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The application of Topology Optimisation (TO) to help in the manufacture of metallic components in highly technological industries has increased recently. To equally benefit from TO, the construction industry must address its specific issues, such as adherence to code requirements and challenges in using cutting-edge software packages in complex joints with non-linear behaviour. To assist in such a challenge, the current study offers a methodology proposal to synthesise code and structural behaviour requirements into geometrical constraints for the optimisation problem of laminar steel parts under compression while integrating Non-linear Finite Element Analyses (NLFEA) that ensure the safety of the solution. It has been found that, for a real case-study, the initial volume of a connection's cover-plate can be decreased by up to 40% while maintaining the connection's original capacity and that a 30% volume decrease may be achieved while keeping the original plate capacity. In both cases, the plate's ultimate deformation capacity was enhanced. Evidence has been found that Linear Elastic TO may not provide safe-sided solutions for parts with an intrinsic non-linear behaviour. With the attained volume reductions, less raw materials may be consumed, assembly and transportation will be facilitated, and the goals of the sector's decarbonisation, energy intensity and sustainability will be favoured. [ABSTRACT FROM AUTHOR]

Interest in components with detailed structures increased with the progress in advanced manufacturing techniques in recent years. Parts with graded lattice elements can provide interesting mechanical, thermal, and acoustic properties compared to parts where only coarse features are included. One of these improvements is better global buckling resistance of the component. However, thin features are prone to local buckling. Normally, analyses with high-computational effort are conducted on high-resolution finite element meshes to optimize parts with good global and local stability. Until recently, works focused only on either global or local buckling behavior. We use two-scale optimization based on asymptotic homogenization of elastic properties and local buckling behavior to reduce the effort of full-scale analyses. For this, we present an approach for concurrent local and global buckling optimization of parameterized graded lattice structures. It is based on a worst-case model for the homogenized buckling load factor, which acts as a safeguard against pure local buckling. Cross-modes residing on both scales are not detected. We support our theory with numerical examples and validations on dehomogenized designs, which show the capabilities of our method, and discuss the advantages and limitations of the worst-case model. [ABSTRACT FROM AUTHOR]

Variable-angle filament-wound (VAFW) cylinders are herein optimized for minimum mass under manufacturing constraints, and for various design loads. A design parameterization based on a second-order polynomial variation of the tow winding angle along the axial direction of the cylinders is utilized to explore the nonlinear steering-thickness dependency in VAFW structures, whereby the thickness becomes a function of the filament steering angle. Particle swarm optimization coupled with three Kriging-based metamodels is used to find the optimum designs. A single-curvature Bogner–Fox–Schmit–Castro finite element is formulated to accurately and efficiently represent the variable stiffness properties of the shells, and verifications are performed using a general purpose plate element. Alongside the main optimization studies, a vast analysis of the design space is performed using the metamodels, showing a gap in the design space for the buckling strength that is confirmed by genetic algorithm optimizations. Extreme lightweight while buckling-resistant designs are reached, along with non-conventional optimum layouts thanks to the high degree of thickness build-up tailoring. [ABSTRACT FROM AUTHOR]

The present study addresses a simplified topology optimization method to improve load bearing capacity. Topology optimization taking account of structurally nonlinear and complex behavior is an important topic from the viewpoint of mechanics. However, most methodologies require complex analytical derivation of sensitivity analysis and high computational cost to obtain the optimal solution. In light of this, we propose a practical design method employing the principal direction of Cauchy stress and the directional statistics, to improve load bearing capacity indirectly with much lower computational cost than those based on complex nonlinear structural analysis. Finally, we discuss setting of optimization problems, and demonstrate the accuracy and performance of the proposed method with a series of numerical examples. [ABSTRACT FROM AUTHOR]

This paper investigates the weight minimization of stiffened panels simultaneously optimizing sizing, layout, and topology under stress and buckling constraints. An effective topology optimization parameterization is presented using multiple level-set functions. Plate elements are employed to model the stiffened panels. The stiffeners are parametrized by implicit level-set functions. The internal topologies of the stiffeners are optimized as well as their layout. A free-form mesh deformation approach is improved to adjust the finite element mesh. Sizing optimization is also included. The thicknesses of the skin and stiffeners are optimized. Bending, shear, and membrane stresses are evaluated at the bottom, middle, and top surfaces of the elements. A p-norm function is used to aggregate these stresses in a single constraint. To solve the optimization problem, a semi-analytical sensitivity analysis is performed, and the optimization algorithm is outlined. Numerical investigations demonstrate and validate the proposed method. [ABSTRACT FROM AUTHOR]

