Local and global Pareto dominance applied to optimal design and material selection of composite structures

The optimal design of hybrid composite structures considering sizing, topology and material selection is addressed in a multi-objective optimization framework. The proposed algorithm, denoted by Multi-objective Hierarchical Genetic Algorithm (MOHGA), searches for the Pareto-optimal front enforcing population diversity by using a hierarchical genetic structure based on co-evolution of multi-populations. An age structured population is used to store the ranked solutions aiming to obtain the Pareto front. A self-adaptive genetic search incorporating Pareto dominance and elitism is presented. Two concepts of dominance are used: the first one denoted by local non-dominance is implemented at the isolation stage of populations and the second one called global non-dominance is considered at age structured population. The age control emulates the human life cycle and enables to apply the species conservation paradigm. A new mating and offspring selection mechanisms considering age control and dominance are adopted in crossover operator applied to age-structured population. Application to hybrid composite structures requiring the compromise between minimum strain energy and minimum weight is presented. The structural integrity is checked for stress, buckling and displacement constraints considered in the multi-objective optimization. The design variables are ply angles and ply thicknesses of shell laminates, the cross section dimensions of beam stiffeners and the variables associated with the material distribution at laminate level and structure level. The properties of the proposed approach are discussed in detail.     

A unified approach to nonlinear buckling optimization of composite structures

A unified approach to nonlinear buckling fiber angle optimization of laminated composite shell structures is presented. The method includes loss of stability due to bifurcation and limiting behaviour. The optimization formulation is formulated as a mathematical programming problem and solved using gradient-based techniques. Buckling of a well-known cylindrical shell benchmark problem is studied and the solutions found in literature are proved to be incorrect. The nonlinear buckling optimization formulation is benchmarked against the traditional linear buckling optimization formulation through several numerical optimization cases of a composite cylindrical shell panel which clearly illustrates the advantage and potential of the presented approach.     

Shape optimization of shallow domes subjected to external pressure

Shallow domes subjected to external pressure are extensively used in missile structures. The critical failure mode for these domes is buckling due to external pressure. Different closed form solutions are available to evaluate buckling pressure of dome shapes like ellipsoid and torisphere. The torisiphere dome is the optimum dome shape among conventional domes. Shape optimization is carried out to find the optimal dome shape among shallow domes subjected to external pressure. Dome geometry is generalized by cubic bezier polynomials. For carrying out shape optimization, a low fidelity model is preferred which can predict the critical buckling pressure of a general dome shape. Towards this a unified model is proposed which meets the above requirement. Using this unified model, shape optimization of dome for minimization of mass is carried out subjected to buckling constraint. The study yielded a dome shape different from conventional dome shapes with a mass saving of 6% over torispherical dome while meeting the buckling constraint. The results of unified model are also validated with high fidelity Finite Element Analysis.

     

基于在线感知Pareto前沿划分目标空间的多目标进化优化

多目标进化优化是求解多目标优化问题的可行方法.但是, 由于没有准确感知并充分利用问题的Pareto前沿, 已有方法难以高效求解复杂的多目标优化问题.本文提出一种基于在线感知Pareto前沿划分目标空间的多目标进化优化方法, 以利用感知的结果, 采用有针对性的进化优化方法求解多目标优化问题.首先, 根据个体之间的拥挤距离与给定阈值的关系感知优化问题的Pareto前沿上的间断点, 并基于此将目标空间划分为若干子空间; 然后, 在每一子空间中采用MOEA/D (Multi-objective evolutionary algorithm based on decomposition)得到一个外部保存集; 最后, 基于所有外部保存集生成问题的Pareto解集.将提出的方法应用于15个基准数值函数优化问题, 并与NSGA-Ⅱ、RPEA、MOEA/D、MOEA/DPBI、MOEA/D-STM和MOEA/D-ACD等比较.结果表明, 提出的方法能够产生收敛和分布性更优的Pareto解集, 是一种非常有竞争力的方法.     

