Evolutionary Structural Optimization Research Articles

Topological design of microstructured materials using energy-based homogenization theory and proportional topology optimization with the level-set method

In this study, we propose a novel approach to designing microstructures of porous material with extreme mechanical properties to maximize the bulk or shear modulus. The approach integrates energy-based homogenization theory and proportional topology optimization with the level-set method (HPTO-LSM). A level-set function (LSF) evolution strategy based on the nodal element mutual energies (EME) proportion is proposed and drives the LSF’s evolution. An EME proportion filtering technique is proposed and reduces the redundant branch structure. And a linear scaling strategy is applied to the HPTO-LSM to enhance the topological change ability. The effectiveness of the HPTO-LSM for designing microstructures of cellular materials to maximize the bulk or shear modulus is demonstrated using several 2D and 3D cases. The effects (changes in objective function values or topologies) of the EME proportion filtering scheme and scaling strategy on the HPTO-LSM are discussed. In addition, we compared the HPTO-LSM method with the solid isotropic material with penalization (SIMP) and bi-directional evolutionary structural optimization (BESO) methods through numerical examples. As shown by the results, effective objective function values and 2D and 3D microstructural topologies with clear interfaces and smooth boundaries can be acquired using the new approach. Applying the EME proportion filtering scheme and scaling strategy in the HPTO-LSM not only enhanced the topological variation of microstructures but also facilitated the acquisition of microstructures conducive to engineering applications. Compared to the SIMP and BESO methods, the HPTO-LSM method has significant advantages in obtaining better objective function values and better microstructural topologies.

Additive manufacturing-driven simultaneous optimization of topology and print direction for thermoelastic structures considering strength failure

Abstract This paper presents a strength-based simultaneous optimization method for optimizing thermoelastic structural topology and print direction in the presence of anisotropy induced by additive manufacturing. The approach utilizes the bi-directional evolutionary structural optimization framework and defines design variables including element density and print-off angle. Firstly, an anisotropic thermoelastic constitutive model is established for finite element analysis. By introducing the Tsai–Hill failure criteria, the strength constraint to evaluate the stress level of additively manufactured anisotropic components is formulated. The P-norm aggregation function is employed to approximate the maximum strength failure coefficient. Then, the aggregated strength constraint is augmented to the optimization objective through a Lagrange multiplier. Sensitivity analysis of the new objective function with respect to the elemental design variables is performed, and an analytical approach is proposed to optimize the print-off angle. To improve the stability of the optimization procedure, a series of numerical algorithms and parameter updating strategies are developed. The effectiveness of our proposed method is demonstrated through typical numerical examples, highlighting a desirable match between the structural topology and the print direction can greatly improve the structural performance.

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.

Personalized prosthesis design in all-on-4® treatment through deep learning-accelerated structural optimization

Background/purposeThe All-on-4® treatment concept is a dental procedure that utilizes only four dental implants to support a fixed prosthesis, providing full-arch rehabilitation with affordable cost and speedy treatment courses. Although the placement of all-on-4® implants has been researched in the past, little attention was paid to the structural design of the prosthetic framework. Materials and methodsThis research proposed a new approach to optimize the structure of denture framework called BESO-Net, which is a bidirectional evolutionary structural optimization (BESO) based convolutional neural network (CNN). The approach aimed to reduce the use of material for the framework, such as Ti–6Al–4V, while maintaining structural strength. The BESO-Net was designed as a one-dimensional CNN based on Inception V3, trained using finite element analysis (FEA) data from 14,994 design configurations, and evaluated its training performance, generalization capability, and computation efficiency. ResultsThe results suggested that BESO-Net accurately predicted the optimal structure of the denture framework in various mandibles with different implant and load settings. The average error was found to be 0.29% for compliance and 11.26% for shape error when compared to the traditional BESO combined with FEA. Additionally, the computational time required for structural optimization was significantly reduced from 6.5 h to 45 s. ConclusionThe proposed approach demonstrates its applicability in clinical settings to quickly find personalized All-on-4® framework structure that can significantly reduce material consumption while maintaining sufficient stiffness.

Evolutionary topology optimization approach to design multiphase soundproof systems with poroelastic media

SummaryWith the constant development of cities, noise sources have become increasingly present inside and outside living environments. Consequently, soundproof systems comprised of porous materials have been widely adopted as filling fabric of closed‐space structures, such as in the components of buildings, airplanes or automobiles. However, in many situations, simply filling spaces may not be the most effective approach. In that scenario, this work introduces a multiphase acoustic topology optimization methodology to design closed‐space structures for sound attenuation. Based on the Bi‐directional Evolutionary Structural Optimization (BESO) algorithm, the proposed approach combines Biot's poroelasticity equations, expressed in the mixed /p form, and the Unified Multiphase (UMP) technique to fully describe the multiphysics involved in the acoustic, poroelastic and elastic model relations. An objective function contemplating different combinations of structural, viscous and thermal dissipated powers is maximized over multiple frequencies. Volume constraints in each material phase and a novel material interpolation scheme are also considered. The resultant topologies present enhanced dissipated power levels and manufacturability, even when compared with various baseline configurations of similar volume fractions.

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.

Topology Optimization using Nonlinear Finite Element Analysis

Topology optimization is a structural-mechanical investigation that structures optimization function values in an optimization iteration. Topology optimization with nonlinear static behavior is difficult due to several design considerations. Due of significant integrate distortion, low-density finite elements provide significant numerical challenges in the prevailing element density focused topology planning taking nonlinear dynamics under factor. Iterative procedures are used in the Bi-directional Evolutionary Structural Optimization (BESO) technique for reducing waste from a structure while concurrently adding efficient material using a finite element-based topology optimization strategy. Integrating the fully-connected Deep Neural Network (DNN) with the norm-level-set techniques yields a powerful method for optimizing structural topology. Hence, BESO-DNN has been designed spatial optimization of material distribution inside a specified area is the focus of topology optimization is a mathematical approach for minimizing a certain cost function while meeting a set of predefined restrictions. Topology optimization of geometrically non-linear systems is advantageous because of the solutions' lack of intermediate-density components and their great processing efficiency of high-resolution barrier representation can be effective. As a result, topology optimization is ensuring effective and providing fresh and efficient designs, understanding that outcomes needs an integration of credible decision-making, domain knowledge, and numerical analytic skills. The outcome of finite element analysis has to understand and anticipate an object's performance under various physical conditions using mathematical models, and testing.

Reinforcement layout design for deep beams based on bi-objective topology optimization

Determining an effective and efficient reinforcement layout for reinforced concrete deep beams is still a challenging problem. This paper proposes a bi-objective evolutionary structural topology optimization method for the reinforcement layout design. Based on a discrete model for steel bars, the optimization aims to achieve a uniform stress distribution of steel rebars while promoting the peak rebar stress. A lexicographic method is used to solve the bi-objective optimization. Two numerical examples are presented to demonstrate the proposed algorithm's feasibility, stability and universal applicability. It is shown the reinforcement stresses in both tension and compression zones are more uniformly distributed and, on average, closer to the yield strength than those of the single-objective optimization. Compared with the design based on the conventional strut-and-tie method, the deep beams optimized by the proposed method use less reinforcement steel, provide a higher ultimate load capacity, and, more interestingly, fail in a ductile failure mode.

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.

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.

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.

Evolutionary structural optimization for designing delicate holders with uniform or profiled contact pressure

Evolutionary structural optimization for designing delicate holders with uniform or profiled contact pressure

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.

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.

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.