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Reliability-based design optimization of rotating FGM cylindrical shells with temperature-dependent probabilistic frequency constraints
Abstract
In this study, the reliability-based design optimization (RBDO) of rotating functionally graded cylindrical shells (FGCSs) subjected to temperature dependent probabilistic frequency constraints is investigated. The uncertain parameters such as FG material properties and the shell thermo-mechanical loads are considered as random variable in RBDO. To decrease model uncertainty, the effect of initial thermal stresses are efficiently included in elasticity-based vibration equations of the shell and the powerful differential quadrature method (DQM) is employed to accurately determine the shell frequency parameters. An efficient RBDO framework based on the hybrid weighted average simulation method (WASM) and DQM together with the particle swarm optimization (PSO) method is then presented for design optimization of the FGCSs. A key feature of proposed hybrid DQ-WASM-PSO is that only one simulation run is required for WASM during entire optimization process of the FGCS, even if the distribution type of input variables and/or the system target reliability level be changed. The influence of temperature rise, temperature-dependence of FGM properties, annular velocity, PDFs of the random variables, and convection heat transfer coefficient of the shell inside fluid on the RBDO results are carried out. Parametric study indicates that exact evaluation of the initial thermal stresses, temperature-dependence of the FGM properties and convection heat transfer coefficient have considerable effect on optimization results.
... In Eq. (1), the subscripts m and c, respectively, refer to the metal and ceramic constituents; (≥ 0) is the material graded index; { = −1,0,1,2,3} are unique for the constituent materials (Safaeian et al., 2017); and = ( ) denotes temperature (in Kelvin). In the absence of the heat generation source within the shell, the temperature distribution can be obtained by solving the following tow-dimensional heat transfer equation: ...
... The differential quadrature method (DQM) is used to discretize the shell governing equations. According to DQM, the derivatives of a function are simply approximated as a weighted sum of its values at a given sampling points (Safaeian et al., 2017;Malekzadeh andSafaeian, 2013, 2016). Since the computational domain of DQM is a rectangular one, the linear geometric transformation (Eq. ...
... In numerical calculations, Biot number (Bio = ℎ × h / ) is used as suitable non-dimensional parameter for convective thermal boundary condition (Malekzadeh and Safaeian, 2013;Safaeian et al., 2017). ...
Functionally graded materials (FGMs) are inhomogeneous materials with continues variation of the constituents, usually metal and ceramic, in a given direction that are commonly useful in thermal environment (Malekzadeh and Safaeian, 2013, 2016). Rotating FGM conical shells are widely used as structural components in different engineering fields such as aeronautics, nuclear engineering, etc. Therefore, exact thermo-elastic analysis of these structures is crucial. In the present study, thermo-elasticanalysisofrotating FGMtruncatedconicalshells with temperature dependent material properties and subjected to convection heat transfer effects between the internal hot fluid and the shell inner surface is investigated based on the elasticity theory.
... All rights reserved. challenges, let alone drive the nanofillers spreading in a specific design distributions [9]; (2) material uncertainties due to manufacturing processes and defects [10][11][12]. Such as the material and geometrical properties (width, length and thickness) of nanofillers, porosity coefficient and matrix properties, the amount of reinforcements and pentagon defects [13]. ...
... (8), and then combining with Eq. (4), the parameters c t1 , c t2 , c t3 , and c t4 of porous nanocomposite beams for different combinations of porosity distributions and GPL dispersion patterns can be derived from (10) where e m is the mass density coefficient of metal foams. Before solving e m , the relationship between Young's modulus and mass density is introduced by adopting Gaussian random field model (11) where ρ(z)/ρ max ∈ (0.15, 1) in the assumption. Then by substituting the third equation of Eq. (1) into Eq. ...
... (11), one can obtain (12) Due to ρ(z)/ρ max ∈ (0.15, 1), the range of porosity coefficient e 0 ∈ (0, 0.9618) is calculated from Eq. (12). Furthermore, the coefficient of porosity for even porosity distribution ϕ 0 can be derived from Eq. (1), (11), as follows ...
This paper presents the first attempt to study the probabilistic stability characteristics of functionally graded (FG) graphene platelets (GPLs) reinforced beams by taking into account the multidimensional probability distributions, such as stochastic porosity and GPL distribution patterns as well as random ma-terial properties. For this purpose, a non-inclusive Chebyshev metamodel (CMM), which is implemented on deterministic analysis using discrete singular convolution (DSC) method with excellent computational efficiency and accuracy, is proposed and used to obtain both deterministic and probabilistic results in-cluding probability density functions (PDFs), cumulative density functions (CDFs), means and standard deviations of the critical buckling load. The present analysis is rigorously validated through direct com-parisons against the results obtained by a direct quasi-Monte Carlo simulation (QMCS) method and those available in open literature. The influences of material properties, porosity distribution, GPL dispersion pattern and boundary condition on probabilistic buckling behaviour of the FG-GPL beam are comprehen-sively investigated. The global sensitivity analysis is also conducted. The results suggest that the critical buckling load of the FG-GPL beam is most sensitive to porosity distribution, followed by porosity coeffi-cient and GPL weight fraction.
... Nonetheless, the main drawback of the simulation-based MORBDO could be high computational cost, specifically when double-loop approaches are employed. Recently, a computationally efficient weighted simulation-based design method (WSDM) is proposed by Rashki et al. [43] that provides accurate results for problems with highly non-linear and non-differentiable probabilistic constraints [1,[42][43][44][45] . WSDM [43] has the flexibility to provide multi-level design solutions by performing only one simulation run, but its' efficiency is limited to small dimension single-objective RBDOs with continuous random design variables [1,42,43] . ...
... WSM can estimate the failure probability with much lower samples than MCS. However, its efficiency will be decreased by increasing the number of random variables [1,42,44] . Okasha [42] proposed an improved WSM (IWSM) that can estimate the failure probability more efficiently than WSM. ...
... A reliable interval selection procedure is to use MCS to generate random samples with regard to PDF of each random variable. The bounds of sampling should be computed based on the strategy presented in Ref. [1,44] noticing that the bounds are related to the order of magnitude of the reliability constraint. ...
In the present study, two new simulation-based frameworks are proposed for multi-objective reliability-based design optimization (MORBDO). The first is based on hybrid non-dominated sorting weighted simulation method (NSWSM) in conjunction with iterative local searches that is efficient for continuous MORBDO problems. According to NSWSM, uniform samples are generated within the design space and, then, the set of feasible samples are separated. Thereafter, the non-dominated sorting operator is employed to extract the approximated Pareto front. The iterative local sample generation is then performed in order to enhance the accuracy, diversity, and increase the extent of non-dominated solutions. In the second framework, a pseudo-double loop algorithm is presented based on hybrid weighted simulation method (WSM) and the Non-dominated Sorting Genetic Algorithm II (NSGA-II) that is efficient for problems including both discrete and continuous variables. According to hybrid WSM-NSGA-II, proper non-dominated solutions are produced in each generation of NSGA-II and, subsequently, WSM evaluates the reliability level of each candidate solution until the algorithm converges to the true Pareto solutions. The valuable characteristic of presented approaches is that only one simulation run is required for WSM during entire optimization process, even if solutions for different levels of reliability be desired. Illustrative examples indicate that NSWSM with the proposed local search strategy is more efficient for small dimension continuous problems. However, WSM-NSGA-II outperforms NSWSM in terms of solutions quality and computational efficiency, specifically for discrete MORBDOs. Employing global optimizer in WSM-NSGA-II provided more accurate results with lower samples than NSWSM.
... Besides, some researchers have studied the thermoelastic model and dynamic characteristics of cylindrical shell, other shell, and cup-shaped resonators. The vibration equations of a shell considering thermal stress are investigated based on elasticity and its frequency parameters are accurately determined [25]. Moreover, in [26], the stress effect of trimmed cupped resonators due to temperature variations is analyzed based on the finite element model. ...
