Performance Analysis of Radial Basis Function Metamodels for Predictive Modelling of Laminated Composites

High-fidelity structural analysis using numerical techniques, such as finite element method (FEM), has become an essential step in design of laminated composite structures. Despite its high accuracy, the computational intensiveness of FEM is its serious drawback. Once trained properly, the metamodels developed with even a small training set of FEM data can be employed to replace the original FEM model. In this paper, an attempt is put forward to investigate the utility of radial basis function (RBF) metamodels in the predictive modelling of laminated composites. The effectiveness of various RBF basis functions is assessed. The role of problem dimensionality on the RBF metamodels is studied while considering a low-dimensional (2-variable) and a high-dimensional (16-variable) problem. The effect of uniformity of training sample points on the performance of RBF metamodels is also explored while considering three different sampling methods, i.e., random sampling, Latin hypercube sampling and Hammersley sampling. It is observed that relying only on the performance metrics, such as cross-validation error that essentially reuses training samples to assess the performance of the metamodels, may lead to ill-informed decisions. The performance of metamodels should also be assessed based on independent test data. It is further revealed that uniformity in training samples would lead towards better trained metamodels.


Introduction
Composites are one the most widely used materials of the 21st century. Laminated composite structures have become an inevitable component of modern structural, marine and aerospace applications. Accurate estimation of the static and dynamic performance of such structures is an important task. With modern computing facilities and a plethora of highly accurate numerical strategies and mathematical theories, composite structures are now being extensively analyzed in silico. Despite the high accuracy of numerical approaches, such as the finite element method (FEM), their time-intensiveness often hinders their widespread applications, especially when multiple re-runs are required (e.g., for different sets of ply angles or global optimization tasks). In this regard, metamodels can provide a remarkable saving in computational effort and cost. However, the metamodel by virtue of being an approximate model of the actual one (e.g., FEM, computational fluid dynamics), would lead to some loss in accuracy. The accuracy of a metamodel usually depends on various factors, like type, size and complexity of the problem, training data characteristics, the algorithm used, etc.
Several researchers have already adopted various metamodeling algorithms ranging from polynomial regression (PR) [1,2] to genetic programming (GP) [3] to the artificial RBF and polynomial neural network metamodels in surrogate modelling of laminated structures. However, it should be noted that in most stochastic modelling studies, the range of design variables is generally very small.
As pointed out by Amouzgar and Stromberg [28], the performance of RBF metamodels had been compared in the past with several other algorithms and their efficacy in predictive modelling of engineering problems had been well established. However, no comprehensive analysis has been carried out on the utility of RBFs in large domain problems. Moreover, the effect of training data uniformity on RBFs has not been studied so far. Thus, in this paper, a comprehensive analysis of RBF metamodels is carried out to ascertain the utility of various basis functions in metamodeling of large domain problems, like laminated structures modelled with respect to ply angles.
The rest of this paper is structured as follows: The objectives of this paper along with the test problems considered are stated in Section 2. Various methods used for sampling, data generation, metamodeling and its evaluation are presented in Section 3. The results of the two test problems are separately discussed in detail in Section 4. Finally, conclusions based on this comprehensive analysis are drawn in Section 5.

Objectives and Problem Description
The prime objective of this paper is to comprehensively investigate the viability of RBF metamodels as a reliable surrogate for high-fidelity analysis of laminated composite structures. The second objective is to assess the effect of training data sampling technique on the overall predictive performance of the RBF metamodel. In this regard, three different data sampling techniques, i.e., random sampling (RS), Latin hypercube sampling (LHS) and Hammersley sampling (HS) are considered here. In Figure 1, a comparison of typical sample sets generated by RS, LHS and HS for a typical 2-variable problem is exhibited. It can be noted from this figure that HS has the maximum uniformity in data generation, while RS has the least uniformity in the considered 2-dimensional design space. Since, different performance metrics (like R 2 , leave-one-out cross-validation, n-fold cross-validation, mean squared error etc.) have often been proposed by the past researchers for quantification of accuracy of the metamodels, it would be an interesting exercise to investigate the influences of uniformity of the training data on the behaviour of different performance metrics. The third objective is thus set to explore the effects of dimensionality and complexity of the problem on the RBF metamodel's predictive power. In this direction, two different problems, i.e., a low-dimensional (LD) problem (2 design variables) and a high-dimensional (HD) problem (16 design variables) are considered. For each problem, three different responses (first frequency (λ 1 ), second frequency (λ 2 ) and third frequency (λ 3 ) of laminated composite plates) of varying complexity are taken into account.

