Bi-directional Evolutionary Structural Optimization Method Research Articles

Dynamic Resistance and Energy Absorption of Sandwich Beam via a Micro-Topology Optimization

The current research of sandwich structures under dynamic loading mainly focus on the response characteristic of structure. The micro-topology of core layers would sufficiently influence the property of sandwich structure. However, the micro deformation and topology mechanism of structural deformation and energy absorption are unclear. In this paper, based on the bi-directional evolutionary structural optimization method and periodic base cell (PBC) technology, a topology optimization frame work is proposed to optimize the core layer of sandwich beams. The objective of the present optimization problem is to maximize shear stiffness of PBC with a volume constraint. The effects of the volume fraction, filter radius, and initial PBC aspect ratio on the micro-topology of the core were discussed. The dynamic response process, core compression, and energy absorption capacity of the sandwich beams under blast impact loading were analyzed by the finite element method. The results demonstrated that the over-pressure action stage was coupled with the core compression stage. Under the same loading and mass per unit area, the sandwich beam with a 20% volume fraction core layer had the best blast resistance. The filter radius has a slight effect on the shear stiffness and blast resistances of the sandwich beams. But increasing the filter radius could slightly improve the bending stiffness. Upon changing the initial PBC aspect ratio, there are three ways for PBC evolution: The first is to change the angle between the adjacent bars, the second is to further form holes in the bars, and the third is to combine the first two ways. However, not all three ways can improve the energy absorption capacity of the structure. Changing the aspect ratio of the PBC arbitrarily may lead to worse results. More studies are necessary for further detailed optimization. This research proposes a new topology sandwich beam structure by micro-topology optimization, which has sufficient shear stiffness. The micro mechanism of structural energy absorption is clarified, it is significant for structural energy absorption design.

Evolutionary topology optimization of fiber reinforced composite laminates for maximum stiffness

In this study, a topology optimization technique based on the bi-directional evolutionary structural optimization (BESO) method is developed to maximize the stiffness of fiber reinforced composite laminates. The elastic properties of single-layer and multi-layer composite laminates are characterized and a composite material interpolation scheme is introduced. The effectiveness of the developed BESO method for the single-layer and multi-layer composite laminates are verified by three classical examples, namely a cantilever, a Michell-type structure, and an L-bracket. Then, based on the analysis of the topological results, a selection suggestion and verification criteria for layers with appropriately matched fiber angles are proposed and verified by two additional examples. This study proves that the BESO method is an effective technique for the topology optimization of composite laminates for maximum stiffness, and the proposed suggestion and criteria are useful for avoiding large computational costs and empirical selection of layers.

Enhancing structural stability in civil structures using the bi-directional evolutionary structural optimization method

Topology optimization techniques are increasingly utilized in structural design to create efficient and aesthetically pleasing structures while minimizing material usage. Many existing topology optimization methods may generate slender structural members under compression, leading to significant buckling issues. Consequently, incorporating buckling considerations is essential to ensure structural stability. This study investigates the capabilities of the bi-directional evolutionary structural optimization method, particularly its extension to handle multiple load cases in buckling optimization problems. The numerical examples presented focus on three classical cases relevant to civil engineering: maximizing the buckling load factor of a compressed column, performing buckling-constrained optimization of a frame structure, and enhancing the buckling resistance of a high-rise building. The findings demonstrate that the algorithm can significantly improve structural stability with only a marginal increase in compliance. The detailed mathematical modeling, sensitivity analyses, and optimization procedures discussed provide valuable insights and tools for engineers to design structures with enhanced stability and efficiency.

Topology optimization structure design of shape memory alloy with multiple constraints

As an emerging functional material, shape memory alloy (SMA) exhibits remarkable mechanical properties and finds diverse applications across industries. This paper presents a topology optimization framework based on the bi-directional evolutionary structural optimization (BESO) method for designing SMA structures, which maximizes structural stiffness under multiple constraints of specified volume fraction, displacement, and fundamental frequency. A phenomenological constitutive model is utilized to simulate the mechanical behavior of SMA accurately. The unit virtual load method is employed to determine sensitivities. Several optimized SMA beam structures and simply-supported cube structures are designed under different thermal-mechanical loads, and their displacement, mean compliance, and fundamental frequency are evaluated throughout the optimization process. The results demonstrate that the proposed framework successfully customizes the SMA topology structure with adjustable displacement and fundamental frequency, and the optimized schemes exhibit more considerable deformation and more uniform mechanical properties than their initial counterparts. The proposed framework has higher computational efficiency than the traditional SIMP-based SMA topology optimization design method.

