Bi-directional Evolutionary Structural Optimization Research Articles

Topology Design of 3D Printing Continuous Fiber‐Reinforced Structure Considering Strength and Non‐Equidistant Fiber

The utilization of continuous fiber‐reinforced composites (CFRC) technology in 3D printing opens up new avenues for designing composite products. This paper introduces a novel parallel topology optimization framework tailored for CFRC 3D printing. The framework incorporates strength constraints and uses topology, fiber volume fraction, and fiber orientation as essential design variables. A fiber material interpolation model for multiple design variables is proposed based on the law of mixtures for composites. Bidirectional evolutionary structural optimization and solid isotropic material with penalization are used for optimization of topology and fiber volume fraction, respectively, while the fiber orientation herein is determined using the principal stress method. Strength constraint is considered according to Tsai–Hill criterion. To match manufacturing process demands, a nonequidistant continuous fiber path optimization method based on Hermite interpolation function is introduced. The optimization framework introduced herein is successfully used for optimal design of L‐shaped beam, cantilever beam, and Michell beam. It is concluded that the design space of composite structure can be further broadened using the proposed method. To validate the viability of the proposed optimization method and fiber path design approach, the optimization results are printed using a custom‐developed continuous fiber 3D printer.

Optimizing structural topology design through consideration of fatigue crack propagation

This paper presents an advanced approach for structural topology optimization by incorporating fatigue crack propagation analysis. The extended finite element method (X-FEM) is employed to model initial crack propagation, while the Paris model serves as the basis for simulating fatigue crack growth. The proposed methodology aims to optimize the structural design by minimizing compliance while considering volume and fatigue constraints. The proposed method employs the developed bi-directional evolutionary structural optimization (BESO) algorithm. The accuracy of the proposed technique is validated through the solution of benchmark problem and is further demonstrated in its effectiveness and robustness by examining several numerical examples. The optimization process considers various crack conditions, including the absence of cracks, horizontal and vertical cracks of different lengths. The optimized topologies obtained through the proposed algorithm clearly demonstrate the impact of crack presence, crack direction, and crack length on the material distribution. Furthermore, the convergence histories of the objective function, represented by mean compliance, highlight the influence of crack length on the stiffness and converged compliance of the structure. The results demonstrate its ability to adapt the material distribution based on fatigue cracks propagation conditions and achieve optimal topologies that balance structural integrity and performance.

Soft- and Hard-Kill Hybrid GPU-based Bi-Directional Evolutionary Structural Optimization

Abstract Bi-directional evolutionary structural optimization (BESO) is a well-recognized method for generating optimal topologies of structures. Its soft-kill variant has a high computational cost, especially for large-scale structures, whereas the hard-kill variant often faces convergence issues. Addressing these issues, this paper proposes a hybrid BESO model tailored for graphics processing units (GPUs) by combining the soft-kill and hard-kill approaches for large-scale structures. A GPU-based algorithm is presented for dynamically isolating the solid/hard elements from the void/soft elements in the finite element analysis (FEA) stage. The hard-kill approach is used in the FEA stage with an assembly-free solver to facilitate the use of high-resolution meshes without exceeding the GPU memory limits, whereas for the rest of the optimization procedure, the soft-kill approach with a material interpolation scheme is implemented. Furthermore, the entire BESO method pipeline is accelerated for both the proposed hybrid and the standard soft-kill BESO. The comparison of the hybrid BESO with the GPU-accelerated soft-kill BESO using four benchmark problems with more than two million degrees of freedom reveals three key benefits of the proposed hybrid model: reduced execution time, decreased memory consumption, and improved FEA convergence, all of which mitigate the major computational issues associated with BESO.

Parametric topology optimization design and analysis of additively manufactured joints in spatial grid structures

This paper presents a complete computer-aided workflow for the parametric topology optimization (TO) design, numerical analysis and additive manufacturing (AM) of steel joints in spatial grid structures. A fully parametric TO design framework for steel joints in spatial grid structure was developed based on the Grasshopper platform. The joint models were parametrically established based on subdivision surface technology and further topology optimized through the bi-directional evolutionary structural optimization (BESO) algorithm with the support of cloud computing server. The structural performance of the optimized joints was thoroughly investigated via the parametric finite element analysis. Variables of loading conditions and TO parameters (target volume and filter radius) that affect the compliance, maximum displacement, maximum stress and stress distribution were taken into account. Analysis showed that the target volume governed the joint structural behaviour while the filter radius affected the geometric details of optimized joint. The practicability of proposed workflow was demonstrated through a parametric optimization design framework of a double-layer spatial grid structure with hundreds of steel joints. A typical optimized joint made of 316L stainless steel was additively manufactured using selective laser melting. This workflow is highly automatic and featured by high design flexibility and integration degree as well as good transplanted ability.