Problem 1: Low-Dimensional (LD) Problem
For the LD problem, a 4-ply symmetric square composite laminate is considered. The ply angles are treated as the design variables with the entire range of possible ply angles (i.e., ±90°) as the design space. However, only discrete values with 1° increment within the ±90° range are considered for training and testing data. The thickness-to-side ratio for the composite plate is taken as 0.005. The boundary condition of the composite plate is assumed to be simply supported on all of its sides. In this paper, graphite-epoxy

Problem 1: Low-Dimensional (LD) Problem
For the LD problem, a 4-ply symmetric square composite laminate is considered. The ply angles are treated as the design variables with the entire range of possible ply angles (i.e., ±90 • ) as the design space. However, only discrete values with 1 • increment within the ±90 • range are considered for training and testing data. The thickness-to-side ratio for the composite plate is taken as 0.005. The boundary condition of the composite plate is assumed to be simply supported on all of its sides. In this paper, graphite-epoxy composite laminates with the following material properties [29] are considered for analysis. E 1 = 138 GPa, E 2 = 8.96 GPa, G 12 = G 13 = 7.1 GPa, G 23 = 3.9 GPa, υ 12 = 0.3, υ 21 = 0.0195. Using a finite element (FE) approach, the first three natural frequencies (λ 1 , λ 2 and λ 3 ) of the composite plates are calculated in the non-dimensional form as λ = ωa 2 ρh/D 0 , Here, ω, a, ρ and h represent frequency, side width, density and thickness of the laminate. These three natural frequencies are treated here as the responses. The detailed formulation of the FEM used in this paper can be available in [30][31][32]. The FE formulation adopts a 9-node isoparametric element and is based on first-order shear deformation theory (FSDT).

Problem 2: High-Dimensional (HD) Problem
In the case of the HD problem, a 32-ply symmetric square composite laminate is considered. Thus, the total number of design variables is 16. The thickness-to-side ratio is taken as 0.04. All the other boundary and geometric conditions are the same as those of the LD problem.

Sampling Schemes
The application of any metamodeling technique starts with setting up a training dataset. The utility of a metamodel as an accurate and effective predictive tool thus largely depends on how well it has been trained using the given dataset. Hence, the training dataset must adequately represent the underlying features of the design space. Therefore, it makes sense to constitute a training dataset drawn from the design space without any bias. In this paper, three different sampling strategies i.e., RS, LHS and HS are adopted to constitute the training dataset. Sample size also plays an important role in influencing the metamodeling process. Several studies have been conducted in the past to ascertain the best possible sample size. In this paper, the recommendations of Jin et al. [33] are considered to determine the sample size (n), which can be expressed as a function of the number of input variables (p) and a scaling parameter (l) [34].
For LD problem: For HD problem: Considering l = 2, the sample sizes (n) for LD and HD problems are considered as 72 and 612 respectively.

Random Sampling
The RS is one of the most commonly employed sampling strategies because it can generate new sample points without taking into consideration the previously generated sample points. Moreover, the total number of sample points to be generated needs not be specified beforehand. In this paper, n sample points are randomly generated between the upper and lower bounds of p input variables.

Latin Hypercube Sampling
The LHS, originally introduced by McKay et al. [35], is a statistical technique to generate near-random sample points. A Latin hypercube is analogous to a Latin square in p-dimensional space. A Latin square may be defined as an n × n array that is filled up with n different parameter values such that each parameter value occurs only once in each row and in each column. Thus, a Latin hypercube for n samples in p dimensions is a matrix with n rows and p columns. In LHS, if the variables are continuous, the number of levels for the variables becomes equal to the total number of samples, thereby each level occurring only once in each dimension. However, in the case of discrete variables, the occurrence of each value is equally possible.

Hammersley Sequence Sampling
The HS [36] is a pseudo-random, low-discrepancy sequence based on the representation of a decimal number in the inverse radix format. The radix values are chosen as the first (p − 1) prime numbers. The Hammersley sequence generates a highly uniform sample of n data points in p-dimensional space. In HS, uniformity in the sample space is maintained in all p dimensions, which is not possible to achieve in RS or LHS. However, in HS, it is not necessary that each variable level must occur only once. Some values may occur multiple times while some may be skipped.

