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This paper presents a new approach to the topology optimization of columns exposed to a loss of stability. The idea is to replace a conventional maximization of a buckling load by a locally formulated topology optimization problem based on compliance minimization. In order to do this, the standard instability analysis of a compressed column is perf...
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... order to illustrate this methodology the following example which makes use of the above idea is presented. A simply supported column of rectangular cross-section, presented in Fig. 1 has been chosen. The transverse loading is applied in one plane which stands also for the buckling one (Fig. 2). The well known solution of the Euler buckling problem leads to instability mode described by a sine function. This refers to both the deflection and the bending moment. The optimal topology has been generated for refe- rence column which consists of 14 × 14 × 200 cells (1 cm × 1 cm × 1 cm). The Young modulus of the column material is E = 200 GPa and the volume fraction of the final topology is set to 0.5. The minimal compliance column is given in Fig. 3. For the obtained topology the critical load 12592 kN has been calculated numerically using finite element system. The buckling mode is presented in Fig. 4. It is worth noting that the critical load calculated for prismatic column of the same volume is 3790 kN. While considering the possibility of buckling in both planes the above solution found for only one plane taken into account may not be correct if the critical load referring to out-of-plane buckling has a lower value. Indeed for the considered example out-of-plane buckling load is 5521 kN. In order to eliminate this drawback the formulation of topology optimization problem is extended so as to take into consideration the loadings applied in both planes, see Fig. 5. As the result the topology shown in Fig. 6 has been generated, for which critical load value is 11701 kN. To make details of the final topology better recognized, a section view is added in Fig. 7, together with a magnified view of a part of generated topology shown in Fig. 8. It is worth noting that assumption of doubly symmetrical column cross-section implies that instability considerations are limited to buckling in principal planes. The above presented introductory example shows that it is possible to achieve the effect of buckling load maximization via minimization of compliance performed for a specially matched loading scheme. It has been assumed for the above that the buckling load is unimodal. At the same time it is worth pointing out that for clamped-clamped column the unimodal formulation of the problem may be not sufficient to obtain optimal solution. If such situation occurs the modified - bimodal formulation of the problem has to be implemented. In order to generate optimal topologies the efficient algorithm is needed. The one used in this paper is based on Cellular Automata rules. The rules of Cellular Automata are briefly described in the next section. As it has been mentioned earlier in this paper the problem of topology optimization is formulated in a manner allowing to solve it with Cellular Automata method. The performance of CA algorithms, reported in literature, is based mostly on heuristic local rules. Similarly, in the present paper the efficient heuristic algorithm, introduced by Bochenek and Tajs-Zielińska (2012) has been implemented. The power law approach defining solid isotropic material with penalization (SIMP) with design variables being relative densities of a material has been utilized. The elastic modulus of each cell element is modelled as a function of relative density d i using power law, according to (4). This power p penalizes intermediate densities and drives design to a solid/void ...
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Citations
... strain energy to identify high-order pseudo modes and optimized the topology layout of the structure utilizing the solid isotropic material with penalization (SIMP) material model. Bochenek and Tajs-Zielińska [16] devised a novel approach for topology optimization of columns to prevent instability, where the buckling problem subjected to compression is substituted with a compliance problem subjected to transverse loads. Cheng and Xu [17] proposed a design method to design thin bending plates for achieving maximum out-of-plane buckling load, where the topologies of both macrostructure and microstructure are optimized. ...
Structures that are subjected to compressive loads may experience not only local fractures but also buckling deformations due to instability. Furthermore, the performance of the structure can also be influenced by uncertain factors. This paper develops a surrogate model-based method for reliability-oriented buckling topology optimization under random field load uncertainty. To reduce the number of reliability constraints, the compliance and buckling responses are synthesized into a hybrid response using the P-norm to construct the limit state function. The compressed load random field is characterized by Karhunen-Loéve expansion. To avoid the complexity of sensitivity analysis in reliability evaluation, the polynomial chaos expansion is employed to estimate the probabilistic constraints in the performance measure approach. The sensitivity of the limit state function with respect to design variables is derived using the chain rule and adjoint method. Three classical design examples are tested to show the effectiveness of the presented methodology. The optimized results demonstrate that the presented methodology is effective and that random field load uncertainty has a significant influence on the final design.
... Tovar and co-workers have published a series of papers regarding the implementation of Cellular Automata rules inspired by the phenomenon of functional adaptation observed in bones [40][41][42]. An efficient CA algorithm was also proposed and then developed by Bochenek and Tajs-Zielińska [43][44][45][46]. This paper takes a step further and presents a novel concept of flexible Cellular Automata rules and their implementation into structural topology optimization. ...
