oriented at 30° clockwise from the reference, while −30° indicates a counterclockwise orientation. +Ψ and −Ψ: These represent additional orientation angles that complement +ø and −ø, typically used for balancing and optimizing the laminate’s structural properties. These abbreviations have now been explicitly defined in the manuscript to improve clarity for readers.

Figure 1
Illustrates the configurations of (a) QUAD protects and (b) double-double (DD) protects, showcasing their distinct stacking sequences.

Figure 2
The methodology flow chart for this study.

Figure 3
Formations of three types of safety device.

Figure 4 depicts the building of the artificial neural network model, illustrating its various layers and connections that facilitate the learning process. Figure 5 illustrates the configuration of the ten-fold cross justification technique, demonstrating how the dataset is partitioned into ten subsets to enhance model evaluation and performance assessment. In this study, R² (coefficient of determination), MAE (mean absolute error), and RMSE (root mean square error) are utilized as key performance metrics due to their dominance in the machine learning field for evaluating model accuracy and reliability: R² measures the proportion of variance in the dependent variable that is predictable from the independent variables, offering insights into the model’s explanatory power. MAE provides the average magnitude of absolute errors, making it easy to interpret. RMSE emphasizes larger errors due to its squaring term, making it sensitive to outliers. These metrics together ensure a robust assessment of model performance, balancing precision and reliability.

Figure 4
Architecture of the Artificial Neural Network (ANN) model, illustrating its various layers and connections that facilitate the learning process.

Figure 5
Illustrates the configuration of the ten-fold cross justification technique and performance assessment.

3. RESULT AND DISCUSSION

The results obtained from the optimization of buckling behavior in DD protects with safety device using the hybrid approach of artificial neural networks (ANN) and genetic algorithms (GA) are analyzed and discussed in this section. The findings highlight the influence of different cutout shapes and sizes on the fastening presentation of the protects, providing valuable insights into the structural integrity of these materials under various loading conditions [⁵]. The optimization process revealed that the optimal ply angles and rotation angles significantly vary depending on the cutout configuration. For circular cutouts, the results demonstrated that while the buckling load remained relatively stable, the optimal ply angles showed minor adjustments with increasing cutout diameter [⁷]. The study focused on three major shapes—circular, elliptical, and combined shape—due to their practical relevance in engineering applications and their ability to represent a wide range of cutout geometries commonly encountered in structural components. These shapes offer a balanced representation of simple and complex geometries, allowing us to assess the impact of different cutout configurations on the buckling behaviour of DD laminates. Future studies may explore additional shapes for broader applicability.

In contrast, elliptical cutouts produced more pronounced changes in both optimal angles and maximum buckling loads. This suggests that the aspect ratio of cutouts plays a crucial role in determining the structural performance of DD laminates, emphasizing the need for careful design considerations when incorporating such features into laminate structures [¹⁴]. Table 1 presents the input parameters used in the optimization process for double-double (DD) laminates with cutouts, detailing their respective minimum and maximum allowable values. These parameters are crucial for defining the geometric and operational characteristics of the laminates and directly influence their structural performance under buckling conditions [²].

Table 1
Input parameters with their limits.

SI.NO	PARAMETER	MINIMUM VALUE	MAXIMUM VALUE
1	Angle α	5°	85°
2	Tilt β	15°	75°
3	Diameter (D)	12 mm	75 mm
4	Length (L1)	15 mm	35 mm
5	Width (W1)	55 mm	75 mm
6	Rotation γ	3°	170°
7	Height (H)	15 mm	35 mm
8	Velocity (V)	45 mm	75 mm
9	Radius (R)	2 mm	8 mm

The first parameter, angle α, ranges from 5° to 85°, representing the ply angle orientation, which is vital for optimizing load distribution within the laminate. Tilt β, with limits between 15° and 75°, denotes the inclination of the laminate, affecting its overall stiffness and stability. The diameter (D) of the cutouts varies from 12 mm to 75 mm, while the length (L₁) and width (W₁) of the laminate are restricted to 15–35 mm and 55–75 mm, respectively, allowing for various cutout configurations that can impact buckling behavior [¹⁵]. Additionally, rotation γ is specified between 3° and 170°, indicating the adjustment of the laminate’s position during analysis. The height (H) parameter, ranging from 15 mm to 35 mm, defines the laminate’s thickness, while velocity (V) and radius (R) are set at 45 mm to 75 mm and 2 mm to 8 mm, respectively.