Finite Element Method
FEM is a numerical approach for solving various types of engineering problems [30,37,38]. To solve a given problem, it discretizes the problem domain into small parts called elements. Each element is connected with its neighboring elements through nodes.
In this paper, the composite laminates are discretized using a 9-node isoparametric plate bending element. It has been well-established that the choice of plate bending theory has a significant influence on the accuracy of the derived results. Thus, in this paper, FSDT is considered to study the effects of rotary inertia and transverse shear deformation. Due to the paucity of space, the FEM formulation employed in this paper is not explained, but can be found in the previous research works by . The detailed FEM formulation is also included in the companion data descriptor to this paper.
Based on past research works on the same finite element formulation [30][31][32], it is revealed that an 18 × 18 mesh discretization of the composite plate using 9-node isoparametric elements is sufficiently accurate for the considered problems. To demonstrate the accuracy of the current FEM formulation, a validation example is included in Figure A1, Appendix A. It has been exhibited that the application of FSDT can yield results with more than 98% accuracy of the exact solutions and higher-order shear deformation theory-based results. Thus, all the training and testing data employed in this paper are generated based on the aforementioned FE formulation.

Radial Basis Function
The RBFs are commonly used as metamodels in many engineering applications. When provided with a training dataset, it can approximate the underlying model by mapping the outputs as functions of the input variables. It was first applied by Hardy [39] in geophysical research to develop metamodels for topographical contours of geographical data. Since then, it has been applied to diverse problems from almost all domains of engineering [40][41][42].
Any general metamodel may be stated as a global approximation function f (x) developed from a set of given data points x i ∈ m (i = 1, 2, .., n) and their corresponding response function f (x) value. Interpolation of metamodels would generally yield accurate response surfaces while satisfying the following condition: Equation (2) indicates that the function f and the approximating function f are equal at all the prescribed n data points.
An RBF is a function φ whose values are real numbers and depend on the distance from the origin, such that φ(x) = φ(||x||). The value of RBF may also depend on distance from the center c, such that: 6 of 22 where || || is any l p norm. In this paper, l 2 norm is considered. The generalized metamodel can be expressed as a linear combination of the basis functions across all the data points: where f (x) can be represented as the sum of n radial basis functions, each associated with a different center x i . Values of the coefficient (w i ) can be calculated by enforcing the constraints given in Equation (4), to form a linear system of equations: Equation (5) can also be expressed in form of matrix as follows: The above system of equations can be solved to obtain a unique vector {w}. This would to an RBF metamodel, as stated in Equation (5), which interpolates all the training data points. Various functional forms of basis function can be adopted to simulate the RBF metamodel. Linear: Cubic: Gaussian: where ε is shape parameter. Multi-quadratic (MQ): Inverse multi-quadratic (IMQ): Thin plate spline (TPS): In this paper, two different ε values (1 and 2) for Gaussian, MQ and IMQ are tried. Hereafter, Gaussian, MQ and IMQ RBFs with ε = 2, are referred to as Gauss-2, MQ-2 and IMQ-2, respectively.

Low-Dimensional Problem
The RBF metamodels are now trained using nine different basis functions (linear, cubic, Gaussian, MQ, IMQ, TPS, Gauss-2, MQ-2 and IMQ-2) on three different training datasets (RS, LHS and HS). During the training phase, the 10-fold cross-validation error is estimated while randomly dividing the training dataset into 10 subsets. Figure 2 depicts the best values of 30 trials of 10-fold cross-validation error of various RBF metamodels in the LD problem. It can be revealed that in general, the 10-fold cross-validation error is the least for RBF metamodels trained on the RS dataset and the maximum for the HS dataset. Further, for first frequency and second frequency estimations, MQ-RBF performs the best, while the performance of linear-RBF is the worst. However, in the case of third frequency estimation, MQ-2-RBF performs the best, whereas Gauss-RBF exhibits the maximum 10-fold cross-validation error.

