Machine Learning Based Uncertain Free Vibration Analysis of Hybrid Composite Plates

1  Introduction

Reinforced hybrid composite materials offer significant advantages over metallic alloys, largely in the aerospace industry, owing to its superior stiffness, impact resistance, corrosion resistance, and lightweight nature. Unlike metallic materials, hybrid composites can be customized to achieve specific structural performances by manipulating design variables such as fiber positioning and ply heaping sequences [1]. This customization often aims for the minimum weight design while adhering to various mechanical, geometrical, and high-tech requirements. However, in spite of progression in founding technologies, manufacturing flawless composite edifices remains a challenge. As a result, the full benefits of these materials have yet to be realized [2]. The finite element method, combined with a multivariate adaptive regression splines surrogate model is used to quantify the uncertainty of the free vibration of functionally graded materials in a cantilever plate [3]. The stochastic sensitivity analysis of FGM plates is crucial for assessing their performance under free vibration and low-energy impact, allowing us to pinpoint the most grave constraints in these investigations [4]. Several works have used a finite element FE technique based on RBF to convincingly depict the stochastic natural frequencies of cantilever plates built from FGM [5]. Furthermore, in order to ensure accurate and dependable findings, the robustness of the projected computation, for stochastic natural frequency analysis of composite plates has been carefully checked and confirmed against the original FEM in conjunction with MCS [6]. An innovative model designed to accurately analyze the stochastic characteristic vibration and the equivalent emitted power of rectangular coated plates. This model efficiently represents elastic parameters, natural frequencies, and acoustic power density using generalized polynomial chaos expansions with variable random bases [7]. The application of MCS through our established regression model enables us to derive essential statistical characteristics of structural natural frequencies, including mean values, standard deviations, probability density functions, and cumulative distribution functions [8]. The development of traditional low-dimensional analytical modeling techniques enables the rapid and efficient extraction of natural frequencies in flexible assembled structures [9]. This comprehensive strategy not only enhances understanding but ultimately provides a robust framework for making informed decisions in structural engineering [10]. The use of artificial intelligence and hybrid knowledge–data-driven approaches for modeling flexible assembled structures has recently gained significant attention [11]. Surrogate models have been widely used in the optimization and prediction of solutions for computationally extensive problems, like as delamination detection [12]. Diverse surrogate replicas tend to execute well under varying circumstances and are integrally influenced by the specific characteristics of the current issues [13]. In order to compare different substitute structures based on a number of performance criteria, such as accuracy, toughness, effectiveness, precision, and conceptual simplicity, researchers built ensembles of meta-models with optimum weight factors [14]. Specifically, Kriging surrogate models have been implemented for delamination finding in fused laminates [15]. Recent comparisons between ANN and Kriging surrogate models for delamination detection have revealed a significant advantage for Kriging models which excel in both prediction accuracy and processing speed [16]. In the realm of engineering, various studies have also explored different surrogate models tailored for specific applications, such as radial basis neural networks in the design of liquid rocket injectors, supersonic turbines, and the shapes of prototypes [17]. While researchers have been focused on selecting the optimal surrogate model, the investigation of ensemble techniques still has a significant deficit [18]. Such methods could be particularly promising for tackling the delamination detection challenge in composite laminates. Ultimately, the evidence suggests that Kriging models consistently stand out as the most effective solution across various problems [19]. Surrogate-based uncertainty quantification has emerged as a crucial approach in the past decade for accurately measuring uncertain universal responses in composite laminates, effectively addressing uncertainties in both geometric and material parameters [20]. To robustly quantify uncertainty, whether probabilistic or non-probabilistic in the dynamic analyses of composite structures, researchers have successfully integrated surrogate and machine learning-based methodologies, including enhancements to lower-order theories [21]. The