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.

A hole-filling based approach to controlling structural complexity in topology optimization

Shape and topology optimization problems are usually associated with geometrical restrictions. Effective control of structural complexity during the optimization process is important for various considerations, e.g., functionality, manufacturability, and aesthetics. Most existing approaches characterize structural complexity as the number of cavities. However, for three-dimensional structures, the genus (i.e., the number of tunnels) should also be considered as an additional topological constraint. In this paper, a hole-filling method is integrated into the bi-directional evolutionary structural optimization (BESO) computational framework to control the number and size of existing cavities and tunnels during the form-finding process. In the hole-filling method, excess cavities are filled with solid material and excess tunnels are covered by building sheet-like patches. The minimum size of the truss-like components and the minimum thickness of the sheet-like components can be controlled separately. Several two- and three-dimensional compliance minimization problems are presented to demonstrate the effectiveness and potential applications of the proposed approach. The structural performances of the optimized structures with and without complexity control are compared and analyzed. The results show that the developed methodology can produce structurally efficient designs with controllable topologies. It is also demonstrated that the generation of sheet-like components can considerably increase the stiffness of the optimized structure. The proposed approach is capable of generating diverse and competitive designs for architects and engineers to achieve a fine balance between architectural novelty and structural efficiency.

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.

Novel Paradigm for Steel Joints in Gridshell Structures: from Structural Optimization to Advanced Construction

AbstractNovel integrated computer‐aided paradigm/workflow for the topology optimization (TO) and additive manufacturing (AM) of steel joints in gridshell structures is proposed. The architecture scheme of gridshell structure was first established and numerically analyzed through Grasshopper parametric platform. Based on the extracted joint geometric parameters and the corresponding loading conditions, the initial joint structures were parametrically modelled using the subdivision surface method. Further parametric TO design was accomplished by the bi‐directional evolutionary structural optimization (BESO) method on the basis of cloud computing server. A series of mesh smoothing and reverse modelling algorithms were applied to the optimized joint to obtain a closed smooth surface in the NURBS form, which was then output as STL file for additive manufacturing. One typical optimized joint was additively manufactured using selective laser melting. The proposed workflow can be extended to a much wider range of spatial structures with various joint types.

Study on the bending and shear properties of quasi-honeycomb sandwich structures considering the variable-density core design

The quasi-honeycomb cell has better coplanar bearing capacity and stability in comparison with a honeycomb cell. The topology optimization can achieve the best distribution of materials within a core composed of cells. Hence, this paper makes an effort to provide a clear understanding about the enhancement effect of the core’s variable-density topology optimization on the mechanical properties of a quasi-honeycomb sandwich structure. The SR (i.e., the initials of SIMP and RAMP) interpolation function, obtained by weighting the Solid Isotropic Material with Penalization (SIMP) and Rational Approximation of Material Properties (RAMP) functions, is modified by adding the minimum elastic modulus term. The Optimal Classification (OC) algorithm is revised by adopting the volume constraint of Bi-directional Evolutionary Structural Optimization (BESO). And then taking into account the mapping between the topology elements and cells, the material distribution optimization of the quasi-honeycomb core is performed to minimize the structural compliance. The failure behavior, strength, stiffness, and energy absorption of three-dimensional (3D) printed uniform and variable-density quasi-honeycomb sandwich structures are studied by three-point bending tests, and the evolution processes and distribution laws on the displacement and strain of cores are analyzed using the 3D Digital Image Correlation (DIC) system. The comparisons between the experimental and numerical results are also carried out for the displacement and energy absorption of the sandwich structure, the bending strength of the face sheet, and the displacement and strain fields of the core. The results show that the bending strength, bending stiffness, shear strength, and energy absorption are enhanced after the variable-density topology optimization of the core, and the numerical values agree well with the experimental data. This study provides new ideas and important guidance for the variable-density design of quasi-honeycomb cores and the mechanical performance improvement of sandwich structures.

