Single-level composite wing optimization based on flexural lamination parameters

An optimization procedure based on flexural lamination parameters is used to integrate unstiffened composite panel design and wing structural design. The lamination parameters are constrained to a hexagonal domain when the amounts of 0°, ±45°, and 90° plies are given. The single-level optimization based on continuous flexural lamination parameters for the minimization of wing weight is compared with a two-level optimization using response surfaces of maximal buckling load for a simple wing box design example. Reasonable agreement between the two procedures indicates that the two-level approach leads to near-optimal designs.     

A general approach forcing convexity of ply angle optimization in composite laminates

This paper deals with optimization of laminated composite structures in which the ply angles are taken as design variables. One of the major problems when using ply-angles as design variables, is the lack of convexity of the objective function and thus the existence of local optima, which implies that usual gradient based optimization procedures may not be effective. Therefore, a new general approach that avoids the abovementioned problems of nonconvexity when ply-angles are used as design variables is proposed. The methodology is based upon the fact that the design space for an optimization problem formulated in lamination parameters [introduced by Tsai and Pagano (1968)] is proven to be convex, because the laminate stiffnesses are expressed linearly in terms of the lamination parameters. However, lamination parameters have at least two major shortcomings: as yet, for the general case involving membrane-bending coupling, the constraints between the lamination parameters are not completely defined; also, for a prescribed set of lamination parameters physically realizable composite laminates (e.g. laminates with equal thickness plies) may not exist. The approach here, uses both lamination parameters and ply-angles and thereby uses the advantages of both and eliminates the shortcomings of both.In order to illustrate this approach, several stiffness optimization examples are provided.     

Stacking sequence optimization for constant stiffness laminates based on a continuous optimization approach

In this paper, an optimization procedure based on multi-phase topology optimization is developed to determine the optimal stacking sequence of laminates made up of conventional plies oriented at ?45°, 0°, 45 and 90°. The formulation relies on the SFP (Shape Functions with Penalization) parameterization, in which the discrete optimization problem is replaced by a continuous approach with a penalty to exclude intermediate values of the design variables. In this approach, the material stiffness of each physical ply is expressed as a weighted sum over the stiffness of the candidate plies corresponding to ?45°, 0°, 45 and 90° orientations. In SFP, two design variables are needed for each physical ply in the laminate to parameterize the problem with respect to the 4 candidate orientations. Even if only constant stiffness laminates of constant thickness are considered in this paper, specific design rules used in aeronautics for composite panels (i.e., no more than a maximum number of consecutive plies with the same orientation in the stacking sequence) are however formulated and taken into account in the optimization problem. The methodology is demonstrated on an application. It is discussed how the different design rules can affect the solution.     

Note on singular optima in laminate design problems

This paper studies the design of laminates subject to restrictions on the ply strength. The minimum weight design is considered. It is shown that this formulation includes singular optima, which are similar to the ones observed in topology optimization including local stress constraints. In laminate design, these singular optima are linked to the removal of ‘zero thickness’ plies from the stacking sequence. It is shown how the fiber orientation variables can circumvent the singularity by relaxing the strength constraints related to such vanishing plies. This demonstrates the key role of fiber orientations in the optimization of laminates and the need for their efficient treatment as design variables.     

Determining the real lay-up of a laminate corresponding to optimal lamination parameters by genetic search

A genetic algorithm search (GA) is employed for tailoring the thermomechanical properties of a laminated plate. Maximization of the strain energy, minimization of certain displacements and maximization of the first buckling factor due to given boundary temperatures and maximization of the lowest natural frequency of a graphite-epoxy plate with respect to the lay-up of the laminate are considered. Instead of computing these values during the GA iteration, lamination parameters giving the optima referred to above are employed in genetic searches. The fitness (objective) function consists of differences between target lamination parameters and lamination parameters of the current design, and thus no time-consuming global solution is needed. Two ways of coding the layers are applied: either orientations and relative thicknesses are coded separately, or commercial uni/multiaxial plies are coded. Received May 12, 1999?Communicated by J. Sobieski     