value of surrogate-based methodologies in stochastic dynamic and stability analysis of composites made of laminates is thoroughly described in a new monograph by distinguished researchers [22]. Surrogate models are the preferred choice in uncertainty quantification; they dramatically reduce the need for extensive finite element simulations, which typically require thousands of function evaluations in traditional MCS based methods [23]. The lifespan of a mechanical system necessitates several extensive and intricate finite element (FE) simulations. However, the traditional approach of comprehensive simulations is evolving. Today, low- and mid-fidelity FEA have taken center stage, updating swiftly and effectively based on real-time experimental measurements of the system’s condition [24]. The rise of the Big Data era has been profoundly influenced by machine learning (ML) algorithms. These algorithms excel at processing large datasets automatically, uncovering trends, drawing insightful conclusions, and predicting future outcomes with remarkable precision [25]. Essentially, through digitalization and connectivity, data is transformed into actionable information by machine learning [26]. In their research, Jang and Smith showcased the impact of temperature dispersal on normal vibrations through a full-scale FE model and simulation study. Notably, FE simulations can also serve as effective training grounds for ML models, thereby sidelining the necessity of rerunning complex simulations [27]. Additionally, two innovative active learning strategies were presented, integrating artificial neural networks and Kriging models to enhance reliability analysis. This approach not only streamlines the process but also elevates the accuracy of assessments, making it an obvious choice for mechanical system analysis in the future [28]. Two finite element problems and illustrative examples were used to demonstrate the effectiveness and precision of the proposed methods. A paradigm for design optimization based on uncertainty and based on the multi-fidelity polynomial chaos. To address the issues of low accuracy and high sensitivity of surrogate predictions in the presence of uncertainties, the Kriging surrogate model was developed [29]. This framework is particularly effective for complex aerodynamic applications. To determine the strain responses of columns based on how high-rise buildings behave under wind impact, RF created a sustainable strain-sensing model using an ANN [30]. They employed an RBF neural network for training, combined with a genetic algorithm for evolutionary learning to create their ANN model [31]. In another study, a meta-model for estimating the final strength of trusses was developed using the SVM technique through direct analysis. The study took into account a number of kernel functions for the SVM model, such as RBF, sigmoid, linear, and polynomial. A novel formulation for extracting important features pertaining to the kinds and locations of braces based on machine learning results was put forth using the SVM with an RBF kernel [32]. Additionally, an effective SVM assisted FE technique was proposed for the stochastic dynamic characterization of FG shells. To achieve a comprehensive probabilistic description of natural frequencies, an ML-based FE algorithmic paradigm was integrated with MCS [33]. The stochastic technique considered both the individual and combined effects of depth-wise source uncertainty in the material properties of functionally graded shells. It investigated the influence of several crucial parameters, including temperature, thickness, power-law exponent, and variations in shell geometries. Utilizing deep learning, Feng and Prabhakar [34] developed the first-ever stress distribution tool, which significantly reduced the computational cost of predicting stress distributions in heterogeneous media using FEA. They focused on areas with discontinuities and high concentrations of stress within their neural network frameworks, which are based on engineering and statistics [35]. Different finite element analysis model geometries and stress values have previously been used as direct inputs to train neural networks. Numerous studies have attempted to apply artificial neural networks to understand the constitutive behavior of composite laminates, particularly focusing on load-displacement and stress-strain curves [36]. The application of ANN techniques has enabled researchers to analyze the effects of several significant variables, which can be categorized into broad groups such as plate characteristics, material properties, geometric details, fracture locations, and porosity distribution [37]. Deep Learning models enhance feature extraction and enable complex data representation learning by stacking multiple layers in a neural network [38]. Without a doubt, neural networks are a popular algorithm across all domains of ML, consistently delivering impressive results for