An improved GBESO method and application for engineering structures

For engineering structures, the evolutionary-type optimizations are usually conducted to obtain the topologies by deleting and restoring a few of their elements in each iteration. As for the Genetic Bi-directional Evolutionary Structural Optimization (GBESO) among them, five major improvements are made in this research: regroup only the remained elements into three groups in each iteration; perform evolution and punishments on the specific elements during each iteration; use a constant probability throughout the crossover; add a global mutation with a very small probability, and adjust half of the genetic codes of the deleted elements from 0 to 1 as the restoration, whose probabilities are determined by the sensitivity of the elements around them. These improvements are intended to increase the participation of probabilistic ideology, and then to reduce the possibility of obtaining the local optimal solutions. As a result, within a large scope all the time, the improved GBESO keeps searching for global optimal solutions more consistent with the optimization objective and helps establish superior strut-and-tie models (STMs) with a reasonable process, indicating its stronger ability on global optimization.

Reliability based bi-directional evolutionary topology optimization of geometric and material nonlinear analysis with imperfections

This paper aims to present a novel computational technique for using reliability-based design while taking into account the effect of geometrically and materially nonlinear imperfect analysis. Consequently, a new bi-directional evolutionary structural optimization scheme is developed. A comparison is made between perfect geometrically and materially nonlinear analysis and imperfect geometrically and materially nonlinear analysis topology optimization designs for both deterministic and probabilistic analysis. In the case of probabilistic analysis, relevant parameters such as volume fraction (including manufacturing imprecisions), material properties, and geometrical imperfections (for stability calculations) are assumed to be random variables that follow a normal distribution to represent the uncertainties. The considered numerical examples have successfully illustrated that the proposed method can find the optimal topology for a reliability-based design using perfect geometrically and materially nonlinear analysis and imperfect geometrically and materially nonlinear analysis. Additionally, the results of the topology optimization according to the mean stress and the final optimized shapes have been influenced by introducing a reliability-based design, considering the reliability index as a bound governing the process.

Non-parametric optimization for lightweight and high heat conductive structures under convection using metaheuristic structure binary-distribution method

In this paper, we propose a novel Metaheuristic Structure Binary-Distribution (MSB) method for attaining lightweight and high thermal conductive structure. MSB combines metaheuristic search with gradient descent optimization, offering a robust and efficient approach. The paper also introduces a connectivity filtering approach to enhance additive manufacturability and eliminate segregated materials in thermal structures, improving integrity and minimizing materials wastage. The performance of the MSB method is compared to other topology optimization methods, namely bi-directional evolutionary structural optimization (BESO), solid isotropic materials with penalization (SIMP), and the parametrized level set method (PLSM). Results consistently demonstrate the better performance of MSB in weight reduction and minimizing thermal compliance. Even at significant weight reductions (50%, 60%, and 70%), MSB outperforms other methods, achieving lower thermal compliance. Investigations on mesh impact and the effectiveness of MSB with suggested connectivity filtering show a successful reduction of islands to solid elements to 0%, improving additive manufacturability and decreasing thermal compliance. The feasibility and suitability of the optimized designs for additive manufacturing are validated through 3D printing, confirming the practicality of the MSB method.

New alternative techniques for strengthening deep beams with circular and rectangular openings

Numerous deep beam design models have been proposed; however, even the most recent design manuals provide limited guidance for deep beam design, especially when the beams have complicated features such as web openings. The openings cause a geometric discontinuity, which makes the distribution of nonlinear stress along the depth of the beam more complicated. In this paper, the structural behavior of the deep beams with openings was investigated experimentally via a total of twenty concrete beam specimens and analytically in four themes. In the first theme, eleven concrete beam specimens with circular and rectangular openings, including the reference solid specimen, were tested to examine how the shape, size, and position of the openings affected the structural behavior of the deep beams. According to earlier studies, the crack patterns and modes of failure for the specimens were discussed in the first theme, and various techniques for strengthening the openings, whether circular or rectangular, were suggested in the second theme. Six concrete beam specimens with rectangular openings were strengthened; three of them were internally strengthened using various thickness plates around the openings, and the other three were strengthened using external fasteners subjected to torque moments. As for the concrete beam specimens with circular openings, these openings were strengthened with three different reinforcement techniques in the three tested beams. The load capacity, deflection, and crack patterns for the strengthened and unstrengthened specimens were recorded and discussed. The experimental results indicated that the load capacity and stiffness of the tested beams were significantly reduced by the presence of openings. Additionally, there was a distinct variety in how the size, position, and shape of the openings affected the beams performance. The results also demonstrated that concrete beam specimens with rectangular openings that were strengthened externally by the fasteners were the best, whereas the efficiency of the remaining strengthening was lower than expected. In the third theme, a numerical study using the nonlinear finite element model (NLFE), the program ANSYS V-19.2, 2018 was conducted on some of the tested strengthened and unstrengthened specimens. The analytical study's findings demonstrated the effectiveness of the NLFE model by comparing with load carrying capacity and load-deflection curve of the tested specimens. In the fourth theme, based on Bi-directional Evolutionary Structural Optimization (BESO), an algorithm developed in MATLAB and integrated with the FEA solver SAP software was used to build the optimal strut and tie model (OSTM) for two concrete beam specimens with rectangular and circular openings. This was undertaken with the objective of seeking and finding a strengthening technique that is more effective and is dependent on a clear methodology, not on trial and expectation. In agreement with the final obtained OSTM, new alternative strengthening techniques were proposed and evaluated using NLFE. In comparison to the other techniques that were described in this research, the proposed strengthening techniques based on BESO was the best and most effective, they increased the ultimate load in some cases by 72.7 %. BESO can be used efficiency for strengthening the deep beams with openings.