Optimal laminate design subject to single membrane loads

In this paper we investigate optimization of laminates for maximal membrane stiffness under single in-plane loads. The design parameters are the relative ply thicknesses and fiber orientations of an arbitrary number of plies. The design is allowed to vary in a pointwise fashion throughout the structure.From prior work on lamination parameters (Hammeret al. 1997), it is known that the optimal design is given by either some sort of two ply lay-up in special strain situations or otherwise by just a single rotated ply. This is exploited in the present analysis to derive analytically the unique parameters of the optimal design (cross-ply, angle-ply or single ply) as expressions of the membrane forces. Both high and low shear stiffness material are treated. Furthermore the analysis covers all possible local strain or membrane force situations.Finally, it is shown how these expressions for the optimal configuration of the laminate also appear as bounds on the principal membrane forces in order to obtain alignment between the numerically largest principal membrane force and principal strain.     

Composite stacking sequence optimization for aeroelastically tailored forward-swept wings

A method for stacking sequence optimization and aeroelastic tailoring of forward-swept composite wings is presented. It exploits bend-twist coupling to mitigate aeroelastic divergence. The method proposed here is intended for estimating potential weight savings during the preliminary aircraft design stages. A structural beam model of the composite wingbox is derived from anisotropic shell theory and the governing aeroelastic equations are presented for a spanwise discretized forward swept wing. Optimization of the system to reduce wing mass is undertaken for sweep angles of ?35° to 0° and Mach numbers from 0.7 to 0.9. A subset of lamination parameters (LPs) and the number of laminate plies in each pre-defined direction (restricted to {0°,±45°, 90°}) serve as design variables. A bi-level hybrid optimization approach is employed, making use of a genetic algorithm (GA) and a subsequent gradient-based optimizer. Constraints are implemented to match lift requirements and prevent aeroelastic divergence, excessive deformations, airfoil stalling and structural failure. A permutation GA is then used to match specific composite ply stacking sequences to the optimum design variables with a limited number of manufacturing constraints considered for demonstration purposes. The optimization results in positive bend-twist coupling and a reduced structural mass. Results are compared to an uncoupled reference wing with quasi-isotropic layups and with panel thickness alone the design variables. For a typical geometry and a forward sweep of ?25° at Mach 0.7, a wingbox mass reduction of 13 % was achieved.     

Approximation of the critical buckling factor for composite panels

This article is concerned with the approximation of the critical buckling factor for thin composite plates. A new method to improve the approximation of this critical factor is applied based on its behavior with respect to lamination parameters and loading conditions. This method allows accurate approximation of the critical buckling factor for non-orthotropic laminates under complex combined loadings (including shear loading). The influence of the stacking sequence and loading conditions is extensively studied as well as properties of the critical buckling factor behavior (e.g concavity over tensor D or out-of-plane lamination parameters). Moreover, the critical buckling factor is numerically shown to be piecewise linear for orthotropic laminates under combined loading whenever shear remains low and it is also shown to be piecewise continuous in the general case. Based on the numerically observed behavior, a new scheme for the approximation is applied that separates each buckling mode and builds linear, polynomial or rational regressions for each mode. Results of this approach and applications to structural optimization are presented.     

Stacking sequence optimization and blending design of laminated composite structures

The stacking sequence optimization problem for multi-region composite structures is studied in this work by considering both blending and design constraints. Starting from an initial stacking sequence design, unnecessary plies can be removed from this initial design and layer thicknesses of necessary plies are optimally determined. The existence of each ply is represented with discrete 0/1 variables and ply thicknesses are treated as continuous variables. A first-level approximate problem is constructed with branched multipoint approximate functions to replace the primal problem. To solve this approximate problem, genetic algorithm is firstly used to optimize discrete variables, and meanwhile, a blending design scheme is proposed to generate a blended structure. Starting from the thinnest region, this scheme shares all layers of current thinnest region with its adjacent regions. For non-shared layers in the adjacent regions, local mutation is implemented to add or delete plies to make them efficient designs. The whole process is repeated until the blending rule is satisfied. After that, a second-level approximate problem is built to optimize the continuous variables of ply thicknesses for retained layers. Those procedures are repeated until the optimal solution is obtained. Numerical applications, including a two-patch panel and a corrugated central cylinder in a satellite, are conducted to demonstrate the efficacy of the optimization strategy.