a variety of real-world challenges. In supervised learning problems, the mean squared error between the actual values and the predictions made by the neural network typically serves as the loss function [39]. In recent years, the integration of machine learning techniques into the stochastic dynamic analysis of hybrid composite plates has attracted significant attention. This interest is driven by the growing need for enhanced performance and reliability in various engineering applications. Carbon fiber reinforced polymeric composites, such as polylactic acid, are particularly well-suited for industries like automotive and aviation due to their lightweight and superior mechanical properties [40]. The use of fused filament fabrication for manufacturing a composite material made of polylactic acid PLA reinforced with carbon fiber composite structures has been explored, revealing that optimal settings for parameters such as fiber orientation, nozzle temperature, and bed temperature can significantly influence mechanical performance [41]. The application of machine learning, specifically Classification and regression trees, has proven effective in predicting tensile strength metrics, showcasing the potential of ML in optimizing composite material properties [42]. The review of existing literature from 2014 to 2024 indicates that significant advancements have been made in utilizing ML for analyzing the dynamic behaviors of composite plates, shells, and beams, thereby providing a valuable resource for researchers aiming to leverage these methods in their work [43]. The stochastic effects on functionally graded plates under dynamic loading conditions have also been a focal point of research. Variabilities in geometric and material properties can significantly impact the performance of FG plates during free vibration and impact loading scenarios. The use of support SVM to construct surrogate models has been shown to enhance the accuracy and computational efficiency of these analyses [44]. The output parameters derived from such analyses, including peak contact force and natural frequencies, are essential for understanding the dynamic response of hybrid composite structures. Moreover, the localization of low-velocity impacts on composite plates is another critical area where machine learning has been effectively applied [45]. The development of the Binary Dynamic Stochastic Search algorithm combined with support vector regression has demonstrated significant improvements in feature selection and localization accuracy for LVIs on carbon fiber reinforced plastic carbon fiber reinforced plastic plates [46]. This innovative approach not only reduces the dimensionality of impact features but also optimizes the SVR model’s parameters, thereby enhancing the overall performance of impact detection systems. Finally, the review of data-driven techniques for dynamic load identification underscores the challenges and prospects in this field [47]. The reliance on indirect identification methods due to difficulties in measuring dynamic loads directly necessitates the use of model-free approaches that are independent of structural characteristics [48]. By employing various data-driven techniques, including SVM and deep learning methods, researchers can effectively address issues related to load localization and reconstruction, paving the way for more accurate and reliable dynamic analyses of hybrid composite plates. In summary, a promising area for the study of hybrid composite plates is the nexus of stochastic dynamic analysis and machine learning [49]. The integration of advanced ML techniques not only enhances the understanding of mechanical and dynamic behaviors but it makes it easier to optimize composite materials for a variety of uses. Future investigations should continue to explore these synergies to further advance the field [50]. Zhuang et al. [51] proposed DAEM demonstrates strong accuracy and efficiency in bending, vibration, and buckling analysis of Kirchhoff plates. The results highlight its potential as a robust and generalizable machine learning framework for energy-based structural analysis. Samaniego et al. [52] demonstrates that DNNs, when guided by the energetic format of PDEs, can serve as a powerful alternative for solving mechanical problems. The results confirm their potential as flexible and efficient function approximators for computational mechanics applications. Liu et al. [53] proposed interpretable stochastic machine learning–multiscale framework demonstrates a robust and efficient strategy for predicting the thermal conductivity of graphene-based polymer nanocomposites under uncertainty. By combining homogenization-based FEM with explainable predictive modeling, this approach not only improves accuracy and computational efficiency but also provides deeper insights into parameter influence, paving the way for rational design of next-generation thermal management materials.