Minimum cost design of a welded orthogonally stiffened cylindrical shell

In this study the optimal design of a cylindrical orthogonally stiffened shell member of an offshore fixed platform truss, loaded by axial compression and external pressure, is investigated. Ring stiffeners of welded box section and stringers of halved rolled I-section are used. The design variables considered in the optimization are the shell thickness as well as the dimensions and numbers of stiffeners. The design constraints relate to the shell, panel ring and panel stringer buckling, as well as manufacturing limitations. The cost function includes the cost of material, forming of plate elements into cylindrical shape, welding and painting. In the optimization a number of relatively new mathematical optimization methods (leap-frog - LFOPC, Dynamic-Q, ETOPC, and particle swarm - PSO) are used, in order to ensure confidence that the finally computed optimum design is accurately determined, and indeed corresponds to a global minimum. The continuous optimization procedures are adapted to allow for discrete values of the design variables to be used in the final manufacturing of the truss member. A comparison of the computed optimum costs of the stiffened and un-stiffened assemblies, shows that significant cost savings can be achieved by orthogonal stiffening, since the latter allows for considerable reduction of the shell thickness, which results in large material and manufacturing cost savings.     

Multi-objective optimization of fiber reinforced composite laminates for strength,stiffness and minimal mass

We present a methodology for the multi-objective optimization of laminated composite materials that is based on an integer-coded genetic algorithm. The fiber orientations and fiber volume fractions of the laminae are chosen as the primary optimization variables. Simplified micromechanics equations are used to estimate the stiffnesses and strength of each lamina using the fiber volume fraction and material properties of the matrix and fibers. The lamina stresses for thin composite coupons subjected to force and/or moment resultants are determined using the classical lamination theory and the first-ply failure strength is computed using the Tsai–Wu failure criterion. A multi-objective genetic algorithm is used to obtain Pareto-optimal designs for two model problems having multiple, conflicting, objectives. The objectives of the first model problem are to maximize the load carrying capacity and minimize the mass of a graphite/epoxy laminate that is subjected to biaxial moments. In the second model problem, the objectives are to maximize the axial and hoop rigidities and minimize the mass of a graphite/epoxy cylindrical pressure vessel subject to the constraint that the failure pressure be greater than a prescribed value.     

Multi-objective optimization for design under uncertainty problems through surrogate modeling in augmented input space

Multi-objective design under uncertainty problems that adopt probabilistic quantities as performance objectives and consider their estimation through stochastic simulation are examined in this paper, focusing on development of a surrogate modeling framework to reduce computational burden for the numerical optimization. The surrogate model is formulated to approximate the system response with respect to both the design variables and the uncertain model parameters, so that it can simultaneously support both the uncertainty propagation and the identification of the Pareto optimal solutions. Kriging is chosen as the metamodel, and its probabilistic nature (its ability to offer a local estimate of the prediction error) is leveraged within different aspects of the framework. To reduce the number of simulations for the expensive system model, an iterative approach is established with adaptive characteristics for controlling the metamodel accuracy. At each iteration, a new metamodel is developed utilizing all available training points. A new Pareto front is then identified utilizing this surrogate model and is compared, for assessing stopping criteria, to the front that was identified in the previous iteration. This comparison utilizes explicitly the potential error associated with the metamodel predictions. If stopping criteria are not achieved, a set of refinement experiments (new training points) is identified and process proceeds to the next iteration. A hybrid design of experiments is considered for this refinement, with a dual goal of global coverage and local exploitation of regions of interest, separately identified for the design variables and the uncertain model parameters.

     

Buckling optimization of fiber-composite laminate shells considering in-plane shear nonlinearity

The buckling strengths of fiber-composite laminate shells with a given material system are maximized with respect to fiber orientations using a sequential linear programming method together with a simple move-limit strategy. While a modified Riks nonlinear solution algorithm is utilized to analyse the buckling and postbuckling behaviour of composite shells, both linear and nonlinear in-plane shear formulations are employed to form the finite-element constitutive matrix for fiber-composite laminae. Results of the optimization study for simply supported composite cylindrical shells using both linear and nonlinear in-plane shear formulations are presented.     

Evolutionary multi criteria design optimization of robot grippers

This paper explores the use of intelligent techniques to obtain optimum geometrical dimensions of a robot gripper. The optimization problem considered is a non-linear, complex, multi-constraint and multicriterion one. Three robot gripper configurations are optimized. The aim is to find Pareto optimal front for a problem that has five objective functions, nine constraints and seven variables. The problem is divided into three cases. Case 1 has first two objective functions, the case 2 considers last three objective functions and case 3 deals all the five objective functions. Intelligent optimization algorithms namely Multi-objective Genetic Algorithm (MOGA), Elitist Non-dominated Sorting Genetic Algorithm (NSGA-II) and Multi-objective Differential Evolution (MODE) are proposed to solve the problem. Normalized weighting objective functions method is used to select the best optimal solution from Pareto optimal front. Two multi-objective performance measures (solution spread measure (SSM) and ratio of non-dominated individuals (RNIs)) are used to evaluate the strength of the Pareto optimal fronts. Two more multi-objective performance measures namely optimizer overhead (OO) and algorithm effort are used to find the computational effort of MOGA, NSGA-II and MODE algorithms. The Pareto optimal fronts and results obtained from various techniques are compared and analyzed.     

