Microsoft Word - numero_66_art_06_4256.docx A. Anjum et alii, Frattura ed Integrità Strutturale, 66 (2023) 112-126; DOI: 10.3221/IGF-ESIS.66.06 112 Analysis of damage control of thin plate with piezoelectric actuators using finite element and machine learning approach Asraar Anjum Department of Mechanical and Aerospace Engineering, Faculty of Engineering, International Islamic University Malaysia, P.O. Box 10, 50725 Kuala Lumpur, Malaysia asraar.anjum@live.iium.edu.my Abdul Aabid Department of Engineering Management, College of Engineering, Prince Sultan University, PO BOX 66833, Riyadh 11586, Saudi Arabia aaabid@psu.edu.sa Meftah Hrairi* Department of Mechanical and Aerospace Engineering, Faculty of Engineering, International Islamic University Malaysia, P.O. Box 10, 50725 Kuala Lumpur, Malaysia meftah@iium.edu.my ABSTRACT. In recent studies, piezoelectric actuators have been recognized as a practical and effective material for repairing cracks in thin-walled structures, such as plates that are adhesively bonded with piezoelectric patches due to their electromechanical effects. In this study, we used the finite element method through the ANSYS commercial code to determine the stress intensity factor (SIF) at the crack tip of a cracked plate bonded with a piezoelectric actuator under a plane stress model. By running various simulations, we were able to examine the impact of different aspects that affect this component, such as the size and characteristics of the plate, actuator, and adhesive bond. To optimize performance, we utilized machine learning algorithms to examine how these characteristics affect the repair process. This study represents the first-time machine learning has been used to examine bonded PZT actuators in damaged structures, and we found that it had a significant impact on the current problem. As a result, we were able to determine which of these parameters were most helpful in achieving our goal and which ones should be adjusted to improve the actuator's quality and reduce significant time and costs. KEYWORDS. Damaged structure, Piezoelectric actuators, Finite element method, Machine learning. Citation: Anjum, A., Aabid, A., Hrairi, M., Analysis of damage control of thin plate with piezoelectric actuators using finite element and machine learning approach, Frattura ed Integrità Strutturale, 66 (2023) 112-126. Received: 30.04.2023 Accepted: 02.08.2023 Online first: 05.08.2023 Published: 01.10.2023 Copyright: © 2023 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. https://youtu.be/JgCrshbMbEU A. Anjum et alii, Frattura ed Integrità Strutturale, 66 (2023) 112-126; DOI: 10.3221/IGF-ESIS.66.06 113 INTRODUCTION any structural health monitoring researchers are interested in this topic because there is a rising need for an efficient, affordable, and trustworthy monitoring system to guarantee the functioning and safety of such structures [1–4]. Due to cyclic stresses and a corrosive operating environment, aircraft are susceptible to fracture over time. For example, fatigue cracks can form in corroded rivet holes and must be discovered and corrected before they cause catastrophic failure [5]. In early studies, damaged structures were studied by determining the fracture parameter such as stress intensity and stress concentration factor by using mathematical modelling [6] and the finite element method [7]. Later on, the same approach was used to compute SIF with bonded laminate structures [8,9]. After successfully computing the fracture parameter this continued with changing the parameter [10], damage propagation mode [11,12], and single/double-sided composite effects [13]. As an active repair method, the PZT was used to repair a notched beam under dynamic loading conditions by the electromechanical characteristic to induce a local moment [14]. With the use of PZT actuator patches, Wu and Wang [15] numerically and analytically restored the delaminated plate under uniaxial tensile strain. Dawood et al., [16] used LS-DYNA explicit FE code to explore delamination control in structures with PZT actuators by producing low-velocity impact. In recent studies, Aabid et al., [17] modelled the different repair configurations of a center-holed aluminium plate with a bonded PZT actuator. It has been determined stresses in a high-stress zone on a circular hole and repaired by increasing the voltage for a certain limit. It suggests a system for quickly identifying and