In this research work, efforts are made to compare the MCS model with various machine learning surrogate methods across different sample sizes, focusing on uncertainty in the 1st, 2nd, and 3rd natural frequencies. Additionally, scatter graphs are plotted to analyze how closely the results align with those of the MCS model. This is the first attempt, as far as the authors are aware, to measure the degree of uncertainty in the vibration characteristics of Nitinol hybrid constructions and develop the related probability density function visuals using a successful approach. Fig. 1 represents the geometric view of hybrid composite plate.

Figure 1: Geometric View of SMA based composite plate

2  Theoretical Formulation

It is possible to express the dynamic system for the equation of motion as [54]:

[M(C)]{δ¨}+[D(C)]{δ˙}+[K(C)]{δ}={F}(1)

where (C) specifies the degree of Stochasticity, {δ} stands for the displacement, {F} symbolizes externally applied load, while the other denomination like [M(C)], [D(C)], [K(C)] are the random mass, damping, and the last is stiffness matrix, correspondingly. Due to undamped free vibration, Eq. (1) becomes:

[M(C)]{δ¨}+[K(C)]{δ}=0(2)

The natural frequencies (ωk) as well as mode shapes (Sk) of the eigenvalue predicament can be solved to determine the structure.

[K(C)]Sk=ωk2[M(C)]Sk(3)

where, k=1,2,3,...,n. Divergent laws for the change of material properties fall within a variety of types. The modeling of the material in the hybrid composite plates will be covered in this section.

E11=Es(ζ)Vs+E1m(1−Vs)(4)

E22=E2m[[(1−Vs)+Vs1−Vs(1−E2mEs(ζ))]](5)

G12=G13=G12m×[1−Vs+Vs1−Vs(1−G12mGs(ζ))](6)

G23=G23m1−Vs(1−G23mGs(ζ))(7)

ν12=ν12SVS+ν12m(1−Vs)(8)

α1=VSαsEs(ζ)+(1−Vs)α1mE1mE11(9)

where, VS= Volume faction of hybrid composite, E = Young’s modulus, G = Shear modulus, ν = Poisson ratio, α = Thermal expansion coefficient. Fig. 2 represents the flowchart of surrogate based uncertain free vibration analysis of hybrid composite plate.

Figure 2: Steps of surrogate based uncertain free vibration analysis

3  Machine Learning Approach

3.1 Radial Basis Function

Radial basis function is an alternative surrogate archetypal which is relatively wide spread amid researchers. RBF is habitually cast-off to achieve the exclamation of dispersed multivariate numbers [55]. The meta-model looks in a direct mixture of Euclidean distances, which may be articulated as:

g^(X)=∑wkψk(X,xk)(10)

here n is the numeral of specimen themes, wk is the weight resolute by the least-squares technique and ψk(X,xk) is the k-th basis role gritty at the selection point xk. In numerable proportional centrifugal occupations are rummage-sale as foundation role. The radiated functions for RBF model are illustrated as [56],

Rf(X)=exp(−(X−c)T(X−c)r2)(For Gaussian)(11)

Rf(X)=1+(X−c)T(X−c)r2(For multi−quadratic)(12)

Rf(X)=11+(X−c)T(X−c)r2(For inverse multi−quadratic)(13)

Rf(X)=11+(X−c)T(X−c)r2(For Chachy)(14)

Gaussian radial basis function is used in this study, as it provided the best trade-off between accuracy and stability for the stochastic free vibration problem considered. It’s imperative to message that, contrasting other contraption learning approaches and the response surface method, Radial Basis Function is not a relapse system [57]. In its place, RBF can be sketchily watched as an exclamation method. Subsequently, disparate regression techniques, RBF delivers exact results at the sample points [58]. To date, RBF has been commonly pragmatic in the field of physical steadfastness and indecision evaluation [59]. For vagueness quantification, the method is heightened with an automatic system based on twist, an adaptive specimen scheme, equivalent in fill, and a multi-response principle. It has also been established that the stochastic RBF-based surrogate surpasses popular surrogates like Kriging. Additionally, a hybridized RBF model has been proposed [60]. For structural steadfastness investigation, the hybridized RBF model has been industrialized by relieving the RBF learning network with a SVM. This amendment allows for leveraging the benefits of the SVM, including higher oversimplification and worldwide optimization proficiencies [61]. Comparative valuations have shown that the projected technique outstrips both RBF and the SVM. Additional requirements for RBF include, but are not restricted to, incorporating RBF into the FORM procedure and creating a reliability analysis method based on performance measures [62]. Nonetheless, RBF primarily produces satisfactory outcomes for delays that are lined or only slightly nonlinear. The efficiency of RBF model arises from the use of localized radial kernels that interpolate training points exactly, avoiding the need for global regression fitting. This property makes RBF particularly accurate for capturing nonlinear variations in natural frequencies due to stochastic material properties.