Low-Dimensional Problem
The RBF metamodels are now trained using nine different basis functions (linear, cubic, Gaussian, MQ, IMQ, TPS, Gauss-2, MQ-2 and IMQ-2) on three different training datasets (RS, LHS and HS). During the training phase, the 10-fold cross-validation error is estimated while randomly dividing the training dataset into 10 subsets. Figure 2 depicts the best values of 30 trials of 10-fold cross-validation error of various RBF metamodels in the LD problem. It can be revealed that in general, the 10-fold cross-validation error is the least for RBF metamodels trained on the RS dataset and the maximum for the HS dataset. Further, for first frequency and second frequency estimations, MQ-RBF performs the best, while the performance of linear-RBF is the worst. However, in the case of third frequency estimation, MQ-2-RBF performs the best, whereas Gauss-RBF exhibits the maximum 10fold cross-validation error. To further understand the behavior of the 10-fold cross-validation error in relation to uniformity of the training dataset, 30 independent training datasets, each for RS, LHS and HS, are generated and the 10-fold cross-validation errors are subsequently recorded. Figure 3 exhibits the box plots of the 10-fold cross-validation errors of 30 trials in the LD problem. In general, the RBF metamodels trained on the HS dataset show the least variation. It can be further noted that the overall variability of the 10-fold cross-validation error is the highest for RS. As a whole, MQ-RBF and IMQ-RBF metamodels show the least variability, while Gauss-RBF metamodels have the maximum variability. uniformity of the training dataset, 30 independent training datasets, each for RS, LHS and HS, are generated and the 10-fold cross-validation errors are subsequently recorded. Figure 3 exhibits the box plots of the 10-fold cross-validation errors of 30 trials in the LD problem. In general, the RBF metamodels trained on the HS dataset show the least variation. It can be further noted that the overall variability of the 10-fold cross-validation error is the highest for RS. As a whole, MQ-RBF and IMQ-RBF metamodels show the least variability, while Gauss-RBF metamodels have the maximum variability. The performance of the metamodels is also validated using the leave-one-out crossvalidation approach. Because of the similar nature of the leave-one-out cross-validation approach to the 10-fold cross-validation approach, the performance characteristics of various metamodels in Figure 4 are noticed to be similar to those in Figure 2.
Thus, it becomes unveiled from the study of the performance of RBF metamodels that cross-validation approaches (like 10-fold and leave-one-out metrics) essentially reusing training data to assess the model accuracy would in general, show better performance when trained on randomly drawn (RS) datasets. This is because of the inherent mechanism of the cross-validation approaches, which randomly select and segregate some portion of the training data to later act as a validation subset. However, in sampling techniques, like LHS and HS, the overall training dataset is so selected that it is (as much as possible) uniformly spread over the entire design space to be modeled. Therefore, withholding some data (for calculation of cross-validation errors) from these training datasets would have detrimental effects on the training and subsequently on the performance of the metamodel. Thus, any typical sample point of the HS dataset has more influence on the overall predictive capability of the metamodel as compared to a typical sample point of the RS dataset. This is why it can be noticed that as the uniformity of the training dataset increases, the performance of metamodels on cross-validation approaches drops. The performance of the metamodels is also validated using the leave-one-out crossvalidation approach. Because of the similar nature of the leave-one-out cross-validation approach to the 10-fold cross-validation approach, the performance characteristics of various metamodels in Figure 4 are noticed to be similar to those in Figure 2. To further assess the performance of RBF metamodels, a 20-sample point random sampling test dataset is generated using FEM formulation. The performance of all the metamodels is validated based on this test dataset using root mean squared error (RMSE) and mean absolute percentage error (MAPE) as the metrics, as portrayed in Figures 5 and  6 respectively. It can be noticed from Figure 5, for all the RBF metamodels, the value RMSE is the least for HS data in the case of first and second frequency estimations. However, in the case of third frequency estimation, in general, the RSME value of all the RBF metamodels trained on RS data is the least. It is also interesting to note that similar to the 10-fold cross-validation error plot ( Figure 2c) and leave-one-out cross-validation plot (Figure 4c), the values of RMSE ( Figure 5c) and MAPE (Figure 6c) of Gaussian-RBF metamodels are unusually high for third frequency estimation.  