Sub-regional design of the bionic bone scaffolds using macrostructural topology

With the increasing demand for bone repair, the bionic bone scaffolds have become a research hotspot. A sub-regional design method of the bionic bone scaffolds, using macrostructural topology, is proposed in this paper, aiming to provide a functionally enhanced region division method for the gradient design. The macrostructural topology was carried out by the bi-directional evolutionary structural optimization (BESO), dividing the predefined design domain into sub-region A and sub-region B. Subsequently, a combined probability sphere model and a distance-to-scale coefficient mapping model are established to implement the graded porosification based on the Voronoi tessellation. This approach takes geometric and mechanical continuity into fully account and assures a reasonable distribution of characteristic parameters, yielding to improve the mechanical strength under specific stress conditions. Finally, the scaffolds were fabricated by the laser powder bed fusion (LPBF) process using the Ti-6Al-4V powder. The results of compression tests are satisfactory, showing that the as-built specimens implement sub-regional functionality. The apparent elastic modulus and the ultimate strength range, respectively, between 1.50 GPa and 7.12 GPa (for the first module) and between 38.55 MPa and 268.03 MPa (for the second module), which conform to the required level of natural bone, providing a possibility for clinical application.

Additive manufacturing oriented parametric topology optimization design and numerical analysis of steel joints in gridshell structures

Additive manufacturing (AM) oriented joint topology optimization (TO) is a novel integrated computer-aided structural design and construction paradigm for spatial structures. This research proposed a complete computer-aided workflow for the TO design, numerical analysis and AM of steel joints in gridshell structures. A fully parametric TO design method for joints in gridshell structures was established based on the Grasshopper platform. The joints were automatically generated based on subdivision surface technology and then optimized using the bi-directional evolutionary structural optimization (BESO) algorithm. The structural performance of the optimized joints in terms of compliance, maximum displacement and maximum stress was studied through finite element analysis. The influences of various design parameters, including the loading condition, target volume and filter radius, on the structural performance of the optimized joints were investigated. The target volume was confirmed to have significant effect on the joint structural behaviour while the filter radius was found to affect the branch features of optimized joint. A practical design example of steel joints in single-layer cylindrical gridshell structure was presented to demonstrate the feasibility and validity of the proposed workflow. A typical optimized joint made of 316L stainless steel was additively manufactured using selective laser melting.

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.

Phase field modelling combined with optimization algorithm for maximizing the resistance in two-phase composites

Recently, the phase field modelling is widely used to model and simulate material damage. In present work, we present a framework of the topology optimization combined with the phase-field modelling with/ without interfacial damage for optimizing the damage resistance of the inclusion-matrix composites. The first phase field method with the interfacial damage described by the phase field variable d(x) and an interfacial phase field variable β(x), thus the crack occurs in the interaction between the bulk fracture and the interfacial one; the second phase field method without the interfacial damage describes the crack by using only a damage variable d(x), thus the crack initiates at the points where stress concentration occurs (damage only occurring in the phases). Extended bi-directional evolutionary structural optimization algorithm (BESO) is used to optimize the inclusion distribution in order to reduce its volume while keeping the fracture resistance value of the initial design unchanged. Moreover, the strain tensor orthogonal decompositions are implemented into the phase field methods to improve the inaccuracy in the mechanical behavior of the materials. We compare the inclusion optimal distribution, the crack propagation and the fracture resistance between the proposed models through several numerical examples.

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.