... Substituting Eq. (31) into Eqs. (23) and (25) yields the results of , , and , and substituting the results into the coefficients 0 , 45 , 0 , and 45 and Eq. (27) obtains the thermal output error model caused by nonuniform thermal deformation. ...
This paper proposes an accurate thermoelastic model of a hemispherical shell resonator (TM-HSR) taking into consideration thermal stress and reveals the effect of its thermal deformation on the hemispherical resonator gyroscope (HRG) output error for the first time. First, a set of thermoelastic dynamic equations of the TM-HSR are derived by leveraging the thermoelastic shell theory, and the relationship between temperature and the equivalent thermal load is determined. Next, an approximate analytical solution of thermal deformation of HSRs is derived using Bubnov-Galerkin method and its accuracy is evaluated by comparing its result with ANSYS simulation results of HSRs with different structures. The approximate analytical solution is found to be very accurate, which is important for temperature testing of HRGs and application to various vibratory sensors with hemispherical shell structures. Subsequently, the effects of thermal deformations under uniform and nonuniform temperature distributions on the HRG output error are revealed by using a thermal output error model. Comparative simulation and experimental results demonstrate the effectiveness and practicability of the proposed model. The proposed model can be used for accurate error analysis and developing a compensation scheme for the thermal output error of HRGs.
... Among all global search algorithms, PSO has been applied and proven useful on a wide range of engineering COPs such as the optimal design of truss structures [42][43][44][45], structural damage detection [46], topology optimization [47][48][49], and reliability-based design optimization [50][51][52]. PSO algorithm can efficiently handle non-linear, non-convex, and non-differentiable design spaces since it does not require prior knowledge about the search space, internal variable transformations, or other manipulations to handle constraints [53]. ...
... where is the swam size; D is the dimension of the search space, k is the current iteration number; ( ) and ( ) are the position and velocity of the i-th particle ( = 1, 2, … , ) in the j-th dimension, respectively ( = 1, 2, … , ); is the constriction factor; 1 and 2 , respectively, are the personal (cognitive) and social learning constant; 1 and 2 are random numbers between [0-1], and is the inertia weight factor used to keep a balance between the exploration and exploitation power of the algorithm. The linearly decreasing inertia weight factor is applied in this study [51,52]. ...
A hybrid algorithm is presented that combines strong points of Particle Swarm Optimization (PSO) and Generalized Reduced Gradient (GRG) algorithm to keep a good compromise between exploration and exploitation. The hybrid PSO-GRG quickly approximates the optimum solution using PSO as a global search engine in the first phase of the search process. The solution accuracy is then improved during the second phase of the search process using the GRG algorithm to probe locally for a proper solution(s) in the vicinity of the current best position obtained by PSO. The k-nearest neighbors (k-NN)-based Purely Uniform Distributed (PUD) initial swarm is also applied to increase the convergence speed and reduce the number of function evaluations (NFEs). Hybridization between both algorithms allows the proposed algorithm to accelerate throughout the early stages of optimization using the high exploration power of PSO whereas, promising solutions will possess a high probability to be exploited in the second phase of optimization using the high exploitation ability of GRG. This prevents PUD-based hybrid PSO-GRG from becoming trapped in local optima while maintaining a balance between exploration and exploitation. The competence of the algorithm is compared with other state-of-the-art algorithms on benchmark optimization problems having a wide range of dimensions and varied complexities. Appraising offered algorithm performance revealed great competitive results on the Multiple Comparison Test (MCT) and Analysis of Variance (ANOVA) test. Results demonstrate the superiority of hybrid PSO-GRG compared to standard PSO in terms of fewer NFEs, fast convergence speed, and high escaping ability from local optima.
... Wang et al. [29] applied first-order shear deformation theory to investigate the free vibration of rotating FG spherical-cylindrical-conical shells with different boundaries under the temperature field. Hamzehkolaei et al. [30] optimized the design of FG cylindrical shells with material temperature dependence. Hu and Yang [31,32] applied magnetoelastic theory to establish the electromagnetic force models for ferromagnetic FG shells under the transverse magnetic field. ...
The free vibration and traveling wave stability of a rotating ferromagnetic functionally graded cylindrical shell in magnetic and temperature fields are explored. The geometric and physical equations are determined within the framework of the Love’s theory. The expressions of kinetic energy and strain energy are obtained by considering temperature and rotation effects. The magnetoelastic theory is employed to establish a model of magnetic force. By using Hamilton’s principle and Galerkin truncation, the governing equations are obtained. The effects of different parameters on the natural frequencies of forward and backward waves are determined. It is found that the natural frequency undergoes separation of forward and backward waves due to the Coriolis force; with increase in the circumferential wave number, the frequency shows a trend of first decreasing and then increasing. The coupling effect of the magnetic induction intensity, rotational speed, power–law index, and thickness-to-diameter ratio leads to the nonlinear trend of frequencies. In addition, the influence of different parameter variations on the traveling wave stability is discussed.
... The bending behavior of advanced functionally graded (AFG) plates, considering various material parameters' effect was investigated using an integral plate model by Al-Osta et al. 26 In the case of optimization methods for composite material plates and shells, the discrete material optimization for composite plate and shell structures with complex geometry and material properties was evaluated by Ma et al. 27 Maalawi 28 conducted the optimization of dynamic properties of FG closed structures with different composites. Hamzehkolaei et al. 29 used the PSO approach to optimize the temperature-dependent vibration characteristics of rotating composite shells. Hammou et al. 30 used the genetic algorithm to maximize the frequency of CNTenhanced FG shells with different geometrical configurations. ...
In this study, an improved multiobjective optimization approach is proposed for optimal structural-vibrational design solution of bi-directional thermal-dependent functionally graded cylindrical shells (FGCS). The first-order shear deformation theory is first applied for the dynamic modeling, and the eigenvalue equations are obtained by the Rayleigh–Ritz method. The obtained results are compared with the literature to verify the accuracy of the analytical method. On this basis, characterization studies are carried out to obtain the important influencing parameters and their value ranges. Subsequently, the numerical experiments are carried out using Design Expert to obtain multivariate polynomials for several objective functions along with the important design variables (fundamental frequency, mass and cost). In order to avoid settling into local optima, a hybrid PSO-GWO technique is used to search for Pareto optimum solutions in space. The results show that the bi-directional graded index ranging from 0 to 1 significantly influences the fundamental frequency of the system. The graded indexes and material distribution are identified as key design variables. The ideal bi-directional graded index and material distribution for functionally graded ceramic shells (FGCS) in a thermal environment are determined. The optimization process aims to balance the conflicting objectives of maximizing frequency while simultaneously minimizing both mass and cost. By addressing these trade-offs, the final design achieves an effective compromise that meets the desired performance goals and economic constraints.
... The aleatory uncertainty is used to characterize the system inherent variation, and is usually quantified as random variable or stochastic process based on the sufficient sample information. Up to now, a lot of probabilistic reliability analysis and optimization methods have been well investigated and successfully applied in many engineering problems [6][7][8][9]. Using the random information * Corresponding author. ...
In aerospace engineering, the incomplete environment conditions have been realized to be a significant factor affecting the system safety assessment. Considering the hybrid random and fuzzy uncertainties in system inputs, the paper proposes a novel numerical procedure for reliability analysis and reliability-based optimization. Random parameters are adopted to denote the aleatory uncertainties with sufficient sample information; whereas fuzzy parameters are used to quantify the epistemic uncertainties associated with expert opinions. Using the level-cut operation, fuzzy parameters are converted into interval variables, and a satisfaction degree-based interval ranking strategy is utilized to precisely quantify the interval safety possibility. Then the system safety possibility is calculated by the multiple integral, where cut levels of different fuzzy parameters are treated as independent variables. Subsequently based on the given safety index, a hybrid reliability optimization model is established. To avoid the huge computational burden caused by nested-loop optimization, a modified interval Monte Carlo method (MIMC) is proposed for limit state function evaluation. Eventually, a numerical example about the refractory ceramics tile of spacecraft verifies the feasibility of proposed method for hybrid reliability analysis and optimization design in practical aerospace engineering.