     

Optimization of anisotropic composite panels with T-shaped stiffeners including transverse shear effects and out-of-plane loading

A two-step method to optimize anisotropic composite panels with T-shaped stiffeners, including a new formulation of the transverse shear properties and an approximation of the ply contiguity (blocking) constraints as functions of the lamination parameters is provided. At the first step, a representative element of the stiffened panel (superstiffener) is optimized using mathematical programming and lamination parameters subjected to combined loading (in-plane and out-of-plane) under strength (laminate or ply failure), buckling and practical design constraints. Ply blocking constraints are imposed at this step to improve convergence towards practical laminates. At the second step, the actual superstiffener’s laminates are obtained by using a genetic algorithm. Results, for the case considered, show that the inclusion of transverse shear effects has an associated 2.5% mass penalty and that neglecting its effects might invoke earlier buckling failure. In addition, the influence of designing for failure strength at laminate or ply level is assessed.     

The influence of failure criteria on the design optimization of stacked-ply composite flywheels

This paper presents some approaches to the optimal design of stacked-ply composite flywheels. The laminations of the disk are constructed such that the principal fiber direction is either tangential or radial. In this study, optimization problems are formulated to maximize the energy density of the flywheel. This is accomplished by allowing arbitrary, continuous, variation of the orientation of the fibers in the radial plies. The paper compares designs based on minimizing cost functions related to the (1) the maximum stress, (2) the maximum strain, and (3) the Tsai–Wu failure criteria. It is shown that the optimized designs provide an improvement in the flywheel energy density when compared to a standard stacked-ply design. The results also show that, for a given disk design, the estimate of the energy density can vary greatly depending on the failure criteria employed.     

Optimization of composite stiffened panels subject to compression and lateral pressure using a bi-level approach

Composite stiffened panel optimization is typically a mixed discrete-continuous design problem constrained by buckling and material strength. Previous work applied a bi-level optimization strategy to the problem by decomposing the mixed problem to continuous and discrete levels to reduce the optimization search space and satisfy manufacturing constraints. A fast-running optimization package, VICONOPT, was used at the continuous optimization level where the buckling analysis was accurately and effectively performed. However, the discrete level was manually adjusted to satisfy laminate design rules. This paper develops the strategy to application on continuously long aircraft wing panels subjected to compression and lateral pressure loading. The beam-column approach used to account for lateral loading for analysis during optimization is reported. A genetic algorithm is newly developed and applied to the discrete level for automated selection of laminated designs. The results that are presented show at least 13% weight saving compared with an existing datum design.     

An ant colony optimization approach to multi-objective optimal design of symmetric hybrid laminates for maximum fundamental frequency and minimum cost

An ant colony optimization algorithm for optimum design of symmetric hybrid laminates is described. The objective is simultaneous maximization of fundamental frequency and minimization of cost. Number of surface and core layers made of high-stiffness and low-stiffness materials, respectively, and fiber orientations are the design variables. Optimal stacking sequences are given for hybrid graphite/epoxy-glass/epoxy laminated plates with different aspect ratios and number of plies. The results obtained by ant colony optimization are compared to results obtained by a genetic algorithm and simulated annealing. The effectiveness of the hybridization concept for reducing the weight and keeping the fundamental frequency at a reasonable level is demonstrated. Furthermore, it is shown that the proposed ant colony algorithm outperforms the two other heuristics.     

