•Sensitivity analysis (SA) approach is used to quantify the effects of correlated input parameters on model outputs.•Penalized spline regression model is used to approximate complex data.
We provide ...a sensitivity analysis toolbox consisting of a set of Matlab functions that offer utilities for quantifying the influence of uncertain input parameters on uncertain model outputs. It allows the determination of the key input parameters of an output of interest. The results are based on a probability density function (PDF) provided for the input parameters. The toolbox for uncertainty and sensitivity analysis methods consists of three ingredients: (1) sampling method, (2) surrogate models, (3) sensitivity analysis (SA) method. Numerical studies based on analytical functions associated with noise and industrial data are performed to prove the usefulness and effectiveness of this study.
•A new method, namely, nonlocal strain gradient method is applied to the FG-CNTRC nanoplates.•Natural frequencies of the nanostructures are the intermediate step to find the responses or ...characteristics.•This study focuses on obtaining the mathematical formulas of natural frequencies.•The influence of material length scale and nonlocal parameter on vibrational responses is investigated.•The correlation between the geometry and material length scale is discussed.
In this research paper, as initial endeavors, the vibrational responses of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) nanoplates taking into account the effect of nonlocal parameter and strain gradient coefficient are investigated. The study aims at developing mathematical modeling via an analytical solution to FG-CNTRC nanoplate structure with allowance for the nonlocal strain gradient effect. The four types of CNT distribution are used and compared in the context of the vibration of nanoplate in the presence of the small length scale effects, namely the (a) UD, (b) FG-V, (c) FG-O, and (d) FG-X. Some theoretical equations based on the first-order shear deformation plate theory (FSDT) are presented to provide a lucid understanding of how the small length-scale influences the FG-CNTRC nanoplate. For the vibrational analysis of a nanoplate, which is simply supported boundary condition, Navier solutions are obtained. Also, in contrast to earlier studies, an analytical approach is used to establish the governing equations of the FG-CNTRC nanoplate. Some specific numerical examples are given and compared with the results presented in the literature. In the section of numerical results, the influence of the nonlocal parameter, strain gradient coefficient, geometric parameters and vibrational modes on the non-dimensional natural frequency are investigated and discussed in detail. These could be useful to analysts and designers to estimate the fundamental natural frequencies in each of the four CNT distributions that the FG-CNTRC nanoplate possesses.
•A discrete variable technique is integrated into ICDE to give Discrete-ICDE.•Discrete-ICDE is then applied for the truss layout optimization problems.•Numerical results show that Discrete-ICDE is ...robust, effective and reliable.
Recently, an improved (μ+λ) constrainted differential evolution (ICDE) has been proposed and proven to be robust and effective for solving constrainted optimization problems. However, so far, the ICDE has been developed mainly for continuous design variables, and hence it becomes inappropriate for solving layout truss optimization problems which contain both discrete and continuous variables. This paper hence fills this gap by proposing a novel discrete variables handling technique and integrating it into original ICDE to give a so-called Discrete-ICDE (D-ICDE) for solving layout truss optimization problems. Objective functions of the optimization problems are minimum weights of the whole truss structures and constraints are stress, displacement and buckling limitations. Numerical examples of five classical truss problems are carried out and compared to other state-of-the-art optimization methods to illustrate the reliability and effectiveness of the proposed method. The D-ICDE’s performance shows that it not only successfully handles discrete variables but also significantly improves the convergence of layout truss optimization problem. The D-ICDE is promising to extend for determining the optimal solution of other structural optimization problems which contain both discrete and continuous variables.