3.2 Polynomial Neural Network (PNN)

A PNN is a cutting-edge form of the assemblage scheme of statistics treatment [63]. This technique employs in numerable multi nominal forms, counting rectilinear, revised quadratic and three-dimensional polynomials. By electing the utmost vital mutable and fit multinomial formulae, the finest part descriptions can be attained over cautious assortment of lumps and deposits. The way endures to increase coat sup until the best concert is accomplished. This tactic can yield a finest PNN edifice. The input-output statistics is providing as shadows [64]:

Ai,Bi=(a1i,a2i,a3i,…aki,bi)(15)

Some place i = 1, 2, 3, …, n. For separately duos of an input capricious ai, and aj, the multi nominal lapse reckoning is intended, and the harvest of model\(B\) is stated by:

Bi=C0+C1ai+C2aj+C3ai2+C4aj2+C5aiaj(16)

where i, j = 1, 2, 3, …, n, C0,C1,C2,C3,C4,C5 are the quantity of multinomial equivalence. This elasticities n(n − 1)/2 sophisticated instruction variable quantity for forecasting the yield B in place of inventive n variables (a1,a2,a3,a4…an) [65]. The appraised production B can be expressed as:

B^=f^(a1,a2,a3…an)(17)

=c0+∑i=1nC1iai+∑i=1n∑j=1nC2ijaiaj+∑i=1n∑j=1n∑k=1nC3ijkaiajak=…(18)

Somewhere i, j, k = 1, 2, 3, …, n and (a1,a2,a3…an) are the input variables C0,C1i,C2ij,C3ijk trajectories epitomize the constants, and the apparatuses of the involvement course can embrace self-governing variables, restricted modification footings, or purpose full arrangements. This procedure empowers the synchronized fortitude of mutually the model edifice and its yield for the most substantial structure contributions [66]. At the outset, it is necessary to establish the input variables; defining them is a prerequisite as ai = 1, 2, 3, …, n [67]. This involves associating with the output variable B and normalizing the input data. Subsequently, a training and testing dataset is created, dividing the input-output data into two segment, exercise data (ntrain) and data (ntest) someplace n = (ntrain + ntest). The drill dataset (ntrain) is operated to physique the PNN model and to approximation the constants of the polynomial descriptions for the lumps at each deposit of the PNN. The challenging dataset is then rummage-sale to evaluate the recital of the assessed PNN model [68]. The PNN edifice necessity be preferred founded on the numeral of involvement variables and the directive of limited metaphors in apiece film. There are dual kinds of PNN assemblies: elementary PNN, where the numeral of participation variables for limited metaphors is reliable athwart all films, and improved PNN, where this amount varies from unity layer to alternative [69]. The numeral of participation variables and the polynomial command used to fashion a limited account of the data have been recognized [70]. The multinomial reversion context is somewhat corresponding to the edifice of a polynomial neural network. It is essential to indicate the contribution variables for individually bulge from n input variables a1, a2, a3, …, an. The entire quantity of fractional metaphors in the existing film varies reliant on the amount of input variables selected as of the swellings in the aforementioned layer.

k=n!/(n−r)!r!(19)

here, r signifies the numeral of selected input variables. The next step is to guesstimate the quantities of the unfinished imageries. The trajectory of feature Ci can be attained by curtailing the mean sharpened error amid Bi and B^i:

PI=1ntrain∑i=1ntrain(Bi−B^i)2(20)

The competent statistics set is used to obtain the typical of recti linear equivalencies in accordance with:

B=∑i=1nAiCi(21)