... However, there are various uncertain factors in most practical engineering applications, such as input loads, structure sizes, material properties and manufacturing tolerances. To decrease the influence of uncertainties and obtain the optimum structure with the minimum failure probability, reliability analysis methods, such as the first order reliability method (FORM) (Low and Tang 2007;Cho 2013;Hu and Du 2015;Pedroso 2017), the second order reliability method (SORM) (Zhang and Du 2010;Chan and Low 2012;Okasha 2016) and the Monte Carlo method (Cardoso et al. 2008;Kostić, Vasović, and Marinković 2017;Stern, Song, and Work 2017) have been developed and widely applied in structural reliability analysis (Jiang et al. 2011), thermal reliability analysis (Hamzehkolaei, Miri, and Rashki 2017), time-dependent reliability analysis (Jiang, Huang, et al. 2014;Zhang et al. 2017), etc. ...
There are differences among sampling data and representation types of uncertain statistical variables, sparse variables and interval variables, which increase the complexity of structure reliability analysis. Therefore, a hybrid first order reliability analysis method considering the three types of uncertain variables is demonstrated in this article. First, distribution types and distribution parameters of sparse variables are identified and probabilistically estimated. Secondly, interval variables are transformed into probabilistic types using a uniformity approach. Thirdly, a unified hybrid reliability calculation method considering these uncertain variables simultaneously is demonstrated. The most probable point (MPP) is searched for using the first order reliability method, and then a linear approximation function of performance function is constructed in the neighbourhood of the MPP. Finally, the belief and plausibility measures of the reliability index are efficiently calculated using the theoretical analytical method. Three examples are investigated to demonstrate the effectiveness of the proposed method.
High temperatures in a gas turbine may lead to severe blade rubbing failure for the bladed thin drum rotor. It is essential to demonstrate such rubbing features. This paper established a bladed drum rotor model with blade rubbing induced by high temperatures. The analytical function of a coupled axial-radial temperature in the drum according to the actual thermal field analysis is obtained. The equations of motion for this rotor are derived. The dynamic model and its solution method are verified through the natural frequency comparison and the rub-impact response analysis. Thereafter numerical simulations are carried out. Results show that the heat at the turbine is transferred from its outer surface to its inner surface, then to the compressor’s inner surface along the axial direction, and finally from the compressor’s inner surface to its outer surface. This is a novel coupled axial-radial thermal effect for the gas turbine, which causes special axial and radial thermal gradients. The effect is induced by high temperatures in a gas turbine and intensifies a blade rubbing fault. Increasing the exhaust temperature aggravates the coupled axial-radial thermal effect, which causes more severe blade rubbing. Fortunately, introducing a lower temperature on the drum’s inner surface can prevent blade rubbing caused by this thermal effect.
By leveraging the capabilities of machine intelligence, metaheuristics can be enhanced to achieve improved performance and convergence. This chapter presents an investigation into the dynamic scenarios for utilizing machine intelligence in metaheuristics, focusing on their application to optimization problems. The main contribution of this research lies in the proposal and evaluation of five distinct methods within the dynamic intelligence (DI) scenarios for optimization using machine learning (ML) and cross-entropy (CE) framework. These methods include ML cooperation in full or fixed number of iterations, adaptive usage, performance-based decision-making, and randomized intelligence usage. Through extensive experimentation, the DICE framework is applied to solve various benchmark truss optimization problems involving frequency constraints and size/shape variables. Prototypes such as the 10-bar planner truss, 72-bar space truss, 120-bar truss dome, 37-bar truss bridge, and 52-bar truss dome are considered. Comparative evaluations against competing techniques are conducted to assess the effectiveness and efficiency of the proposed DICE methods. The findings indicate the efficiency of the proposed methods in fast and optimum convergence.
This study proposes an intelligent hybridization between the multi-level cross-entropy optimizer (MCEO) and moth-flame optimization (MFO) algorithms to keep a good compromise between exploration and exploitation. The proposed hybrid multi-level cross-entropy-based moth-flame (MCMF) algorithm uses MCEO as a global search engine during the first phase of the optimization process that allows for fast approximation of the global best position (BP). The boundaries of the search space are then adaptively confined within the effective region around the current BP by applying the proposed search space boundaries confining factor (SSBCF). The modified MFO with two different moth generation patterns is then employed as a local search engine to simultaneously probe for new and proper BPs within the confined and overall search space. This prevents MCMF from becoming trapped in local optima while maintaining a balance between exploration and exploitation. The hybridization between both algorithms allows MCMF to accelerate throughout the early steps of the search process using the high exploration ability of MCEO, whereas, in the later stages of optimization, promising solutions will possess a high probability to be exploited using the higher exploitation power of MFO in the confined space. The competence of the MCMF is compared with other well-known state-of-the-art algorithms on 15 unconstrained benchmark functions and 5 constrained engineering design problems having a wide range of dimensions and varied complexities. The statistical results on the benchmark functions and the solved engineering examples verify that the proposed algorithm can provide very competitive and promising results. The results demonstrate the comprehensive superiority of the hybrid MCMF compared to both MCEO and MFO in terms of fewer function calls, high escaping ability from local optima, and fast convergence speed.
This paper develops a non-probabilistic system reliability-based design optimization method for composite laminated structures under mechanical - thermal coupling environments. This method improves the traditional method of roughly considering the temperature effect through the humid-thermal amplification factor and precisely considers the thermal stress and temperature effects. In addition, a system reliability analysis method and structural failure criterion for composite laminated structures based on the theory of last-ply failure (LPF) are proposed. The method can calculate the system reliability of composite laminated structures by considering the correlation of multiple failure modes. Finally, non-probabilistic system reliability-based design optimization (NSRBDO) of composite laminated structure system in a thermal-mechanical coupling environment is completed by means of the two-level optimization method. After the proposed optimization strategy is described in detail, two examples of the laminated structure are illustrated, and the results are compared to the traditional design method using the humid-thermal amplification factor, demonstrating the advantages and validity of the proposed method.
Uncertainty propagation analysis is a key issue for the safety and optimal design of spacecraft structures. This paper proposes an interval reduced basis method (IRBM) for predicting static response of spacecraft structures with unknown-but-bounded uncertainties. Meanwhile, an integrated framework on the basis of IRBM is established by coupling the current finite element codes, which lays the foundation for development of an easy-to-use interval finite element software. In this paper, the novel idea is to approximate the static response using a linear combination of interval basis vectors with undetermined coefficients. The reduced order equations for computing the undetermined coefficients are derived based on Galerkin scheme. For the second-order approximation in IRBM, the static response is easily converted into a family of quadratic programming problems subject to box constraints. The difference of convex functions algorithm is employed to solve these quadratic programming problems. The proposed method can improve the accuracy of the traditional first-order approximation and perform the efficient computation of the second-order approximation of the structural response. Two numerical examples are presented to show the good performance of the proposed method. And an engineering example is performed for its application.
Purpose
Reliability-based design optimizations (RBDO) of engineering structures involve complex non-linear/non-differentiable performance functions including both continues and discrete variables. The gradient-based RBDO algorithms are less than satisfactory for these cases. The simulation-based approaches could also be computationally inefficient, especially when the double-loop strategy be employed. In the present study, a pseudo-double loop flexible RBDO is presented that is efficient for the problems including both discrete/continuous variables.
Design/methodology/approach
The method is based on the hybrid improved binary bat algorithm (BBA) and weighed simulation method (WSM). According to this method, each BBA’s movement generates proper candidate solutions and, subsequently, WSM evaluates the reliability levels for design candidates to conduct the swarm in low-cost safe-region.
Findings
The accuracy of the proposed enhanced BBA and also the hybrid WSM-BBA are examined for 10 benchmark deterministic optimizations and also 4 RBDO problems of truss structures, respectively. The solved examples reveal computational efficiency and superiority of the method to conventional RBDO approaches for complex problems including discrete variables.
