Stress Relaxation and Sensitivity Weight for Bi-Directional Evolutionary Structural Optimization to Improve the Computational Efficiency and Stabilization on Stress-Based Topology Optimization

Stress-based topology optimization is one of the most concerns of structural optimization and receives much attention in a wide range of engineering designs.To solve the inherent issues of stress-based topology optimization,many schemes are added to the conventional bi-directional evolutionary structural optimization(BESO)method in the previous studies.However,these schemes degrade the generality of BESO and increase the computational cost.This study proposes an improved topology optimization method for the continuum structures considering stress minimization in the framework of the conventional BESO method.A global stress measure constructed by p-norm function is treated as the objective function.To stabilize the optimization process,both qp-relaxation and sensitivity weight scheme are introduced.Design variables are updated by the conventional BESO method.Several 2D and 3D examples are used to demonstrate the validity of the proposed method.The results show that the optimization process can be stabilized by qp-relaxation.The value of q and p are crucial to reasonable solutions.The proposed sensitivity weight scheme further stabilizes the optimization process and evenly distributes the stress field.The computational efficiency of the proposed method is higher than the previous methods because it keeps the generality of BESO and does not need additional schemes.

A Modified Bi-Directional Evolutionary Structural Optimization Procedure with Variable Evolutionary Volume Ratio Applied to Multi-Objective Topology Optimization Problem

Natural frequency and dynamic stiffness under transient loading are two key performances for structural design related to automotive,aviation and construction industries.This article aims to tackle the multi-objective topological optimization problem considering dynamic stiffness and natural frequency using modified version of bi-directional evolutionary structural optimization(BESO).The conventional BESO is provided with constant evolutionary volume ratio(EVR),whereas low EVR greatly retards the optimization process and high EVR improperly removes the efficient elements.To address the issue,the modified BESO with variable EVR is introduced.To compromise the natural frequency and the dynamic stiffness,a weighting scheme of sensitivity numbers is employed to form the Pareto solution space.Several numerical examples demonstrate that the optimal solutions obtained from the modified BESO method have good agreement with those from the classic BESO method.Most importantly,the dynamic removal strategy with the variable EVR sharply springs up the optimization process.Therefore,it is concluded that the modified BESO method with variable EVR can solve structural design problems using multi-objective optimization.

Topology Optimization with Aperiodic Load Fatigue Constraints Based on Bidirectional Evolutionary Structural Optimization

Because of descriptive nonlinearity and computational inefficiency,topology optimization with fatigue life under aperiodic loads has developed slowly.A fatigue constraint topology optimization method based on bidirectional evolutionary structural optimization(BESO)under an aperiodic load is proposed in this paper.In viewof the severe nonlinearity of fatigue damagewith respect to design variables,effective stress cycles are extracted through transient dynamic analysis.Based on the Miner cumulative damage theory and life requirements,a fatigue constraint is first quantified and then transformed into a stress problem.Then,a normalized termination criterion is proposed by approximatemaximum stress measured by global stress using a P-normaggregation function.Finally,optimization examples show that the proposed algorithm can not only meet the requirements of fatigue life but also obtain a reasonable configuration.

A Smooth Bidirectional Evolutionary Structural Optimization of Vibrational Structures for Natural Frequency and Dynamic Compliance

A smooth bidirectional evolutionary structural optimization(SBESO),as a bidirectional version of SESO is proposed to solve the topological optimization of vibrating continuum structures for natural frequencies and dynamic compliance under the transient load.A weighted function is introduced to regulate the mass and stiffness matrix of an element,which has the inefficient element gradually removed from the design domain as if it were undergoing damage.Aiming at maximizing the natural frequency of a structure,the frequency optimization formulation is proposed using the SBESO technique.The effects of various weight functions including constant,linear and sine functions on structural optimization are compared.With the equivalent static load(ESL)method,the dynamic stiffness optimization of a structure is formulated by the SBESO technique.Numerical examples show that compared with the classic BESO method,the SBESO method can efficiently suppress the excessive element deletion by adjusting the element deletion rate and weight function.It is also found that the proposed SBESO technique can obtain an efficient configuration and smooth boundary and demonstrate the advantages over the classic BESO technique.

