Multilevel optimization of laminated composite structures

A two-stage optimization method aiming at the optimal design of shells and plates made of laminated composites has been developed. It is based on a mixture of sensitivity analysis, optimality criteria and mathematical programming techniques. The design variables are the optimality criteria and mathematical programming techniques. The design variables are the macro-element thicknesses and the layers' angles. Weight minimization with material efficiency maximization are the objectives with constraints on stresses and displacements. Maximization of the material efficiency is performed at one level using the conjugated method applied to the angles of the macro-element layers keeping the thicknesses constant. The other level is dedicated to weight reduction using optimality criteria and using as variables the macro-element thicknesses with the angles of the macro-element layers constant.     

Optimal design of laminated composite beam structures

This work deals with design sensitivity analysis and optimal design of composite structures modelled as thin-walled beams. The structures are treated as a torsion-bending resistant beams. The analysis problem is discretized by a finite element technique. A two-node Hermitean beam element is used. The beam sections are made from an assembly of elements that correspond to flat layered laminated composite panels. Optimal design is performed with respect to the lamina orientations and thickness of the laminates. The structural weight is considered as the objective function. Constraints are imposed on stresses, displacements, critical load and natural frequencies. Two failure criteria are used to limit the structural strength: Tsai-Hill and maximum stress. The Tsai-Hill criterion is also adopted to predict the first-ply-failure loads. The design sensitivity analysis is analytically formulated and implemented. An adjoint variable method is used to derive the response sensitivities with respect to the design. A mathematical programming approach is used for the optimization process. Numerical examples are performed on three-dimensional structures.     

Damping optimization of viscoelastic laminated sandwich composite structures

Recent developments on the optimization of passive damping for vibration reduction in sandwich structures are presented in this paper, showing the importance of appropriate finite element models associated with gradient based optimizers for computationally efficient damping maximization programs. A new finite element model for anisotropic laminated plate structures with viscoelastic core and laminated anisotropic face layers has been formulated, using a mixed layerwise approach. The complex modulus approach is used for the viscoelastic material behavior, and the dynamic problem is solved in the frequency domain. Constrained optimization is conducted for the maximization of modal loss factors, using gradient based optimization associated with the developed model, and single and multiobjective optimization based on genetic algorithms using an alternative ABAQUS finite element model. The model has been applied successfully and comparative optimal design applications in sandwich structures are presented and discussed.     

Coordinative optimization method of composite laminated structures based on system reliability

This paper develops the coordinative optimization method based on system reliability for laminated structures. The proposed method improves the rough RBO based on first layer failure (FLF) criterion for composite laminates, and the coupling optimization method of thickness and sequence in traditional RBO strategy based on last layer failure criterion (LLF) is improved. In this paper, the finite element analysis is used to obtain the response for the failure based on two-dimension Hashin failure criterion (the limit function). Obviously, the stiffness of composite materials will decline due to destruction of elements. Therefore, stiffness degradation is considered to describe the process of damage evolution. Subsequently, combining with the branch-bound method (B&B), we can complete the search of main failure sequences and calculate the system reliability with the help of the second-order upper bound theory. In order to guarantee the efficiency and accuracy of optimization, the adaptive GA algorithm is introduced in the whole optimization procedure. After the proposed optimization policy is given in detail, two laminated structures are presented and the results are compared with the traditional optimal method based on safety factor, which demonstrates the validity and reasonability of the developed methodology.

     

Reliability based design with a degradation model of laminated composite structures

Uncertainties in deviations of physical properties lead to a probabilistic failure analysis of the composite materials. The proposed optimization model for laminate composites is based on reliability analysis considering the ultimate failure state. To avoid difficulties associated with the complete analysis of the failure modes, bounds are established for the failure probability of the structural system. These bounds are related with theintact and degraded configurations of the structure. Using thefirst ply failure and thelast ply failure theories and a degradation model for the mechanical properties with load sharing rules we obtain the failure probabilities corresponding to the two above configurations. The failure probability of each configuration is obtained using level 2 reliability analysis and the Lind-Hasofer method.The optimization algorithm is developed based on the problem decomposition into three subproblems having as objectives the maximization of the structural efficiency atintact and degraded configurations of the structure and weight minimization subjected to allowable values for the structural reliability. Additionally, the search for the initial design is performed introducing a weight minimization level. It is expected to explore the remaining load capacity of the structures afterfirst ply failure as a function of the anisotropic properties of the composites. The design variables are the ply angles and the thicknesses of the laminates. The structural analysis for the model developed is performed through the finite element method mainly using the isoparametric degenerated shell finite element. The sensitivities are obtained using the discrete approach through the adjoint variable method. In order to show the performance of the analysis two examples are presented.     