... Tsavdaridis et al. [8] used segmental optimization method (SOM) by superimposing 40 standard load combinations to obtain the optimal performance Al section. Bochenek et al. [9] transformed the compression bar maximization buckling load problem into a local flexibility minimization based on the cellular automata method (CA) topology optimization problem. Kai et al. [10] used the KS aggregation function to aggregate the structural load bearing surface node displacements to convert the restricted load bearing surface deformation problem into a standard quadratic programming subproblem to effectively control the local maximum deformation of the structure. ...
To solve the problems of low bending and torsional stiffness and high weight of the all-steel body-in-white (BIW), an aluminum (Al) alloy profile with optimized cross-section is used to replace the original sheet metal threshold beam. Firstly, considering the feasibility of extrusion process, the Al alloy profile cross-section is designed based on the original threshold beam cross-section. Then, we extracted the equivalent load acting on the center of mass of the threshold beam section during the calculation of the bending and torsional stiffness of the BIW, and carried out the planar topology optimization with the design boundary conditions to minimize the cross-sectional flexibility, resulting in an Al alloy section with superior performance. The optimization results show that the bending stiffness and torsional stiffness of the BIW are improved by 2.1% and 3.4%, respectively, after replacing the Al alloy foundation beam with internal reinforcing ribs. Additionally, there is a 5.0% enhancement in first-order bending mode and a 2.7% improvement in first-order torsional mode, accompanied by a slight weight reduction of 0.1%.
... Faramarzi and Afshar [39] used a hybrid version of CA and Linear programing (LP) to solve truss optimization problems involving displacement constraints. Bochenek and Tajs-Zieliń ska [40] used CA for the solution of the column buckling design problem where the conventional maximization of the buckling load (global quantity) was replaced by a locally formulated topology optimization problem based on the compliance minimization. ...
The conventional Cellular Automata (CA) approach has the disadvantage of being particularly suited for design optimization of uniform truss ground structures with identical cells. Moreover, the conventional CA algorithms, which are intrinsically local, have to be modified in order to be able to tackle problems involving displacement constraints. This paper presents a new non-uniform Cellular Automata algorithm, which is based on non-identical cells and a modified FSD/FUD approach. The new algorithm can solve the minimum weight optimization problem of truss structures including both stress and displacement constraints. The efficiency and accuracy of the proposed methodology are proved on a number of benchmark truss design problems.
... By drawing a preliminary structure on the top of base topology, the actual shape optimization model is developed. Bochenek et al. [8] used a minimal compliance objective approach to optimize extruded columns with different support constraints (pinned and fixed) and subjected to a transverse distributed load representing buckling in one or two planes. Optimal topologies and their corresponding critical buckling loads were obtained. ...
Steering knuckle connects steering system, suspension system and braking system to the chassis. The steering knuckle contributes a significant weight to the total weight of a vehicle. Increasing the efficiency of an automobile without compromising the performances is the major challenge faced by the manufacturers. This paper presents an effective topology optimization of steering knuckle used in a vehicle with the primary objective of minimizing weight. The study on optimization of knuckle is divided into two phases, the first phase involves making of a computer-aided design model of the original steering knuckle and carry out finite element analysis on the knuckle by estimating the loads, which are acting on the component. In the second phase, design optimization of the model of steering knuckle is carried out, and excess material is removed at the region where induced stress is negligible as obtained in finite element analysis assuming standard boundary and loading conditions. The paper describes a research work carried out to optimize structural topology giving the essential details. The methodology may be applied to optimize structural components used in applications where the ratio of desired properties to the cost, generally in terms of weight, is to be optimized. In the case of automobiles, strength to weight ratio has to be maximized. New researchers working in the area will have an understanding of the procedures, and further, the techniques may be applied to design in general.
... STO has been used in order to create optimised beams and columns [4,15,23,24,25]. In Grekavicius et al. [7], the complimentary STO study, the sections optimised using the SIMP technique with minimum weight design. ...
... Recently, Thomsen et al. (2018) presented works on the design of periodic microstructures with respect to multi-scale buckling conditions, laying the foundations for plenty of future applications within multi-scale structural and material design. Other recent works and applications can be found where we wish to mention Zhou (2004), Lund (2009), Bochenek and Tajs-Zieliṅska (2015), Cheng and Xu (2016), and Chin and Kennedy (2016). ...
We review some features of topology optimization with a lower bound on the critical load factor, as computed by linearized buckling analysis. The change of the optimized design, the competition between stiffness and stability requirements and the activation of several buckling modes, depending on the value of such lower bound, are studied. We also discuss some specific issues which are of particular interest for this problem, as the use of non-conforming finite elements for the analysis, the use of inconsistent sensitivity that may lead to wrong signs of sensitivities and the replacement of the single eigenvalue constraints with an aggregated measure. We discuss the influence of these practices on the optimization result, giving some recommendations.
