Design and analysis of laminated composite fuselage to maximize fundamental frequency using finite element analysis and artificial neural networks

ABSTRACT

This work presents an optimization framework for the design of a laminated composite aircraft fuselage aimed at maximizing its fundamental natural frequency to mitigate resonance-induced structural failures during flight. While Finite Element Analysis (FEA) and Artificial Neural Networks (ANN) have been widely used for vibration studies, their direct application in large-scale design optimization remains computationally intensive. To overcome this limitation, the present work integrates a Genetic Algorithm (GA) with an ANN surrogate model to efficiently explore laminate stacking sequences under in-plane strength and weight constraints. The ANN, trained using high-fidelity FEA data, enables rapid prediction of natural frequencies across a wide design space, significantly reducing computation time. The integrated FEA–ANN–GA framework successfully identifies optimal ply configurations that enhance structural integrity, vibration resistance, and passenger comfort. The results demonstrate the potential of this hybrid computational approach to improve fuselage dynamic performance and reliability, thereby supporting the growing transition toward composite-based aerospace structures.

Keywords:
Composite fuselage; Finite Element Analysis; Artificial Neural Network; Genetic Algorithm; Frequency Optimization.

1. INTRODUCTION

Fiber-reinforced polymer (FRP) composites have become indispensable in aerospace structures due to their superior strength-to-weight ratio, high stiffness, corrosion resistance, and fatigue performance. Modern aircraft increasingly rely on laminated composites to replace metallic components and achieve significant reductions in structural mass while maintaining or improving strength and durability.

Zhanghao WANG et al. [¹] reported that FRP composites are favoured over conventional metals for fuselage applications because of their lightweight nature, corrosion resistance, and design flexibility. Ahmad ALHAJAHMAD and MITTELSTEDT [²] further highlighted that graphite–epoxy panels have largely replaced metallic fuselage skins owing to their superior stiffness and fatigue characteristics. Experimental investigations by LIAN et al. [³] demonstrated the impact resilience of T800/M21C carbon fiber fuselage panels, while ALBANESI et al. [⁴] recommended glass–epoxy sandwich structures as a cost-effective alternative for large fuselage sections.

Additional studies by MEMMOLO et al. [⁵] and BÖNISCH et al. [⁶] explored debonding phenomena and stress distributions at skin–stringer junctions in composite fuselages. ALHAJAHMAD and MITTELSTEDT [⁷] proposed optimized grid-stiffened fuselage designs, while MIRANDA et al. [⁸] emphasized the crashworthiness of composite fuselages under dynamic impact. Similarly, RODRIGUES et al. [⁹] integrated sensor data for probabilistic damage detection in CFRP panels. Collectively, these investigations confirm the growing importance of laminated composite materials in achieving lighter, safer, and more durable aircraft structures.

Vibration remains a critical concern in the design of aircraft fuselage structures, directly influencing fatigue life, passenger comfort, and flight safety. Fuselage vibrations arise from multiple sources, including engine operation, aerodynamic forces, turbulence, and dynamic loads during take-off and landing. If the natural frequencies of the structure coincide with excitation frequencies, resonance can lead to excessive deformation or catastrophic structural failure. Recent research has focused on mitigating vibration effects in composite structures. LI and LU [¹⁰] implemented active vibration control techniques to suppress helicopter fuselage oscillations, while SONG et al. [¹¹] examined the vibration behaviour of composite wing-boxes under hygrothermal environments, demonstrating the sensitivity of composite stiffness to temperature and moisture. ZHAO et al. [¹²] analysed the rigid–flexible coupling behaviour in hydrogen-electric aircraft fuselage panels, emphasizing the role of structural coupling on vibration modes. XU et al. [¹³], WANG et al. [¹⁴], and BROWN et al. [¹⁵] each studied the influence of geometry, boundary conditions, and material properties on vibration characteristics of composite shells and wings. LV et al. [¹⁶] employed multi-view vibration measurements to characterize structural dynamics in full-scale fuselage tests. These studies collectively underline that a comprehensive understanding of fuselage vibrations is essential to ensure structural integrity and comfort in modern aircraft.

The fundamental natural frequency plays a crucial role in preventing resonance in aircraft fuselages. A structure with a higher fundamental frequency can withstand dynamic loads and environmental disturbances without entering resonant conditions. Consequently, optimizing laminate design to maximize this frequency has become a prominent research objective. YANG and LIANG [¹⁷] used Bayesian optimization to enhance the fundamental frequency of laminated composite plates. KHAJAH and NATARAJAN [¹⁸] applied a differential evolution algorithm for tow-steered composites, improving flutter resistance. PAL et al. [¹⁹] used a grey wolf optimization approach for laminated shells, while INNAMI et al. [²⁰], WANG et al. [²¹], and JING et al. [²²] optimized stacking sequences and fibre orientations in composite panels to improve vibration resistance. All these works affirm that maximizing the fundamental frequency through tailored stacking sequences and fibre orientations is critical for designing vibration-resistant composite fuselages.