Parameter estimation of a pressure swing adsorption model for air separation using multi-objective optimisation and support vector regression model

In order to successfully estimate parameters of a numerical model, multiple criteria should be considered. Multi-objective Differential Evolution (MODE) and Multi-objective Genetic Algorithm (MOGA) have proved effective in numerous such applications, where most of the techniques relying on the condition of Pareto efficiency to compare different solutions. We describe the performance of two population based search algorithms (Nondominated Sorting Differential Evolution (NSDE) and Nondominated Sorting Genetic Algorithm (NGAII)) when applied to parameter estimation of a pressure swing adsorption (PSA) model. Full PSA mode is a complicated dynamic processing involving all transfer phenomena (mass, heat and momentum transfer) and has proven to be successful in a wide of applications. The limitation of using full PSA models is their expensive computational requirement. The parameter estimation analysis usually needs to run the numerical model and evaluate the performance thousands of times. However, in real world applications, there is simply not enough time and resources to perform such a huge number of model runs. In this study, a computational framework, known as v-support vector regression (v-SVR) PSA model, is presented for solving computationally expensive simulation problems. Formulation of an automatic parameter estimation strategy for the PSA model is outline. The simulations show that the NSDE is able to find better spread of solutions and better convergence near the true Pareto-optimal front compared to NSGAII-one elitist MOGA that pays special attention to creating a diverse Pareto-optimal front.     

Fiber composite thin shells subjected to thermal buckling loads

The results of parametric studies to assess the effects of various parameters on the buckling behavior of angle-ply, laminated thin shells in a hot environment are presented in this paper. These results were obtained by using a three-dimensional finite element analysis. An angle-ply, laminated thin shell with fiber orientation of [θ/ −θ]₂ was subjected to compressive mechanical loads. The laminated thin shell has a cylindrical geometry. The laminate contained T300 graphite fibers embedded in an intermediate-modulus, high-strength (IMHS) matrix. The fiber volume fraction was 55% and the moisture content was 2%. The residual stresses induced into the laminated structure during the curing were taken into account. Parametric studies were performed to examine the effect on the critical buckling load of the following parameters: cylinder length and thickness, internal hydrostatic pressure, different ply thicknesses, different temperature profiles through the thickness of the structure, and different layup configurations and fiber volume fractions. In conjunction with these parameters the ply orientation varied from 0° to 90°. Seven ply angles were examined: 0°, 15°, 30°, 45°, 60°, 75°, and 90°. The results show that the ply angle θ and the laminate thickness had significant effects on the critical buckling load. The fiber volume fraction and the internal hydrostatic pressure had important effects on the critical buckling load. The cylinder length had a moderate influence on the buckling load. The thin shell with [θ/−θ]₂ or [θ/−θ]^s angle-ply laminate had better buckling-load performance than the thin shell with [θ]₄ off-axis laminate. The temperature profiles through the laminate thickness and various laminates with the same thickness but with the different ply thickness had insignificant effects on the buckling behavior of the thin shells.     

Structural Optimization of Automotive Body Components Based on Parametric Solid Modeling

Parametric modeling was used to build several models of an automotive front structure concept that utilizes carbon fiber composite materials and the corresponding molding processes. An ultra-lightweight aluminum body front structure was redesigned to include an all-composite front structure. Two alternative concepts were studied which represent the structure as a bonded assembly of shells. Closed sections result from two pieces – an inner and outer. Parametric modeling was found to be a useful tool for building and modifying models to use in optimization concept studies. Such models can be built quickly and both the sketch dimensions and location dimensions are particularly useful for making the adjustments necessary to fit the various body pieces together. The parametric models then must be joined together as one geometric solid model in order to obtain a surface mesh. Structural optimization input data can then be seamlessly and quickly created from the parametric-model-based finite element model to begin the tradeoff studies. This integrated process in which parametric modeling was coupled with structural optimization was used to carry out design studies on the lightweight body front structure. Several carbon fiber material combinations were studied to determine mass reduction potential of certain types of carbon fiber products considered to be lower cost than typical carbon fiber materials used in the past. Structural optimization was used to compare several composite constructions for the design of the bonded front structure. Eight cases were studied using various materials and composite lay-ups. Mass savings estimates from 45–64% over steel were obtained. The most reasonable design consisted of a combination of relatively low cost chopped carbon fiber and woven carbon fiber and using a 20 mm balsa core in the top of the shock tower area. This design had a maximum thickness of 7 mm and a mass reduction over steel of approximately 62%.     