Originality/value
Unlike other RBDO approaches, proposed method is such organized that only one simulation run suffices during optimization process. The flexibility future of the proposed RBDO framework enables designer to present multi-level design solutions for different arrangements of the problem by using the results of the only one simulation for WSM that is very helpful to decrease computational burden of the RBDO. In addition, a new suitable transfer function is introduced that enhanced convergence rate and search ability of the original BBA.
To make multi-objective reliability-based design optimization (MORBDO) more effective for complex structure with multi-failure modes and multi-physics coupling, multiple response surface method (MRSM)-based artificial neural network (ANN) (ANN-MRSM) and dynamic multi-objective particle swarm optimization (DMOPSO) algorithm are proposed based on MRSM and MOPSO algorithm. The mathematical model of ANN-MRSM is established by using artificial neural network to fit the multiple response surface function. The DMOPSO algorithm is proposed by designing dynamic inertia weight and dynamic learning factors. The proposed approach is verified by the MORBDO of turbine blisk deformation and stress with respect to fluid-thermal-structure interaction from probabilistic analysis. The optimization design results show that the proposed approach has the promising potentials to improve computational efficiency with acceptable computational precision for the MORBDO of turbine blisk deformation and stress. Moreover, Pareto front curve and a set of viable design values of turbine blisk are obtained for the high-reliability high-performance design of turbine blisk. The presented efforts provide an effective approach for MORBDO of complex structures, and enrich mechanical reliability design theory as well.
The main aim of the present paper is to solve numerically the free vibration problem of sandwich shell structures with variable thickness and made of Functionally Graded Materials (FGMs). Several Higher-order Shear Deformation Theories (HSDTs), defined by a unified formulation, are employed in the study. The FGM structures are characterized by variable mechanical properties due to the through-the-thickness variation of the volume fraction distribution of the two constituents and the arbitrary thickness profile. A four-parameter power law expression is introduced to describe the FGMs, whereas general relations are used to define the thickness variation, which can affect both the principal coordinates of the shell reference domain. A local scheme of the Generalized Differential Quadrature (GDQ) method is employed as numerical tool. The natural frequencies are obtained varying the exponent of the volume fraction distributions using higher-order theories based on a unified formulation. The structural models considered are two-dimensional and require less degrees of freedom when compared to the corresponding three-dimensional finite element (FE) models, which require a huge number of elements to describe the same geometries accurately. A comparison of the present results with the FE solutions is carried out for the isotropic cases only, whereas the numerical results available in the literature are used to prove the validity as well as accuracy of the current approach in dealing with FGM structures characterized by a variable thickness profile.
For tailoring the non-uniform axial compression, each sub-panel of stiffened shells should be designed separately to achieve a high load-carrying efficiency. Motivated by the challenge caused by numerous variables and high computational cost, a fast procedure for the minimum weight design of non-uniform stiffened shells under buckling constraint is proposed, which decomposes a hyper multi-dimensional problem into a hierarchical optimization with two levels. To facilitate the post-buckling optimization, an efficient equivalent analysis model of stiffened shells is developed based on the Numerical Implementation of Asymptotic Homogenization Method. In particular, the effects of non-uniform load, internal pressure and geometric imperfections are taken into account during the optimization. Finally, a typical fuel tank of launch vehicle is utilized to demonstrate the effectiveness of the proposed procedure, and detailed comparison with other optimization methodologies is made.
There are available in the literature several papers on the development of methods to decouple the reliability analysis and the structural optimization to solve RBDO problems. Most of them focused on strategies that employ the First Order Reliability Method (FORM) to approximate the reliability constraints. Despite of all these developments, one limitation prevailed: the lack of accuracy in the approximation of the reliability constraints due to the use of FORM. Thus, in this paper, a novel approach for RBDO is presented in order to overcome such a limitation. In this approach, we use the concept of shifting vectors, originally developed in the context of the Sequential Optimization and Reliability Assessment (SORA). However, the shifting vectors are found and updated based on a novel strategy. The resulting framework is able to use any technique for the reliability analysis stage, such as Monte Carlo simulation, second order reliability methods, stochastic polynomials, among others. Thus, the proposed approach overcomes the aforementioned limitation of most of RBDO decoupling techniques, which required the use of FORM for reliability analysis. Several examples are analyzed in order to show the effectiveness of the methodology. Focus is given on examples that are poorly solved or even cannot be tackled by FORM based approaches, such as highly nonlinear limit state functions comprised by a maximum operator or problems with discrete random variables. It should be remarked that the proposed approach was not developed to be more computationally efficient than RBDO decoupling strategies based FORM, but to allow the utilization of any, including more accurate, reliability analysis method.
The enhanced weighted simulation-based design method in conjunction with particle swarm ptimization (PSO) is developed as a pseudo double-loop algorithm for accurate reliability-based design ptimization (RBDO). According to this hybrid method, generated samples of weighed simulation method (WSM) are considered as initial population of the PSO. The proposed population is then employed to valuate the safety level of each PSO swarm (design candidates) during movement. Using this strategy, there is no required to conduct new sampling for reliability assessment of design candidates (PSO swarms). Employing PSO as the search engine of RBDO and WSM as the reliability analyzer provide more accurate results with few samples and also increase the application range of traditional WSM. Besides, a shift strategy is also intro-
duced to increase the capability of the WSM to investigate general RBDO problems including both deterministic and random design variables. Several examples are investigated to demonstrate the accuracy and robustness of the method. Results demonstrate the computational efficiency and superiority of the proposed method for practical engineering problems with highly nonlinear and implicit probabilistic
constrains.
A robust design optimization (RDO) approach for minimum weight and safe shell composite structures with minimal variability into design constraints under uncertainties is proposed. A new concept of feasibility robustness associated to the variability of design constraints is considered. So, the feasibility robustness is defined through the determinant of variance–covariance matrix of constraint functions introducing in this way the joint effects of the uncertainty propagations on structural response. A new framework considering aleatory uncertainty into RDO of composite structures is proposed. So, three classes of variables and parameters are identified: deterministic design variables, random design variables and random parameters. The bi-objective optimization search is performed using on a new approach based on two levels of dominance denoted by Co-Dominance-based Genetic Algorithm (CoDGA). The use of evolutionary concepts together sensitivity analysis based on adjoint variable method is a new proposal. The examples with different sources of uncertainty show that the Pareto front definition depends on random design variables and/or random parameters considered in RDO. Furthermore, the importance to control the uncertainties on the feasibility of constraints is demonstrated. CoDGA approach is a powerfully tool to help designers to make decision establishing the priorities between performance and robustness.
The Imperialist Competitive Algorithm (ICA) that was recently introduced has shown its good performance in optimization problems. In this paper, a novel optimization algorithm based on ICA is used for optimizing the generalized formulation of critical buckling load on plates made of Functionally Graded Materials (FGMs) with variation in thickness. The formulation, which is applicable to thin Functionally Graded (FG) plates, can also be used to analyze and optimize FGMs in each layer of the plates. At last, the results of ICA are compared with those obtained from Genetic Algorithm (GA). The outcome shows the success of ICA in optimizing the buckling load of FG plates.
Reliability-based design optimization (RBDO) is a powerful tool for design optimization when considering the probabilistic characteristics of the design variables. However, its use in practical applications is hindered by the huge computational cost during the evaluation of structure reliability. A number of algorithms have been proposed, such as reliability index approach (RIA), performance measure approach (PMA), decoupled approach and single-loop approach. In this paper, the adaptive decoupling approach (ADA) is developed. It adopts a new update angle strategy and a novel feasibility-checking method to improve the efficiency of probabilistic optimization. The update angle is utilized to reduce the number of performance function evaluations. And the probabilistic constraints are divided into feasible, active and violated categories, and only violated and active probabilistic constraints are evaluated precisely. The computation capability of ADA is demonstrated using five examples which are widely used to test RBDO methods. The comparison results show that ADA has the similar accuracy as existing methods, and it is also very efficient.