Finite Element Analysis (FEA) has become an indispensable tool for analysing the dynamic behaviour of laminated composite structures. Its capability to simulate complex geometries, heterogeneous material properties, and boundary conditions makes it the preferred method for accurate vibration prediction in aerospace applications. Subham PAL et al. [²³] employed Finite Element Analysis (FEA) to investigate the free vibration behaviour of laminated composite shell structures with central cut-outs based on First-Order Shear Deformation Theory (FSDT). while CHEN et al. [²⁴] used a quasi-three-dimensional model to analyse pre-twisted laminated shells. Fan and LEZGY-NAZARGAH [²⁵] developed advanced shell theories validated through FEA, and MUC and FLIS [²⁶] studied supersonic flutter in laminated panels. Additional contributions by PARVEZ and KHAN [²⁷], ZHAO et al. [²⁸], LIU et al. [²⁹], and TONG et al. [³⁰] confirmed FEA’s reliability for predicting modal behaviour under diverse loading and geometric conditions. Collectively, these investigations demonstrate that FEA is the most reliable approach for capturing the dynamic response of composite fuselage structures.

Artificial Neural Networks (ANN) provide a powerful data-driven alternative to traditional numerical simulations, capable of learning complex nonlinear relationships between material parameters and structural responses. Their ability to rapidly predict outcomes once trained makes them an efficient surrogate for computationally expensive FEA simulations. HOANG and THANH [³¹], PRUSTY et al. [³²], and ZHANG et al. [³³] showed that ANN models accurately predict vibration and buckling characteristics of composite components. MAJUMDER and MISHRA [³⁴] used ANN-based meta-modelling for strength prediction in conical shells, while LIANG et al. [³⁵] developed ANN-based fatigue life estimators for FRP laminates. Studies by SREEKANTH et al. [³⁶] and KALLANNAVAR et al. [³⁷] confirmed ANN’s capability in delamination detection and vibration prediction under environmental effects.

A comprehensive review by LIU et al. [³⁸] concluded that ANN-based methods outperform conventional analytical models for complex composite analyses. NICHOLAS et al. [³⁹] successfully demonstrated ANN-assisted stacking sequence optimization for wind turbine blades, highlighting its potential for aerospace structures. These findings confirm that ANN can complement FEA by providing accurate and computationally efficient predictions for complex composite designs.

Genetic Algorithms (GAs) have proven highly effective for optimizing laminated composite structures because of their ability to navigate large, nonlinear, and discrete design spaces. Inspired by evolutionary principles, GAs perform global searches and avoid local minima, making them ideal for multi-objective optimization. Recent research includes NICHOLAS et al. [⁴⁰], who used NSGA-II for mass minimization and strength maximization of laminated panels under manufacturing constraints, and LIU et al. [⁴¹], who combined GA with ANN surrogates for efficient weight optimization and buckling load prediction. These studies illustrate GA’s strength in balancing multiple performance criteria, including stiffness, strength, and vibration resistance, while satisfying structural and manufacturing constraints. The integration of GA with ANN enables high-speed exploration of complex design spaces, thereby offering an efficient and accurate route to optimize laminated composite fuselage structures. Duc Tien NGUYEN et al. [⁴²] focused ANN-based hybrid neural analytical surrogate model for efficient and accurate prediction of graded graphene nanoplatelet-reinforced composite (FG-GPLRC) plates subjected to vibration analysis with structured Design of Experiments based dataset generation approach. Nguyen Cong TAN et al. [⁴³] optimized linear and nonlinear vibration behaviour of graphene nanoplatelet-reinforced composites ­complex-shaped One-Variable Edge Plates (OVEPs) using Physics-Informed Neural Network (PINN) integrated with FEA aims to reduce computational cost and prediction time with high prediction accuracy for real-time structural dynamic analysis. NGUYEN and NINH [⁴⁴] employed heteroscedastic Gaussian process (HGP) framework to predict the natural vibration frequency of FG-CNT nanocomposite shells associated with stochastic geometric uncertainties. ANNs are highly effective method for complex nonlinear large structures with relationship between ply orientations, material properties, geometric parameters and structural responses such as natural frequency. Duc Tien NGUYEN et al. [⁴⁵] Studied the forward and inverse vibrational response of functionally graded carbon nanotube reinforced composite (FG-CNTRC) cylindrical shells to determine the unknown excitation forces and optimize the shell thickness with the L-BFGS-B optimization algorithm for enhanced vibration control and structural efficiency in advanced engineering applications.Nguyen Manh DZUNG et al. [⁴⁶] investigated the nonlinear stochastic dynamic behavior of graphene nanoplatelet-reinforced composite (FG-GPLRC) plates to evaluate the effects of geometric parameters, GPL weight fraction and GPL distribution patterns on natural frequencies