Failure analysis of laminated composite cylindrical/spherical shell panels subjected to low-velocity impact

A 4-noded, 48 d.o.f. doubly curved quadrilateral shell finite element based on Kirchhoff–Love shell theory, is used in the nonlinear finite element analysis to predict the damage of laminated composite cylindrical/spherical shell panels subjected to low-velocity impact. The large displacement stiffness matrix is formed using Green's strain tensor based on total Lagrangian approach. An incremental/iterative scheme is used for solving resulting nonlinear algebraic equations by Newton–Raphson method. The damage analysis is performed by applying Tsai–Wu quadratic failure criterion at all Gauss points and the mode of failure is identified using maximum stress criteria. The modes of failure considered are fiber breakage and matrix cracking. The progressive failure analysis is carried out by degrading the stiffness of the material suitably at all failed Gauss points. The load due to low-velocity impact is treated as an equivalent quasi-static load and Hertzian law of contact is used for finding the maximum contact force. After evaluating the nonlinear finite element analysis thoroughly for typical problems, damage analysis was carried out for cross-ply and quasi-isotropic cylindrical/spherical shell panels.     

多目标优化Knee前沿搜索方法研究进展

多目标优化算法是近年来进化计算研究领域的一个热点,大多数的多目标优化算法试图找到问题的完整的Pareto前沿.然而,随着待优化问题目标个数的增加,算法需要更大的种群规模才能合理地描绘出完整的Pareto前沿.显然这样不仅增加了算法的运行时间,更增加了(决策者)最终解的选择难度.因此,聚焦于搜索Pareto前沿上的特定区域显得尤为重要,近年来也得到了越来越多学者的关注. Knee点指的是Pareto前沿上具有最大边际效用的点,在这个点附近,一个目标值的微小提升将带来至少一个其他目标值的巨大衰退,因此该点通常被认为是在没有特殊偏好的情况下对决策者更具吸引力的点.本文旨在对多目标优化中Knee前沿搜索相关的方法进行总结,包括Knee的检测方法、保留策略、测试问题等,并对多目标优化的Knee前沿搜索未来研究工作进行展望.     

Multicriteria optimization of cylindrical sandwich shells under combined loads

The paper is about multicriteria optimization of thin-walled cylindrical shells subjected to simple loads, such as axial compression and external pressure, and combined loads (axial compression and pressure). The optimization problem is given as a bicriterial one, with the weight of the shell as the first objective, and the flexibility of the shell as the second. The set of constraints includes the stability condition, strength conditions for each layer, technological and constructional requirements, and so on. Numerical calculations were obtained with the help of the program MOST. MOST is designed to solve multicriteria optimization problems for nonlinear engineering models with discrete and continuous decision variables. In MOST a concept of Pareto optimum is introduced for generating a set of optimal compromise solutions. The best optimal solution must be chosen from the Pareto optimal set with the help of the preference functions. Results of numerical calculations are presented in the form of tables and diagrams.     

Postbuckling analysis of hydrostatic pressurized FGM microsized shells including strain gradient and stress-driven nonlocal effects

Herein, with the aid of the newly proposed theory of nonlocal strain gradient elasticity, the size-dependent nonlinear buckling and postbuckling behavior of microsized shells made of functionally graded material (FGM) and subjected to hydrostatic pressure is examined. As a consequence, the both nonlocality and strain gradient micro-size dependency are incorporated to an exponential shear deformation shell theory to construct a more comprehensive size-dependent shell model with a refined distribution of shear deformation. The Mori–Tanaka homogenization scheme is utilized to estimate the effective material properties of FGM nanoshells. After deduction of the non-classical governing differential equations via boundary layer theory of shell buckling, a perturbation-based solving process is employed to extract explicit expressions for nonlocal strain gradient stability paths of hydrostatic pressurized FGM microsized shells. It is observed that the nonlocality size effect causes to decrease the critical hydrostatic pressure and associated end-shortening of microsized shells, while the strain gradient size dependency leads to increase them. In addition, it is found that the influence of the internal strain gradient length scale parameter on the nonlinear instability characteristics of hydrostatic pressurized FGM microsized shells is a bit more than that of the nonlocal one.