classifying concrete crack/non-crack features that combines picture binarization and a Fourier-based 1D DL model. DL training and testing using image binarization removes the plane structural backdrop and identifies potential Crack Candidate Regions (CCR) [18]. However, to regulate the centralised HVAC system of a multizone office complex, the heating, ventilation, and air conditioning (HVAC) system created a model- free end-to-end dynamic HVAC control approach based on a recently presented deep reinforcement learning framework [19]. It recommends utilising a simulated acceleration response on a nominal RC4 power car travelling over a 15m simply supported reinforced concrete railway bridge to train, test, and optimise a deep convolutional neural network to identify damage [20]. Similarly, to remotely gather thorough photos of road cracks, an omnidirectional mobile robot based on virtual reality technology is created. Next, a deep convolutional neural network (DCNN) model is trained and tested using an image dataset made up of various crack images gathered by the mobile robot [21]. And it introduces a piezoelectric sensor that uses frequency scanning technology and machine learning techniques to measure the thickness of ice and water coating on road surfaces. This sensor vibrates to identify ice and water using a constant elasticity alloy plate and a three-electrode piezoelectric transducer disc [22]. The tracking precision and robustness of traditional feedforward (FF) compensators against unmodeled dynamics and perturbation uncertainties were both increased by the reinforcement learning (RL) controller [23]. It introduces RL- Controller and proposes a revolutionary RL-based approach for creating active controllers. It suggests a framework that is simple to train for a benchmark building with five stories. In a study comparing the suggested model-free algorithm to the LQG active control approach, it is shown that the latter learns more effective actuator forcing schemes [24]. Case studies of damage detection of the model bridge and real bridge structures employing Digital twin (DT) technology or DL algorithms, with high accuracy of 92 percent, are used to show the viability of the suggested framework [25]. In the latest studies, piezoelectric materials adhesively bonded to structures were also found in some other cases such as in composite patches [26] which is used to evaluate the vibration excitation and separation in laminates and concrete beams which is used for damage evaluation of reinforced concrete structures at lap splices of tensional steel bars [27]. Additionally, the performance of PZT actuators was found in the effects of a viscoelastic bonding layer bonded to an elastic structure [28]. In some cases, surface-bonded piezoelectric sensors and actuators' performance on the host structures has been evaluated for PZT itself using FE simulations to check the adhesive layer effects, including non-uniform thickness [29]. However, no studies have been reported on reducing the crack damage propagation in a thin plate over the last four years of work excluding Aabid et al. [30] and Abuzaid et al. [31–35]. Therefore, the current study aims to reduce the SIF factor bonded with PZT actuators, and we introduce a novel methodology to determine the results through a machine learning approach. The study focuses on several parameters, including the position of the actuator, actuator cross-sectional area, actuator thickness, and adhesive thickness. To achieve this, we simulated various cases based on these parameters and levels, utilizing machine learning algorithms that have proven to be effective in this type of analysis. By identifying the most crucial factors that influence the result, we can improve actuator performance by determining its optimal size. M A. Anjum et alii, Frattura ed Integrità Strutturale, 66 (2023) 112-126; DOI: 10.3221/IGF-ESIS.66.06 114 METHODOLOGY Problem definition damaged rectangular plate with a piezoelectric actuator under uniform stress of 1 MPa is shown in Fig. 1. A thin- walled structure is utilized to simulate the current work. The distance of the actuator from the damaged edge affects the stress transmission to the crack surfaces because of the effectiveness of the repair. Therefore, given that the high stresses created by the integrated structure act at the piezoelectric actuator's edge, it is anticipated that the edge will be close to the