The comprehensive literature survey establishes that laminated composite materials have become integral to modern aircraft fuselage design due to their superior specific strength, stiffness, and fatigue resistance. However, vibration-induced failure remains a persistent challenge that threatens structural integrity, passenger safety, and ride comfort. Consequently, increasing the fundamental natural frequency of composite fuselage structures is a key design objective to mitigate resonance effects. Finite Element Analysis (FEA) continues to serve as the benchmark tool for accurately simulating the dynamic response of composite laminates. Nevertheless, its computational intensity limits its direct use in large-scale optimization where numerous design iterations are required. Artificial Neural Networks (ANN) complement FEA by acting as reliable surrogate models trained on numerical or experimental data, providing fast and accurate predictions of vibration characteristics and damage indicators.

Genetic Algorithms (GA) have emerged as robust global optimizers capable of handling discrete and nonlinear design spaces. They effectively explore lamination parameters such as ply orientation, stacking sequence, and layer thickness to enhance stiffness, reduce weight, and maximize frequency under realistic manufacturing and operational constraints. Despite substantial research combining FEA and ANN for composite vibration analysis, there remains a critical gap in efficiently optimizing fuselage laminates for maximum fundamental frequency while maintaining computational feasibility.

Most of the researchers mainly focused their investigations on simplified composite plates, cylindrical shells and other static structural components rather than aircraft fuselage structure subjected to various vibration constraints. ANN based surrogate modelling is incorporated to FEA framework to develop a non-linear relationship between the laminate design parameters and the structural dynamic responses based on FEA generated datasets. The trained ANN model predicts the vibration characteristics of new laminate configurations rapidly, with a high degree of accuracy and a considerable reduction in computational effort. Moreover, the Genetic Algorithm (GA) is used to efficiently search the global design space and obtain the optimal stacking sequences for maximum vibration resistance and structural efficiency, while simultaneously satisfying the constraints of strength and weight. Thus, the coupling of FEA with ANN and GA has considerable advantages like reduction in optimization time, increase in computational efficiency, robustness in predicting the dynamic response and ability to handle complex composite fuselage optimization problem for engineering applications. The present work addresses this gap by integrating GA with an ANN surrogate model trained on FEA-generated datasets. This hybrid approach accelerates the optimization of laminate stacking sequences, achieving high accuracy at dramatically lower computational cost. The developed framework thus provides a practical pathway for designing vibration-resilient composite fuselages capable of sustaining operational loads without resonance-induced degradation.

2. MATERIALS AND METHODS

This study focuses on enhancing the dynamic performance of a laminated composite aircraft fuselage through maximization of its fundamental natural frequency while keeping overall structural weight unchanged. Increasing this frequency minimizes the risk of resonance-induced vibration, thereby improving safety and comfort. The fuselage geometry considered is derived from the Embraer E190 regional jet [⁴², ⁴³]. A 3 m-long representative section is modelled as shown in Figure 1 using detailed cross-sectional dimensions to capture realistic stiffness and mass distribution.

Figure 1
Embraer 190 Fuselage cross section.

Since fiber-reinforced polymer (FRP) composites are orthotropic, the laminate stacking sequence critically influences their vibrational behaviour. Hence, the design variables in this study are the ply orientations of a symmetric 10-layer laminate. The aim is to identify the orientation set that yields the maximum fundamental frequency without violating in-plane strength or weight constraints.

The optimization objective is expressed as

(1) $$\underset{\theta }{\max }{f}_1(\theta )$$

Subject to the following engineering constraints:

(2) $${\sigma }_{max}{}_{(\theta )}\le {\sigma }_{allowable}$$

The in-plane strength constraint that ensures the maximum induced stress does not surpass material limits, guaranteeing structural safety. Where, σ_max(θ) is the maximum in-plane stress induced in the laminate under applied loads and σ_mallowable is the allowable stress of the chosen material

(3) $$W_{(\theta )}\le W_{max}$$

The weight constraint that ensures that structural weight remains within the allowable design limit. Where, W_(θ) is the total weight of the fuselage section and W_max is the maximum allowable weight

(4) $${\theta }_{full}=[{\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4,{\theta }_5,{\theta }_5,{\theta }_4,{\theta }_3,{\theta }_3,{\theta }_1]$$

The above constraint says that laminate is symmetric in order to eliminate coupling effects and simplify manufacturing.