Stacking sequence design of a composite wing under a random gust using a genetic algorithm

The layup optimization by genetic algorithm (GA) for the composite wing subject to random gust is presented. The aim of optimization is to maximize the strength of wing and the failure index of Tsai-Hill criterion is used as the objective function. The failure index is calculated by Monte Carlo simulation because the external loading and the material properties have random characteristics. The optimization results are validated by comparing the failure probability of the initial and optimal designs. In addition, the optimum by maximum stiffness criterion is also obtained to show that current objective function is appropriate for the design of composite wing.     

Optimum weight design of composite laminated plates

Optimum designs for the minimum weight of composite laminated plates subjected to size, displacement, buckling and natural frequency constraints are investigated by a technique of combining finite element method and mathematical programming, in which the structural analysis is based on the YNS theory. The recurrence relation based on the feasible direction method (FDM) and the scaling step is used to modify the design variables (ply-thicknesses and ply-orientations) during the iterative procedure. Grouping technique is engaged in the procedure in order that the number of design variables can be greatly reduced to make the problem more practical. Illustrative examples are given to show that the present technique is quite efficient and reliable.     

Efficient strategies for reliability-based design optimization of variable stiffness composite structures

This study investigates efficient design optimization frameworks for composite structures with uncertainties related to material properties and loading. The integration of two decoupled reliability-based design optimization methodologies with a decoupled discrete material optimization is proposed to determine material and fiber orientation for three-dimensional composite structures. First, a deterministic and decoupled discrete material optimization is used for baseline comparison. The objective is to minimize the cost of composite structures with the design variables comprising of the piecewise patch orientations and material properties of the fiber reinforced composites. The reliability-based design optimization includes a hybrid method, and also the sequential optimization and reliability assessment method. In the sequential optimization and reliability assessment method, the inverse reliability analysis is evaluated using a stochastic response surface method and a first order reliability approach. Comparing the methods based on the optimal material and fiber orientations, the uncertainties in loads and material properties lead to different optimal layouts compared to the deterministic solutions. The numerical results also reveal that the hybrid method applied in reliability based designs results in negligible additional computational cost.     

Cross-section optimal design of composite laminated thin-walled beams

This paper aims to perform optimal design of cross-section properties of thin-walled laminated composite beams. These properties are expressed as integrals based on the cross-section geometry, on the warping functions for torsion, shear bending and shear warping, and on the individual stiffness of the laminates constituting the cross-section. The finite element method is used in discretizing the theory. For design sensitivity calculations, the cross-section is modelled throughout design elements. Geometrically, these elements may coincide with the laminates that constitute the cross-section. The developed formulation is based on the concept of adjoint structure. After a warping function is calculated for the cross-section, an adjoint problem may be formulated for each of the properties and a corresponding adjoint warping is determined. It can be applied in a unified way to open, closed or hybrid cross-sections. Design optimization is performed by nonlinear programming techniques. Laminate thickness and lamina orientations are considered as design variables.     

MPI-based parallel synchronous vector evaluated particle swarm optimization for multi-objective design optimization of composite structures

This paper presents a decentralized/peer-to-peer architecture-based parallel version of the vector evaluated particle swarm optimization (VEPSO) algorithm for multi-objective design optimization of laminated composite plates using message passing interface (MPI). The design optimization of laminated composite plates being a combinatorially explosive constrained non-linear optimization problem (CNOP), with many design variables and a vast solution space, warrants the use of non-parametric and heuristic optimization algorithms like PSO. Optimization requires minimizing both the weight and cost of these composite plates, simultaneously, which renders the problem multi-objective. Hence VEPSO, a multi-objective variant of the PSO algorithm, is used. Despite the use of such a heuristic, the application problem, being computationally intensive, suffers from long execution times due to sequential computation. Hence, a parallel version of the PSO algorithm for the problem has been developed to run on several nodes of an IBM P720 cluster. The proposed parallel algorithm, using MPI's collective communication directives, establishes a peer-to-peer relationship between the constituent parallel processes, deviating from the more common master-slave approach, in achieving reduction of computation time by factor of up to 10. Finally we show the effectiveness of the proposed parallel algorithm by comparing it with a serial implementation of VEPSO and a parallel implementation of the vector evaluated genetic algorithm (VEGA) for the same design problem.     

