Aeroelastic tailoring using fiber orientation and topology optimization

This work presents a structural optimization aided design methodology for composite laminated plates subject to fluid-structure interaction. The goal of the optimization procedure is to increase the flutter speed onset through the maximization of natural frequencies related to the vibration modes involved in the phenomenon. The aeroelastic stability analysis is performed using ZAERO software system, which includes ZONA 6 unsteady lifting surface method. The finite element method is applied to solve the structural model equilibrium equations, the eigenvalues sensitivities with respect to design variables are calculated analytically, and sequential linear programming is applied. The maximization is accomplished using two methods; the first method uses an aeroelastic analysis to determine which eigenmode causes the flutter onset, and its eigenvalue is then maximized. In the second method, a forward finite difference method is applied and the flutter speed sensitivities with respect to the eigenvalues are calculated. This sensitivity is used to guide the optimization process. Finally, a topology optimization problem is formulated to reduce the plate mass under a minimum flutter velocity constraint, using density distribution as the design variable.     

The genetic algorithm applied to stiffness maximization of laminated plates: review and comparison

The design of laminated structures is highly tailorable owing to the large number of available design variables, thereby requiring an optimization method for effective design. Furthermore, in practice, the design problem translates to a discrete global optimization problem which requires a robust optimization method such as the genetic algorithm. In this paper, the genetic algorithm, based on the real variable coding, is applied to the strain energy minimization of rectangular laminated composite plates. The results for both a point load and uniformly distributed load compare well with those achieved using trajectory methods for continuous global optimization.     

Topology optimization for minimum compliance under multiple loads based on continuous distribution of members

A method to minimize the compliance of structures subject to multiple load cases is presented. Firstly, the material distribution in design domain is optimized to form a truss-like continuum. The anisotropic composite is employed as the material model to simulate the constitutive relation of the truss-like continuum. The member densities and orientations at the nodes are taken as design variables. The member densities and orientations at any point in an element vary continuously. Then, parts of members, which are formed according to the member distribution field, are chosen to form the nearly optimum discrete structure. Lastly, the positions of the nodes and the cross-sectional areas of the members are optimized. In the above process, numerical instabilities such as checkerboard and mesh dependencies disappear without any additional technique. The sensitivities of the compliance are derived. Examples are presented to demonstrate the capability of the proposed method.     

Shape optimization of axisymmetric shells using a higher-order shear deformation theory

In this paper structural and sensitivity analysis for the optimization of laminated axisymmetric shells subjected to static constraints and arbitrary loading is presented. The shell thickness, radial coordinate of a nodal point, lamina thickness and the angle of orientation of the fibers are the design variables. The objective of the design optimization is the minimization of the volume of the shell or the strain energy. The model is based on a three-node axisymmetric finite element with 24 degrees of freedom. A higher-order theory is developed for the nonlinear distribution of the meridional displacement component through the thickness of the shell. The sensitivities of the discrete model developed are evaluated analytically using a symbolic manipulator. The efficiency and accuracy of the proposed model is discussed with reference to the applications.     

Optimization formulations for the maximum nonlinear buckling load of composite structures

This paper focuses on criterion functions for gradient based optimization of the buckling load of laminated composite structures considering different types of buckling behaviour. A local criterion is developed, and is, together with a range of local and global criterion functions from literature, benchmarked on a number of numerical examples of laminated composite structures for the maximization of the buckling load considering fiber angle design variables. The optimization formulations are based on either linear or geometrically nonlinear analysis and formulated as mathematical programming problems solved using gradient based techniques. The developed local criterion is formulated such it captures nonlinear effects upon loading and proves useful for both analysis purposes and as a criterion for use in nonlinear buckling optimization.     

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.     

Characterization of delamination effect on composite laminates using a new generalized layerwise approach

A dynamic analysis method has been developed to investigate and characterize the effect due to presence of discrete single and multiple embedded delaminations on the dynamic response of composite laminated structures with balanced/unbalanced and arbitrary stacking sequences in terms of number, placement, mode shapes and natural frequencies. A new generalized layerwise finite element model is developed to model the presence of multiple finite delamination in laminated composites. The new theory accurately predicts the interlaminar shear stresses while maintaining computational efficiency.     

A laminate parametrization technique for discrete ply-angle problems with manufacturing constraints

In this paper we present a novel laminate parametrization technique for layered composite structures that can handle problems in which the ply angles are limited to a discrete set. In the proposed technique, the classical laminate stiffnesses are expressed as a linear combination of the discrete options and design-variable weights. An exact $\ell _{1}$ penalty function is employed to drive the solution toward discrete 0–1 designs. The proposed technique can be used as either an alternative or an enhancement to SIMP-type methods such as discrete material optimization (DMO). Unlike mixed-integer approaches, our laminate parametrization technique is well suited for gradient-based design optimization. The proposed laminate parametrization is demonstrated on the compliance design of laminated plates and the buckling design of a laminated stiffened panel. The results demonstrate that the approach is an effective alternative to DMO methods.     

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.     

Optimization strategies for discrete multi-material stiffness optimization

Design of composite laminated lay-ups are formulated as discrete multi-material selection problems. The design problem can be modeled as a non-convex mixed-integer optimization problem. Such problems are in general only solvable to global optimality for small to moderate sized problems. To attack larger problem instances we formulate convex and non-convex continuous relaxations which can be solved using gradient based optimization algorithms. The convex relaxation yields a lower bound on the attainable performance. The optimal solution to the convex relaxation is used as a starting guess in a continuation approach where the convex relaxation is changed to a non-convex relaxation by introduction of a quadratic penalty constraint whereby intermediate-valued designs are prevented. The minimum compliance, mass constrained multiple load case problem is formulated and solved for a number of examples which numerically confirm the sought properties of the new scheme in terms of convergence to a discrete solution.     