An Eigenfrequency-Constrained Topology Optimization Method with Design Variable Reduction

The dynamic response of structures heavily relies on eigenfrequency, so the optimization of eigenfrequency is valuable in various working conditions. The bi-directional evolutionary structural optimization (BESO) method has been widely applied due to its ability to eliminate grayscale elements. Based upon BESO, this paper introduces a topology optimization method that incorporates eigenfrequency constraints and reduces the number of design variables. In this method, the optimization objective was to minimize compliance. The Lagrange multiplier was used to introduce eigenfrequency constraints, allowing for coordinated control of compliance and eigenfrequency. To prevent oscillation during the optimization process, the sensitivity was normalized. Additionally, to achieve faster convergence, the variables were reduced after meeting volume constraints. The numerical examples demonstrate the effectiveness of this method in increasing the eigenfrequency of the structure and avoiding resonance.

Plastic-limit probabilistic structural topology optimization of steel beams

This work presents a novel geometrically nonlinear analysis with imperfections reliability-based topology optimization (RBTO) approach, considering the volume fraction and geometric imperfections as random variables due to their crucial connections to the manufacturability of web openings in steel I-beams. The objective is achieved by controlling the plastic behavior through limit analysis, which imposes a limit on the plastic ultimate load multipliers. The suggested method is developed by imposing constraints related to the available material volume. The bi-directional evolutionary structural optimization (BESO) method is utilized to fulfill the objectives of this paper. The adequacy of the proposed technique is demonstrated by comparing the results of the algorithm with benchmark steel beams that have conventional web openings. Based on the numerical examples, it appears that the selected technique has the potential to increase the load capacity of steel beams while utilizing the same quantity of material within the design domain. Moreover, the improved beams also exhibit better performance in terms of stress levels. The results of this research suggest that the proposed technique enhances the load capacity of steel beams while maintaining the same quantity of material and improving their performance in terms of stress levels.

Topological Optimization of Bi-Directional Progressive Structures with Dynamic Stress Constraints under Aperiodic Load

The topology optimization of dynamic stress constraints is highly nonlinear and singular and has been little studied. Dynamic stress based on progressive structural optimization is only available by applying the modal iteration method, but due to the nonlinear limitations of the modal superposition method, there is an urgent need to develop a progressive structural optimization method based on dynamic stress sensitivity under direct integration. This method is for the dynamic stresses under non-periodic loading with iterative cycle updating variations. This article proposes a topological optimization method of continuum structures with stress constraints under an aperiodic load based on the Bi-directional Evolutionary Structural Optimization Method (BESO). First, the P-norm condensation function was used to obtain the global stress to approximate maximum stress. By introducing the Lagrange multiplier, the design goal was to increase the P-norm stress on the basis of the smallest volume. After that, based on the dynamic finite element theory, the sensitivity of each cell formula of the objective function and the constraint conditions of the design variables were strictly derived. Then, the performance evaluation index was put forward based on volume and stress, and the convergence criterion based on the performance evaluation index was defined. This method solves the topology optimization problem of stress constraints under a non-periodic load and the topology optimization problem of stress constraints under a periodic load, such as a simple harmonic load.

Practical application of multi-material topology optimization to performance-based architectural design of an iconic building

With the rapid development of architectural design methods and structural analysis techniques, structural performance-based design approaches have become increasingly important in architectural design. Among these approaches, topology optimization is a significant design method. However, most existing topology optimization research and related applications are based on single-material topology optimization techniques, which have limitations when applied to the design of complex and challenging structures. Building upon our previously proposed multi-material bi-directional evolutionary structural optimization (multi-material BESO) method, this paper presents the approaches for achieving diverse and competitive structural designs and then presents a practical application of the multi-material BESO method to the structural design of the core structure of a long-cantilevered building project called “Xiong’an Wings”. After an evolutionary design process based on the multi-material BESO method, an efficient frame with lower overall material costs, better static and dynamic performance compared to the initial design by experienced engineers is achieved. As the first practical application of the multi-material BESO method to a large-scale architectural design, the example of the “Xiong’an Wings” building demonstrates the tremendous prospects for the application of multi-material topology optimization technology to the generative design of innovative and efficient building structures.