Thus, it becomes unveiled from the study of the performance of RBF metamodels that cross-validation approaches (like 10-fold and leave-one-out metrics) essentially reusing training data to assess the model accuracy would in general, show better performance when trained on randomly drawn (RS) datasets. This is because of the inherent mechanism of the cross-validation approaches, which randomly select and segregate some portion of the training data to later act as a validation subset. However, in sampling techniques, like LHS and HS, the overall training dataset is so selected that it is (as much as possible) uniformly spread over the entire design space to be modeled. Therefore, withholding some data (for calculation of cross-validation errors) from these training datasets would have detrimental effects on the training and subsequently on the performance of the metamodel. Thus, any typical sample point of the HS dataset has more influence on the overall predictive capability of the metamodel as compared to a typical sample point of the RS dataset. This is why it can be noticed that as the uniformity of the training dataset increases, the performance of metamodels on cross-validation approaches drops.
To further assess the performance of RBF metamodels, a 20-sample point random sampling test dataset is generated using FEM formulation. The performance of all the metamodels is validated based on this test dataset using root mean squared error (RMSE) and mean absolute percentage error (MAPE) as the metrics, as portrayed in Figures 5 and 6 respectively. It can be noticed from Figure 5, for all the RBF metamodels, the value RMSE is the least for HS data in the case of first and second frequency estimations. However, in the case of third frequency estimation, in general, the RSME value of all the RBF metamodels trained on RS data is the least. It is also interesting to note that similar to the 10-fold cross-validation error plot (Figure 2c) and leave-one-out cross-validation plot (Figure 4c    Further, the performance of the RBF metamodels is assessed based on the entire population of the 2-variable LD problem. For this, 32,761 sample points are generated considering the LD problem of 4-ply symmetric laminate design with discrete angles of 1°. The performance of the RBF metamodels with respect to mean absolute error (MAE) and mean squared error (MSE) values for the entire population is depicted in Figure 7 and Further, the performance of the RBF metamodels is assessed based on the entire population of the 2-variable LD problem. For this, 32,761 sample points are generated considering the LD problem of 4-ply symmetric laminate design with discrete angles of 1 • . The performance of the RBF metamodels with respect to mean absolute error (MAE) and mean squared error (MSE) values for the entire population is depicted in Figures 7 and 8 respectively. From Figures 7a and 8a, it can be observed that the HS trained RBF metamodels comprehensively outperform the LHS and RS trained RBFs for first frequency estimation.
Among different basis functions, MQ and IMQ show the most promising results for first frequency estimation. For second frequency estimation (Figures 7b and 8b), the linear-RBF metamodel trained on the LHS dataset has the least error. In case of third frequency estimation (Figures 7c and 8c), linear-RBF metamodel trained on the HS dataset records the lowest deviation from the true values. The performance of the metamodels with respect to different error metrics is presented in Appendix B through Tables A1-A3. The 2D contour plots depicting the prediction performance of various RBFs trained with different datasets for first frequency estimation are shown in Figure A2 along with the corresponding FEM solutions. It is clear that in general, the HS trained RBFs have better encapsulated the intricacies in the design space. The 2D contours of HS-trained MQ and IMQ RBFs are observed to be almost identical to the FEM solutions. the lowest deviation from the true values. The performance of the metamodels with respect to different error metrics is presented in Appendix B through Tables A1-A3. The 2D contour plots depicting the prediction performance of various RBFs trained with different datasets for first frequency estimation are shown in Figure A2 along with the corresponding FEM solutions. It is clear that in general, the HS trained RBFs have better encapsulated the intricacies in the design space. The 2D contours of HS-trained MQ and IMQ RBFs are observed to be almost identical to the FEM solutions.