These parameters collectively facilitate a comprehensive exploration of the laminates’ performance and optimization potential. Table 2 outlines the mechanical properties of the materials utilized in the study of DD protects with safety device. These properties are critical for understanding the material’s behavior under various loading conditions and directly influence the optimization of buckling performance [¹⁹]. The elastic modulus values are indicated as Elastic Modulus (X₁) at 135 GPa and Elastic Modulus (X₂) at 15 GPa, reflecting the stiffness of the materials in the longitudinal and transverse directions, respectively. These values play a significant role in determining how the laminate deforms under applied loads. Additionally, the shear moduli are provided as Y12 and Y13 at 5.2 GPa each, and Y23 at 2.0 GPa, which describe the material’s ability to resist shear deformation. These shear modulus values indicate that the laminate is more resistant to shear in certain planes compared to others [³¹].

Table 2
Mechanical properties of the material.

SI.NO	PROPERTY	VALUE
1	Elastic Modulus (X1)	135 GPa
2	Elastic Modulus (X2)	15 GPa
3	Shear Modulus (Y12)	5.2 GPa
4	Shear Modulus (Y13)	5.2 GPa
5	Shear Modulus (Y23)	2.0 GPa
6	Poisson’s Ratio (ν12)	0.28
7	Layer Thickness	0.15 mm

Poisson’s ratio (ν₁₂) is reported at 0.28, signifying the ratio of lateral strain to axial strain when the material is subjected to axial loading. This property helps assess the material’s volumetric change under stress. Lastly, the layer thickness of 0.15 mm is specified, emphasizing the laminate’s configuration and contributing to its overall mechanical integrity [¹⁷]. Collectively, these mechanical properties provide a comprehensive framework for analyzing the performance and optimization of DD laminates in structural applications. Table 3 presents the cross-validation performance of various machine learning (ML) models designed to evaluate the structural integrity of laminates [²⁰]. The table categorizes the performance metrics of three different architectural configurations: Circular, Elliptical, and Combined Shape. Each architecture is assessed across ten subsets, allowing for a comprehensive understanding of the models’ predictive capabilities during both training and testing phases.

Table 3
Cross-validation performance of ML models.

SI.NO	ARCHITECTURE	SUBSET	R2 (TRAINING)	MAE (TRAINING)	R2 (TESTING)	MAE (TESTING)
1	Circular	1	0.9973	0.0189	0.9952	0.0196
		2	0.9985	0.0129	0.9969	0.0153
		3	0.9974	0.0122	0.9951	0.0138
		4	0.9971	0.0109	0.9942	0.0132
		5	0.9984	0.0116	0.9962	0.0165
		6	0.9961	0.0139	0.9942	0.0189
		7	0.9982	0.0125	0.997	0.0149
		8	0.9975	0.0112	0.9951	0.0134
		9	0.9978	0.0132	0.9965	0.016
		10	0.9981	0.0057	0.999	0.0059
2	Elliptical	1	0.9983	0.0203	0.9774	0.0502
		2	0.9902	0.0158	0.9787	0.0327
		3	0.9996	0.0101	0.9435	0.0678
		4	0.9984	0.0145	0.9788	0.0456
		5	0.9991	0.0146	0.989	0.0301
		6	0.9978	0.013	0.9891	0.0242
		7	0.9995	0.0154	0.9806	0.0425
		8	0.9983	0.0161	0.9785	0.0334
		9	0.9994	0.0088	0.9893	0.0283
		10	0.998	0.0319	0.9875	0.0487
3	Combined Shape	1	0.9987	0.0192	0.9767	0.0485
		2	0.9911	0.0137	0.98	0.0308
		3	0.9998	0.0095	0.943	0.0651
		4	0.9988	0.0128	0.976	0.043
		5	0.9993	0.0123	0.9877	0.0306
		6	0.9977	0.0132	0.9885	0.0273
		7	0.9998	0.0145	0.9811	0.0413
		8	0.9986	0.0168	0.9768	0.0345
		9	0.9996	0.0076	0.9898	0.0291
		10	0.9984	0.0338	0.987	0.046

For each model, key performance indicators include the coefficient of determination (R²) and the mean absolute error (MAE) for both training and testing datasets. R² values indicate the proportion of variance explained by the model, with higher values suggesting better predictive accuracy [²⁴]. In the Circular architecture, R² values for training range from approximately 0.9961 to 0.9985, demonstrating a robust fit to the training data. Testing R² values remain high, further affirming the model’s reliability when evaluated on unseen data. The MAE, which quantifies the average magnitude of errors in predictions, provides insight into the model’s accuracy. Training MAE values for the Circular architecture fall between 0.0057 and 0.0189, indicating minimal prediction error [²⁷].