Multi-Scale Concurrent Topology Optimization Based on BESO, Implemented in MATLAB

In multi-scale topology optimization methods, the analysis encompasses two distinct scales: the macro-scale and the micro-scale. The macro-scale refers to the overall size and dimensions of the structural domain being studied, while the micro-scale pertains to the periodic unit cell that constitutes the macro-scale. This unit cell represents the entire structure or component targeted for optimization. The primary objective of this research is to present a simplified MATLAB code that addresses the multi-scale concurrent topology optimization challenge. This involves simultaneously optimizing both the macro-scale and micro-scale aspects, taking into account their interactions and interdependencies. To achieve this goal, the proposed approach leverages the Bi-directional Evolutionary Structural Optimization (BESO) method. The formulation introduced in this study accommodates both cellular and composite materials, dealing with both separate volume constraints and the utilization of a single volume constraint. By offering this simplified formulation and harnessing the capabilities of the multi-scale approach, the research aims to provide valuable insights into the concurrent optimization of macro- and micro-scales. This advancement contributes to the field of topology optimization and enhances its applications across various engineering disciplines.

Topological optimization design on constrained layer damping treatment for vibration suppression of thin-walled structures via improved BESO method

In this paper, an improved bi-directional evolutionary structural optimization (BESO) technique is proposed, to investigate topological optimization design on constrained layer damping (CLD) treatment for vibration suppression of experiment rack backplane. First, the optimization model for vibration reduction of CLD treated backplane is established, with volume fraction of CLD material as constraint condition, and the maximization of weighted value of the first four order modal loss factors of the structure as objective function. An improved evolutionary rate considering the CLD element sensitivity coefficient of variation of each iteration is presented to increase optimization efficiency. Then, the BESO with improved evolutionary rate is utilized to search the optimal CLD layout of the experiment rack backplane. CLD layouts of three optimization schemes with different volume fraction of CLD material are obtained, and are highly consistent with the stress distribution of the backplane viscoelastic layer, which verifies the rationality of the optimization results. Visualization of iterative optimization process, including elements removal/addition and dynamic change of CLD elements sensitivity in each iteration, is demonstrated, respectively. Further, the effect of improved evolutionary rate on optimization process is discussed. Compared with the conventional BESO method, the improved BESO technique can not only obtain a more reasonable optimized CLD layout, but also improve the optimization efficiency and accuracy. Finally, the validity of the optimized CLD layout is analyzed from three following aspects: natural frequency, modal loss factor and vibration response under order excitation. Numerical and experimental results show that the proposed BESO technique is effective and efficient in the application of CLD material, and the small amount of CLD material can be retained under the premise of ensuring damping.

A strategy for improving the safety and strength of topologically optimized multi-material structures

This paper presents a new strategy to distribute two different materials for multi-material topology optimization. Extended from the bi-directional evolutionary structural optimization (BESO) method for a single material, the multi-material bidirectional evolutionary structural optimization (MBESO) method has been developed, which can effectively handle the topology optimization problems involving two materials like steel and concrete. However, in some special cases, overloading of part of the compressed material occurs in the multi-material structures designed by using the MBESO method. Aimed to solve this critical problem, a simple but effective strategy is proposed in this paper. In steel-concrete composite structures, for instance, the overloaded compressed concrete elements with exceedingly high stress are replaced with steel material. The small amount of steel material added to the highly compressed region can effectively reduce the maximum compressive stress of the concrete material to a safe level. The comparison between the original MBESO method and the improved strategy based on a series of two-dimensional and three-dimensional examples clearly demonstrates the effectiveness of the proposed strategy in enhancing the structural safety and strength of the topologically optimized composite structures. This distinctly different material distribution strategy shows its potential and value in multi-material topology optimization research and applications.

Topology optimization for thermal structures considering design-dependent convection boundaries based on the bidirectional evolutionary structural optimization method

Abstract. Based on the bidirectional evolutionary structural optimization (BESO) method, the present article proposes an optimization method for a thermal structure involving design-dependent convective boundaries. Because the BESO method is incapable of keeping track of convection boundaries, virtual elements are introduced to assist in identifying the convection boundaries of the structure. In order to solve the difficult issue of element assignment under a design-dependent convection boundary, label matrixes are employed to modify the heat transfer matrix and the equivalent temperature load vector of elements over topology iterations. Additionally, the optimization objective is set to minimize the maximum temperature of the structure in order to deal with the objective reasonableness, and the p-norm method is then used to fit the objective function to calculate sensitivity. Finally, several cases, including 2D and 3D structures under various heat transfer boundary conditions, are provided to illustrate the effectiveness and good convergence of the proposed method.