(5) $$\text{The design variables are}{\theta }_i\in =\{-90°,-89°,\dots ,0°,\dots ,+89°,+90°\},i=1,2,\dots ,5$$

This allows a fine search over the entire feasible orientation space. This formulation maintains physical realism, symmetry, and manufacturability while enabling exhaustive exploration of the design space.

The Finite Element Analysis (FEA) is used to determine the natural frequencies and mode shapes of the fuselage structure under fixed-end boundary conditions. FEA provides the high-fidelity training data for the surrogate model. Artificial Neural Network (ANN) serves as a predictive model for fundamental frequency, trained on FEA outputs to approximate frequency responses for new stacking sequences. Genetic Algorithm (GA) performs global optimization of ply orientations using the ANN model to accelerate evaluations. Promising configurations are subsequently verified by FEA. This integrated GA–ANN–FEA approach ensures accuracy comparable to full FEA-based optimization while reducing computational cost by several orders of magnitude.

3. FINITE ELEMENT ANALYSIS

The finite element method (FEM) was employed to analyse the dynamic behaviour of the laminated composite fuselage modelled as a thin-walled cylindrical shell. All analyses were performed using ABAQUS, with the entire workflow that includes geometry generation, material assignment, meshing, boundary condition setup, and frequency extraction are automated through Python scripting. This automation enabled rapid simulation of hundreds of stacking sequences generated by the optimization algorithm.

A 3 m-long fuselage segment from the Embraer E190 model [⁴⁷, ⁴⁸] was selected for the study. The model was discretized using S4R shell elements, which are four-node, reduced-integration elements suitable for layered composite analysis. The laminate consisted of ten symmetric plies, each modelled as an orthotropic elastic lamina. Transverse shear deformation was neglected because of the shell’s high slenderness ratio. Both ends of the fuselage section were fully fixed to replicate realistic boundary conditions encountered during ground vibration testing and in-service constraints.

Each ply was defined by its orthotropic stiffness matrix [Q], that encapsulates the material’s direction-dependent properties. For every ply i, the local stiffness matrix is transformed into global coordinates based on the ply orientation θ_i. The transformation is given by

(6) $$\overline{Q_{IJ}}({\theta }_i)=T^{-1}({\theta }_i)\cdot Q_{ij}{T}^{-T}({\theta }_i)$$

where Q_ij and Q_ij ̅ represents the ply stiffness in local and global coordinates, and T^–1 (θ_i) is the transformation matrix for orientation θ_i.

The structural dynamic equilibrium for free vibration is represented as:

(7) $$[M]\left\{\ddot{u}\right\}+[K]\left\{u\right\}=\left\{\varphi \right\}$$

where [M] and [K] denote the global mass and stiffness matrices, {u} is the displacement vector, and {ü} the acceleration vector. For undamped free vibration, this leads to the classical eigenvalue problem:

(8) $$[K]-{\omega }^2[M]\left\{\phi \right\}=\left\{\varphi \right\}$$

where, ω is the natural angular frequency (rad/s) and {φ} is the corresponding mode shape. The fundamental frequency f₁ is calculated as:

(9) $$f_1=\raisebox{1ex}{${\omega }_1$}\!\left/ \!\raisebox{-1ex}{$2\pi $}\right.$$

For composite laminates, the stiffness matrix is derived from Classical Laminate Plate Theory (CLPT):

(10) $$[K]={\displaystyle \iint ([A]}+z[B]+z^2[D])dz$$

Where [A], [B], [D] are the extensional, coupling, and bending stiffness matrices, respectively. Since the laminate is symmetric, the coupling matrix [B]is zero, ensuring decoupled in-plane and bending behaviours. [B]. A structured quadrilateral mesh was adopted to achieve numerical accuracy while maintaining computational efficiency. Mesh convergence studies confirmed that the chosen element size captured the first three modes accurately with minimal error. Both ends of the fuselage were modelled as fixed (clamped–clamped boundary condition).

The free vibration analysis was carried out without external loads, and the first fundamental frequency was extracted automatically from the simulation output (.ODB) using Python scripts. This FEA framework provided the high-fidelity dataset for training the ANN and validating the optimization results.