The reliability based optimization (RBO) issue of composite laminates under fundamental frequency constraint is studied. Consid-ering the uncertainties of material properties, the frequency constraint reliability of the structure is evaluated by the combination of re-sponse surface method (RSM) and finite element method. An optimization algorithm is developed based on the mechanism of laminate frequency characteristics, to optimize the laminate in terms of the ply amount and orientation angles. Numerical examples of composite laminates and cylindrical shell illustrate the advantages of the present optimization algorithm on the efficiency and applicability respects. The optimal solutions of RBO are obviously different from the deterministic optimization results, and the necessity of considering mate-rial property uncertainties in the composite structural frequency constraint optimization is revealed.
In this study, an efficient and robust algorithm of non-probabilistic reliability-based design optimization (NRBDO) is proposed based on convex model. In this double-nested optimization model, the inner loop concerns a Min–max problem for the evaluation of reliability index, where the target performance approach is applied to substitute the Min–max problem. To improve the convergence rate, an enhanced chaos control (ECC) method is developed on the basis of chaotic dynamics theory, which can check and re-update the control factor by the Wolfe–Powell criterion. To further enhance the optimization efficiency, a novel NRBDO algorithm is developed based on the proposed ECC, where HL–RF algorithm is applied at the initial stage of this algorithm, while ECC is used to improve the robustness once the oscillation or chaotic behavior is identified. Three mathematical examples, one numerical example and one complex engineering problem, i.e. axially compressed stiffened shells in launch vehicles, are utilized to demonstrate the robustness and efficiency of the proposed method by comparison with other existing methods. Results indicate that the proposed method is particularly suitable for complicated engineering problems without prior knowledge of uncertainty distributions, which is expected to be utilized in the structural design of future launch vehicles.
In this article, free vibration of the hard-coating cantilever cylindrical shell is investigated considering the elastic constraints at the clamped end. Love's first approximation theory and Rayleigh–Ritz method are applied to build the analytical model of hard-coating cylindrical shell. In the modeling process, orthogonal polynomials are used as admissible displacement functions to formulate the displacement field, and the elastic constraints are simulated by constrained springs whose stiffness values are determined using model updating technique. The developed model has been validated by the comparison between the natural frequencies obtained by analytical calculation and by experiment respectively. Finally, the influences of hard-coating parameters, including thickness, Young's modulus and loss factor, on the vibration characteristics of the cylindrical shell are studied.
This article is devoted to the study of the optimal design of fibers orientation in a composite specimen with the objective to minimize the displacement. The composite specimen considered is within the scope of aerospace and mechanical applications. The objective function associated with the composite design is computed based on a static analysis of a finite element solid model, which allows one to define (or control) the fibers orientation. The recent global and local optimization using direct search methods (GLODS) is used for the optimization process. To validate and compare the numerical and optimized results, the specimens were manufactured and tested experimentally. The orientation of the layers that minimize the maximum displacement is achieved through the computational interaction of the optimization program, GLODS, in loop with the finite element program, ANSYS. It is shown that the optimized lamination schemes found by GLODS minimized about 60% of the displacement compared to the nonoptimized specimens.
We present a computational framework for robust and reliability based design optimization which combines stochastic expansion methods, namely polynomial chaos expansion, with design sensitivity analysis. It is well known that the statistical moments and their gradients with respect to design variables can be readily obtained from the polynomial chaos expansion. However, the evaluation of the failure probabilities of the cost and constraint functions and their gradients, requires integrations over failure regions. To simplify this we introduce an indicator function into the integrand, whereby the integration region becomes the known range of random variables and to alleviate the non-differentiable property of the indicator function, a smooth approximation is adopted to facilitate the sensitivity analysis. Both intrusive and non-intrusive polynomial chaos approaches for uncertainty propagation are employed in the design optimization of linear elastic structures. Guidelines to assess the computational costs associated with both polynomial chaos approaches are also presented.
An efficient optimization framework of cylindrical stiffened shells with reinforced cutouts by curvilinear stiffeners is proposed in this study. First, an adaptive method to determine the near field around the cutout and far field away from the cutout is presented. Then, a novel hybrid model is established to reduce the computational efforts of postbuckling analysis; the numerical implementation asymptotic homogenization method is used to smear out the stiffeners in the far field, and curvilinear stiffeners are adopted to improve the loading path and thus local stiffness of the near field, which can provide a type of flexible stiffener configurations for cutout reinforcement. After that, the optimization of curvilinear stiffeners is performed by a novel bilevel strategy based on the hybrid model. In the first level, a stiffener distribution function is used to reduce the number of active variables, and then stiffener layout, stiffener number, and section profile are optimized simultaneously. In the second level, the stiffener number and section profile are held constant, and local optimization is then performed for each curvilinear stiffener location. An illustrative example demonstrates the effectiveness of the proposed framework, when compared with traditional optimizations.
The paper proposes a comparison between a three-dimensional (3D) exact solution and several two-dimensional (2D) numerical solutions. Numerical methods include classical 2D finite elements (FEs), and classical and refined 2D generalized differential quadrature (GDQ) solutions. The free vibration analysis of two different configurations of functionally graded material (FGM) plates and cylinders is proposed. The first configuration considers a one-layered FGM structure. The second one is a sandwich configuration with external classical skins and an internal FGM core. Low and high order frequencies are analyzed for thick and thin simply supported structures. Vibration modes are investigated to make a comparison between results obtained via the 2D numerical methods and those obtained by means of the 3D exact solution. The 3D exact solution is based on the differential equations of equilibrium written in general orthogonal curvilinear coordinates. This exact method is based on a layer-wise approach where the continuity of displacements and transverse shear/normal stresses is imposed at the interfaces between the layers of the structure. The 2D finite element results are obtained by means of a well-known commercial FE code. Classical and refined 2D GDQ models are based on a generalized unified approach which considers both equivalent single layer and layer-wise theories. The differences between 2D numerical solutions and 3D exact solutions depend on the considered mode, the order of frequency, the thickness ratio of the structure, the geometry, the embedded material and the lamination sequence.
Differential evolution optimization is used to find the volume fraction that maximizes the first natural frequency for a functionally graded beam. A formulation using three parameters is used to describe volume fraction. Beams with different ratios of material properties are considered. Two methods are used to compute the natural frequencies, analytical and meshless numerical method. Results show that differential evolution is capable to find distributions for volume fraction that increase the natural frequency of beams. It was also found that the RBF numerical method can be used with differential evolution to solve problems related to maximization of natural frequencies in functionally graded beams.
Discrete layer approach coupled with the differential quadrature method (DQM) is employed to temperature dependent analyze the laminated functionally graded annular plates under mechanical loading in thermal environment. The formulations are derived based on the elasticity theory, which includes the effects of the initial thermal stresses and two-parameter elastic foundation. The material properties are assumed to be temperature-dependent and graded in the thickness direction. In order to accurately evaluate the effect of thermal environment, the initial thermal stresses are obtained by solving the thermoelastic equilibrium equation. Comparison studies with the available solutions in the literature for FG plates are performed. Then as an application, three common types of FG sandwich plates, namely, the sandwich with homogeneous face sheets and FG core and the sandwich with FG face sheets and homogeneous metal (soft) and ceramic (hard) core are analyzed. The influences of temperature rise, temperature-dependence of material properties, layers lay-out, foundation stiffness parameters, material graded index and geometrical parameters on the solution are carried out. The new results can be used as benchmark solutions for future researches.
Variations of manufacturing process parameters, environment aspects, and imperfections may significantly affect the quality and performance of stiffened shells. The reliability-based design optimization (RBDO) of stiffened shells, considering all these uncertainty factors simultaneously, is extremely time-consuming, even if the surrogate-based technology is used. Therefore, a hybrid bi-stage framework for RBDO of stiffened shells is presented to release the computational burden, where two main sources of uncertainties are considered: variations of material properties and geometric dimensions are described as random variables, while various forms of imperfections of stiffened shells are covered by the single perturbation load approach. The basic idea of the proposed method is to combine the efficiency of smeared stiffener method with the accuracy of finite element method, and then narrow the design window efficiently with little accuracy sacrifice. The adaptive chaos control method is used to ensure the robustness of the search process of the most probable target point. The numerical example illustrates the advantage of the proposed method over other RBDO approaches from the point of view of computational cost, accuracy and robustness of the result.
