This study presents a semi-analytical solution that can calculate the free vibration frequencies of functionally graded nanobeams with three distinct pore distributions under both deformable and rigid boundary conditions, based on nonlocal elasticity and Levinson beam theories. The novelty lies in the incorporation of transverse springs at both ends of porous functionally graded nanobeams and introducing a general eigenvalue problem dependent on the stiffness of these springs. This solution provides vibrational frequencies considering Levinson beam theory, non-local elasticity theory, spring stiffnesses, porosity coefficients, and temperature change. Additionally, the vibrational behavior of these porous nanobeams is explored through machine learning (ML) techniques. Four ML models namely artificial neural network (ANN), support vector regression (SVR), decision tree (DT), and extreme gradient boosting (XGB) are trained to predict the natural frequencies of nanobeams with varying pore distributions. The Sobol quasi-random space-filling method is employed to generate samples by altering input feature combinations for different porous nanobeam distributions. Model performance is evaluated using different performance indicators and visualization tools, with optimal hyperparameters determined via a Bayesian optimization algorithm. Results underscore the efficacy of ML models in predicting natural frequencies, with SVR and ANN demonstrating superior performance compared to XGB and DT. Notably, SVR and ANN exhibit exceptional R² values of 0.999, along with the lowest MAE, MAPE, and RMSE values among the models assessed. At the end of the study, the effects of various parameters on porous gold (Au) nanobeams using the solution of the presented eigenvalue problem are discussed.

This study investigates the free vibration behavior of Euler–Bernoulli beams made of viscoelastic materials using nonlocal theory. The mechanical properties of the nanobeam are functionally graded through its thickness, and the viscoelastic effects on energy damping are considered. Furthermore, micro- and nano-scale structural effects are incorporated into the model using nonlocal elasticity theory. Based on this, a semi-analytical solution method is developed to determine the natural frequencies and damping ratios of the beam under elastic boundary conditions. The effects of various parameters such as geometry, material grading, viscoelastic properties, and nonlocality on the dynamic behavior of beam are studied using this solution, and the results are compared with other studies in literature. Subsequently, a space-filling sampling technique is used to generate well-distributed samples of input parameters uniformly across an input space. The generated dataset is used to train various machine learning (ML) models such as k-nearest neighbor, decision tree regression, extreme gradient boosting, and light gradient boosting. Various hyperparameter optimization techniques including metaheuristic algorithms (particle swarm and genetic algorithms) and model-based methods (Bayesian optimization with Gaussian process and tree-structured Parzen estimator) are explored. A detailed study is conducted to identify the most efficient optimization technique with the most robust ML model. It is found that the decision tree regression incorporated into Bayesian optimization with tree-structured Parzen estimator) achieves the best performance in terms of computational cost and accuracy. This hybrid model requires only 11.64 s to train and perfectly predicts vibration frequencies with coefficient of determination (R ²) of 1. The model's robustness is further validated using comprehensive statistical and graphical evaluations.

Macroscopic mechanical properties of fibrous materials are often characterized by modeling their microscale behavior using micromechanical techniques. This process typically involves using a Representative Volume Element (RVE) and finite element simulations to obtain the macroscopic behavior through homogenization. However, these micromechanical simulations can be computationally demanding, especially for 3D models with discrete material microstructures. This paper uses boosting machine learning algorithms to predict the homogenized macroscopic material behavior of heterogeneous composites. These models are trained on the micromechanical simulation results generated by varying the constitutive parameters of local phases and microscopic parameters such as fiber volume fraction. The Bayesian optimization is used to determine the best hyperparameters of the considered boosting models, which include adaptive boosting (AdaB), gradient boosting (GBR), light gradient boosting (LGB), and extreme gradient boosting (XGB). The performances of trained models are assessed using various metrics such as R ² , MAE, MAPE, and RMSE and using various plots such as scatter plots, Taylor plots, radar plots, and bar plots. The comparative assessment showed that all the models predicted the homogenized stiffness matrix of the RVE successfully, with R ² values between 0.94 and 0.99. The XGB model presented the best overall performance. This work contributes to the field of composites by presenting a new and computationally efficient approach to predict the macroscopic behavior of RVEs using boosting models.