The factors of the PD for the dispensation protuberances in separately sheet are found in the method:

Ci=[AiTAi]−1AiTB(22)

B=[b1,b2,b3,b4…btrain]T(23)

Ai=[a1i,a2i,a3i…Aki…Atraini]T(24)

AkiT=[aki1,aki2,aki3…akin…aki1m…aki2m…akinm]T(25)

Ci=[C0iC1iC2i…Cn′i]T(26)

here, I signifies the node numeral, k signifies the information numeral, and n represents the tally of designated involvement variables, (ntrain) is the numeral of exercise information subsection, m is the supreme directive, and n′ is the numeral of predictable factors [71]. In the subsequent stage, the fractional imageries with the sturdiest prophetic presentation are preferred by employing together the exercise and testing numbers cliques [72]. Therefore, it gauges separately limited portrayal and likens the outcomes to recognize the ones that deliver the best prognostic recital for the yield variable. If W signifies the predefined amount of partial metaphors to choice, then indicating W is based on the succeeding circumstances [73].

(a)   If n!/(n − r)!r! < W formerly the numeral of the PDs reserved for the following sheet is equivalent to n!/(n − r)!r!

(b)   If n!/(n − r)!r! ≥ W then for the next film, the numeral of the engaged PDs is alike to W.

Afterward, it’s important to verify the discontinuing principles. The ending ailment must safeguard that the previous layer has an optimal PNN model, allowing the modeling process to be concluded. If PIj>PI∗ formerly finest PNN prototypical will be attained [74], where PIj is the negligible mistake in existing layer and PI∗ is the trifling fault in the preceding film. The designer might also specify the stopping condition based on the number of iterations [75]. New input variables are eventually accepted for the following paper. The creation of the previous layer as input for the subsequent layer is how the model must inflate if the ending condition is not met [76]. The algorithm leverages polynomial expansions to represent nonlinear input–output relationships explicitly. By progressively selecting the most significant polynomial terms, PNN reduces dimensionality and minimizes computational cost while maintaining high accuracy. This is especially effective in modeling the combined variability of multiple material properties in hybrid laminates.

3.3 Multivariate Adaptive Regression Splines (MARS)

In a MARS based method, the association amongst the scheme’s input and output rejoinders is gritty by choosing illustrations using a definite procedure [77]. The affiliation amid autonomous and reliant on variable quantity is resolute consuming a set of coefficients and basic roles derivative from relapse data, rather than a predefined affiliation. A nonparametric reversion algorithm dividers the input interstellar into provinces, and output retorts are approached using a customary of rudimentary occupations nominated over a forward and backward method [78]. A relentless base role is primarily rummage-sale to generate a modest model. Afterward, supplementary basic roles are merged to progress a extra compound model. Lastly, a backward method is pragmatic to eradicate inconsequential complicated functions after the model. The efficiency of MARS lies in its nonparametric regression and piecewise adaptive basis functions, which automatically capture local nonlinearities. The forward–backward pruning strategy ensures that only the most relevant basis functions are retained, improving both accuracy and interpretability without overfitting. The MARS model can be epitomized as below [79].

A=∑n=1NαnSnf(yi)(27)

here A,αnSnf(yi) denote the approximation function, coefficients of enlargement, and the multivariate projection basic occupations, correspondingly. Input planetary is alienated into N numeral of sections. Eq. (27) can be converted as: [80].

