On compliance and buckling objective functions in topology optimization of snap-through problems

This paper deals with topology optimization of static geometrically nonlinear structures experiencing snap-through behaviour. Different compliance and buckling criterion functions are studied and applied for topology optimization of a point loaded curved beam problem with the aim of maximizing the snap-through buckling load. The response of the optimized structures obtained using the considered objective functions are evaluated and compared. Due to the intrinsic nonlinear nature of the problem, the load level at which the objective function is evaluated has a tremendous effect on the resulting optimized design. A well-known issue in buckling topology optimization is artificial buckling modes in low density regions. The typical remedy applied for linear buckling does not have a natural extension to nonlinear problems, and we propose an alternative approach. Some possible negative implications of using symmetry to reduce the model size are highlighted and it is demonstrated how an initial symmetric buckling response may change to an asymmetric buckling response during the optimization process. This problem may partly be avoided by not exploiting symmetry, however special requirements are needed of the analysis method and optimization formulation. We apply a nonlinear path tracing algorithm capable of detecting different types of stability points and an optimization formulation that handles possible mode switching. This is an extension into the topology optimization realm of a method developed, and used for, fiber angle optimization in laminated composite structures. We finally discuss and pinpoint some of the issues related to buckling topology optimization that remains unsolved and demands further research.     

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.     

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.     

Sensitivity analysis and optimization of thin laminated structures with a nonsmooth eigenvalue based criterion

This paper presents a discrete model for the design sensitivity analysis of thin laminated angle-ply composite structures using a plate shell element based on a Kirchhoff discrete theory for the bending effects. To overcome the nondifferentiability of multiple eigenvalues, which may occur during a structural optimization involving free vibrations or buckling design situations, a nonsmooth eigenvalue based criterion is implemented. Angle-ply design variables and vectorial distances from the laminated midle surface to the upper surface of each layer are considered as design variables. The design sensitivities and the directional derivatives are evaluated analytically. The efficiency and accuracy of the model developed is discussed with two illustrative cases which show the need to compute sensitivities of multiple eigenvalues as directional derivatives for laminated composite structures.     

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     

Crippling analysis of composite stringers based on complete unloading method

This paper addresses a nonlinear finite element method for the crippling analysis of composite laminated stringers. For the finite element modeling, a nine-node laminated shell element based on the first order shear deformation theory is used. Failure-induced stiffness degradation is simulated by the complete unloading method. A modified arc-length algorithm is incorporated in the nonlinear finite element method to trace the post-failure equilibrium path after a local buckling. Finite element results show excellent agreement with those of previous experiment. A parametric study is performed to assess the effect of the flange-width, web-height, and stacking sequence on the buckling, local buckling, and crippling stresses of stringers.     

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.     

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.     

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.

     

Discrete Material and Thickness Optimization of laminated composite structures including failure criteria

This work extends the Discrete Material and Thickness Optimization approach to structural optimization problems where strength considerations in the form of failure criteria are taken into account for laminated composite structures. It takes offset in the density approaches applied for stress constrained topology optimization of single-material problems and develops formulations for multi-material topology optimization problems applied for laminated composite structures. The method can be applied for both stress- and strain-based failure criteria. The large number of local constraints is reduced by the use of aggregate functions, and the developed approach is demonstrated for optimization problems involving both constant and varying thickness laminated composites.     

Benchmark case studies in optimization of geometrically nonlinear structures

A structural optimization algorithm is developed for truss and beam structures undergoing large deflections against instability. The method combines the nonlinear buckling analysis using the displacement control technique, with the optimality criteria approaches. Several benchmark case studies illustrate the procedure and the results are compared with examples reported in the literature. It is shown that a design based on the generalized eigenvalue problem (linear buckling) highly underestimates the optimum mass or overestimates the buckling load for these types of structures, so a design based on the linear buckling analysis may result in catastrophic failure. The effect of geometrical nonlinearities and element imperfections has also been studied.     