fracture point [36]. The rectangular plate is 1 mm thick and has the following measurements: W = 40 mm, H = 200 mm. The plate features a 20 mm-long fracture that is open to an external load of 1 MPa. The actuator is 0.5 mm thick and measures Hp = 0.1H and 2Wp = 0.2H. The piezoelectric actuator is positioned at 1 mm from the crack tip and denoted as S. Tab. 1 lists the properties of the cracked plate, piezoelectric actuator, and adhesive bond. Figure 1: Center-cracked aluminum plate with piezoelectric actuators. Parameter Cracked Plate PIC151 Patch Adhesive Density 2715 kg/m3 7800 kg/m3 1160 kg/m3 Poisson’s Ratio 0.33 0.345 Young’s Modulus 68.95 GPa 5.1 GPa Shear Modulus 1.2 GPa Compliance Matrix 4 S11 = 15.0×10-12m2/N S33 = 19.0×10-12m2/N Electric Permittivity Coefficient 5 T 11ε  = 1977 T 33ε  = 2400 PZT strain coefficient 9 d31 = - 2.10×10-10m/V d32 = - 2.10×10-10m/V Table 1: Properties of the cracked plate, PIC151, and adhesive. Finite Element Simulation and Analysis The present work employs a coupled-field analysis approach to investigate the interaction between structural and electric fields, with a specific focus on piezoelectric analysis. To model the piezoelectric actuator, we used the coupled element SOLID226 [37], which contains 20 nodes and can have up to 5 degrees of freedom. Meanwhile, the SOLID186 element A A. Anjum et alii, Frattura ed Integrità Strutturale, 66 (2023) 112-126; DOI: 10.3221/IGF-ESIS.66.06 115 was used for the cracked plate and adhesive layer, a higher-order 20-noded element that is suited for solid structure analysis and recommended for linear application. To optimize run time and processor requirements, only one-quarter of the plate was considered due to symmetry, and a fine mesh was created around the fracture front to determine the SIF. The interaction integral was employed to compute the SIF, using the virtual crack extension principles technique, which produced accurate results with fewer mesh requirements. To create a singularity in stresses and strains around the crack tip, singular elements were used, ideally oriented in the same direction as the crack. Fig. 2 displays the complete set of FE models for all aspects of the quarter model generated based on the given boundary conditions. The crack front was modelled using ten singular elements, while the PZT actuator was modelled using 4,999 coupled-field elements. The damaged plate was modelled using 12,393 high-order reduced integration solid elements, and the adhesive bond was modelled using 2,499 high- order reduced integration solid elements.   Figure 2. FE mesh model of the repaired plate. Machine learning approach Machine learning (ML) is a type of artificial intelligence that involves the use of algorithms to automatically learn from data, identify patterns, and make decisions without explicit instructions. ML is a powerful tool for data analysis that can automate the development of analytical models. However, there is no one-size-fits-all approach to using machine learning to solve real-world problems, as different problems require different methods. Fig. 3 provides an overview of the general process of using machine learning to analyses data.   Figure 3. Machine Learning Process. In this study, we examined the impact of actuator position, cross-sectional area, thickness, and adhesive thickness on the SIF using finite element simulations. Our dataset includes 4 features and 1 target value. Tab. 2 presents the features X1, X2, X3, and X4, which correspond to actuator position, cross-sectional area, thickness, and adhesive thickness, respectively. The target value, Y1, represents the SIF and is the goal parameter of our machine learning approach. Tab. 2 provides an overview of these characteristics and the desired values. We collected 27 data points through numerical simulations for our dataset. A. Anjum et alii, Frattura ed Integrità Strutturale, 66 (2023) 112-126; DOI: 10.3221/IGF-ESIS.66.06 116 Feature ID Property Notations Levels Range (mm) Data type X1 Position of Actuator S 3 0.75 – 1.75 Numeric X2 Actuator Cross- sectional Area A 3 225 – 625 Numeric X3 Actuator Thickness Tp 3 0.5 – 1.0 Numeric X4 Adhesive Thickness Ta 3 0.0025 – 0.0035 Numeric Table 2: Data set numerical model. Selected machine learning models Machine learning algorithms have become popular in solving problems in aeronautical, civil, and mechanical engineering due to their ability to predict optimal