Mixed least-squares finite element model for the static analysis of laminated composite plates

This paper presents a mixed finite element model for the static analysis of laminated composite plates. The formulation is based on the least-squares variational principle, which is an alternative approach to the mixed weak form finite element models. The mixed least-squares finite element model considers the first-order shear deformation theory with generalized displacements and stress resultants as independent variables. Specifically, the mixed model is developed using equal-order C⁰ Lagrange interpolation functions of high p-levels along with full integration. This mixed least-squares-based discrete model yields a symmetric and positive-definite system of algebraic equations. The predictive capability of the proposed model is demonstrated by numerical examples of the static analysis of four laminated composite plates, with different boundary conditions and various side-to-thickness ratios. Particularly, the mixed least-squares model with high-order interpolation functions is shown to be insensitive to shear-locking.     

Maximum stiffness and minimum weight optimization of laminated composite beams using continuous fiber angles

This paper deals with identification of optimal fiber orientations and laminate thicknesses in maximum stiffness and minimum weight design of laminated composite beams. The structural response is evaluated using beam finite elements which correctly account for the influence of the fiber orientation and cross section geometry. The resulting finite element matrices are significantly smaller than those obtained using equivalent finite element models. This modeling approach is therefore an attractive alternative in computationally intensive applications at the conceptual design stage where the focus is on the global structural response. An optimization strategy is presented which aims at enabling the use of fiber angles as continuous design variables albeit the problems may have many local minima. A sequence of closely related problems with an increasing number of design variables is treated. The design found for a problem in the sequence is projected to generate the starting point for the next problem in the sequence. Numerical results are presented for cantilever beams with different geometries and load cases. The results indicate that the devised strategy is well suited for finding optimal fiber orientations and laminate thicknesses in the design of slender laminated composite structures.     

Optimum design of structures with high-quality approximation of frequency constraints

Optimum design of structures is achieved with multiple frequency constraints. A two-point approximation is employed to approximate the frequencies. Thus, in the specified move limits, the eigenvalue analysis of the structure is not required. To enhance the quality of approximation, the Rayleigh quotient is approximated in terms of some intermediate variables, using the two-point approximation. The analytical derivatives of the eigenvalues are used in the design process. Examples are offered and the results are compared with previous work.     

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.     

Optimal design of infinite repetitive structures

An approach for designing optimal repetitive structures under arbitrary static loading is presented. It is shown that the analysis of such infinite structures can be reduced to the analysis of the repeating module under transformed loading and boundary conditions. Consequently, both the design parameters and the analysis variables constitute a relatively small set which facilitates the optimization process. The approach hinges on the representative cell method. It is based on formulating the analysis equations and the continuity conditions for a sequence of typical modules. Then, by means of the discrete Fourier transform this problem translates into a boundary value problem of a representative cell in transformed variables, which can be solved by any appropriate analytical or numerical method. The real structural response any-where in the structure is then obtained by the inverse transform. The sensitivities can also be calculated on the basis of the sensitivities of the representative cell. The method is illustrated by the design for minimum compliance with a volume constraint of an infinite plane truss. It is shown that by employing this analysis method within an optimal design scheme one can incorporate a reduced analysis problem in an intrinsically small design space.     

基于假定应变法的协同转动四边形复合材料壳单元

为对复合材料层合板壳结构进行精确的大变形数值模拟,提出一种采用假定应变法的能分析层合结构大转动问题的协同转动四边形壳单元.该方法在建立有限元公式时引入假定应变法以克服膜闭锁和剪切闭锁的不利影响.与其他能分析大转动问题的复合材料壳单元相比,在新的协同转动框架中采用矢量型转动变量,可大大降低在非线性增量求解过程中更新转动变量的难度,且能得到对称的单元切线刚度矩阵,提高单元的计算效率.分析两个典型算例,并与其他学者的结果进行对比,结果表明在计算层合结构大转角问题时拥有较好的精度和收敛性.     

Optimal design of hybrid laminated composite plates with dynamic and static constraints

Optimization procedures are presented that consider the static and dynamic characteristic constraints for laminated composite plates and hybrid laminated composite plates subject to a concentrated load on the center of the plate. The design variables adopted are ply angle or ply thickness. Considered constraints are deflection, natural frequency and specific damping capacity. Using a recursive linear programming method, nonlinear optimization problems are solved, and by introducing the design scaling factor, the number of iterations is reduced significantly. Relating interactive optimization procedures with the finite element method analysis, various hybrid composite plates with arbitrary boundary conditions can be designed optimally. In the optimization procedure, verification of analysis and design of the laminated composite plates are compared with a previous paper. Various design results are presented on laminated composite plates and hybrid laminated composite plates.     

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.     

DMTO – a method for Discrete Material and Thickness Optimization of laminated composite structures

This paper presents a gradient based topology optimization method for Discrete Material and Thickness Optimization of laminated composite structures, labelled the DMTO method. The capabilities of the proposed method are demonstrated on mass minimization, subject to constraints on the structural criteria; buckling load factors, eigenfrequencies, and limited displacements. Furthermore, common design guidelines or rules, referred to as manufacturing constraints, are included explicitly in the optimization problem as series of linear inequalities. The material selection and thickness variation are optimized simultaneously through interpolation functions with penalization. Numerical results for several parameterizations of a finite element model of a generic main spar from a wind turbine blade are presented. The different parameterizations represent different levels of complexity with respect to manufacturability. The results will thus give insight into the relation between potential weight saving and design complexity. The results show that the DMTO method is capable of solving the problems robustly with only few intermediate valued design variables.