As a first endeavor, the differential quadrature method in conjunction with the genetic algorithms (GAs) is applied to obtain the optimum (maximum) buckling temperature of laminated composite skew plates. The material properties are assumed to be temperature dependent and the governing equations are based on the first-order shear deformation plate theory. After discretizing the governing equations and the related boundary conditions, a direct iterative method in conjunction with GAs is used to determine the optimum fiber orientation for the maximum buckling temperature. The applicability, rapid rate of convergence, and high accuracy of the method are established by solving various examples and by comparing the results with those in the existing literature. Then, the effects of the temperature dependence of the material properties, boundary conditions, length-to-thickness ratio, number of layers, and skew angle on the maximum buckling temperature of the laminated skew plates are presented.
In the present study, a rotating functionally graded cylindrical shell (FGM) with imperfectly surface bounded functionally graded piezoelectric material (FGPM) subjected to an axisymmetric hygrothermo-electro-mechanical loading is considered. The shell is simply supported and could be rested on an elastic foundation. The material properties of FGM and FGPM are assumed to be exponentially graded in the radial direction. The Fourier series expansion method through the longitudinal direction and the differential quadrature method (DQM) across the thickness direction are used for solving governing differential equations. To check the validity of the present work, comparisons with the previous results are performed. Finally, numerical results are shown to clarify the effects of important parameters on the behavior of the smart shell.
Functionally graded materials (FGMs) have been proposed to be potential structural materials applied in high-speed spacecraft and power generation industries. In this study, an interface shape optimization method for designing FG sandwich structures with two different materials is proposed to minimize the compliance of FG sandwich structures under the volume constraint. Using the material derivative and adjoint methods, the shape gradient function is derived to determine the optimized interface shapes between different materials in FG sandwich structures without requiring shape design parameterization. FG sandwich structures with two sets of metal-ceramic materials, aluminum-alumina and aluminum-zirconia, are given as examples to verify the validity of the proposed optimization method, respectively. The results show that the compliance of FG sandwich structures with large difference between Young’s modulus of the component materials can be significantly reduced after optimized by the proposed method.
This paper presents an analytical approach to investigate the nonlinear dynamic response and vibration of imperfect eccentrically stiffened functionally graded thick circular cylindrical shells surrounded on elastic foundations using both the first order shear deformation theory and stress function with full motion equations (not using Volmir's assumptions). Material properties are graded in the thickness direction according to a sigmoid power law distribution (S-FGM) in terms of the volume fractions of constituents with metal–ceramic–metal layers. The S-FGM shells are subjected to mechanical and damping loads. Numerical results for dynamic response of the shells are obtained by Runge–Kutta method. The results show the influences of geometrical parameters, the volume fractions of metal–ceramic–metal layers, imperfections, the elastic foundations, eccentrically stiffeners, pre-loaded axial compression and damping loads on the nonlinear dynamic response and nonlinear vibration of functionally graded cylindrical shells. The proposed results are validated by comparing with other results reported in the literature.
A survey of several methods under the heading of SFEM (Strong Formulation Finite Element Method) is presented. These approaches are distinguished from classical termed Weak Formulation Finite Element Method (WFEM). The main advantage of the SFEM is that it uses differential quadrature method (DQM) for the discretization of the equations and the mapping technique for the coordinate transformation from the Cartesian to the computational domain. Moreover the element connectivity is performed by using kinematic and static conditions so that displacements and stresses are continuous across the element boundaries. Numerical investigations integrate this survey by giving graphical details on the subject.
We investigate recovery of through-the-thickness transverse normal and shear strains and stresses in statically deformed functionally graded (FG) doubly-curved sandwich shell structures and shells of revolution using the generalized zigzag displacement field and the Carrera Unified Formulation (CUF). Three different through-the-thickness distributions of the volume fractions of constituents and two different homogenization techniques are employed to deduce the effective moduli of linear elastic isotropic materials. The system of partial differential equations for different Higher-order Shear Deformation Theories (HSDTs) is numerically solved by using the Generalized Differential Quadrature (GDQ) method. Either the face sheets or the core is assumed to be made of a FGM. The through-the-thickness stress profiles are recovered by integrating along the thickness direction the 3-dimensional (3-D) equilibrium equations written in terms of stresses. The stresses are used to find the strains by using Hooke’s law. The computed displacements and the recovered through-the-thickness stresses and strains are found to compare well with those obtained by analyzing the corresponding 3-D problems with the finite element method and a commercial code. The stresses for the FG structures are found to be in-between those for the homogeneous structures made of the two constituents of the FGM.
Reliability-based design optimization (RBDO) is a topic of interest in the design of economical structures. It allows designers to effectively reach a balanced cost-safety configuration in the design of structures. In this study, a simulation-based method is presented for RBDO problems in which the design variables are treated as random variables. The method works by uniformly distributing samples in the design space and employing a feature that allows the designer to obtain the optimum design solution by performing only one simulation run. Moreover, the proposed feature also helps the designer to use the results of aforementioned run to provide multi-level design solutions when the arrangement of the design problem is changed. The robustness and accuracy of the method are examined by solving design problems with highly nonlinear constraints and comparing with the results of common RBDO methods. The results confirm the robustness of the method for highly nonlinear problems with different design arrangements.
This paper presents the effects of thermal environment and temperature-dependence of the material properties on axisymmetric bending of functionally graded (FG) circular and annular plates. The material properties are assumed to be temperature-dependent and graded in the thickness direction. In order to accurately evaluate the effect of thermal environment, the initial thermal stresses are obtained by solving the thermoelastic equilibrium equations. Governing equations and the related boundary conditions, which include the effects of initial thermal stresses, are derived using the virtual work principle based on the elasticity theory. The differential quadrature method (DQM) as an efficient and robust numerical tool is used to obtain the initial thermal stresses and response of the plate. Comparison studies with some available results for FG plates are performed. The influences of temperature rise, temperature-dependence of material properties, material graded index and different geometrical parameters are carried out.
This paper deals with optimisation of three-parameter power-law distribution of functionally graded (FG) beam. The main goal of the optimisation problem is to determine the optimum volume fraction relation for maximising the first natural frequency of FG beam. Since the search space is large, the optimisation processes become very complicated and too time consuming. Thus, a novel meta-heuristic called imperialist competitive algorithm (ICA), which is a socio-politically motivated global search strategy is applied to find the optimal solution. Applying the proposed algorithm to some of benchmark cost functions, it shows its ability in dealing with different types of optimisation problems. A proper and accurate artificial neural network (ANN) is trained by training data sets obtained from generalised differential quadrature method and then is applied as the objective function in ICA. The ANN improves the speed of optimisation process by a considerable amount by reproducing the fundamental natural frequency of the structure. The performance of ICA is evaluated in comparison with other nature-inspired technique genetic algorithm. Comparison shows the success of combination of ANN and ICA for design of material profile of beam. Finally the optimised material profile for the optimisation problem is presented.
In this study, the stability reliability of the piezoelectric delaminated cylindrical shell with random variables is investigated. By introducing the Heaviside step function into the displacement components and using the Rayleigh–Ritz method, the buckling governing equations for the piezoelectric delaminated cylindrical shells is obtained. The least-square support vector machine is adopted to predict the expression of the safety margin of the piezoelectric delaminated cylindrical shell. The probability of failure is computed by Monte Carlo simulation. In numerical examples, the effects of voltage, load, delamination, and geometrical parameters on the probability of failure of piezoelectric delaminated cylindrical shells are discussed in detail.