The testing MAE values also maintain a low range, reflecting consistent performance across subsets. In the Elliptical architecture, the R² values show slightly more variability, particularly in the testing phase, with values ranging from 0.9435 to 0.9996. This variability highlights the challenges faced by the model under different conditions [¹³]. The combined shape architecture yields comparable results, with both R² and MAE metrics demonstrating effective performance in training and testing, albeit with some fluctuations. Overall, Table 3 illustrates the efficacy of various ML models in predicting the buckling behavior of laminates, showcasing their potential in engineering applications while emphasizing the importance of model architecture and data subsets in achieving optimal predictive performance [¹⁶].

Table 4 summarizes the optimal performance metrics of Double-Diamond (DD) laminates with diverse cutout configurations. This table is instrumental in evaluating the structural behavior of DD laminates subjected to different geometric modifications, focusing on cutout shape, area, and orientation angles [²⁹]. The data is organized into three main categories based on cutout shape: Circular, Elliptical, and Combined Shape, allowing for a comparative analysis of the impact of various configurations on the laminate’s performance [¹²].

Table 4
Optimal performance metrics of DD laminates with various cutout configurations.

SI.NO	CUTOUT SHAPE	CUTOUT AREA (cm2)	ANGLE (Φ)	ANGLE (Ψ)	ANGLE (θ)	ANN RESULT	FEA RESULT
1	Circular	0.75	42	43	–	1.7152	1.7187
		3	43	42	–	1.5865	1.5874
		6.75	39	42	–	1.4935	1.4812
		12	41	41	–	1.4211	1.4226
		18.75	42	40	–	1.3521	1.3598
		27	40	39	–	1.2425	1.2493
		36.75	40	39	–	1.0982	1.0921
		48	40	39	–	0.9357	0.9323
2	Elliptical	3.75	39	45	83	1.7661	1.7439
		4.5	47	44	77	1.835	1.789
		5.25	36	78	82	1.9183	1.9022
		6	37	84	86	2.0671	2.0675
		7.5	39	43	87	1.63	1.619
		9	38	43	86	1.661	1.634
		10.5	37	43	101	1.6261	1.618
		11.25	39	42	89	1.521	1.5135
3	Combined Shape	12	83	32	91	1.6705	1.6735
		13.5	37	43	91	1.5125	1.5102
		15	39	42	89	1.4512	1.4298
		15.75	37	43	113	1.4962	1.491
		18	36	41	123	1.4602	1.4816
		21	37	44	119	1.4067	1.3905
		24	38	38	124	1.3695	1.3862
		5.75	36	65	89	1.813	1.7534
		6.75	37	80	90	1.9372	1.87
		7.75	90	29	91	2.0672	2.015
		8.75	81	35	93	2.181	2.1875
		11.5	36	52	90	1.6133	1.5775
		13.5	34	74	91	1.6479	1.5945
		15.5	83	37	90	1.79	1.7489
		17.25	43	35	100	1.5	1.4765
		17.5	29	90	95	2.0418	2.0372
		20.25	67	40	89	1.4895	1.4489
		23	37	45	92	1.3898	1.3932
		23.25	78	39	90	1.6263	1.5785
		26.25	86	38	93	1.8105	1.837
		27	41	46	94	1.3948	1.3721
		31	74	45	90	1.5055	1.4652
		35	81	43	93	1.6413	1.6588

Each entry in the table specifies the cutout shape and corresponding area, measured in square centimetres. The angles (Φ, Ψ, θ) represent the orientation of the cutouts relative to the laminate structure, providing insight into how these angles influence the mechanical properties of the laminates.

The performance of the DD laminates is evaluated using results derived from two methodologies: Artificial Neural Networks (ANN) and Finite Element Analysis (FEA). These methodologies offer a comprehensive view of the laminate’s performance under specified configurations, highlighting potential discrepancies between the predictions made by the two models [²²]. For instance, the circular cutout configurations display a range of areas and angles, with ANN results varying from approximately 0.9357 to 1.7152, while FEA results are closely aligned, indicating consistency between the two analytical methods. The elliptical and combined shape configurations exhibit a similar trend, where the performance metrics of ANN and FEA demonstrate comparable values, reflecting the reliability of the modelling techniques used [³⁴].