Effectiveness of optimal design with respect to computational models for laminated composite structures weakened by holes

The present work deals with the shape optimization of laminated structures via semianalytical sensitivity anslysis, based on linear programming. Particular attention is focused on the concise and correct finite element formulation of the problem taking into account explicit differentiation with respect to the control parameters. A series of numerical examples shows the influence of orthotropy parameters (treated as the level of anisotropy), stacking sequences, FE models and the employed 2-D plate theories of structures on the resulting optimal shapes.     

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.     

Stacking sequence optimization with genetic algorithm using a two-level approximation

We propose a new method for laminate stacking sequence optimization based on a two-level approximation and genetic algorithm (GA), and establish an optimization model including continuous size variables (thicknesses of plies) and discrete variables (0/1 variables that represent the existence of each ply). To solve this problem, a first-level approximate problem is constructed using the branched multipoint approximate (BMA) function. Since mixed-variables are involved in the first-level approximate problem, a new optimization strategy is introduced. The discrete variables are optimized through the GA. When calculating the fitness of each member in the population of GA, a second-level approximate problem that can be solved by the dual method is established to obtain the optimal thicknesses corresponding to the each given ply orientation sequence. The two-level approximation genetic algorithm optimization is performed starting from a ground laminate structure, which could include relatively arbitrarily discrete set of angles. The method is first applied to cylindrical laminate design examples to demonstrate its efficiency and accuracy compared with known methods. The capacity of the optimization strategy to solve more complex problems is then demonstrated using a design example. With the presented method, the stacking sequence in analytical tools can be directly taken as design variables and no intermediate variables need be adopted.     

Dynamic instability of composite laminated rectangular plates and prismatic plate structures

Periodic dynamic loadings may cause dynamic instability of a structure through parametric resonance. In this paper, a B-spline finite strip method (FSM) is presented for the dynamic instability analysis of composite laminated rectangular plates and prismatic plate structures, based on the use of first-order shear deformation plate theory (SDPT). The equations of motion of a structure are established by using Lagrange's formulation and they are a set of coupled Mathieu equations. The boundary parametric resonance frequencies of the motion are determined by using the method suggested by Bolotin through a novel development which incorporates the Sturm sequence method and the multi-level substructuring technique to achieve reliability, efficiency and accuracy. Various loading patterns, arbitrary lamination and general boundary conditions are accommodated. A variety of numerical applications is presented to test the developed method and to study the dynamic instability behaviour of single plates and of complicated plate structures under various types of dynamic loading. A dynamic instability index (DII) is devised to measure the degree of instability against certain parameters which include the thickness-to-length ratio, the degree of orthotropy, the fibre orientation, the loading pattern and the boundary conditions.     

Nonlinear buckling optimization of composite structures

The paper presents an approach to nonlinear buckling fiber angle optimization of laminated composite shell structures. The approach accounts for the geometrically nonlinear behaviour of the structure by utilizing response analysis up until the critical point. Sensitivity information is obtained efficiently by an estimated critical load factor at a precritical state. In the optimization formulation, which is formulated as a mathematical programming problem and solved using gradient-based techniques, a number of the lowest buckling factors are included such that the risk of “mode switching” during optimization is avoided. The presented optimization formulation is compared to the traditional linear buckling formulation and two numerical examples, including a large laminated composite wind turbine main spar, to clearly illustrate the pitfalls of the traditional formulation and the advantage and potential of the presented approach.     

Artificial Bee Colony (ABC) for multi-objective design optimization of composite structures

In this paper, we present a generic method/model for multi-objective design optimization of laminated composite components, based on Vector Evaluated Artificial Bee Colony (VEABC) algorithm. VEABC is a parallel vector evaluated type, swarm intelligence multi-objective variant of the Artificial Bee Colony algorithm (ABC). In the current work a modified version of VEABC algorithm for discrete variables has been developed and implemented successfully for the multi-objective design optimization of composites. The problem is formulated with multiple objectives of minimizing weight and the total cost of the composite component to achieve a specified strength. The primary optimization variables are the number of layers, its stacking sequence (the orientation of the layers) and thickness of each layer. The classical lamination theory is utilized to determine the stresses in the component and the design is evaluated based on three failure criteria: failure mechanism based failure criteria, maximum stress failure criteria and the tsai-wu failure criteria. The optimization method is validated for a number of different loading configurations—uniaxial, biaxial and bending loads. The design optimization has been carried for both variable stacking sequences, as well fixed standard stacking schemes and a comparative study of the different design configurations evolved has been presented. Finally the performance is evaluated in comparison with other nature inspired techniques which includes Particle Swarm Optimization (PSO), Artificial Immune System (AIS) and Genetic Algorithm (GA). The performance of ABC is at par with that of PSO, AIS and GA for all the loading configurations.