4. GENETIC ALGORITHM (GA)

A Genetic Algorithm (GA) was implemented to identify the optimal ply stacking sequence that maximizes the fundamental frequency under defined strength and weight constraints. GAs is particularly effective for laminate optimization problems due to their robustness in navigating nonlinear and discrete design spaces [⁴⁰, ⁴¹].

Each laminate configuration was represented as a chromosome, with individual genes corresponding to ply orientation angles. The design variables {θ₁, θ₂, θ₃, θ₄, θ₅} are defined as the symmetric 10-layer laminate as [θ₁, θ₂, θ₃, θ₄, θ₅, θ₅, θ₄, θ₃, θ₂, θ₁]. The GA initialized a population of random chromosomes within the specified bounds –90° to 90° with an increment of 1°. The fitness function was defined as the fundamental frequency predicted by either ANN model or FEA, subject to the constraints specified in Eqs. (2) to (4). Penalty functions were introduced to eliminate infeasible designs that violated stress or weight limits.

The GA evolution process involved selection, crossover and mutation operators to reproduce the offspring. The roulette wheel selection process was used to favour high-performing chromosomes. The uniform crossover process was applied for genetic diversity. The uniform mutation was performed to explore unexplored regions of the design space. Each GA iteration generated a population of candidate solutions, which were rapidly evaluated using the ANN surrogate model. The best-performing solutions were then validated through FEA to ensure prediction accuracy. The pseudocode of the implemented GA is illustrated in Figure 2. The hybrid GA–ANN scheme provided a computationally efficient optimization strategy, achieving global search capability with drastically reduced FEA computations.

Figure 2
Pseudocode for genetic algorithm driven laminate sequence optimization.

5. ARTIFICIAL NEURAL NETWORK (ANN)

In this investigation, a feedforward Artificial Neural Network (ANN) is designed to efficiently predict the fundamental frequency of laminated composite fuselage structures using input variables representing laminate ply angles. Leveraging high-fidelity finite element analysis (FEA) results as the training data source, the ANN acts as a rapid surrogate model to accelerate the optimization process.

The ANN architecture comprises four layers: an input layer with five neurons (corresponding to distinct ply orientations for the symmetric laminate), two hidden layers (with neuron counts optimized via trial-and- error to balance learning capacity and computational efficiency), and a single-neuron output layer predicting the fundamental frequency. This moderate network depth ensures flexibility for capturing complex, non­linear relationships between laminate parameters and dynamic response, without risking overfitting or excessive computation.

High fidelity Finite Element Analysis Datasets of 1000 Nos were generated using a structured Design of Experiments strategy based on Latin Hypercube Sampling (LHS). The LHS approach ensures uniform sampling method for adequate coverage of the multidimensional stacking-sequence design which prevents clustering of samples and exposing the ANN to diverse composite laminate configurations under dynamic responses.

ANN surrogate model performance is evaluated using Mean Squared Error (MSE),

(11) $$MSE=\frac{1}{N}{\displaystyle {\sum }_{i=1}^N{(y_i-{\widehat{y}}_i)}^2}$$

Training is performed using the Levenberg–Marquardt (LM) algorithm, selected for its recognized speed and accuracy, especially in function approximation problems with moderate data size. The LM method offers fast convergence by adaptively interpolating between gradient descent and Newton-type updates, making it especially suitable for engineering regression tasks. Backpropagation is used to propagate and minimize the error between predicted and true frequencies by adjusting network weights during learning.

The activation functions are tailored for the task: a hyperbolic tangent sigmoid (tansig) in the input and hidden layers to capture nonlinearity, and a linear output function for continuous frequency prediction. Model performance is measured using Mean Squared Error (MSE), a standard criterion in regression, offering intuitive interpretation of prediction accuracy.

Nonlinear activation functions tanh/ReLU were used in the hidden layers to effectively capture the nonlinear relationship between laminated composite fuselage structure parameters and dynamic response. Linear activation function was adopted in the output layer for laminated composite fuselage fundamental frequency prediction for continuous regression. This moderate network depth was selected to ensure sufficient modelling flexibility without causing excessive computational cost or overfitting.

This moderate ANN training process incorporated validation-based monitoring and repeated training with different neuron combinations to avoid overfitting and improve generalization capability. The parameters including the number of hidden neurons, learning rate and training iterations were tuned to achieve the optimal solution between stability and accuracy.