Layout optimization of steel reinforcement in concrete structure using a truss-continuum model

Owing to advancement in advanced manufacturing technology, the reinforcement design of concrete structures has become an important topic in structural engineering. Based on bi-directional evolutionary structural optimization (BESO), a new approach is developed in this study to optimize the reinforcement layout in steel-reinforced concrete (SRC) structures. This approach combines a minimum compliance objective function with a hybrid truss-continuum model. Furthermore, a modified bi-directional evolutionary structural optimization (M-BESO) method is proposed to control the level of tensile stress in concrete. To fully utilize the tensile strength of steel and the compressive strength of concrete, the optimization sensitivity of steel in a concrete–steel composite is integrated with the average normal stress of a neighboring concrete. To demonstrate the effectiveness of the proposed procedures, reinforcement layout optimizations of a simply supported beam, a corbel, and a wall with a window are conducted. Clear steel trajectories of SRC structures can be obtained using both methods. The area of critical tensile stress in concrete yielded by the M-BESO is more than 40% lower than that yielded by the uniform design and BESO. Hence, the M-BESO facilitates a fully digital workflow that can be extremely effective for improving the design of steel reinforcements in concrete structures.

An Investigation of the Recent Developments in Reliability-based Structural Topology Optimization

Optimizing a structure's topology involves finding the best possible distribution of material and connections within a given design space. It has a significant effect on its performance, which is why there has been a meteoric rise in the number of articles published on the topic over the last two decades. This work offers an investigation of reliability-based topology optimization of structures, in light of the recent development of several topology optimization techniques for both linear and nonlinear systems. Therefore, the emphasis of this study is on the latest advancements, enhancements, and applications of reliability-based topology optimization. This paper's primary objective is to provide an overview of the latest advancements in the reliability-based topology optimization of structures, with a particular emphasis on the recent improvement of integrating reliability-based design into the bi-directional evolutionary structural optimization (BESO) method, which accounts for the topological optimization of geometric and material nonlinearity as well as thermoelastic problems.

Concurrent topology optimization of multiscale structure under uncertain dynamic loads

This study proposes a robust concurrent topology optimization method with considering dynamic load uncertainty for the design of structures composed of periodic microstructures based on the bi-directional evolutionary structural optimization (BESO) method. The objective function is formulated as the summation of the mean and standard deviation of the structural dynamic compliance modulus. The constraints are imposed on the macrostructure and material microstructure volumes, respectively. The hybrid dimension reduction method and Gauss integral (HDRG) method is proposed to quantify and propagate load uncertainty to estimate the objective function. By the HDRG method, robust topology optimization with uncertainty modeled by probabilistic methods can be handed uniformly. To reduce the computational burden, a decoupled sensitivity analysis method is proposed to calculate the sensitivities of objective function with respect to the microstructure design variables. Five numerical examples are used to validate the effectiveness of the proposed robust concurrent topology optimization method and demonstrate the influence of load uncertainty on the design results. Results illustrate that the proposed methods can obtain the clear topologies of macro and micro structures, and the dynamic load uncertainty has a significant impact on the design results.

Topology Optimization of Stiffened Steel Plate Shear Wall Based on the Bidirectional Progressive Structural Optimization Method

Many studies on structural topology optimization of steel plate shear walls have been conducted. However, research on topology optimization using the bidirectional evolutionary structural optimization method is limited. Accordingly, this study optimized the topology of the stiffening effect of steel plate shear walls (SPSWs) based on this method. A finite element model of the SPSW was established using Abaqus software through the “sandwich” modeling method. An optimization region was expanded into two optimization regions. As the optimization targets, SPSWs with different aspect ratios were selected. Elastoplastic optimization of a single-layer SPSW was performed through the horizontal displacement cyclic loading, and the distribution law of the stiffening effect was obtained. The stiffeners on the SPSW were arranged according to the SPSW-A075 scheme. Monotonic and reciprocating loading simulation tests were performed on the unstiffened SPSW and common transverse and longitudinal stiffeners to analyze their mechanical properties. The results show that the optimized layout of the stiffened SPSW demonstrated better seismic performance and energy dissipation capacity. The buckling bearing capacity increased by 2.17–2.61 times, and the stiffness and initial stiffness improved significantly.

