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.     

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.     

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.     

Buckling of cylindrical panels under axial compression

This paper investigates the effects of boundary conditions and panel width on the axially compressive buckling behavior of unstiffened, isotropic, circular cylindrical panels. Numerical results are presented for eight different sets of boundary conditions along the straight edges of the panels. For all sets of boundary conditions except one (SSI), the results show that the panel buckling loads monotonically approach the complete cylinder buckling load from above as the panel width is increased. Low buckling loads, sometimes less than half the complete cylinder buckling load, are found for simply supported panels with free in-plane edge displacements (SSI). The SSI buckling loads are below the complete cylinder load even for ‘360° panels’. It is also observed that the prevention of circumferential edge displacement is the most important in-plane boundary condition from the point of view of increasing the buckling load, and that the prevention of edge rotation (i.e. clamping) in the circumferential direction also significantly increases the buckling load. Parametric studies are also performed to determine the effects of variations in panel length and thickness on the buckling loads.     

Buckling of eccentrically stringer-stiffened cylindrical panels under axial compression

This paper investigates the effects of boundary conditions and panel width on the axially compressive buckling behavior of eccentrically stringer-stiffened circular cylindrical panels. Numerical results are presented for eight different sets of boundary conditions along the straight edges of the panels. As the panel width is increased, the results show that the complete cylinder buckling load is reached only for one set of boundary conditions (SS3, classical simple support conditions). However, for 180° and wider panels, the panel buckling loads are within ± 10% of the complete cylinder load for all cases except SS1 panels (free in-plane edge displacements) with outside stringers. Low buckling loads, as low as half the complete cylinder load, are found for some SS 1 panels. It is also observed that the prevention of circumferential edge displacement is the most important in-plane boundary condition from the point of view of increasing the buckling load, and that the prevention of edge rotation (i.e. clamping) in the circumferential direction is more effective in increasing the buckling loads of panels with free circumferential edge displacement υ that it is for panels with υ = 0. From stringer-eccentricity studies, it is shown that buckling loads are generally at least 40–50% higher for the case of outside stringers, and that eccentricity effects are generally similar for clamped and simply supported panels with the same in-plane boundary conditions.     

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.     

Design optimization and sensitivity analysis on buckling stability of piezoelectric truss structures

The design optimization of buckling behavior is studied for piezoelectric intelligent truss structures. First, on the basis of mechanical–electric coupling equation and considering electric load and mechanical loads together, the finite element model of piezoelectric trusses has been built up. Then, the computational formula has been derived for the design sensitivities of critical buckling load factor of the structure with respect to size and shape design variables. The electric voltage is taken as a new kind of design variable and the calculation method of critical load buckling factor with respect to the electric voltage variables is proposed. Particularly, the variations of the loads and pre-buckling inner forces with design variables have been accounted for. Finally, the sequential linear programming algorithm is employed to solve the optimization problem, and a new method of controlling structural buckling stability by optimizing the voltages of piezoelectric active bars is proposed. Numerical examples given in the paper have demonstrated the effectiveness of the methods presented.     

Maximization of buckling loads of composite panels using flexural lamination parameters

Unstiffened composite panels are optimized by using flexural lamination parameters as continuous design variables for the case in which the amounts of 0°, ±45°, and 90° plies are given. It is shown that for this case, the lamination parameters are located in a hexagonal domain. Continuous optimization is compared with genetic optimization for the stacking sequence that accounts for the discreteness of the design space and constraints on the number of contiguous plies of the same orientation. It is shown that only for very thin panels with low aspect ratios is there a significant difference between the continuous and discrete solutions.     

Aerothermoelastic topology optimization with flutter and buckling metrics

This work develops a framework for SIMP-based topology optimization of a metallic panel structure subjected to design-dependent aerodynamic, inertial, elastic, and thermal loads. Multi-physics eigenvalue-based design metrics such as thermal buckling and dynamic flutter are derived, along with their adjoint-based design derivatives. Locating the flutter point (Hopf-bifurcation) in a precise and efficient manner is a particular challenge, as is outfitting the optimization problem with sufficient constraints such that the critical flutter mode does not switch during the design process. Results are presented for flutter-optimal topologies of an unheated panel, thermal buckling-optimal topologies, and flutter-optimality of a heated panel (where the latter case presents a topological compromise between the former two). The effect of various constraint boundaries, temperature gradients, and (for the flutter of the heated panel) thermal load magnitude are assessed. Off-design flutter and thermal buckling boundaries are given as well.     