A new optimization strategy for eigenfrequency optimization of functionally graded (FG) structures within the framework of isogeometric analysis (IGA) approach is introduced. The proposed methodology which utilizes a concurrent procedure by combining the shape and material composition optimization of these structures employs an extended form of the standard IGA method by allowing for gradation of material properties through patches. The distribution of the graded material properties are considered as imaginary surfaces over the computational domain and captured in a fully isogeometric formulation using the same NURBS-based parameterization which is employed for the geometry modeling as well as the solution approximation. Considering the in-plane coordinates of the control points defining the design boundary surfaces as well as the applicates of all the control points describing the variations of material properties as design variables, we subsequently adopt a mathematical programming algorithm to simultaneously find the optimum shape and material composition of FG structures. A couple of illustrative numerical examples in 2D elasticity with eigenfrequencies as their either constraints or objective function are presented to demonstrate the high performance of the proposed methodology. It will be seen that the obtained results by this concurrent optimization procedure have much better dynamic performance compared to the optimal results of the simple shape or material composition design.
Functionally graded materials are a type of composite materials which are tailored to provide continuously varying properties, according to specific constituentʼs mixing distributions. These materials are known to provide superior thermal and mechanical performances when compared to the traditional laminated composites, because of this continuous properties variation characteristic, which enables among other advantages, smoother stresses distribution profiles. Therefore the growing trend on the use of these materials brings together the interest and the need for getting optimum configurations concerning to each specific application.
In this work it is studied the use of particle swarm optimization technique for the maximization of a functionally graded sandwich beam bending stiffness. For this purpose, a set of case studies is analyzed, in order to enable to understand in a detailed way, how the different optimization parameters tuning can influence the whole process.
It is also considered a re-initialization strategy, which is not a common approach in particle swarm optimization as far as it was possible to conclude from the published research works. As it will be shown, this strategy can provide good results and also present some advantages in some conditions.
This work was developed and programmed on symbolic computation platform Maple 14.
For cylindrical stiffened shells under internal pressure and nonuniform axial compression, a load-controlled loading pattern based on the nonlinear explicit dynamic analysis is employed. Because of the nonuniform load, numerous variables are involved in structural design, whereas it is practically impossible to perform the optimization for the overall variables simultaneously, even if a surrogate model is adopted. The framework of a bistep surrogate-based optimization with adaptive sampling is presented in this study, which aims to build relatively high-fidelity surrogate models with less computational cost for complex engineering designs. A three-dimensional rigid frame is first established to validate the proposed method. Results indicate that the optimum design obtained by the proposed method is potentially close to the global optimum design. Then, a typical fuel tank configured for use in a launch vehicle is used to illustrate the proposed method. Comparisons of direct surrogate-based optimization and the proposed method for two examples show that the bistep surrogate-based optimization with adaptive sampling can achieve a larger weight saving with less computational cost. Consequently, it can be concluded that the proposed method provides an efficient and practical approach to finding the optimum design for structure with numerous variables and various variable types.
In this paper, Reliability-Based Multidisciplinary Design Optimization (RBMDO) of a two-stage solid propellant expendable launch vehicle (LV) is investigated. Propulsion, weight, aerodynamics (geometry) and trajectory (performance) disciplines are used in an appropriate combination. Throw weight minimization is chosen as objective function. Design variables for system level optimization are selected from propulsion, geometry and trajectory disciplines. Mission constraints contain the final velocity, the height above ground, and flight path angle. The constraints that appear during the flight are also considered. Assuming a normal distribution for the uncertain variables, Latin Hypercube Sampling (LHS) method selects the sample values for simulation runs which are eventually utilized for calculating probability density function of constraints and their reliability at each design point. Sequential Quadratic Programming (SQP) technique is used to achieve the optimal solution. Although the launch vehicle throw weight is increased negligibly in comparison with deterministic optimization, results show that the reliability-based method satisfied desired reliability of the constraints.
An accurate and efficient solution procedure based on the elasticity theory is employed to investigate the thermoelastic behavior of rotating laminated functionally graded (FG) cylindrical shells in thermal environment. The material properties are assumed to be temperature dependent and graded in the thickness direction. In order to accurately model the variation of the field variables across the thickness, the shell is divided into a set of mathematical layers. The differential quadrature method (DQM) is adopted to discretize the governing differential equations of each layer together with the related boundary and compatibility conditions at the interface of two adjacent layers. Using the DQM enables one to accurately and efficiently discretize the partial differential equations, especially along the graded direction, and also implement the boundary and compatibility conditions in their strong forms. After demonstrating the convergence and accuracy of the presented approach, the effects of material and geometrical parameters and also temperature dependence of material properties on the stresses and displacement components of rotating laminated FG cylindrical shells are studied.
Functionally graded material (FGM) has a continuous and functional distribution of volume fractions of constituent particles, which leads to superior thermo-mechanical performance to classical laminated composite materials. Since the thermo-mechanical characteristics of an FGM depend on the volume fraction distribution, it is important to tailor appropriate volume fraction distribution that satisfies the desired performance requirements under given loading and boundary conditions. Even though numerical optimization technique may serve as an excellent material tailoring tool, the capacity of current manufacturing techniques of FGM may not yield the target volume fraction. To deal with uncertainty in the manufacturing process, a reliability-based design optimization (RBDO) for FGM composite is proposed. In RBDO, a finite number of volume fractions of homogenized FGM layers and material properties are considered as random variables, with statistical information such as mean, standard deviation, and statistical distributions. Design of experiments and response surface models are used to obtain explicit forms of thermal stresses for RBDO formulation. It is observed through the numerical experiment that the RBDO finds the optimized volume fraction distribution with high reliability, such that the graded layers do not fail in the presence of manufacturing uncertainty.
A three-dimensional (3D) free vibration analysis of the functionally graded (FG) truncated conical shells subjected to thermal environment is presented. The material properties are assumed to be temperature-dependent and graded in the radius direction, which can vary according to a simple power law distribution. The initial thermal stresses are obtained accurately by solving the thermoelastic equilibrium equations and by considering the two-dimensional axisymmetric temperature distribution in the shell. The differential quadrature method (DQM) as an efficient and accurate numerical tool is adopted to solve the thermal and thermo-mechanical governing equations. For this purpose, a mapping technique is employed to transform the cross section of the shell into the computational domain of DQM. The convergence behavior of the method is numerically demonstrated and comparison studies with the available solutions in the literature are performed. The effects of temperature dependence of material properties, geometrical parameters, material graded index, thermal and mechanical boundary conditions on the frequency parameters of the FG truncated conical shells are carried out.
In this paper, a new simulation method for approximating the probability of failure and the most probable point of failure is proposed. The method works by generating uniformly distributed samples in a design space for all random variables and applying the probability density value as a weight index at any sample. The result is a simple approximation of the probability of failure without any limitation becoming feasible. The probability of failure is defined as the ratio of the sum of the weight indices in the failure domain over the sum of the indices in the entire domain. High accuracy in estimating small values of the probability of failure as well as the need for few samples are advantages of this method. Moreover, the possibility of estimating the point and the region with the highest failure probability for different types of limit state functions can be considered as another important advantage of the proposed method. The efficiency and robustness of the method are investigated by solving several examples. The results are compared with the results of common reliability methods, and they demonstrate the efficiency and robustness of the proposed method.
A three-dimensional (3D) discrete layer approach coupled with the differential quadrature method (DQM) is developed for the free vibration analysis of the laminated functionally graded (FG) annular plates subjected to thermal environment. The formulations are derived based on the elasticity theory, which includes the effects of the initial thermal stresses. After demonstrating the convergence behavior and accuracy of the method, two common types of FG sandwich plates, namely, the sandwich with homogeneous face sheets and FG core and the sandwich with FG face sheets and homogeneous core are analyzed. The influences of different parameters on the solution are carried out.
Composite materials, especially composite laminates, play a significant role in the modern industry. Laminate is a material
built by joining two materials and it usually consists of two phases: the matrix and the reinforcement. The laminate is typically
build of many plies (laminas) having different ply angles. Laminates are popular due to two main reasons: i) the high weight-strength
ratio (in comparison with the conventional materials); ii) the possibility to tailor the material properties to the designer
requirements by manipulating several parameters like: components material, stacking sequence, fibres orientation or layer
thickness [1]. If laminas are composed of the different materials the laminate is called a hybrid one.