Bi-directional evolutionary structural optimization with buckling constraints

Buckling is a critical phenomenon in structural members under compression, which could cause catastrophic failure of a structure. To increase the buckling resistance in structural design, a novel topology optimization approach based on the bi-directional evolutionary structural optimization (BESO) method is proposed in this study with the consideration of buckling constraints. The BESO method benefits from using only two discrete statuses (solid and void) for design variables, thereby alleviating numerical issues associated with pseudo buckling modes. The Kreisselmeier-Steinhauser aggregation function is introduced to aggregate multiple buckling constraints into a differentiable one. An augmented Lagrangian multiplier is developed to integrate buckling constraints into the objective function to ensure computational stability. Besides, a modified design variable update scheme is proposed to control the evolutionary rate after the target volume fraction is reached. Four topology optimization design examples are investigated to demonstrate the effectiveness of the buckling-constrained BESO method. The numerical results show that the developed optimization algorithm with buckling constraints can significantly improve structural stability with a slight increase in compliance.

Architectural Form Finding and Computational Design of Tall Building Applying Topology Optimization against Lateral Loads

The hierarchy of a building is created from the hermeneutics aspects changing from a typology to a topology paradigm. As tall buildings are growing rapidly, engineering science is experiencing developments such as the finite-element method. The present research creates a platform for architects using the topology optimization method as an assistant designer. This platform serves as a starter for a common language of these two professions in the early stages of design. Architects can use this platform to produce the final topological optimized form by defining their shape plans, lateral loads, height, and the target ratio of models as platform inputs. This paper highlights the importance of the bidirectional evolutionary structural optimization method simultaneously in architectural and structural contexts to reduce the total weight of the structure while keeping the displacement of the structure in an acceptable range. The reduction will be made by eliminating the structural material zones with lower stress with an effective method for achieving the structural efficiency of a tall building, considering architectural aspects.

Reliability based topology optimization of thermoelastic structures using bi-directional evolutionary structural optimization method

The aim of this paper is to propose a novel computational technique of applying reliability-based design to thermoelastic structural topology optimization. Therefore, the optimization of thermoelastic structures' topology based on reliability-based design is considered by utilizing geometrical nonlinearity analysis. For purposes of introducing reliability-based optimization, the volume fraction parameter is viewed as a random variable with a normal distribution having a mean value and standard deviation. The Monte Carlo simulation approach for probabilistic designs is used to calculate the reliability index, which is used as a constraint related to the volume fraction constraint of the deterministic problem. A new bi-directional evolutionary structural optimization scheme is developed, in which a geometrically nonlinear thermoelastic model is applied in the sensitivity analysis. The impact of changing the constraint of a defined volume of the required design in deterministic problems is examined. Additionally, the impact of altering the reliability index in probabilistic problems is investigated. The effectiveness of the suggested approach is shown using a benchmark problem. Additionally, this research takes into account probabilistic thermoelastic topology optimization for a 2D L-shaped beam problem.

An BESO Approach for Optimal Retrofit Design of Steel Rectangular-Hollow-Section Columns Supporting Crane Loads

In this paper, we propose a cost-effective optimal-topology retrofitting technique for hollow-steel-section columns to sufficiently support industrial running cranes. A so-called bi-directional evolutionary structural optimization (BESO) method was encoded within the MATLAB modeling framework, with a direct interface with an ANSYS commercial finite-element analysis program, to determine the optimal topology of double external steel plates connected to columns in a 3D space. For the initial ground structure, we have adopted standard uniform double U-shaped external stiffener plates located at the top and bottom flange layers of an I-beam to box-column connection (IBBC) area. The influences of inelastic materials and the incorporated nonlinear geometry can effectively describe the premature (local buckling) failures of the columns in an IBBC area. The applications of the proposed optimal-topology BESO-based stiffening method are illustrated through the retrofitting of three hollow-steel-section columns, characterized by non-slender and slender compression sections. Some concluding remarks are provided on the pre- and post-retrofitted responses of the columns, with the results showing both the accuracy and robustness of the proposed external stiffening schemes.