results using available information. In this study, FE results were used as input data and the objective was to reduce the SIF factor. Different variables such as dimensions and mechanical properties of the plate, adhesive bond, and bonded piezoelectric patch were chosen to achieve this objective. To predict the optimum results, Gaussian Process Regression (GPR) and Support Vector Machine (SVM) techniques with various kernel functions were implemented. The GPR is a probabilistic supervised machine learning framework that presents a probability distribution over potential functions that fit a set of points and computes the means and variances to show how confident the forecasts are [38]. The Gaussian processes model is a commonly used probabilistic framework in supervised machine learning for both classification and regression tasks. This model provides a probability distribution over a set of functions that can fit a given set of data points. By computing the means of these functions, we can obtain predictions for new data points, while variances provide a measure of confidence in these predictions, since they reflect the probability distribution over all possible functions [39]. Models based on the Gaussian process regression (GPR) kernel are nonparametric probabilistic models. Take into account the training set   , ; 1, 2, ...,i ix y i n , where ix d , and  ,iy   are taken from an unidentified distribution. With the fresh input vector xnew and the training data, a GPR model attempts to predict the value of a response variable ynew . The formula for a linear regression model is    ,Ty x where    20, .N The coefficients  and error variance  2 are estimated using the data. A GPR model uses explicit basis functions, h , and latent variables,   , 1, 2,..., ,if x i n from a Gaussian process (GP), to describe the response. The smoothness of the response is captured by the covariance function of the latent variables, and basic functions project the inputs x into a p -dimensional feature space [40,41]. A GP is a collection of random variables that all have the same joint Gaussian distribution for any finite number of them. Given n observations  1, 2,..., ,x x xn , the joint distribution of the random variables       1 , 2 , ...,f x f x f xn is Gaussian if   , df x x  is a GP. A GP is described by its covariance function, k (x, x ′), and mean function, m(x). In other words, if the function   , df x x  is a Gaussian process, then     E f x m x and                        , ' ' ' , ' .Cov f x f x E f x m x f x m x k x x The Gaussian model is represented by     h x T f x where     ~ 0, , 'f x GP k x x indicates that  f x are from a zero mean GP with covariance function,  , 'k x x .  x in d is changed into the new feature vector    h x in pR by a collection of basis functions called  h x . A basis function coefficients vector of size p by 1 is known as  . A. Anjum et alii, Frattura ed Integrità Strutturale, 66 (2023) 112-126; DOI: 10.3221/IGF-ESIS.66.06 117 One way to model a response y instance is as         ,  ~ , 2i i i i iP yi f x x N y h x T f x∣ ∣ Therefore, a GPR model is a probabilistic model. The GPR model is nonparametric since a latent variable  if x is introduced for each observation ix . In the GPR model, the latent variables    1 , 2 ,..., (x f x f xn have the following joint distribution:     ~ 0, , ,P f X N f K X X∣ ∣ which resembles a linear regression model, where  ,K X X                                           1, 1 1, 2 1, , 2, 1 2, 2 2, , 1 , 2 , k x x k x x k x xn K X X k x x k x x k x xn k xn x k xn x k xn xn Typically, a set of kernel parameters or hyperparameters ,   are used to parameterize the covariance function  ,  'k x x and is frequently written as  , 'k x x∣ to explicitly state the reliance on  . The kernel parameters are based on the signal standard deviation  f and the characteristic length scale  l for many of the standard kernel functions. The characteristic length scales provide a brief definition of the minimum distance at which the input values ix must be separated before the response values become uncorrelated. The various kernel functions used in this study are listed in Tab. 3.   