Computational Simulations of Bone Remodeling under Natural Mechanical Loading or Muscle Malfunction Using Evolutionary Structural Optimization Method

Live bone inherently responds to applied mechanical stimulus by altering its internal tissue composition and ultimately biomechanical properties, structure and function. The final formation may structurally appear inferior by design but complete by function. To understand the loading response, this paper numerically investigated structural remodeling of mature sheep femur using evolutionary structural optimization method (ESO). Femur images from Computed Tomography scanner were used to determine the elastic modulus variation and subsequently construct finite element model of the femur with stiffest elasticity measured. Major muscle forces on dominant phases of healthy sheep gait were imposed on the femur under static mode. ESO was applied to progressively alter the remodeling of numerically simulated femur from its initial to final design by iteratively removing elements with low strain energy density (SED). The computations were repeated with two different mesh sizes to test the convergence. The elements within the medullary canal had low SEDs and therefore were removed during the optimization. The SEDs in the remaining elements varied with angle around the circumference of the shaft. Those elements with low SED were inefficient in supporting the load and thus fundamentally explained how bone remodels itself with less stiff inferior tissue to meet load demand. This was in line with the Wolff’s law of transformation of bone. Tissue growth and remodeling process was found to shape the sheep femur to a mechanically optimized structure and this was initiated by SED in macro-scale according to traditional principle of Wolff’s law.

Structural Topology Optimization by Combining BESO with Reinforcement Learning

In this paper,a new algorithm combining the features of bi-direction evolutionary structural optimization(BESO)and reinforcement learning(RL)is proposed for continuum structural topology optimization(STO).In contrast to conventional approaches which only generate a certain quasi-optimal solution,the goal of the combined method is to provide more quasi-optimal solutions for designers such as the idea of generative design.Two key components were adopted.First,besides sensitivity,value function updated by Monte-Carlo reinforcement learning was utilized to measure the importance of each element,which made the solving process convergent and closer to the optimum.Second,ε-greedy policy added a random perturbation to the main search direction so as to extend the search ability.Finally,the quality and diversity of solutions could be guaranteed by controlling the value of compliance as well as Intersection-over-Union(IoU).Results of several 2D and 3D compliance minimization problems,including a geometrically nonlinear case,show that the combined method is capable of generating a group of good and different solutions that satisfy various possible requirements in engineering design within acceptable computation cost.

Topology Optimization in Damping Structure Based on ESO

The damping material optimal placement for the structure with damping layer is studied based on evolutionary structural optimization (ESO) to maximize modal loss factors. A mathematical model is constructed with the objective function defined as the maximum of modal loss factors of the structure and design constraints function defined as volume fraction of damping material. The optimal placement is found. Several examples are presented for verification. The results demonstrate that the method based on ESO is effective in solving the topology optimization of the structure with unconstrained damping layer and constrained damping layer. This optimization method suits for free and constrained damping structures.

Structural Optimization of Hatch Cover Based on Bi-directional Evolutionary Structure Optimization and Surrogate Model Method

Weight reduction has attracted much attention among ship designers and ship owners.In the present work,based on an improved bi-directional evolutionary structural optimization(BESO) method and surrogate model method,we propose a hybrid optimization method for the structural design optimization of beam-plate structures,which covers three optimization levels:dimension optimization,topology optimization and section optimization.The objective of the proposed optimization method is to minimize the weight of design object under a group of constraints.The kernel optimization procedure(KOP) uses BESO to obtain the optimal topology from a ground structure.To deal with beam-plate structures,the traditional BESO method is improved by using cubic box as the unit cell instead of solid unit to construct periodic lattice structure.In the first optimization level,a series of ground structures are generated based on different dimensional parameter combinations,the KOP is performed to all the ground structures,the response surface model of optimal objective values and dimension parameters is created,and then the optimal dimension parameters can be obtained.In the second optimization level,the optimal topology is obtained by using the KOP according to the optimal dimension parameters.In the third optimization level,response surface method(RSM) is used to determine the section parameters.The proposed method is applied to a hatch cover structure design.The locations and shapes of all the structural members are determined from an oversized ground structure.The results show that the proposed method leads to a greater weight saving,compared with the original design and genetic algorithm(GA) based optimization results.