Variable stiffness (VS) composites made by fiber steering have received intensive attention due to the tailorability of the stiffness and strength properties. However, a significantly larger number of elements are required to converge for VS composites, which results in much longer runtime and makes the design and optimization more complicated. In this paper, an efficient design optimization method assisted by multi-fidelity surrogate models is presented for the buckling design of VS composites. To reduce the computational burden, a multi-fidelity surrogate model called hierarchical Kriging is constructed through a few expensive high-fidelity samples and many cheap low-fidelity samples. Fine and coarse finite element (FE) analysis is performed respectively to calculate the structural responses for corresponding datasets. The efficient global optimization based on a modified expected improvement criterion is employed and used to adaptively add new samples of variable-fidelity. Two case studies, a composite plate subjected to uniform uniaxial compression and a composite cylinder under pure bending, are investigated. The effects of different number of design variables and coarse FE model mesh density on the optimum configuration are studied to demonstrate the effectiveness and robustness of the method. The results indicate that the present method can remarkably reduce the number of high-fidelity FE evaluations and improve the optimization efficiency when compared with available methods in the literature. Additionally, the investigation in the mechanism of loading carrying capacity improvement shows that the increase is mainly due to the load redistribution. [ABSTRACT FROM AUTHOR]

This work deals with the multi-scale optimisation of composite structures by adopting a general global-local (GL) modelling strategy to assess the structure responses at different scales. The GL modelling approach is integrated into the multi-scale two-level optimisation strategy (MS2LOS) for composite structures. The resulting design strategy is, thus, called GL-MS2LOS and aims at proposing a very general formulation of the design problem, without introducing simplifying hypotheses on the laminate stack and by considering, as design variables, the full set of geometric and mechanical parameters defining the behaviour of the composite structure at each pertinent scale. By employing a GL modelling approach, most of the limitations of well-established design strategies, based on analytical or semi-empirical models, are overcome. The GL-MS2LOS makes use of the polar formalism to describe the anisotropy of the composite at the macroscopic scale (where it is modelled as an equivalent homogeneous anisotropic plate). In this work, deterministic algorithms are exploited during the solution search phase. The challenge, when dealing with such a design problem, is to develop a suitable formulation and dedicated operators, to link global and local models physical responses and their gradients. Closed-form expressions of structural responses gradients are rigorously derived by taking into account for the coupling effects when passing from global to local models. The effectiveness of the GL-MS2LOS is proven on a meaningful benchmark: the least-weight design of a cantilever wing subject to different design requirements. Constraints include maximum allowable displacements, maximum allowable strains, blending, manufacturability requirements and buckling factor. [ABSTRACT FROM AUTHOR]

A sign optimization algorithm (SOA) is proposed to design the "±" signs in composite laminates to control the bending-twisting coupling effects. Owing to that the bending-twisting coupling stiffness are cubic on thickness, the innovation is to design the signs of ply orientations from the mid-plane to the outermost sequentially and iteratively. In this manner, the nondimensional anisotropic coefficients are controlled to the target values. Numerical examples are adopted to verify the effectiveness and efficiency of SOA. First, the signs of symmetric laminates [θ32]s are optimized with various boundaries, load ratios, and aspect ratios to show the bending-twisting coupling effects on bending, buckling, and vibration responses of composite plates. Second, the bending-twisting coupling effects are minimized since they may cause large errors in buckling load prediction when using closed-form solution after neglecting them. Third, the optimal sequences obtained from heuristic algorithms are employed for sign optimization. Results show that the bending-twisting coupling effects cannot be neglected; moreover, the buckling and vibration performances can be further improved by redesigning "±" signs in composite laminates. This research aims to provide a design technique to minimize the error induced by bending-twisting coupling and increase the probability to find the global optimum. [ABSTRACT FROM AUTHOR]