Kernel Function Covariance function definition Rational Quadratic Kernel               2 2 2, j 1 2i f l r k x x Matern 5/2 GPR                     2 2 2 5 5 5 i, j 1 exp 3f l ll r r r k x x Exponential GPR            2i, j  expf l r k x x Squared Exponential GPR                   2 2 1 i, j  exp 2 T i j i j f l x x x x k x x Table 3. Kernel functions for the GPR technique [37]. where       T i j i jr x x x x is the Euclidean distance between ix and jx and  is a positive-valued scale-mixture parameter. Vladimir Vapnik and his coworkers reported the support vector machine (SVM) analysis in 1992 [42], and it has since become a widely used machine learning technique for classification and regression. As a result of its reliance on kernel functions, SVM regression is regarded as a nonparametric method. There are various kernel functions available for SVM techniques as shown in Tab. 4. In this study, polynomial kernel functions with order 2 and order 3 are used. A. Anjum et alii, Frattura ed Integrità Strutturale, 66 (2023) 112-126; DOI: 10.3221/IGF-ESIS.66.06 118 The performance of regression models is evaluated using various evaluation metrics such as root means square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE) and coefficient of determination (R2). The mathematical formulations, characteristics, applications, advantages, disadvantages, and limitations of these metrics can be found in [43]. Type of SVM Mercer Kernel Description Gaussian or Radial Basis Function (RBF)           2 1 2 1 2 2, exp 2 x x k x x One class learning.  is the width of the kernel Linear   1 2 1 2, Tk x x x x Two class learning. Polynomial     1 2 1 2, 1Tk x x x x  is the order of the polynomial Sigmoid      1 2 0 1 2 1, tanh Tk x x x x 0 , and 1 are constants for which it becomes a mercer kernel function. Table 4. Types of SVM and Kernel Functions [41]. FINITE ELEMENT RESULTS AND DISCUSSION Grid independence test o evaluate the influence of mesh size on computational results, three mesh sizes were chosen and presented in Fig. 4. The adhesive bond and piezoelectric actuator were meshed using the grid-structured mesh with component size based on the mesh type. The elements were separated for the damaged plate by choosing each line, resulting in an unstructured type of mesh. As shown in Fig. 4, refining the mesh from medium to fine led to a slight improvement in the accuracy of the SIF value solution, with a maximum relative difference of 14 percent. However, the medium mesh provided adequate resolution and accuracy while requiring only half the computation time, making it the preferred option for the remaining simulations. Figure 4: Mesh independence results. Validation of the numerical model Fig. 5 provides visual evidence of the strong agreement between the two results, indicating that the computational model is a reliable and accurate representation of the physical experiment. The maximum relative error of about 10% suggests that the differences between the simulation and experimental results are relatively small, considering the complexities involved in conducting physical experiments. Despite the small differences, it is important to note that the accuracy of the experimental results may have been affected by various factors, such as human error in situating and aligning the Coarse Medium Fine 0.00 0.02 0.04 0.06 0.08 0.10 0.12 No. of Elements : 68,920 No. of Nodes : 133,633 CPU run : 1522 sec No. of Elements : 33,452 No. of Nodes : 73,633 CPU run : 600 sec S IF ( M P a) Mesh Type No. of Elements : 11,832 No. of Nodes : 28,692 CPU run : 309 sec T A. Anjum et alii, Frattura ed Integrità Strutturale, 66 (2023) 112-126; DOI: 10.3221/IGF-ESIS.66.06 119 piezoelectric actuator and strain gage on the host plate. These challenges can lead to variations in the experimental results, which may not be present in the simulation results. Additionally, the small notch outside the computational model that initiates the crack may have had an impact on the experimental results. The presence or absence of such factors could explain the minor differences between the two sets of results. Overall, the linear fluctuation of the SIF with the applied electric field remains consistent with the supersession principles, providing further support for the accuracy of the computational model. Figure 5: Validation of current FE findings using experimental data from [31]. Reduction of stress intensity factor The study presented in Fig. 6 shows the reduction in SIF for the patched plate in comparison to the unpatched plate. Based on results it shows SIF decreases linearly with the increase in crack length. The plot represents the maximum reduction of applied negative voltage at 