Detection of delamination in laminated composites with limited measurements combining PCA and dynamic QPSO

This paper presents an output only damage diagnostic algorithm based on frequency response functions and the principal components for health monitoring of laminated composite structures. The principal components evaluated from frequency response data, are employed as dynamical invariants to handle the effects of operational/environmental variability on the dynamic response of the structure. Finite element models of a laminated composite beam and plate are used to generate vibration data for healthy and damaged structures. Three numerical examples include a laminated composite beam, cantilever plate made of carbon–epoxy and a laminated composite simply supported plate. Varied levels of delamination of laminated composite plies and matrix cracking at varied locations in the plies are simulated at different spatial locations of the structure. Numerical investigations have been carried out to identify the spatial location of damage using the proposed principal component analysis (PCA) based algorithm. In order to limit the number of sensors on the structure, an optimal sensor placement algorithm based on PCA is employed in the present work and the effectiveness of the proposed algorithm with a limited number of sensors is also investigated. Finally, the inverse problem associated with the detection of delamination and matrix cracking is formulated as an optimization problem and is solved using the newly developed dynamic quantum particle swarm optimization (DQPSO) algorithm. Studies carried out and presented in this paper clearly indicate that the proposed SHM scheme can robustly identify the instant of damage, spatial location, the extent of delamination and matrix cracking even with limited sensor measurements and also with noisy data.     

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.     

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.     

An efficient facet shell element for corotational nonlinear analysis of thin and moderately thick laminated composite structures

In the present work, an efficient facet shell element for the geometrically nonlinear analysis of laminated composite structures using the corotational approach is developed. The facet element is developed by combining the discrete Kirchhoff-Mindlin triangular bending element (DKMT), and the optimal membrane triangular element (OPT). The membrane-bending coupling effect of composite laminates is incorporated in the formulation, and inconsistent stress stiffness matrix is formulated. Using corotational formulation and the proposed facet element, some example laminated composite structures with geometric nonlinearity are analyzed, and the results are compared with those found using other facet elements.     

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.     

Accurate computation of critical local quantities in composite laminated plates under transverse loading

The design of laminated composite based components requires a detailed analysis of the response of the structure when subjected to external loads. For the analysis of laminated composite plates, several plate theories have been proposed in the literature. Generally, these plate theories are used to obtain certain global response quantities like the buckling load. However, the use of these theories to obtain local response quantities, i.e. point-wise stresses; interlaminar stresses and strains, can lead to significant errors.In this paper, a detailed study of the quality of the point-wise stresses obtained using higher-order shear deformable, hierarchic and layerwise theories is done for a plate under transverse loading. The effect of equilibrium based post-processing on the transverse stress quantities is also studied. From the detailed study it is observed that the layerwise theory is very accurate. However, for all the models proper mesh design is required to capture boundary layer effects, discretization error, etc. Using focussed adaptivity, and post-processed state of stress, accurate representation of the local state of stress can be obtained, even with the higher-order shear deformable theories. Using this approach, the first-ply failure load is obtained with the Tsai–Wu criterion. It is observed that use of an adaptive procedure leads to significantly lower failure loads as compared to those given in the literature.     

Buckling and postbuckling of circular plates containing concentric penny-shaped delaminations

Buckling and postbuckling analyses of circular laminated composite plates with delaminations are presented. An axisymmetric finite element model based on a layer-wise laminated composite plate theory is developed to formulate the problem. Geometric nonlinearity in the sense of von Kármán and imperfections in the form of initial global deflection and initial delamination openings are included. A simple contact algorithm which precludes the physically inadmissible overlapping between delaminated surfaces is proposed and incorporated into the analysis.

Numerical results are obtained addressing the effects of the initial imperfections, the number of delaminations and their sizes on the critical buckling load and buckling mode shapes as well as postbuckling responses.     

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.