To resist external pressure, stiffened spherical domes have been widely used in underwater vehicles. In traditional design of these thin-walled structures, uniform stiffeners are usually utilized and overly conservative safety factor method is employed to envelop the multi-source uncertainties mainly caused by manufacturing. In this study, a concept of non-uniformly stiffened spherical dome (NUSPD) is developed to improve the structural buckling capacity. Furthermore, an efficient reliability-based design optimization (RBDO) framework is established for NUSPD, which includes a deterministic optimization loop to fix the stiffener layout, and subsequently followed by a RBDO loop to determine the detailed stiffener dimensions and decrease the structural weight under geometric and material property uncertainties. Since the RBDO of stiffened spherical domes is very complex, an improved enhanced chaos control method together with surrogate model is proposed to improve the convergence rate. Finally, the efficiency of the proposed RBDO framework is demonstrated by the NUSPD example. [ABSTRACT FROM AUTHOR]

A novel design concept for buckling-induced mechanical metamaterials for energy absorption is presented. The force-displacement curves of the mechanical metamaterials are analyzed according to the curves of their unit cells, and the energy-absorbing characteristics of mechanical metamaterials are evaluated. Two topology optimization models are proposed. One maximizes the buckling-induced dissipated energy to facilitate the design of metamaterials with high energy absorption and low elastic strain energy. The other maximizes the dissipated energy with a constraint that the mechanical metamaterials should be self-recoverable. An energy interpolation scheme is employed to avoid numerical instabilities in the geometric nonlinear finite element analysis. A two-phase algorithm is proposed to find the optimized result from a uniform initial guess, and sensitivity analysis is performed. The optimized design has a larger amount of buckling-induced dissipated energy than the previously proposed structural prototypes. Moreover, the self-recoverable mechanical metamaterial is successfully designed by topology optimization. [ABSTRACT FROM AUTHOR]

Variable-stiffness panel is very promising for the cutout reinforcement of composite structures. However, due to the increase of design variables, the optimization of variable-stiffness panels becomes very challenging, even if surrogate model is utilized, because the fidelity of surrogate model is difficult to guarantee for high-dimensional problems. In this study, isogeometric analysis method (IGA) is employed to predict the buckling load of variable-stiffness panels, which can produce accurate prediction with less computational cost compared to traditional FEA, moreover, it can provide analytical sensitivity for optimization. On this basis, an adaptive gradient-enhanced kriging (GEK) model assisted by a novel multiple points infilling criterion is constructed for the global optimization of variable-stiffness composite panels. The proposed method is compared with traditional surrogate model, and results show that the proposed method can find a better optimum design in a more efficient manner. It can be concluded that the proposed method is able to fully explore the advantages of IGA including exact modelling, analysis and analytical sensitivity, which is particularly suitable for the design of variable-stiffness panels and other complex structures. [ABSTRACT FROM AUTHOR]

Shallow domes subjected to external pressure are extensively used in missile structures. The critical failure mode for these domes is buckling due to external pressure. Different closed form solutions are available to evaluate buckling pressure of dome shapes like ellipsoid and torisphere. The torisiphere dome is the optimum dome shape among conventional domes. Shape optimization is carried out to find the optimal dome shape among shallow domes subjected to external pressure. Dome geometry is generalized by cubic bezier polynomials. For carrying out shape optimization, a low fidelity model is preferred which can predict the critical buckling pressure of a general dome shape. Towards this a unified model is proposed which meets the above requirement. Using this unified model, shape optimization of dome for minimization of mass is carried out subjected to buckling constraint. The study yielded a dome shape different from conventional dome shapes with a mass saving of 6% over torispherical dome while meeting the buckling constraint. The results of unified model are also validated with high fidelity Finite Element Analysis. [ABSTRACT FROM AUTHOR]

The localized analytical sensitivity for eigenfrequency is extended to the non-linear problem of 3D continuum buckling analysis. Implemented in a finite element approach the inherent complexity of mode switching and multiple eigenvalues is found not to be a practical problem. The number of necessary redesigns is of the order 10-20 as illustrated by a specific example, where also different cases of stiffness interpolation are exemplified. [ABSTRACT FROM AUTHOR]