150 V [31]. As the crack length decreases, it was observed that the SIF increases and the percentage reduction falls. This is due to the requirement of a greater negative electric field to decrease the SIF to the same level for shorter crack lengths, owing to the larger stress gradient in the crack tip region. For instance, a voltage of 150 V applied to a crack length of 5 mm resulted in a reduction of approximately 52.3% in the SIF, while a crack length of 12.5 mm resulted in a reduction of only 45.8%. The observed linear relationship between the crack length and the reduction of SIF is an essential finding that can help researchers design more effective patch configurations to reduce SIF in practical applications. The results also suggest that the effectiveness of the patch decreases as the crack length reduces, indicating that the patch's ability to mitigate damage is more effective for larger cracks. However, it should be noted that the reduction in SIF is not the only factor to consider in practical applications, and other factors such as the cost, weight, and durability of the patch should also be considered in the design process. Overall, the findings from this study provide valuable insights into the use of piezoelectric patches for reducing SIF in damaged structures. MACHINE LEARNING RESULTS AND DISCUSSION n the current study, six different models were used to predict the results (Tab. 5) using the regression learner app of release 2022a of MATLAB® [44]. Simulated FE results are predicted using six machine-learning models. Fig. 7 shows the responses using various methods, Fig. 8 shows the comparison of true response vs the predicted response by various models and Fig. 9 shows residuals of various models. Tab. 6 provides the simulated values vs the predicted values for each parametric variation. From Fig. 10 model 3 and model 6 give us the best fit, which results in lower RMSE values and R2 values nearer to 1. 20 30 40 50 60 70 80 90 100 110 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 N S IF Applied Voltage (V) Present Results Abuzaid et al., 2018 I A. Anjum et alii, Frattura ed Integrità Strutturale, 66 (2023) 112-126; DOI: 10.3221/IGF-ESIS.66.06 120 Figure 6. Comparison of patched and unpatched plates. Model Number Model Model Parameters Model 1 Quadratic SVM Kernel function-Polynomial; Kernel polynomial order-2; Solver-Sequential Minimal Optimization; Epsilon-0.005 Model 2 Cubic SVM Kernel function-Polynomial; Kernel polynomial order-3; Solver-Sequential Minimal Optimization; Epsilon-0.005 Model 3 Rational Quadratic GPR Kernel function-Rational Quadratic; Optimizer-Quasi Newton; Method – Gaussian Process Regression Model 4 Matern 5/2 GPR Kernel function-Matern 5/2; Optimizer-Quasi Newton; Method – Gaussian Process Regression Model 5 Exponential GPR Kernel function- Exponential; Optimizer-Quasi Newton; Method – Gaussian Process Regression Model 6 Squared Exponential GPR Kernel function-Squared Exponential; Optimizer-Quasi Newton; Method – Gaussian Process Regression Table 5. Model descriptions. Finite element simulations require a mesh study to achieve accurate results. Typically, a fine mesh is preferred for accurate results. After conducting mesh sensitivity analysis, a fine mesh was chosen for this study. As a result, the FE and ML work achieved less than a 1% error, which can be safely ignored. Furthermore, the selected ML models were found to be suitable for predicting and simplifying the current problem. Therefore, this simulation demonstrates that the selected ML models can accurately predict and simplify the current problem. Fig. 7 demonstrates the performance of the GPR and SVM algorithms in predicting the SIF values. Both training and testing results using GPR and SVM algorithms are presented, and the predicted values are compared to the simulation data for the respective data sets. The results reveal that the GPR algorithm using Rational Quadratic and Squared Exponential kernel functions is effective in predicting the SIF values for these data sets, and the algorithm fits well with the simulation data. The GPR algorithm was further optimized to fit the validation samples with the simulation data, which resulted in good agreement with the FE data for the tested data. This study is noteworthy as it is the first to utilize six different regression models for the regression analysis of solid mechanics and structure problems. Additionally, this study highlights the potential of GPR for the analysis of highly linear and non-linear numeric data, as the results indicate that the SIF values are significantly dependent on the patch shear strength, which was successfully predicted by GPR for the testing points. This study demonstrates the potential of GPR for various fields where regression analysis of highly linear and non-linear numeric data is required. 