... One can observe over the last two decades the implementation of various specific methods ranging from gradient-based approaches, e.g., Reference [6], where mathematical models are derived to calculate the sensitivities of design variables to non-gradient-based ones, where the material is redistributed using various, usually heuristic techniques. In what follows, the generation of optimal topologies involves, among others, evolutionary ESO/BESO structural optimization [11,12], genetic algorithms [13,14], other biologically inspired algorithms [15,16], the material cloud method [17], spline based topology optimization [18], level set method [19,20], cellular automata [21][22][23][24][25][26] or the moving morphable components approach [27,28]. It can be seen from the above that traditional gradient-based mathematical programming algorithms are more readily replaced by novel and efficient heuristic methods inspired by biological, chemical or physical phenomena. ...
Although well-recognized in the fields of structural and material design and widely present in engineering literature, topology optimization still arouses a high interest within research communities. Moreover, it is observed that the development of innovative, efficient and versatile methods is one of the most important issues stimulating progress within the topology optimization area. Following this activity, in the present study, a concept of a hybrid algorithm developed in order to generate optimal structural topologies of minimal compliance is presented. The hybrid algorithm is built based on two existing approaches. The first one makes use of the formal optimality criterion, whereas the second one utilizes a special heuristic rule of design variables updating. The main idea that stands behind the concept of the present proposal is to take the advantage of both algorithms capabilities. In a numerical implementation of the hybrid algorithm, the design variables are updated at each iteration step using both approaches, and the solution with a lower objective function value is selected for the next iteration. The numerical tests of the generation of minimal compliance structures have been performed for chosen structures including a real engineering one. It has been confirmed that the proposed hybrid technique based on switching between the considered rules allows the final structures having lower values of compliance as compared with the results of an application of basic algorithms running separately to be obtained. Moreover, based on promising results of the tests performed, one can consider the proposed concept of a hybrid algorithm as an alternative for other existing topology generators.
... Luo [14] studied the topology optimization problem by using a moving isosurface threshold method, with the maximum linear critical buckling loads being the objective functions and the volume constraint. Bochenek [15] put forward a new idea using the optimization problem based on compliance minimization to replace a conventional maximization of buckling loads for the topology optimization of columns. Cheng [16] presented a two-scale topology optimization scheme for maximizing the out-of-plane buckling load with a given amount of material. ...
This research focuses on the lightweight topology optimization method for structures under the premise of meeting the requirements of stability and vibration characteristics. A new topology optimization model with the constraints of natural frequencies and critical buckling loads and the objective of minimizing the structural volume is established and solved based on the independent continuous mapping method. The eigenvalue equations and composite exponential filter functions are applied to convert the optimization formulation into a continuous, solvable mathematical programming model. In the process of topology optimization, suitable initial values of the filter functions are chosen to avoid local modes, and the dynamic frequency gap constraints are added in the optimal model to prevent mode switches. Furthermore, for the optimal structures with grey elements obtained by the continuous optimization model, the bisection–inverse iteration is applied to search the optimal discrete structures. Finally, a detailed scheme is given for the buckling and frequency topology optimization problem. Numerical examples illustrate that the modelling method of minimizing the economic index with given performance requirements is practical and feasible for multi-performance topology optimization problems.
... A value of 7mm was found to be the most suitable value and has been used throughout the remainder of the study. A methodology to the one described herein has been validated against previous optimisation studies for both the 2D and 3D cases (Bochenek and Tajs-Zielińska, 2015;Höglund and Kosteas, 1994). ...
Aluminium is lightweight and corrosion-resistant; however, its low Young's Modulus predisposes the need for better material distribution across its section to increase stiffness. This paper studies a holistic design optimisation approach with the power of structural topology optimisation aiming to develop novel structural aluminium beam and column profiles. The optimisation methodology includes forty standard loading combinations while the optimisation results were combined through an X-ray overlay technique. This design optimisation approach is referred here to as the Sectional Optimisation Method (SOM). SOM is supported by engineering intuition as well as the collaboration of 2D and 3D approaches with a focus on post-processing and manufacturability through a morphogenesis process. Ten optimised cross-sectional profiles for beams and columns are presented. The shape of one of the best performing optimised sections was simplified by providing cross-section elements with a uniform thickness and using curved elements of constant radius. A second level heuristic shape optimisation was done by creating new section shapes based on the original optimised design. The paper also carries out stub column tests using finite element analysis (FEA) to determine the loacl buckling behaviour of the above-optimised aluminium profiles under compression and to investigate the effectiveness of using the existing classification methods according to codified provisions of Eurocode 9. The herein presented work aims to integrate topology optimisation aspects in cross-sectional design of aluminium alloy element building applications, providing thus a new concept in design procedures of lightweight aluminium structures.