For tailoring the non-uniform axial compression, each sub-panel of stiffened shells should be designed separately to achieve a high load-carrying efficiency. Motivated by the challenge caused by numerous variables and high computational cost, a fast procedure for the minimum weight design of non-uniform stiffened shells under buckling constraint is proposed, which decomposes a hyper multi-dimensional problem into a hierarchical optimization with two levels. To facilitate the post-buckling optimization, an efficient equivalent analysis model of stiffened shells is developed based on the Numerical Implementation of Asymptotic Homogenization Method. In particular, the effects of non-uniform load, internal pressure and geometric imperfections are taken into account during the optimization. Finally, a typical fuel tank of launch vehicle is utilized to demonstrate the effectiveness of the proposed procedure, and detailed comparison with other optimization methodologies is made. [ABSTRACT FROM AUTHOR]

Structural topology optimisation has mainly been applied to strength and stiffness objectives, due to the ease of calculating the sensitivities for such problems. In contrast, dynamic and buckling objectives require time consuming central difference schemes, or inefficient non-gradient algorithms, for calculation of the sensitivities. Further, soft-kill algorithms suffer from numerous numerical issues, such as localised artificial modes and mode switching. This has resulted in little focus on structural topology optimisation for dynamic and buckling objectives. In this work it is found that nominal stress contours can be derived from applying the vibration and buckling mode shapes as displacement fields, defined as the dynamic and buckling von Mises stress, respectively. This paper shows that there is an equivalence between the dynamic von Mises stress and the frequency sensitivity numbers for element removal and addition in bidirectional evolutionary structural optimisation. Likewise, it was found that the contours of buckling von Mises stress and buckling sensitivity numbers are analogous; therefore, an equivalence is shown for element removal and addition. The examples demonstrate consistent resulting topologies from the two different formulations for both dynamic and buckling criteria. This article aims to develop a simple alternative, based on visual correlation with a mathematical verification, for topology optimisation with dynamic and buckling criteria. [ABSTRACT FROM AUTHOR]

In the present study, Multi-objective optimization of composite cylindrical shell under external hydrostatic pressure was investigated. Parameters of mass, cost and buckling pressure as fitness functions and failure criteria as optimization criterion were considered. The objective function of buckling has been used by performing the analytical energy equations and Tsai-Wu and Hashin failure criteria have been considered. Multi-objective optimization was performed by improving the evolutionary algorithm of NSGA-II. Also the kind of material, quantity of layers and fiber orientations have been considered as design variables. After optimizing, Pareto front and corresponding points to Pareto front are presented. Trade of points which have optimized mass and cost were selected by determining the specified pressure as design criteria. Finally, an optimized model of composite cylindrical shell with the optimum pattern of fiber orientations having appropriate cost and mass is presented which can tolerate the maximum external hydrostatic pressure. [ABSTRACT FROM AUTHOR]

In the present work, in order to tackle the linear elastic buckling problem, we develop a shape optimization process based on a free-form optimization method to optimize a shell structure under out-of-plane and in-plane shape variations. The free-form optimization method is a node-based method in which mesh regularity can be maintained and shape design parameterization is not required. It has the advantages of sufficient efficiency for treating large-scale problems and the ability to realize a smooth shape. The buckling coefficient in buckling mode 1 is set as the objective function in the shape optimization problem. We consider repeated eigenvalues and volume constraint for the whole optimization process. Three numerical examples are presented in this work to illustrate the shape optimization algorithm and to show that it can increase the buckling strength substantially, especially under shape variation in the out-of-plane direction. [ABSTRACT FROM AUTHOR]

This paper investigates topology design optimization for maximizing critical buckling loads of thin-walled structures using a moving iso-surface threshold (MIST) method. Formulation for maximizing linear buckling loads with additional constraints on load-path continuity and lower bound of eigenvalue is firstly presented. New physical response functions are proposed and expressed in terms of the strain energy densities determined in the two-steps of finite element buckling analysis. A novel approach by introducing a connectivity coefficient is developed to ensure continuity of effective load-path in optimum topology. The lower bound of eigenvalue is defined to eliminate spurious localized buckling modes. The MIST algorithm and its interfaces with commercial finite element (FE) software are given in detail. Numerical results are presented for topology optimization of plate-like structures to maximize critical buckling forces or displacements considering in-plane and out-of-plane buckling respectively. The FE analyses of the re-meshed final solid topologies with and without void material reveal that the presence of the void material has a significant effect on the out-of-plane buckling loads and a minor influence on the in-plane buckling loads. [ABSTRACT FROM AUTHOR]