The performance of the ANN surrogate model was quantitatively evaluated using statistical error metrics such as Root Mean Square Error (RMSE) and the coefficient of determination (R²).

$$RMSE=\sqrt{\frac{1}{N}{{\displaystyle \sum _{i=1}^N(y_i-{\widehat{y}}_i)}}^2}$$

The RMSE measures the average magnitude of error between the ANN outputs and the validated FEA results. Least RMSE values indicate better prediction accuracy and better consistency between the surrogate model and the numerical simulations.

The coefficient of determination (R²) is given by:

$$R^2=1-\frac{{\displaystyle \sum {(y_i-{\widehat{y}}_i)}^2}}{{\displaystyle \sum {(y_i-\widehat{y})}^2}}$$

To ensure robust generalization, the FEA-generated dataset is split into 70% training, 15% testing, and 15% validation subsets. This approach confirms the model’s capacity not only for accurate prediction within known settings but also for handling unseen stacking configurations. A schematic pseudocode detailing the ANN implementation and training workflow is shown in Figure 3.

Figure 3
Pseudocode for ANN-based fundamental frequency prediction.

The in-plane strength constraint was introduced to ensure that the induced stresses remained below the allowable material strength limit:

$$g_{strength}(X)=\frac{{\sigma }_{max}(X)}{{\sigma }_{allow}}-1\le 0$$

Similarly, the weight constraint was imposed as:

$$g_{weight}(X)=\frac{W(X)}{W_{max}}-1\le 0$$

The laminate weight was calculated using:

$$W=\rho \times A\times t$$

This ANN integration dramatically reduces the computational burden during optimization, enabling rapid evaluation of thousands of laminates stacking sequences and empowering the genetic algorithm to more effectively search for globally optimal designs.

During optimization, the ANN rapidly predicted fundamental frequencies for thousands of laminate configurations generated by the GA. Only the most promising candidates were verified using FEA. This integration reduced computational time by over 90% compared to direct FEA-only optimization.

6. RESULTS AND DISCUSSION

6.1. Validation of the finite element model

Before applying the optimization framework, the finite element analysis (FEA) procedure was validated against benchmark studies on laminated composite plates by YANG and LIANG [¹⁷], ZHAO et al. [²⁸], and WANG et al. [²¹]. These works reported non-dimensional frequencies for various stacking sequences and aspect ratios under clamped boundary conditions. The same configurations were reproduced using the present FEA procedure to ensure consistency. Material properties used for validation are listed in Table 1.

Table 1
Material properties of laminated composite plates.

PROPERTY	SYMBOL	VALUE
Longitudinal modulus	E1	250 GPa
Transverse modulus	E 2	10 GPa
In-plane shear modulus	G12	5 GPa
Transverse shear moduli	G13	5 GPa
Transverse shear moduli	G23	2 GPa
Poisson’s ratio	µ12	0.25
Density	ρ	1300 kg/m3

The shell structure was discretized using a structured quadrilateral mesh with S4R elements, as shown in Figure 4(a). The four edges of the plate were fully fixed to simulate a CCCC boundary condition. Since the focus was on fundamental frequency estimation, no external load was applied. The finite element model with boundary conditions is shown in Figure 4(b).

Figure 4
(a) and (b) The finite element model with boundary conditions.

The non-dimensional frequencies obtained in this work were compared with those from previous studies, as summarized in Table 2. The strong agreement validates the adopted mesh density, boundary conditions, and numerical settings. The same modelling strategy was subsequently applied to the fuselage configuration. The variation of natural frequencies with aspect ratio is presented graphically in Figure 5. The resulting frequencies and their variation with aspect ratio closely matched published references, confirming the reliability and accuracy of the FEA setup.

Table 2
Comparison of non-dimensional frequencies under CCCC condition.

LAMINATION SCHEME	b/a	NON-DIMENSIONAL FREQUENCY OBTAINED BY
		PRESENT WORK	YANG ET AL. [17]	ZHAO ET AL. [28]	WANG ET AL. [21]
[15°/–15°/15°/–15°]	1	30.84	30.85	30.915	30.854
	1.5	29.09	29.05	29.103	29.046
	2	29	28.51	28.561	28.506
[30°/–30°/30°/–30°]	1	29.76	29.73	29.786	29.725
	1.5	25.49	25.47	25.518	25.474
	2	24.3	24.11	24.145	24.106
[45°/–45°/45°/–45°]	1	29.40	29.37	29.432	29.372
	1.5	21.51	21.49	21.522	21.489
	2	18.95	18.89	18.909	18.885

Figure 5
Natural frequency of laminated composite plate.