A Multi-Objective Optimal Evolutionary Algorithm Based on Tree-Ranking

Multi-objective optimal evolutionary algorithms (MOEAs) are a kind of new effective algorithms to solve Multi-objective optimal problem (MOP). Because ranking, a method which is used by most MOEAs to solve MOP, has some shortcoming s, in this paper, we proposed a new method using tree structure to express the relationship of solutions. Experiments prove that the method can reach the Pare-to front, retain the diversity of the population, and use less time.

Effectiveness Assessment of the Search-Based Statistical Structural Testing

Search-based statistical structural testing(SBSST)is a promising technique that uses automated search to construct input distributions for statistical structural testing.It has been proved that a simple search algorithm,for example,the hill-climber is able to optimize an input distribution.However,due to the noisy fitness estimation of the minimum triggering probability among all cover elements(Tri-Low-Bound),the existing approach does not show a satisfactory efficiency.Constructing input distributions to satisfy the Tri-Low-Bound criterion requires an extensive computation time.Tri-Low-Bound is considered a strong criterion,and it is demonstrated to sustain a high fault-detecting ability.This article tries to answer the following question:if we use a relaxed constraint that significantly reduces the time consumption on search,can the optimized input distribution still be effective in faultdetecting ability?In this article,we propose a type of criterion called fairnessenhanced-sum-of-triggering-probability(p-L1-Max).The criterion utilizes the sum of triggering probabilities as the fitness value and leverages a parameter p to adjust the uniformness of test data generation.We conducted extensive experiments to compare the computation time and the fault-detecting ability between the two criteria.The result shows that the 1.0-L1-Max criterion has the highest efficiency,and it is more practical to use than the Tri-Low-Bound criterion.To measure a criterion’s fault-detecting ability,we introduce a definition of expected faults found in the effective test set size region.To measure the effective test set size region,we present a theoretical analysis of the expected faults found with respect to various test set sizes and use the uniform distribution as a baseline to derive the effective test set size region’s definition.

Structural-acoustic topology optimization analysis based on evolutionary structural optimization approach

The continuum structural-acoustic topology optimization with external loading is investigated herein. Finite element method （FEM） is used to obtain the structural frequency response and boundary element method （BEM） is adopted to perform exterior acoustic radiation analysis. The evolutionary structural optimization （ESO） is served as an optimization method in structural-acoustic radiation topology analysis. The acoustic radiation optimization of a plate under harmonic excitation is given for example. The numerical results show that using ESO solution to analyze structural-acoustic topology optimization is feasible and effective.

声子晶体能带结构仿真的CS-FEM方法及其在拓扑优化设计中的应用

针对声子晶体拓扑优化设计时能带结构和灵敏度计算效率低的缺陷,构造了一种Cell-based光滑有限元法模型(CS-FEM),并结合双向渐进结构优化法(BESO)实现了声子晶体的带隙最大化设计。基于四边形单元构造光滑子域,将梯度光滑技术与Bloch定理结合,构建了声子晶体能带结构计算的CS-FEM数值模型,并将其用于正问题的仿真模拟。在渐进优化准则下,通过BESO算法完成了声子晶体的优化设计。数值算例表明:CS-FEM能够适当软化离散系统的刚度,提供更加准确、高效的能带结构仿真结果;基于CS-FEM进行正问题的计算,在优化设计中得到了最优的拓扑构型,并有效地提高了优化效率,对于实现声子晶体的高效设计具有重要的参考价值。

COMPUTER PROGRAM FOR DIRECTED STRUCTURE TOPOLOGY OPTIMIZATION

To compensate for the imperfection of traditional bi-directional evolutionary structural optimization, material interpolation scheme and sensitivity filter functions are introduced. A suitable filter can overcome the checkerboard and mesh-dependency. And the historical information on accurate elemental sensitivity numbers are used to keep the objective function converging steadily. Apart from rational intervals of the relevant important parameters, the concept of distinguishing between active and non-active elements design is proposed, which can be widely used for improving the function and artistry of structures directly, especially for a one whose accurate size is not given. Furthermore, user-friendly software packages are developed to enhance its accessibility for practicing engineers and architects. And to reduce the time cost for large timeconsuming complex structure optimization, parallel computing is built-in in the MATLAB codes. The program is easy to use for engineers who may not be familiar with either FEA or structure optimization. And developers can make a deep research on the algorithm by changing the MATLAB codes. Several classical examples are given to show that the improved BESO method is superior for its handy and utility computer program software.