A=α1(28)

here,Snf(y1,y2,y3,…,ym)=1forn=1(29)

If the order of interactions is shown by in, Zi,n=±1, ‘Tr’ epitomizes the role as condensed power purpose, qi,n embodies the knot position of agreeing variable star, ‘g’ denotes the demand of projections, yj(i,n) symbolizes the jth capricious, 1 ≤ j(i, n) ≤ m. Then, the rudimentary purpose is stated as [81]:

Snf(yi)=∏i=1in[Zi,n(yj(i,n)−qi,n)]Trg(30)

The rudimentary occupation may be in ensuing form:

[Zi,n(yj(i,n)−qi,n)]Trg=[Zi,n(yj(i,n)−qi,n)]gfor[Zi,n(yj(i,n)−qi,n)]<0(31)

[Zi,n(yj(i,n)−qi,n)]=0 otherwise

3.4 Finite Element Model

In FE formulation, the shape functions (Sj) are the function of local natural coordinates of the element (1, v). The shape functions can be shown as:

Si=0.25(1+ξξi)(1+ηηi)(ξξi+ηηi−1)…i=1,2,3,4(32)

Si=0.5(1+ηη)(1−ξ2)…i=5,7(33)

Si=0.5(1+ξξ)(1−η2)…i=6,8(34)

The shape functions accuracy is given by:

∑i=18⁡Si=1, ∑i=18⁡∂Si∂ξ=0 and ∑i=18⁡∂Si∂η=0(35)

4  Results and Discussions

In the present research work, cantilever Nitinol based hybrid composite is considered for stochastic free vibration analysis by using the different surrogate model. The combined variation in material properties (±10%) is considered as the input parameters for the uncertainty modeling. The material properties of hybrid composite are determined by using the mathematical formulae. The convergence study of FEM mesh size is carried out as shown in Table 1, and found that with multiple discretization levels to ensure that the results are mesh independent and free from mesh bias. The result of fundamental natural frequency parameter is compared with previous published paper as presented in Table 2. The convergence study with considering different sample size (N) for different surrogate model is presented in Table 3. From Table 1, it can be concluded that with increase in the sample size (N) from 256 to 2048, the accuracy of all the surrogate model is increased. The PDFs are estimated using kernel density estimation applied to the surrogate model outputs over the Monte Carlo sampling, while mean and variance are computed directly from the ensemble statistics. Fig. 3 represents the k-fold cross-validation of RBF model, in which model converged very effectively, achieving an extremely low training error (~3.3 × 10−4) after 200 epochs. The shape of the curve (fast initial drop + slow refinement) is typical of a well-trained network. The three graphs namely Fig. 4 comparing RBF and MCS models for estimating the probability density function of natural frequencies at three different modes (first, second, and third natural frequencies). Each graph displays how closely RBF approximations align with MCS results as the sample size increases from 256 to 2048. The scatter plots comparing the results from the RBF model against the MCS model, for three different natural frequencies. Each scatter plot shows the distribution of results from various sample sizes (256, 512, 1024, and 2048) of the RBF model, plotted against the MCS model results, to evaluate the accuracy of the RBF model at different sampling levels as shown in Fig. 5. In all three graphs, the 2048 sample points are the most densely clustered along the diagonal line, representing the best agreement with the MCS model. The alignment along this line signifies that the RBF model is an accurate approximation of the MCS model for natural frequencies, especially when using larger sample sizes. Beyond the original ±10% variation in material properties, we have now tested the trained surrogate models under wider variations (up to ±20%) as shown in Fig. 6. The Fig. 7a–c shows three probability density function plots comparing the results of a PNN model and a MCS model in predicting the first, second, and third natural frequencies of a hybrid composite plate. Across all three frequencies, 2048 the PNN sample size improves the agreement with the MCS model, indicating that the PNN model’s predictions become more reliable with more data. The third natural frequency shows the most significant improvement with 2048 sample size, while the first natural frequency. These results imply that the PNN model can be a viable alternative to MCS for predicting natural frequencies, especially at 2048 sample size. The Fig. 8a–c presents scatter plots comparing the MCS model and the PNN model predictions for the first, second, and third natural frequencies of a hybrid composite plate. The plots show how well the PNN model predictions correlate with the MCS results at different sample sizes (256, 512, 1024, and 2048). In all three scatter plots, the trend shows that the PNN model predictions improve in accuracy and consistency with the MCS model as sample size increases. The alignment along the diagonal line indicates that the PNN model can approximate the MCS model very closely when sufficient samples are used.