Two aims of the present paper are: i) to identify the material constants of the laminates; ii) to find the optimum stacking
sequence of the laminates. The standard laminates as well as the hybrid ones are considered [2]. Different optimization criteria connected with the modal analysis and free vibrations are taken into account.
To solve global optimization tasks and discrete optimization tasks as well as to avoid difficulties with the objective function
gradient computation, the evolutionary algorithm (EA) is employed as the optimization procedure. To reduce the computation
time, the distributed version of the evolutionary algorithm is used [3].
The finite element method (FEM) professional software package with the laminate modeller is used to solve a direct eigenfrequency
problem for the laminate plates. The numerical examples presenting the efficiency of the proposed attitude are attached.
As it can be seen from the numerical examples, the proposed identification and optimization method gives positive results.
Consequently, this method can be applied to different laminated structures in order to identify the material constants or
to find the desired laminate properties for a given criteria.
We consider thermoelastic bodies composed of two-constituent functionally graded materials under steady-state conditions and address the problem of the optimal choice of composition profile. First, we formulate the problem as a partial-differential-equation constrained optimization problem, where the control function is the composition profile. The formulation includes the temperature-dependence of the constituents’ properties. Next, we derive the objective functional gradient using the continuous adjoint-field approach. Lastly, we use the gradient information into a gradient-based algorithm to optimize a thick-walled functionally graded sphere subjected to thermal gradients. For the numerical data we use, the optimal composition profile obtained is such that in the graded sphere the maximum von Mises stress, here used as a performance index, is about half that in the homogeneous sphere composed of either constituent.
The distributions of properties across the thickness (core) and in the plane (face sheets) that minimise the interlaminar stresses at the interface with the core are determined solving the Euler–Lagrange equations of an optimisation problem in which the membrane and transverse shear energy contributions are made stationary. The bending stiffness is maximised, while the energy due to interlaminar stresses is minimised. As structural model, a refined zig-zag model with a high-order variation of displacements is employed. Simplified, sub-optimal distributions obtainable with current manufacturing processes appear effective for reducing the critical interfacial stress concentration, as shown by the numerical applications.
This paper is a reconsideration and reformulation of the Mori-Tanaka's theory in its application to the computation of the effective properties of composites. Previous applications of the theory in this context continued to be linked with eigenstrain, equivalent inclusion, and back stress concepts, and many times involved energy considerations. In this paper we adopt the ‘direct approach’ of defining and computing effective moduli. By elucidating the nature of the approximation in applying Mori—Tanaka's theory to composites insofar as the ‘concentration-factor’ tensors are concerned, we achieve a straightforward exposition and interpretation of the method which are different than those existing in previous formulations. The analysis is given for two-phase composites with anisotropic elastic constituents and an inclusion phase consisting of aligned or randomly oriented ellipsoidal particles. The derived simple expressions for the predicted stiffness and compliance tensors permit a proof of the self-consistency of the method, a discussion of the predictions' relation to the Hashin-Shtrikman bounds in the case of isotropic constituents and randomly oriented ellipsoidal particles, and finally a derivation of some new results in randomly cracked bodies with penny-shaped cracks.
Natural frequencies and buckling stresses of shallow shells made of functionally graded materials (FGMs) are analyzed by taking into account the effects of transverse shear and normal deformations, and rotatory inertia. The modulus of elasticity of shells is assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional (2D) higher-order theory for rectangular functionally graded (FG) shallow shells is derived through Hamilton’s principle. Several sets of truncated approximate theories are applied to solve the eigenvalue problems of FG shallow shells with simply supported edges. Three types of simply supported shallow shells with positive, zero and negative Gaussian curvature are considered. In order to assure the accuracy of the present theory, convergence properties of the fundamental natural frequency and also buckling stress are examined in detail. Critical buckling stresses of FG shells subjected to in-plane stresses are also obtained and a relation between the buckling stress and natural frequency of simply supported FG shells without in-plane stresses is presented. The modal transverse stresses have been obtained by integrating the three-dimensional (3D) equations of motion in the thickness direction with satisfying the surface boundary conditions of a shell. The present numerical results are also verified by satisfying the energy balance of external and internal works are considered to be sufficient with respect to the accuracy of solutions. It is noticed that the present 2D higher-order approximate theories can predict accurately the natural frequencies and buckling stresses of simply supported FG shallow shells.
The analysis of thermoelastic problem of a rotating functionally graded hollow circular disk is made. The hollow disk is assumed to have varying material properties along the radial direction. An analytical method is presented to investigate steady thermal stresses in a functionally graded circular annulus rotating with constant angular velocity about its central axis. The associated boundary value problem is reduced to a Fredholm integral equation. The thermal stresses and radial displacement are obtained by numerically solving the resulting equation. A comparison of the numerical results with the exact ones for material properties of special power-law profile confirms the effectiveness of the method. For generally varying material parameters, numerical results are presented graphically to show the effects of gradient parameter, temperature change, angular velocity and thickness of the disk on the distribution of thermal stresses and radial displacement.
Volume fraction optimization of Functionally Graded Materials (FGMs) is investigated considering stress and critical temperature. Material properties are assumed to be temperature dependent, and are assumed to be varied continuously in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituent materials. The effective material properties are obtained by applying linear rule of mixtures. The 3-D finite element model is adopted using an 18-node solid element to analyze more accurately the variation of material properties and temperature field in the thickness direction. For the various FGMs volume fraction distributions, mechanical stress analysis and thermo-mechanical buckling analysis are performed to get the critical conditions. Finally, the optimal designs of FGMs panels are investigated for stress reduction and improving thermo-mechanical buckling behavior.
An efficient single-loop approach for series system reliability-based design optimization problems is presented in this paper. The approach enables the optimizer to apportion the system reliability among the failure modes in an optimal way by increasing the reliability of those failure modes whose reliability can be increased at low cost. Furthermore, it identifies the critical failure modes that contribute the most to the overall system reliability. A previously reported methodology uses a sequential optimization and reliability approach. It also uses a linear extrapolation to determine the coordinates of the most probable points of the failure modes as the design changes. As a result, the approach can be slow and may not converge if the location of the most probable failure point changes significantly. This paper proposes an alternative system RBDO approach that overcomes the above difficulties by using a single-loop approach where the searches for the optimum design and for the most probable failure points proceed simultaneously. An easy to implement active set strategy is used. The maximum allowable failure probabilities of the failure modes are considered as design variables. The efficiency and robustness of the method is demonstrated on three design examples involving a beam, an internal combustion engine and a vehicle side impact. The results are compared with deterministic optimization and the conventional component RBDO formulation.
We present a numerical approach for material optimization of metal-ceramic functionally graded materials (FGMs) with temperature-dependent material properties. We solve the non-linear heterogeneous thermoelasticity equations in 2D under plane strain conditions and consider examples in which the material composition varies along the radial direction of a hollow cylinder under thermomechanical loading. A space of shape-preserving splines is used to search for the optimal volume fraction function which minimizes stresses or minimizes mass under stress constraints. The control points (design variables) that define the volume fraction spline function are independent of the grid used in the numerical solution of the thermoelastic problem. We introduce new temperature-dependent objective functions and constraints. The rule of mixture and the modified Mori-Tanaka with the fuzzy inference scheme are used to compute effective properties for the material mixtures. The different micromechanics models lead to optimal solutions that are similar qualitatively. To compute the temperature-dependent critical stresses for the mixture, we use, for lack of experimental data, the rule-of-mixture. When a scalar stress measure is minimized, we obtain optimal volume fraction functions that feature multiple graded regions alternating with non-graded layers, or even non-monotonic profiles. The dominant factor for the existence of such local minimizers is the non-linear dependence of the critical stresses of the ceramic component on temperature. These results show that, in certain cases, using power-law type functions to represent the material gradation in FGMs is too restrictive.