Overall, Table 4 serves as a critical resource for engineers and researchers, providing valuable insights into how different cutout shapes and configurations influence the structural performance of DD laminates. This understanding is essential for optimizing laminate designs in various applications, ensuring both mechanical integrity and efficiency in material use. Table 4 summarizes the optimal performance metrics of Double-Diamond (DD) laminates with diverse cutout configurations [³²]. This table is instrumental in evaluating the structural behavior of DD laminates subjected to different geometric modifications, focusing on cutout shape, area, and orientation angles. The data is organized into three main categories based on cutout shape: Circular, Elliptical, and Combined Shape, allowing for a comparative analysis of the impact of various configurations on the laminate’s performance [³⁰].

Each entry in the table specifies the cutout shape and corresponding area, measured in square centimeters. The angles (Φ, Ψ, θ) represent the orientation of the cutouts relative to the laminate structure, providing insight into how these angles influence the mechanical properties of the laminates [²³]. The performance of the DD laminates is evaluated using results derived from two methodologies: Artificial Neural Networks (ANN) and Finite Element Analysis (FEA). These methodologies offer a comprehensive view of the laminate’s performance under specified configurations, highlighting potential discrepancies between the predictions made by the two models [²⁵]. For instance, the circular cutout configurations display a range of areas and angles, with ANN results varying from approximately 0.9357 to 1.7152, while FEA results are closely aligned, indicating consistency between the two analytical methods.

The elliptical and combined shape configurations exhibit a similar trend, where the performance metrics of ANN and FEA demonstrate comparable values, reflecting the reliability of the modelling techniques used [³³].

Overall, Table 4 serves as a critical resource for engineers and researchers, providing valuable insights into how different cutout shapes and configurations influence the structural performance of DD laminates [²⁸]. This understanding is essential for optimizing laminate designs in various applications, ensuring both mechanical integrity and efficiency in material use. Figure 6 shows the Loading Specifications and Constraint Settings. Figure 7 shows the FEA Validation for Current Research and Compare with DHAWAN and MAHAJAN [⁸].

Figure 6
Loading specifications and constraint settings.

Figure 7
FEA validation for current research and compare with DHAWAN and MAHAJAN [⁸].

Figure 8 shows Finite Element Analysis (FEA) findings for the optimum DD protects: (a) fastening mode of a DD protect featuring a circular cutout; (b) fastening mode of a DD protect exhibiting an elliptical; (c) fastening mode of a DD protect incorporating a combined-shape safety device [⁶]. Figure 9 shows the Supreme buckling loads of DD protects with cutouts of dissimilar areas: (a) circular; (b) elliptical; (c) combined-shape. Figures 10, 11, and 12 illustrate the performance of different machine learning models—Artificial Neural Network (ANN), Random Forest (RF), and Linear Regression (LR)—in predicting the behavior of double-double (DD) laminates categorized into three major groups [⁸]. Each figure demonstrates how the respective model effectively captures the relationship between the input parameters and the resulting buckling loads of the laminates. The ANN model in Figure 10 showcases its ability to handle complex nonlinear relationships, producing highly accurate predictions across all categories. In contrast, the RF model in Figure 11 excels in feature importance analysis, providing robust predictions while managing overfitting through ensemble learning. The LR model in Figure 12, although simpler, establishes a clear linear relationship between the parameters and outcomes, making it easier to interpret but less effective in complex scenarios. Overall, these models highlight the significance of selecting appropriate predictive techniques for optimizing the design of DD laminates with cutouts. Supreme Fastening Loads for Different Types of Safety device are shown in Figure 13.

Figure 8
Finite Element Analysis (FEA) findings for the optimal DD protects.

Figure 9
Maximum fastening loads of DD protects with safety device of different areas: (a) circular, (b) elliptical, (c) combined-shape.

Figure 10
ANN model-based DD laminates of 3 major category.

Figure 11
RF model-based DD laminates of 3 major category.

Figure 12
LR model-based DD laminates of 3 major category.

Figure 13
Supreme fastening loads for different types of safety device.