4 5 6 7 8 9 10 11 12 13 -0.04 0.00 0.04 0.08 0.12 0.16 0.20 0.24 S IF ( M P a. m -1 /2 ) Crack Length (a) Unpatched Patched A. Anjum et alii, Frattura ed Integrità Strutturale, 66 (2023) 112-126; DOI: 10.3221/IGF-ESIS.66.06 121 The comparison of the regression results of SIF using GPR and SVM algorithms is important for predicting the values of the data sets with high accuracy. The Rational Quadratic and Squared Exponential kernel functions used in the GPR algorithm prove to be effective in predicting the data sets and fitting properly with simulation results. This study provides valuable insight into the potential of GPR for regression analysis in the field of solid mechanics and structure problems. Furthermore, the successful prediction of the testing points by GPR demonstrates its suitability for highly non-linear data analysis. The study's novelty lies in its comprehensive comparison of six different regression models, paving the way for future research to explore GPR on topics with highly linear and non-linear numeric data. Runs Simulation Response Output NSIF using Machine Learning Techniques NSIF Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 1 0.443706 0.464191 0.448939 0.446833 0.449311 0.46889 0.446833 2 0.519506 0.512213 0.514377 0.519338 0.519317 0.519487 0.519338 3 0.57479 0.567337 0.570027 0.574677 0.574723 0.574755 0.574677 4 0.422777 0.450991 0.428494 0.423224 0.423039 0.422836 0.423224 5 0.473577 0.496179 0.480527 0.473615 0.473723 0.473615 0.473615 6 0.504582 0.508822 0.510129 0.504501 0.504626 0.504598 0.504501 7 0.43676 0.436135 0.434291 0.436658 0.436658 0.43676 0.436658 8 0.465967 0.460134 0.461338 0.465924 0.465922 0.465959 0.465924 9 0.48866 0.493809 0.490649 0.488752 0.488685 0.488668 0.488752 10 0.479372 0.48213 0.482995 0.479456 0.479428 0.479393 0.479456 11 0.548186 0.529621 0.540968 0.548162 0.548245 0.548157 0.548162 12 0.598832 0.584213 0.592897 0.599111 0.598931 0.598808 0.599111 13 0.453232 0.46596 0.458793 0.45266 0.45299 0.453236 0.45266 14 0.498066 0.510617 0.502352 0.49811 0.498004 0.498083 0.49811 15 0.526072 0.528489 0.530316 0.526229 0.526055 0.526076 0.526229 16 0.464992 0.447835 0.454094 0.458952 0.454308 0.449011 0.458952 17 0.489278 0.477063 0.484282 0.489563 0.489619 0.480984 0.489563 18 0.508227 0.510206 0.505695 0.508132 0.508204 0.508224 0.508132 19 0.505432 0.500228 0.509872 0.505389 0.505418 0.50543 0.505389 20 0.570027 0.547188 0.563585 0.569952 0.569883 0.569986 0.569952 21 0.617714 0.601249 0.614996 0.617507 0.617621 0.617659 0.617507 22 0.474824 0.481089 0.480268 0.475108 0.474972 0.474852 0.475108 23 0.516393 0.525215 0.518556 0.516504 0.516516 0.516426 0.516504 24 0.542765 0.548315 0.548186 0.5428 0.542863 0.542775 0.5428 25 0.484435 0.459694 0.462965 0.473563 0.465459 0.468738 0.473563 26 0.50605 0.49415 0.499598 0.506002 0.505987 0.506027 0.506002 27 0.523015 0.526763 0.516317 0.521929 0.522337 0.525415 0.521929 Table 6. Machine learning outcomes. The residual is the difference between the predicted value and the actual value, and Fig. 8 displays the residual plots for different models. A negative residual value indicates that the predicted value is greater than the actual value, while a positive residual value suggests that the predicted value is less than the actual value. As depicted in Fig. 8, model 3 and model 8 demonstrate the best fit for the dataset. The regression line for the given data with an R-Square value of 0.998 is shown in Fig. 8. To ensure the accuracy of the regression models, it is essential to evaluate the residual plot. The residual plot helps to identify the outliers and the presence of patterns in the data. From the residual plot in Fig. 9, it is observed that model 3 and model 8 have the lowest values of residuals, indicating that they have the best fit to the data. In contrast, models 1, 2, 4, 5, 6, and 7 have higher residuals, which indicates a less accurate fit to the data. The R-Square value of 0.998 for the regression line in Fig. 9 indicates that the regression models