The circular cylindrical shells have been widely used in modern engineering structures, especially in the aerospace industry such as the oil pipeline, the missile, spacecraft hull, storage tanks. In ...recent years, functionally graded carbon nanotube composites (FG-CNTRCs) have emerged, as a promising type of composites. Due to the increasing demands for high structures performance, this research paper proposes a closed-form solution to investigate the nonlinear buckling behavior of the FG-CNTRC cylindrical shells subjected to compressive load. The small initial imperfections of the FG-CNTRC cylindrical shells are also considered through analytical modeling. Effective properties of materials of the shells reinforced by single-walled carbon nanotubes (SWCNTs) are estimated through a micro-mechanical model based on the extended rule of mixtures. The Donnell shell theory and von-Karman nonlinear kinematics are used for nonlinear equilibrium equations. The novelty of this work is to exploit an exact solution via Galerkin procedure and term of the Airy stress function in order to reveal the impacts of the imperfection parameter, different types of CNTs distribution, the volume fraction of CNTs on nonlinear behavior and compressive equilibrium paths of FG-CNTRC cylindrical shells.
•A closed form solution for nonlinear buckling analysis.•Carbon nanotubes (CNTs) distribution are dispersed by UD, FG-A, FG-V and FG-X distributions.•The Donnell shell theory and von-Karman nonlinear kinematics are used. .•The initial geometric imperfections of a cylindrical shell are considered.
This paper proposes an intelligent multi-objective optimization approach using the deep feedforward neural network (DNN) integrated with the speed-constrained multi-objective particle swarm ...optimization (SMPSO) to give the so-called DNN-SMPSO algorithm for solving multi-objective optimization problems of two-dimensional functionally graded (2D-FG) beams under a static load and free vibration. In the proposed approach, a high accurate DNN integrated with an intelligent sampling technique is used as a surrogate model to replace time-consuming numerical models in predicting objectives and constraints during the optimization process. Meanwhile, the SMPSO algorithm is utilized to search a set of Pareto-optimal solutions which show the best trade-off solutions of the required objectives. The ceramic volume fraction values at control points defined by the isogeometric analysis (IGA) framework are taken into account as continuous design variables and input parameters of the DNN model while the objectives and constraints are considered as output signals. In order to avoid the overfitting phenomena and speed up the training process of the DNN model, the state-of-the-art dropout and mini-batch techniques are applied. Additionally, various activation functions, optimizers, and hyper-parameters such as number of hidden layers and hidden units of the DNN model are surveyed. The accuracy, efficiency, and applicability of the proposed method are illustrated through two different multi-objective optimization examples of the 2D-FG beams with various boundary conditions. Optimal results obtained by the DNN-SMPSO method are compared with those of other methods to investigate the reliability of the proposed method. The optimal material distribution of the 2D-FG beams is described by two-dimensional Non-Uniform Rational B-spline (2D-NURBS) basis functions. Through the obtained numerical results, the DNN-SMPSO shows its accuracy, effectiveness, and capability in solving multi-objective optimization problems of engineering structures, especially in aspect of saving the computational cost. In addition, the attained optimal material distribution is useful for the 2D-FG beam fabrication.
•An aeDE is proposed for optimization of truss structures with discrete design variables.•The aeDE algorithm is a newly improved adaptive version of DE with three modifications.•Three improvements ...relate to the mutation phase, selection phase and rounding technique.•The numerical results for six benchmark problems illustrate the effectiveness of aeDE.
This paper proposes an adaptive elitist differential evolution (aeDE) for optimization of truss structures with discrete design variables. The aeDE algorithm is a newly improved version of the differential evolution (DE) algorithm with three modifications. Firstly, in the mutation phase, an adaptive technique based on the deviation of objective function between the best individual and the whole population in the previous generation is proposed to select a suitable mutation operator. This technique helps preserve the balance between global and local searching abilities in the DE. Secondly, in the selection phase, an elitist selection technique which helps choose the best individuals for the next generation is utilized to increase the convergence rate. Finally, a rounding technique is integrated into the aeDE for solving optimization problems with discrete design variables. The efficiency and reliability of the proposed method are demonstrated through six optimization problems of truss structures with discrete design variables. Numerical results reveal that in most of the test cases, the aeDE is more efficient than the DE and some other methods in the literature in terms of the quality of solution and convergence rate.