6.2. Design optimization of composite aircraft fuselage

After model validation, the hybrid optimization methodology was applied to a 3 m fuselage section of the Embraer E190 aircraft [⁴², ⁴³]. The geometric representation and the three-dimensional CAD model are shown in Figure 6. Material properties representative of aerospace-grade carbon/epoxy composites are listed in Table 3.

Figure 6
3D CAD model of the E190 fuselage.

Table 3
Material properties of laminated composite fuselage [⁴³].

PROPERTY	SYMBOL	VALUE
Longitudinal modulus	E1	125 GPa
Transverse moduli	E 2	8.50 Gpa
Shear moduli	G12	3.10 Gpa
Poisson’s ratios	µ12	0.25
Density	ρ	1400 kg/m3

The fuselage shell was discretized using S4R elements to balance accuracy and computational efficiency. Both ends were fixed to simulate realistic boundary constraints. The meshing and boundary conditions are shown in Figures 7a and 7b.

Figure 7
(a) and (b) Meshing and boundary condition of the fuselage.

A simple but robust genetic algorithm (GA), coded in MATLAB, was deployed to explore the large, discrete design space of ply orientations in a symmetric 10-layer laminate (each 1 mm thick, ply angles from –90° to +90° in 1° steps). The GA efficiently handles complex design constraints and enables global search. A coupled Python script executed ABAQUS simulations automatically for each GA-proposed design.

To minimize computation time, a feedforward Artificial Neural Network (ANN) was trained on 500 finite element solutions spanning randomly selected stacking sequences. After validation (85% training, 15% testing), the ANN, with a five-neuron input layer, two hidden layers (tuned by trial and error), and a single-output neuron, predicted fundamental frequency during GA iterations. As shown in Figure 5, the ANN achieved high fidelity with FEA—prediction error is less than 2% across test cases.

For optimization, a simple genetic algorithm (GA) was developed based on the procedure mentioned using MATLAB. The GA was chosen as it efficiently handles large and complex design spaces. A Python script was prepared to implement the validated FEA procedure, which was executed within MATLAB. Additionally, an Artificial Neural Network (ANN) was constructed in MATLAB and trained with solutions obtained from FEA. Once validated, the ANN was employed to predict new solutions during the optimization process in order to significantly reduce computational effort.

The training and testing data set required for the ANN is generated randomly. A symmetric laminate scheme with a total number of 10 layers is considered, where the thickness of each layer is 1 mm. The fibre angle of each layer may vary from –90° to 90° with an increment of 1°. The natural frequency is found using the validated FEA procedure. A total of 500 data points are generated randomly, and 85% of the data are used for training, while the remaining 15% are used for testing the ANN. As the number of inputs to the ANN is five (half of the symmetric stacking sequence), five input layers are used during the construction of the ANN. The control parameters of the ANN, as determined through the trial-and-error method, are listed in Table 4 and used during the construction of the ANN.

Table 4
The control parameters of ANN.

CONTROL PARAMETERS	VALUES
Neurons (layers)	5 – 15 – 10 – 1
Activation functions	tansig (hidden), purelin (output)
Training algorithm	Levenberg–Marquardt
Epochs	Auto-stop at min. MSE
Error tolerance	< 2 %

After the training, the prediction of the ANN is tested by applying it to the randomly generated 50 new samples. Figure 8 compares ANN-predicted and FEA-computed frequencies, showing near-linear correlation, while Figure 9 presents the corresponding percentage error distribution, which remained below 2% for all cases. These results confirm that the ANN surrogate is sufficiently accurate for integration within the optimization loop.

Figure 8
Comparison of results obtained by ANN and FEA.

Figure 9
Percentage of error in ANN prediction.

The validated ANN model was coupled with a simple genetic algorithm (GA) implemented in MATLAB. Control parameters adopted for the optimization are summarized in Table 5. The GA converged smoothly, as shown by the evolution curve of the objective function in Figure 10, where the fundamental frequency progressively increased with generations and stabilized after about 100 iterations.

Table 5
The control parameters of GA.

CONTROL PARAMETERS	VALUES
Number of generations	125
Crossover Probability	0.3
Mutation Probability	0.06
Population size	40

Figure 10
Optimum solutions obtained using GA.

The three best stacking sequences obtained using GA are listed in Table 6. The optimum solution obtained in GA is reconfirmed with FEA in order to avoid the wrong solution, if any mistake is there in the prediction of ANN. The calculation of natural frequency using ANN and FEA is compared in Table 6, which shows that it is very close to the actual one. The graphical variation of frequency obtained for the optimal solution using FEA is shown in Figure 11.

Table 6
Verification of optimum solutions obtained by GA.