Reliability-based optimum design of a symmetric laminated plate subject to buckling

This study is concerned with the buckling reliability maximization of a symmetric laminated composite plate with respect to the mean ply orientation angle. The reliability is evaluated by modelling the buckling failure as a series system consisting of potential eigenmodes. The mode reliability is obtained by the first-order reliability theory (FORM), where material constants and orientation angles of individual layers, as well as the applied loads are treated as random variables. In order to keep track of the intended buckling mode during the reliability analysis, the mode tracking method is utilized. Then, the failure probability of the series system is approximated by Ditlevsen's upper bound. The reliability maximization problem is formulated as a nested problem with two levels of optimization. Through numerical calculations, the reliability-based design is demonstrated to be important for the structural safety in comparison with the deterministic buckling load maximization design.     

Buckling design optimization of complex built-up structures with shape and size variables

The design optimization of buckling behaviour is studied for complex built-up structures composed of various kinds of elements and implemented within JIFEX95, a general-purpose software for finite element analysis and design optimization. The direct and adjoint methods of sensitivity analysis for critical buckling loads are presented with detailed computational procedures. Particularly, the variations of prebuckling stresses and external loads have been accounted for. The design model and solution methods presented in this paper are available for both shape and size optimization, and buckling optimization can also be combined with static, frequency and dynamic response optimization. The numerical examples show the applications of the buckling optimization method and the effectiveness of the methods and the program of this paper. Received February 23, 1999     

On buckling optimization under uncertain loading combinations

This paper proposes a technique to optimize structural components for buckling when the applied loads are partially unknown or unpredictable. As opposed to the traditional buckling optimization situation where the loading configuration is specified, the load ratios are assumed uncertain and are incorporated as variables in the optimization problem formulation. As a result, the optimal designs obtained are insensitive to load variations within an admissible convex set. Additionally, in order to generalize the results and therefore provide a systematic solution procedure, a theorem concerning the shape of the stability boundary of structures whose buckling loads are the solution of linear eigenproblems is stated and proven. Recevied February 2, 2000     

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.

     

Natural frequency and buckling optimization of laminated hybrid composite plates using genetic algorithm and simulated annealing

In this study, genetic algorithm and simulated annealing are used to maximize natural frequency and buckling loads of simply supported hybrid composite plates. The aim of the study is to use two different techniques of optimization on the frequency and buckling optimization of composite plates, and compare the techniques for their effectiveness. The composite plate is made of carbon/epoxy and glass/epoxy hybrid plies, and assumed to be symmetric and balanced. The effect of hybridization is investigated using both techniques. The buckling problem has many maxima in the vicinity of local maxima. The best configurations are identified for different plate aspect ratios. The performance of both techniques is compared in terms of number of function evaluations as well as the capability of finding best configurations.     

Multi-fidelity optimization of laminated conical shells for buckling

Optimum laminate configuration for minimum weight of filament-wound laminated conical shells is investigated subject to a buckling load constraint. In the case of a composite laminated conical shell, due to the manufacturing process, the thickness and the ply orientation are functions of the shell coordinates, which ultimately results in coordinate dependence of the stiffness matrices (A,B,D). These effects influence both the buckling load and the weight of the structure and complicate the optimization problem considerably. High computational cost is involved in calculating the buckling load by means of a high-fidelity analysis, e.g. using the computer code STAGS-A. In order to simplify the optimization procedure, a low-fidelity model based on the assumption of constant material properties throughout the shell is adopted, and buckling loads are calculated by means of a low-fidelity analysis, e.g. using the computer code BOCS. This work proposes combining the high-fidelity analysis model (based on exact material properties) with the low-fidelity model (based on nominal material properties) by using correction response surfaces, which approximate the discrepancy between buckling loads determined from different fidelity analyses. The results indicate that the proposed multi-fidelity approaches using correction response surfaces can be used to improve the computational efficiency of structural optimization problems.     