     

基于ε占优的自适应多目标粒子群算法

针对粒了群算法求解多目标问题极易收敛到伪Pareto前沿(等价于单目标优化问题中的局部最优解),并且收敛速度较慢的问题,提出一种ε占优的自适应多目标粒子群算法(εDMOPSO)..在εDMOPSO算法中,每个粒子的邻居根据粒了的运行动态地组建,且粒了的速度小由其邻居中运行最好的粒予来调整,而是由其所有邻居共同调整.同时...     

Reliability-based optimum design of a welded stringer-stiffened steel cylindrical shell subject to axial compression and bending

The economy of stiffened shells vs the unstiffened version depends on loading, type of stiffening and stiffener profile. The stiffening is economic when the shell thickness can be decreased in such a measure that the cost savings caused by this decreasing is higher than the additional cost of stiffening material and welding. The present work deals with cylindrical shell columns fixed at the bottom and free at the top subject to axial compression and horizontal force acting on the top of the column. The shell is stiffened outside with stringers welded by longitudinal fillet welds. Half rolled I-section (UB) stiffeners are used to reduce welding cost. The cost function to be minimized includes the costs of the materials, forming of shell elements into the cylindrical shape, assembly, welding and painting. The design variables are the shell thickness, number and profile of stiffeners for the stiffened shell, but only the first type of variable in the unstiffened case. Randomness is considered both in loading and material properties. A level II reliability method (first-order reliability method) is employed. Individual reliability constraints related with shell buckling, stringer panel buckling and the limitation of the horizontal displacement of the column top are considered. The overall structural reliability is obtained by using Ditlevsen's method of conditional bounding. The costs of both the stiffened and unstiffened shells designed to ensure a stipulated probability of failure will be compared with the solutions obtained for a code-based method, which employs partial safety factors. Results are given illustrating the influence of the constraint on the horizontal displacement.     

混合分解和强度帕累托多目标进化算法

在多目标进化优化中,使用分解策略的基于分解的多目标进化算法(MOEA/D)时间复杂度低,使用〖BP(〗强度帕累托策略的〖BP)〗强度帕累托进化算法-2(SPEA2)能得到分布均匀的解集。结合这两种策略,提出一种新的多目标进化算法用于求解具有复杂、不连续的帕累托前沿的多目标优化问题(MOP)。首先,利用分解策略快速逼近帕累托前沿;然后,利用强度帕累托策略使解集均匀分布在帕累托前沿,利用解集重置分解策略中的权重向量集,使其适配于特定的帕累托前沿;最后,利用分解策略进一步逼近帕累托前沿。使用的反向世代距离(IGD)作为度量标准,将新算法与MOEA/D、SPEA2和paλ-MOEA/D在12个基准问题上进行性能对比。实验结果表明该算法性能在7个基准问题上最优,在5个基准问题上接近于最优,且无论MOP的帕累托前沿是简单或复杂、连续或不连续的,该算法均能生成分布均匀的解集。     

Optimization of the buckling load for composite structures taking thermal effects into account

This paper deals with optimization of the buckling load for laminated composite structures. A new methodology has been developed where thermal residual stresses introduced in the manufacturing process are included in the buckling analysis. The thermal effects are also included in the calculation of the buckling load sensitivities, and it is therefore possible to “tailor” the thermal residual stresses in order to increase the buckling load. Rectangular plates and circular cylindrical shells subjected to axial compression are considered. The structures are optimized twice; the first time the thermal residual stresses are ignored in the optimization, and the second time the thermal residual stresses are included in the optimization. These two sets of optimizations give two important results. Firstly, it is possible to increase the buckling load for the structures significantly when the thermal residual stresses are taken into account. Secondly, structures which have been optimized ignoring the effects of thermal residual stresses, may have a buckling load which is much less than expected when the effects of the thermal residual stresses are included. Received November 7, 1999