Performance of a bell-curve based evolutionary optimization algorithm

An evolutionary search strategy utilizing two normal distributions to generate children is presented. This Bell-Curve Based (BCB) evolutionary algorithm is similar in spirit to (μ+μ) evolutionary strategies but with fewer parameters to adjust. Extensive tests regarding the sensitivity of BCB parameters to performance are provided. The test suite includes continuous variable constrained hub design problems, mixed discrete and continuous variable constrained hub design problems, and an unconstrained highly multimodal discrete optimization problem. Received March 23, 2000     

Size optimization for hybrid photovoltaic–wind energy system using ant colony optimization for continuous domains based integer programming

In this paper, ant colony optimization for continuous domains (ACO_R) based integer programming is employed for size optimization in a hybrid photovoltaic (PV)–wind energy system. ACO_R is a direct extension of ant colony optimization (ACO). Also, it is the significant ant-based algorithm for continuous optimization. In this setting, the variables are first considered as real then rounded in each step of iteration. The number of solar panels, wind turbines and batteries are selected as decision variables of integer programming problem. The objective function of the PV–wind system design is the total design cost which is the sum of total capital cost and total maintenance cost that should be minimized. The optimization is separately performed for three renewable energy systems including hybrid systems, solar stand alone and wind stand alone. A complete data set, a regular optimization formulation and ACO_R based integer programming are the main features of this paper. The optimization results showed that this method gives the best results just in few seconds. Also, the results are compared with other artificial intelligent (AI) approaches and a conventional optimization method. Moreover, the results are very promising and prove that the authors’ proposed approach outperforms them in terms of reaching an optimal solution and speed.     

Robust eigenstructure assignment combining time- and frequency-domain performance specifications

This paper is concerned with robust eigenstructure assignment for multivariable systems. It combines time-domain performance specifications provided by eigenstructure assignment and robust performance specifications in the frequency domain considered by H^∞ control to realize joint optimal robust control design. A unified parametric solution for state-feedback eigenstructure assignment is derived for both the case where the sets of closed- and open-loop eigenvalues do not intersect and the case where these sets do intersect. This is based on a set of free parameters. All complex operations are converted into the real field so that the algorithm which is developed for the controller design can be easily implemented on computers. It uses a robustness index defined in the frequency domain as the cost function to be optimized. The analytical gradient calculation of the cost function with respect to the free parameters is given. Using gradient-based optimization, the robustness index is minimized by making full use of the freedom provided by eigenstructure assignment. © 1998 John Wiley & Sons, Ltd.     

Optimization of coupled thermal-structural problems of laminated plates with lamination parameters

The behaviour of a laminated plate with given boundary temperatures and displacement constraints is optimized and the optimization problem is expressed in terms of lamination parameters. Because the thermal conductivity and structural properties of a laminate depend on the lamination parameters of the laminate, the analysis of the plate consists of solving a coupled-field problem. The strain energy, or certain displacements of the laminated plate due to given boundary temperatures and displacement boundary conditions, is optimized with respect to in-plane lamination parameters, and also buckling of the plate is considered. The buckling factors for thermal loading are expressed as a function of four in-plane and four bending lamination parameters, and the smallest factor is maximized with respect to these parameters. In addition to these thermal problems, the natural frequencies of the laminated plate are studied. Since transverse shear deformations are taken into account,the natural frequencies can be expressed as functions of two in-plane and four bending lamination parameters, with respect to which the lowest natural frequency of the plate is maximized. The lay-up for the laminate, corresponding to four optimal in-plane or bending lamination parameters, consists of three layers at most and can be determined using explicit equations. Explicit equations are derived for creating a lay-up having optimal bending lamination parameters. Received May 12, 1999     

Orthogonal Methods Based Ant Colony Search for Solving Continuous Optimization Problems

Research into ant colony algorithms for solving continuous optimization problems forms one of the most significant and promising areas in swarm computation. Although traditional ant algorithms are designed for combinatorial optimization, they have shown great potential in solving a wide range of optimization problems, including continuous optimization. Aimed at solving continuous problems effectively, this paper develops a novel ant algorithm termed ＂continuous orthogonal ant colony＂ （COAC）, whose pheromone deposit mechanisms would enable ants to search for solutions collaboratively and effectively. By using the orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently. By implementing an ＂adaptive regional radius＂ method, the proposed algorithm can reduce the probability of being trapped in local optima and therefore enhance the global search capability and accuracy. An elitist strategy is also employed to reserve the most valuable points. The performance of the COAC is compared with two other ant algorithms for continuous optimization -API and CACO by testing seventeen functions in the continuous domain. The results demonstrate that the proposed COAC algorithm outperforms the others.