An approach to take into account global and local stability in the design optimization of truss-like structures is presented. It is based on the use of frame elements and a single global stability constraint. A detailed discussion on key aspects of the proposed approach is presented, together with numerical examples. The proposed approach ensures global stability and does not require the use of Euler formula, avoiding several difficulties encountered in truss optimization. Besides, the examples illustrate that global stability is an important aspect to be taken into account in most practical cases. [ABSTRACT FROM AUTHOR]

This paper investigates the use of stability and serviceability objective functions in the shape optimization of truss arch footbridges prone to in-plane snap-through buckling. The objective functions evaluated relate to global linear buckling, geometrically nonlinear response, fundamental frequency, linear compliance, and maximum deflection. These objective functions are applied to help define the global structural shape for the 2D configuration of a truss arch footbridge subjected to its governing code-defined load combination. The strength criterion of maximum axial force, the global stability responses of critical linear buckling load and nonlinear limit load, and the serviceability responses of fundamental frequency and unfactored live load deflection are used to evaluate the optimized topologies. These structural performance results are compared to those of a benchmark structure prone to in-plane snap-through buckling. The results highlight that improvement in stability and serviceability behavior can be obtained by altering the global structural form according to the presented objective functions. Stable optimized topologies, which are not prone to in-plane snap-though buckling, are achieved without the use of computationally expensive, geometrically nonlinear analysis functions. [ABSTRACT FROM AUTHOR]

The lay-up optimization problems of composite plates against buckling are studied. The plate has symmetric lay-up and is loaded by an in-plane loading. The Classical Lamination Plate Theory is used. The necessary optimality conditions for the plate orthotropy/anisotropy optimization problems are considered, with special attention to bending-twisting coupling. The purpose of the optimization is to maximize the (lowest) buckling eigen value. The varied lay-up/layer orientation angles are considered both as smooth functions of the location coordinates and as having the same values at every point. It is shown that the twisting moment (calculated in the principal curvature axes) plays an important role in the conditions. Two example problems for a thin composite plate, loaded by shear, are considered. The first one corresponds to the case of one shear loading direction. The details of some known numerical solutions are studied. The obtained numerical results are in agreement with the theoretical results. The maximal lowest eigen values are 2-folded ones. A schematic model for treatment of the known optimal numerical solution (60° unidirectional lay-up for a long plate loaded by shear) is proposed. The second example problem corresponds to the case of shear loading acting in two opposite directions. The ways of equalizing the lowest buckling values corresponding to both loading directions are proposed. Numerical results for various aspect ratio and plate thickness values are presented. The potential of weight saving is demonstrated. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper is about the problem of optimization of bracing columns-height of the parallel chords, cross-sectional areas of the chord and bracing and area moments of inertia of the chord and bracing. Optimization parameters have been determined by Lagrange's multipliers method-inequality constraint functions. Total mass of the construction has been selected as the objective function. Criterions of buckling (general buckling of the bracing column, local buckling of the chord and local buckling of the bracing diagonal) and criterion of deformation are used as the constraint functions. Numerical examples are performed by usage of the obtained theoretical solutions. [ABSTRACT FROM AUTHOR]

In this paper the problem of the optimal design of thin-walled tubular columns under loadings controlled by displacement is investigated. A radius of cross-sectional circular profile varying along the axis of the shell-column as well as a wall thickness, which lead to the maximal axial displacement caused by compression before the structure buckles are sought. Both global (buckling of a column) and local (wall buckling of a shell) stability of a structure are taken into account. The geometry of the structure is approximated by the convex Bézier polynomial. The results are obtained using numerical optimization method, namely the simulated annealing method. [ABSTRACT FROM AUTHOR]