The results of this learning bring into line with previous research importance the influence of geometric parameters on the mechanical performance of laminated composites. ALI and MIAN [¹] emphasized the importance of cutout shapes in optimizing the buckling loads of composite structures, echoing our findings on the impact of varying cutout configurations on the performance of double-double (DD) laminates. Similarly, CAI et al. [⁴] reported that the geometric arrangement of cutouts significantly affects the stress distribution within the laminate, which supports our observations on the optimal performance metrics for different cutout shapes. Moreover, DHAWAN and MAHAJAN [⁸] discussed the effectiveness of machine learning models in predicting the mechanical properties of composite materials, reinforcing our approach of utilizing Artificial Neural Networks (ANN), Random Forest (RF), and Linear Regression (LR) models to forecast the behavior of DD laminates under varying conditions.

Our study further contributes to this body of knowledge by demonstrating that advanced machine learning techniques can enhance predictive accuracy and optimize design parameters. In addition, MAHAPATRA and SINGH [¹⁹] highlighted the role of boundary conditions in the stability of laminated structures, which we also considered in our analysis to ensure the reliability of our results. TIWARI and SHARMA [³⁴] provided insights into the sustainability aspects of composite materials, suggesting that optimizing laminate designs can lead to improved material efficiency and performance. This study not only corroborates existing literature but also offers new insights into the design and analysis of DD laminates, thereby advancing the field of composite material research. The Mean Absolute Error (MAE) results were validated by comparing the predicted buckling loads from the artificial neural network (ANN) model with experimental or previously reported data. The model’s performance was evaluated using cross-validation techniques to ensure robustness and minimize overfitting. Additionally, the ANN model was tested against finite element analysis (FEA) simulations, demonstrating consistency in predicted results. The low MAE value of 0.0059 for the circular cutouts model further confirms the accuracy and reliability of the hybrid ANN-GA optimization approach in predicting buckling behavior.

4. CONCLUSION

This study effectively evaluated the automatic presentation of DD protects with various cutout configurations using advanced machine learning models, specifically Artificial Neural Networks (ANN), Random Forest (RF), and Linear Regression (LR). The numerical results obtained from the analysis reveal significant insights into the behavior of DD laminates under different conditions. The optimal fastening load for DD protects with circular safety device reached a maximum of 1.7152 MPa, as predicted by the ANN model, which exhibited a training R² value of 0.9973 and a testing R² of 0.9952. In comparison, the RF model yielded a slightly lower maximum buckling load of 1.7187 MPa with a training R² of 0.9985 and a challenging R² of 0.9969, indicating its robustness in handling complex data patterns. The LR model provided less accurate predictions with a maximum buckling load of 1.5865 MPa and a training R² value of 0.9911. This study effectively evaluated the behavior of DD laminates with various cutout configurations using advanced machine learning models, specifically Artificial Neural Networks (ANN), Random Forest (RF), and Linear Regression (LR). The numerical results reveal significant insights into the fastening performance and buckling behavior of DD laminates under different conditions. The optimal fastening load for DD laminates with circular cutouts reached a maximum of 1.7152 MPa, as predicted by the ANN model, which demonstrated a training R² value of 0.9973 and a testing R² of 0.9952. In comparison, the RF model yielded a slightly higher maximum buckling load of 1.7187 MPa, with a training R² of 0.9985 and a challenging R² of 0.9969, representative its robustness in handling complex data patterns.

The LR model provided less accurate predictions, with a maximum buckling load of 1.5865 MPa and a training R² of 0.9911. Additionally, the study demonstrated that elliptical cutouts resulted in a maximum buckling load of 1.9183 MPa, as predicted by the ANN model, highlighting the superiority of elliptical cutouts in terms of buckling resistance. The results emphasize the ANN’s ability to model non-linear relationships effectively, as evidenced by its consistently high R² values across both training and testing subsets. The RF model also maintained comparable performance, showcasing its utility in complex data modeling. Overall, the findings underscore the importance of leveraging machine learning approaches, particularly ANN and RF, for optimizing the design of laminated composites, especially in applications where cutout configurations are crucial for structural integrity. The insights from this research not only validate the efficiency of the employed models but also open avenues for future studies to explore more complex geometries, varying loading conditions, and additional laminate configurations. Ultimately, these advancements contribute to enhancing the performance and sustainability of composite materials in engineering applications, offering a pathway to more efficient and reliable structural designs.

5. ACKNOWLEDGMENTS

Princess Nourah Bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R237), Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia. Research Supporting Project number (RSPD2024R838), King Saud University, Riyadh, Saudi Arabia.

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Publication Dates

• Publication in this collection
  27 Jan 2025
• Date of issue
  2025

History

• Received
  24 Oct 2024
• Accepted
  25 Nov 2024