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 trusscontinuum 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.

基于双向渐进结构优化算法的预制拼装箱梁剪力键设计研究

为研究单项和多项荷载组合作用下节段预制拼装箱梁接缝处剪力键受力状况,以郑州南四环全预制装配高架桥为背景,采用Abaqus软件和双向渐进结构优化(BESO)算法,建立典型箱梁节段分析模型,研究剪力键在单位轴力、剪力、扭矩、弯矩和不同荷载组合作用下的布置规律。结果表明:轴力作用时剪力键应布置于顶板;剪力作用时剪力键应布置于腹板和底板;扭矩作用时剪力键应布置在顶板、腹板交接处和腹板外侧;横桥向弯矩作用时剪力键应布置于顶板、腹板下侧和底板;竖向弯矩作用时,剪力键应布置于顶板翼缘。轴剪组合作用下,当剪力占比小于50%时,轴力对剪力键布置起控制作用;当剪力占比达95.24%时,剪力对剪力键布置起控制作用。弯剪组合作用下,当剪力占比超过4.76%时,剪力对剪力键布置起控制作用;当剪力占比小于0.50%时,横桥向弯矩对剪力键布置起控制作用。扭剪组合作用下,当剪力占比超过0.10%时,剪力对剪力键布置起控制作用。预制拼装箱梁剪力键布置受荷载类型及荷载组合影响显著,应考虑不同位置剪力键的实际受力状况进行设计。

基于应力及其灵敏度的结构拓扑渐进优化方法

在导出应力灵敏度的基础上,建立了一种改进的基于应力及其灵敏度的结构拓扑双方向渐进优化算法.该方法是对传统ESO和BESO方法的改进和完善.算例表明该方法能较大程度减少解的振荡状态数,获得更佳的拓扑结构.具有概念上的简洁性和应用上的有效性.

面向3D打印体积极小的拓扑优化技术

与传统制造所生产的产品相比,3D打印产品的成本仍相对较高.因此,如何能在不牺牲打印物体表面质量的前提下通过模型优化来减少打印材料消耗,对于降低打印成本至关重要.针对这一问题,借鉴传统渐进结构优化方法,结合Von Mises应力计算,给出一种面向3D打印体积极小的拓扑优化算法.该算法通过模型力学计算所得的最大Von Mises应力与材料允许应力之比来引导模型体积减小进化,直至最大Von Mises应力达到允许应力值为止.同时,引入多分辨率技术,由粗网格再到细网格进行优化计算,有效地提高了计算效率.与现有其他给定结构模式的方法相比,该优化结果能更好地体现模型荷载受力的传递路径.

结构动力学设计优化方法的新进展

阐述结构动力学优化设计的研究背景和意义。根据实践,扼要介绍近十年作者在结构动力学设计研究方面取得的若干新近展,内容包括,随机激励下以均方响应为约束的设计方法、结构动力学形状优化、结构动力学拓扑优化、动力学设计约束的性质和解的存在性以及结构控制一体化优化设计,并对进一步的研究工作做简要展望。

一种基于应力的双方向结构拓扑优化算法

基于传统的渐进结构优化方法,提出了一种基于应力的双方向渐进拓扑优化算法。该方法是对传统方法和目前的双方向法的改进和完善。算例表明该方法能避免目前的双方向法具有的解的振荡问题,具有更高的可靠性,能获得更佳的拓扑结构。

循环对称结构静力学渐进拓扑优化设计

采用双向渐进结构优化方法(BESO)研究循环对称结构拓扑优化设计问题。循环对称结构的几何特征无法划分为均匀一致的网格,造成单元灵敏度计算结果与单元体积直接相关,不同网格剖分后的单元体积分布将导致渐进优化过程中不同的单元去除顺序,最终使得优化结果具有单元体积依赖性。针对循环对称结构的特殊性,提出灵敏度密度新概念与消除非等体积单元棋盘格现象的体积加权灵敏度密度过滤新方法,大量算例表明这是一种有效的方法,同时优化结果也充分显示子结构单胞尺寸效应对优化构型的显著影响。