have a high degree of fit to the data. Overall, the residual plot and R-Square value suggest that models 3 and 8 are the most appropriate regression models for the given dataset. A. Anjum et alii, Frattura ed Integrità Strutturale, 66 (2023) 112-126; DOI: 10.3221/IGF-ESIS.66.06 122 Figure 7: Comparison of responses using various methods. Figure 8: True response vs the predicted. Figure 9: True response vs the residual. A. Anjum et alii, Frattura ed Integrità Strutturale, 66 (2023) 112-126; DOI: 10.3221/IGF-ESIS.66.06 123 Fig. 10 illustrates the evaluation metrics for the six different models used in this study. The metrics include MAE (Mean Absolute Error), MAPE (Mean Absolute Percentage Error), RMSE (Root Mean Squared Error), and R-Square values. From the results, it is observed that model 3 and model 6 exhibit the least values of MAE, MAPE, and RMSE, with the highest R-Square value of 0.997377. These values indicate that these two models have the same accuracy and performance. Conversely, model 1 is found to be the least accurate model among the six models used in the study, with an RMSE of 0.013593. Overall, the evaluation metrics demonstrate that models 3 and 6 perform better than the other models in predicting the outcomes. The evaluation metrics presented in Fig. 10 provide insight into the performance of the different models. The results indicate that models 3 and 6 performed the best among the six models tested. These two models have the lowest values of MAE, MAPE, and RMSE, and the highest value of R-square. The low values of MAE, MAPE, and RMSE suggest that these models have less error and better accuracy in predicting the outcomes. Moreover, the high value of R-square indicates that these models have a strong correlation between the predicted and actual values. Conversely, model 1 had the least accuracy among all six models, with a higher value of RMSE. The results presented in Tab. 7 support the findings of Fig. 10, indicating that models 3 and 6 are the most accurate models in predicting the outcomes of interest. These findings suggest that models 3 and 6 could be preferred over other models for predicting the outcomes in this study. Machine Learning Algorithm Applied R-Squared RMSE Quadratic SVM 0.92406 0.013593 Cubic SVM 0.98045 0.006714 Rational Quadratic GPR 0.99738 0.002484 Matern 5/2 GPR 0.99209 0.004331 Exponential GPR 0.97993 0.006696 Squared Exponential GPR 0.99738 0.002484 Table 7: Comparison of Evaluation metrics for various regression models. Figure 10. Various evaluation metrics for different models. CONCLUSION he study presented a successful approach for identifying the optimal value of SIF for damage control in an aluminum thin plate structure. To achieve this, 27 finite element simulations were performed with varying parameters, and six machine learning models were utilized to predict the SIF. The finite element method was initially used to obtain SIF with four different parameters and three different levels as input parameters/levels. The results showed that machine T A. Anjum et alii, Frattura ed Integrità Strutturale, 66 (2023) 112-126; DOI: 10.3221/IGF-ESIS.66.06 124 learning algorithms are an effective and practical approach to understanding complex phenomena as they consider all components and their interactions. Furthermore, the method allowed for the identification of the actuators and adhesive parameters that resulted in the lowest SIF, and consequently, the enhancement of the actuator design. This study contributes to the advancement of the understanding and control of SIF in thin plate structures and highlights the potential of machine learning in solid mechanics and structural engineering. ACKNOWLEDGEMENT he author Asraar Anjum acknowledges the support of the TFW2020 scheme from the Kulliyyah of Engineering, International Islamic University Malaysia and this research was supported by the Ministry of Education of Malaysia (MOE) through Fundamental Research Grant Scheme (FRGS/1/2021/TK0/UIAM/01/5). Also, the authors acknowledge the support of the Structures and Materials (S&M) Research Lab of Prince Sultan University. CONFLICTS OF INTEREST he authors declare no conflict of interest. AVAILABILITY OF DATA AND MATERIALS he datasets used during the current study are available from the corresponding author upon reasonable request. REFERENCES [1] Aabid, A., Baig, M., Hrairi, M. (2023). 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