S. NO.	OPTIMUM STACKING SEQUENCE OBTAINED USING GA	NATURAL FREQUENCY (HZ)
		ANN	FEA
1	[0/40/-50/70/-70] s	69.59	69.576
2	[-8/39/-48/74/-72] s	69.51	69.543
3	[0/45/-45/75/-60] s	69.42	69.468

Figure 11
The graphical variation of frequency obtained for the optimal solution.

6.3. Discussion

The optimization results demonstrate that the proposed FEA–ANN–GA framework effectively identifies laminate stacking sequences that maximize fuselage natural frequency while satisfying weight and strength constraints. The integrated ANN surrogate significantly reduces the number of costly FEA simulations by more than 90 %, accelerating convergence without compromising accuracy.

Compared with conventional FEA-only approaches, the hybrid method significantly increases the fundamental frequency for the same structural weight. The results also show that dispersed ply angles contribute most to stiffness enhancement in the cylindrical fuselage configuration. These findings corroborate previous optimization studies by WANG et al. [²¹] and LIU et al. [⁴³], confirming that evolutionary algorithms coupled with machine-learning surrogates provide powerful, computationally efficient tools for complex composite design.

The present framework is particularly effective for non-random loading conditions such as free vibration analysis, static bending, harmonic loading and stiffness-based structural optimization. Under these loading conditions, the structural response of composite structure varies smoothly with design parameters, enabling the ANN to learn the nonlinear relationship accurately between stacking sequence and structural performance. However, when random or stochastic loads are considered such as random vibration, turbulent aerodynamic excitation, probabilistic dynamic loading and uncertain operational environments,

The main advantages of the proposed framework are:

• Significant reduction in computational time approximately 90%

• Efficient optimization of large laminate design structures

• Accurate prediction of non linear capability with low ANN error,

• Reduce vibration behaviour performance without increasing structural weight,

• Reduced dependence on repeated high-cost FEA simulations.

The primary limitations include:

• Depend on ANN training data quality and coverage,

• Limited extrapolation ability out of trained design domain,

• Increased computational complexity under stochastic loading conditions,

• Require larger datasets when uncertainty effects are included,

Therefore, the proposed FEA–ANN–GA framework is highly suitable for deterministic vibration and optimization problems for large structures and extended to random loading applications.

7. CONCLUSIONS

This study developed an integrated computational framework combining Finite Element Analysis (FEA), Artificial Neural Networks (ANN), and Genetic Algorithms (GA) to optimize the stacking sequence of laminated composite fuselage structures with the objective of maximizing the fundamental natural frequency. The FEA model, validated against established benchmark studies, demonstrated excellent agreement with reference data, confirming its reliability for dynamic analysis. The ANN surrogate, trained using 500 FEA-generated datasets, accurately predicted fundamental frequencies with less than 2% error, effectively capturing the nonlinear relationship between stacking sequence and dynamic response. Coupling the ANN with a GA significantly reduced computational cost—by more than 90% compared with conventional FEA-only optimization—while maintaining comparable accuracy. The proposed hybrid framework efficiently identified optimal laminate configurations that improved the fundamental frequency without increasing structural weight, thereby enhancing vibration resistance and overall fuselage performance. These results confirm that integrating machine learning and evolutionary algorithms provides a powerful, efficient, and accurate approach for optimizing advanced composite structures.

The results also provide important physical insights into the behaviour of the optimized laminated composite. The study confirms that the optimized stacking sequences are mainly governed by stiffness redistribution without any changes in mass of the composite material. The vibration performance is improved only by the tailoring of the ply orientations within the specified bounds -90° to 90° with an increment of 1°. In addition to that the successful integration of FEA, ANN and GA has demonstrated that machine learning assisted optimization is very effective in solving complex laminated composite structural design problems for various engineering applications.

From this proposed framework provides the combined analysis of FEA, ANN, and GA confirms that the:

• highly accurate prediction capability,

• excellent computational efficiency,

• robust estimation of dynamic response,

• efficient identification of optimal laminate configurations,

• significant increase in vibration resistance without increasing the weight of the structure.

These results show that surrogate-assisted intelligent optimization frameworks can be powerful tools for the design of future lightweight composite fuselage structures under dynamic loading environments. Future research will extend this methodology to consider hygrothermal effects, geometric nonlinearities, and manufacturing variability to achieve robust, real-world applicability in next-generation composite fuselage design.

9. DATA AVAILABILITY

All data supporting the findings of this study are included within the article.

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Publication Dates

• Publication in this collection
  06 July 2026
• Date of issue
  2026

History

• Received
  04 Mar 2026
• Accepted
  26 May 2026