High-Dimensional Problem
The HD problem, as indicated earlier, is a 16-design variable problem, where the variables are the ply angles of 16 laminas of the composite plate with each having a design bound of ±90°. Figure 9 depicts the 10-fold cross-validation error for different RBF metamodels trained on various datasets. It can be observed that the 10-fold crossvalidation error is the minimum for the RS dataset and maximum for the HS dataset. Thus, as expected, higher uniformity in the spread of sample points in a training dataset imparts

High-Dimensional Problem
The HD problem, as indicated earlier, is a 16-design variable problem, where the variables are the ply angles of 16 laminas of the composite plate with each having a design bound of ±90 • . Figure 9 depicts the 10-fold cross-validation error for different RBF metamodels trained on various datasets. It can be observed that the 10-fold crossvalidation error is the minimum for the RS dataset and maximum for the HS dataset. Thus, as expected, higher uniformity in the spread of sample points in a training dataset imparts each sample point with more influence on the training goal as compared to a typical sample point from a less uniform dataset. However, it is interesting to note that the variability of 10-fold cross-validation errors on 30 independent trials (Figure 10) is the lowest for the HS dataset and highest for the RS dataset. Thus, when compared for all solutions, the worst solution may also belong to the RS dataset. The negligible variability of 10-fold cross-validation errors for HS datasets makes them more attractive for such HD problems since any typical HS training solution is more reliable and thereby would not require a large number of independent trials (e.g., 30 trials used in this paper) to ascertain the accuracy of the solutions. In real-world situations, a smaller number of trials directly corresponds to less computational cost and quick model deployment time.
The leave-one-out cross-validation error of the RBF metamodels in the HD problem is presented in Figure 11. For first frequency RBF metamodels, the lowest leave-one-out crossvalidation error is noticed for the LHS dataset, whereas, for second and third frequency modelling, the leave-one-out cross-validation error is the lowest for the RS dataset. The MQ (LHS), IMQ (RS) and IMQ-2 (RS) are the best performing RBF metamodels based on the leave-one-out cross-validation error for the HD problem of first, second and third frequency modelling respectively.     An independent test dataset of 50 random samples is again generated and the accuracy of the RBF metamodels is assessed based on RMSE and MAPE values, as shown in Figures 12 and 13 respectively. Cubic (HS) emerges out as the best RBF metamodel for first frequency modelling. Based on the 10-fold cross-validation error and leave-one-out cross-validation error as shown in Figures 9 and 11, respectively, cubic (HS) RBF is the worst metamodel for modelling of all the frequencies in the considered HD problem. However, it should be noted that in absolute terms of the cross-validation error, cubic (HS) RBF is only 19.5%, 28% and 31.5% worst as compared to the best metamodel for first, second and third frequencies. Contrary to this, the worst metamodel in absolute terms of An independent test dataset of 50 random samples is again generated and the accuracy of the RBF metamodels is assessed based on RMSE and MAPE values, as shown in Figures 12 and 13 respectively. Cubic (HS) emerges out as the best RBF metamodel for first frequency modelling. Based on the 10-fold cross-validation error and leave-one-out crossvalidation error as shown in Figures 9 and 11, respectively, cubic (HS) RBF is the worst metamodel for modelling of all the frequencies in the considered HD problem. However, it should be noted that in absolute terms of the cross-validation error, cubic (HS) RBF is only 19.5%, 28% and 31.5% worst as compared to the best metamodel for first, second and third frequencies. Contrary to this, the worst metamodel in absolute terms of MAPE on testing dataset is 212% (Gauss-2 (LHS)), 193% (Gauss-2 (LHS)) and 203% (cubic (LHS)) worst as compared to the best metamodel for first (cubic (HS)), second (MQ (HS)) and third (IMQ (HS)) frequencies. Further, from Figures 12 and 13, it can be noticed that irrespective of the basis function adopted, RBF metamodels trained on HS datasets are significantly better than the metamodels trained on LHS and RS datasets.
In case of the LD problem, it is unveiled that MQ basis functions perform well in all the examples, while for the HD problem, it is difficult to choose the best RBF metamodel. Therefore, a multi-criteria decision-making approach in the form of TOPSIS is employed here to evaluate and rank various RBF metamodels for all the examples. The rankings of the metamodels using TOPSIS are presented in Table A4. In case of the LD problem, it is unveiled that MQ basis functions perform well in all the examples, while for the HD problem, it is difficult to choose the best RBF metamodel. Therefore, a multi-criteria decision-making approach in the form of TOPSIS is employed here to evaluate and rank various RBF metamodels for all the examples. The rankings of the metamodels using TOPSIS are presented in Table A4.  In case of the LD problem, it is unveiled that MQ basis functions perform well in all the examples, while for the HD problem, it is difficult to choose the best RBF metamodel. Therefore, a multi-criteria decision-making approach in the form of TOPSIS is employed here to evaluate and rank various RBF metamodels for all the examples. The rankings of the metamodels using TOPSIS are presented in Table A4. Figure 13. MAPE of various RBF metamodels in HD problem for testing data. Figure 13. MAPE of various RBF metamodels in HD problem for testing data.