Buckling analysis of laminated composite plates with an elliptical/circular cutout using FEM

In this study, a buckling analysis was carried out of a woven–glass–polyester laminated composite plate with an circular/elliptical hole, numerically. In the analysis, finite element method (FEM) was applied to perform parametric studies on various plates based on the shape and position of the elliptical hole. This study addressed the effects of an elliptical/circular cutout on the buckling load of square composite plates. The laminated composite plates were arranged as symmetric cross-ply [(0°/90°)₂]_s and angle-ply [(15°/−75°)₂]_s, [(30°/−60°)₂]_s, [(45°/−45°)₂]_s. The results show that buckling loads are decreased by increasing both c/a and b/a ratios. The increasing of hole positioned angle cause to decrease of buckling loads. Additionally, the cross-ply composite plate is stronger than all other analyzed angle-ply laminated plates.     

Buckling optimization of laminated composite plates using genetic algorithm and generalized pattern search algorithm

In this paper, genetic algorithm and generalized pattern search algorithm are used for optimal stacking sequence of a composite panel, which is simply supported on four sides and is subject to biaxial in-plane compressive loads. The problem has several global optimum configurations in the vicinity of local optima. The composite plate under consideration is 64-ply laminate made of graphite/epoxy. The laminate is taken to be symmetric and balanced, comprised of two-ply stacks with discrete fiber angles of 0₂, ± 45, 90₂ in the laminate sequence. The critical buckling loads are maximized for several combinations of load case and plate aspect ratio, and are compared with published results. Performance of both algorithms is compared in terms of capability of identifying global optima. It is found that genetic algorithm is efficient for problems with global optima.     

Optimization of fiber orientations near a hole for increased load-carrying capacity of composite laminates

In this paper, a procedure for optimizing the fiber orientations near a hole in a single layer of multilayer composite laminates for increased strength is presented. A symmetric six-layer [(C-0)/+45/-45]_s laminate with central hole is considered. Within the ±45^° layers, the fiber orientations are fixed. The C-0 layer is divided into two fields: a small near field around the hole and a relatively large far field away from the hole. In the far field, the fiber orientations are fixed at 0^°, and in the near field, the fiber orientations are assumed to have continuous distribution represented by piecewise bilinear interpolation functions. The Tsai–Wu-criterion-based failure load magnitudes of [(C-0)/+45/-45]_s and [C-0]₆ designs are maximized by iteratively alternating between a gradient-based search and a genetic algorithm. The results show that the load-carrying capacity of composite laminates with hole can be greatly increased through the optimization of continuous fiber orientation distribution within a small area around the hole in the C-0 layer, and the optimum fiber orientation distribution pattern in the C-0 layer is similar to that of the corresponding principal stress orientation distribution. The [(C-0)/+45/-45]_s design is only a few percent weaker than the [C-0]₆ design, which is important for carrying off-design loadings.     

Conceptual design of aeroelastic structures by topology optimization

A topology optimization methodology is presented for the conceptual design of aeroelastic structures accounting for the fluid–structure interaction. The geometrical layout of the internal structure, such as the layout of stiffeners in a wing, is optimized by material topology optimization. The topology of the wet surface, that is, the fluid–structure interface, is not varied. The key components of the proposed methodology are a Sequential Augmented Lagrangian method for solving the resulting large-scale parameter optimization problem, a staggered procedure for computing the steady-state solution of the underlying nonlinear aeroelastic analysis problem, and an analytical adjoint method for evaluating the coupled aeroelastic sensitivities. The fluid–structure interaction problem is modeled by a three-field formulation that couples the structural displacements, the flow field, and the motion of the fluid mesh. The structural response is simulated by a three-dimensional finite element method, and the aerodynamic loads are predicted by a three-dimensional finite volume discretization of a nonlinear Euler flow. The proposed methodology is illustrated by the conceptual design of wing structures. The optimization results show the significant influence of the design dependency of the loads on the optimal layout of flexible structures when compared with results that assume a constant aerodynamic load.