The Fig. 9a–c shows three PDF plots comparing the results of a MARS model and MCS model in predicting the first, second, and third natural frequencies of a hybrid composite plate. Each plot examines the alignment of the MARS model predictions with the MCS results at different sample sizes (256, 512, 1024, and 2048). For all three natural frequencies, the MARS model’s accuracy in approximating the MCS model improves with larger sample size 2048. The MARS model tends to perform better at higher sample sizes (1024 and 2048), where its predictions align closely with the MCS model, indicating that the MARS model can be an effective alternative to MCS with adequate data. This Fig. 10a–c contains three scatter plots labeled as comparisons between the MARS model and the MCS model. Each plot seems to show how well the MARS model approximates results from MCS for different natural frequencies in a dataset, with varying sample sizes. Each scatter plot represents a different sample size, with larger sample size of 2048 reflects yielding results closer to the diagonal (indicating better agreement between models). There’s a general trend that as the sample size increases, the MARS model’s predictions align more closely with the MCS model for each natural frequency. The present study adopts a purely data-driven surrogate modeling strategy (RBF, PNN, MARS) integrated with Monte Carlo simulations. While this significantly reduces computational costs compared to direct large-scale FE–MCS analyses, the accuracy and generalizability of the models remain dependent on the size and quality of training data. The modeling assumes idealized plate geometry and material property variations derived from established mathematical formulations, without explicitly including damage states or manufacturing defects. The main advantage of this study included the Computational efficiency, straightforward implementation, and flexibility in handling different surrogate models. The framework can be easily adapted for different composite systems and uncertainty descriptions. All simulations in this study are performed on an HP Z2 Tower G9 Workstation equipped with a 12th Gen Intel® Core™ i9-12900 processor (2.40 GHz, 16 cores), 32 GB RAM, and an NVIDIA RTX A2000 GPU with 6 GB VRAM. The workstation was configured with a 477 GB SSD for fast execution and a 1.82 TB HDD for data storage.

Figure 9: PDF plot of MCS Model and MARS model (with considering different sample size of surrogate model) for first, second & third natural frequencies of hybrid composite plate

Figure 10: Scatter plot between MARS and MCS Model for first, second & third natural frequencies of hybrid composite plate with considering different sample size

5  Conclusions

The novelty of this study lies in the development of a computationally efficient framework using RBF, PNN, and MARS surrogate models for uncertainty quantification in hybrid composite plates, enabling significant reductions in sample size compared to traditional Monte Carlo simulation (MCS). The results show that these surrogate models closely match the MCS benchmark (10,000 samples) in terms of mean and standard deviation of the first three natural frequencies, with the RBF method at 2048 samples providing the most accurate approximation. This demonstrates the feasibility of surrogate-based approaches for effectively capturing uncertainty propagation in stochastic dynamic analysis. Notably, the RBF model achieves superior computational efficiency while maintaining accuracy. Although the present work focused on un-damped free vibration with material uncertainties, the proposed framework is adaptable to more complex scenarios, including damping effects, forced vibration, and geometric nonlinearities. Moreover, while demonstrated on hybrid composite plates, the approach is generalizable to other structural forms such as shells and beams with appropriate training datasets. Future research will further strengthen the methodology by integrating physics-informed neural networks (PINNs) with finite element solvers, incorporating sensitivity analysis to identify the most influential parameters, and pursuing experimental validation to enhance credibility and practical relevance.

Acknowledgement: Not applicable.

Funding Statement: The authors received no specific funding for this study.

Author Contributions: Bindi Saurabh Thakkar: Conceptualization of the research framework, development of machine-assisted finite element methodologies, execution of simulations, and analysis of results. Pradeep Kumar Karsh: Supervision, validation of methodologies, interpretation of findings, and revision of the manuscript. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: The data that support the findings of this study are available from the corresponding author, Pradeep Kumar Karsh, upon reasonable request.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest to report regarding the present study.

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