Conclusions
In this paper, a comprehensive analysis is carried out on the utility of RBF metamodels in the predictive modelling of laminated composites while considering nine different basis functions. The effect of problem dimension is analyzed using two different problems, i.e., an LD problem of 2 design variables and an HD problem of 16 design variables. The ply angles of each lamina with a design bound of ±90 • are treated as the design variables. Further, the role of uniformity of the training data on the predictive ability of RBF metamodels is also studied in detail while considering three different sampling strategies, i.e., RS, LHS and HS. Based on the extensive investigations on the FEM generated high-fidelity data, the following conclusions can be drawn: (a) The RBF metamodels trained on RS datasets have the best 10-fold cross-validation error and leave-one-out cross-validation error. However, this excellent prediction on training data does not necessarily correspond to excellent prediction (in terms of MAPE and RMSE) on independent test data. In fact, in all the three responses of LD problem, the worst MAPE and RMSE values are recorded for RBFs trained on the RS dataset. (b) The RBF metamodels trained on HS datasets have the best prediction with respect to MAPE and RMSE on independent test data. However, for all the three responses of both LD and HD problems, HS-data-trained RBFs show the worst 10-fold crossvalidation error and leave-one-out cross-validation error. Nevertheless, in case of the LD problem, for the best (in terms of MAPE and RMSE) HS-data-trained RBF metamodels, the 10-fold cross-validation error is 47% (first frequency), 138% (second frequency) and 95% (third frequency) worst as compared to the overall best RBF metamodels. In case of the HD problem, these deviations are much lower, i.e., 19% (first frequency), 15% (second frequency) and 11% (third frequency). Thus, despite using metrics, like 10-fold cross-validation error and leave-one-out cross-validation error, performance measurement of metamodels on independent test data should be encouraged. (c) In general, irrespective of the sampling strategy and basis function, all RBF metamodels show better performance on the HD problem as compared to the corresponding metamodels for the LD problem. It should be noted that in terms of design variables, the HD problem is 8 times more complex than the LD problem, whereas the training datasets used have a ratio of 8.5:1 for HD and LD problems. Thus, the size of the training dataset has more influence on the metamodel's predictive performance as compared to the number of variables. (d) Using TOPSIS, it can be observed that in general, MQ basis functions perform well for LD problem, whereas, for HD-problem, linear and MQ basis functions perform with high reliability.
Thus, it can be concluded that RBF metamodels can be employed to accurately estimate the frequency parameters of laminated composite structures. Further, for large-dimensional complex problems, low discrepancy sampling methods, like Hammersley is likely to be more effective in developing global metamodels. The scope of this paper may include studying the effect of size of the training datasets as well as to apply other metamodeling approaches, like Kriging, support vector regression etc. for predictive modeling of laminated composite structures.

Data Availability Statement:
The data presented in this study are available in the article.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A. Validation of the FEM Formulation
To demonstrate the accuracy of the FEM formulation used in the current paper, an example from literature is solved and compared with published results by various researchers. An all sides simply supported symmetric four layered cross-ply laminated composite square plate is considered. The material properties are E 1 E 2 = 0.5 and ϑ 12 = ϑ 13 = ϑ 23 = 0.25. The thickness-to-length ratios (h/a) are varied between 0.5 to 0.001. The current FEM results are compared with those of Wu et al. [43], Matsunaga [44], Mindlin [45], Sayyad and Ghugal [46]. While the current work uses FSDT, Wu et al. [43] used a local higher order shear deformation theory (HSDT), Matsunaga [44] used a global HSDT, Mindlin [45] pioneered the FSDT and Sayyad and Ghugal [46] used a trigonometric shear deformation theory (TSDT). Theoretically, HSDTs and TSDT are more accurate than the FSDT. However, in context of this study the current results are seen to be at par with the HSDTs and TSDT. This is because the considered plates in the example are thin or moderately thin plates, on which the effect of transverse shear strain is negligible.
FSDT, Wu et al. [43] used a local higher order shear deformation theory (HSDT), Matsunaga [44] used a global HSDT, Mindlin [45] pioneered the FSDT and Sayyad and Ghugal [46] used a trigonometric shear deformation theory (TSDT). Theoretically, HSDTs and TSDT are more accurate than the FSDT. However, in context of this study the current results are seen to be at par with the HSDTs and TSDT. This is because the considered plates in the example are thin or moderately thin plates, on which the effect of transverse shear strain is negligible. Figure A1. Comparison of non-dimensional fundamental frequencies computed by current FEM and values reported by various researchers [43][44][45][46]. (a) moderately thin plates having h/a between 0.5 to 0.1 (b) thin plates having h/a between 0.08 to 0.001. Figure A1. Comparison of non-dimensional fundamental frequencies computed by current FEM and values reported by various researchers [43][44][45][46]. (a) moderately thin plates having h/a between 0.5 to 0.1 (b) thin plates having h/a between 0.08 to 0.001.   Figure A2. 2D contour plots for first frequency of the LD problem generated by various RBF metamodels.