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.

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.

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.

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.

Shape and Size Optimization of Truss Structures under Frequency Constraints Based on Hybrid Sine Cosine Firefly Algorithm

Shape and size optimization with frequency constraints is a highly nonlinear problem withmixed design variables,non-convex search space,and multiple local optima.Therefore,a hybrid sine cosine firefly algorithm(HSCFA)is proposed to acquire more accurate solutions with less finite element analysis.The full attraction model of firefly algorithm(FA)is analyzed,and the factors that affect its computational efficiency and accuracy are revealed.A modified FA with simplified attraction model and adaptive parameter of sine cosine algorithm(SCA)is proposed to reduce the computational complexity and enhance the convergence rate.Then,the population is classified,and different populations are updated by modified FA and SCA respectively.Besides,the random search strategy based on Lévy flight is adopted to update the stagnant or infeasible solutions to enhance the population diversity.Elitist selection technique is applied to save the promising solutions and further improve the convergence rate.Moreover,the adaptive penalty function is employed to deal with the constraints.Finally,the performance of HSCFA is demonstrated through the numerical examples with nonstructural masses and frequency constraints.The results show that HSCFA is an efficient and competitive tool for shape and size optimization problems with frequency constraints.

TOPOLOGY OPTIMIZATION OF TRUSS STRUCTURE WITH FUNDAMENTAL FREQUENCY AND FREQUENCY DOMAIN DYNAMIC RESPONSE CONSTRAINTS

In this paper, adaptive genetic algorithm （AGA） is applied to topology optimization of truss structure with frequency domain excitations. The optimization constraints include fundamental frequency, displacement responses under force excitations and acceleration responses under foundation acceleration excitations. The roulette wheel selection operator, adaptive crossover and mutation operators are used as genetic operators. Some heuristic strategies are put forward to direct the deletion of the extra bars and nodes on truss structures. Three examples demonstrate that the proposed method can yield the optimum structure form and the lightest weight of the given ground structure while satisfying dynamic response constraints.

Structure optimization of connection frames based on frequency sensitivity in macro-micro motion platforms

High-performance connection frames are of great significance for ultra-high acceleration and ultra-precision positioning in macro-micro motion platforms. This paper first takes the connection frame as a research object,builds a finite element model(FEM) of the natural frequency of the frame, and then verifies the correctness of this model. The frequency sensitivity method is then used to perturb the structural parameters of the FEM of the connection frame, and the sensitivities of the first-order natural frequency and mass of the corresponding structural parameters are obtained by calculation and analysis. The design variables are also determined. The natural frequency is used as the optimization objective, and the design parameters and mass of the connection frame are constrained. The structural parameters of the connecting frame are obtained through optimization, and the model is built and verified by experiments. The results show that the first-order natural frequency of the connecting frame is effectively improved by the frequency sensitivity method, avoids resonance between the connecting frame and the voice coil motor, and realizes the lightweight design of the connection frame. This research provides a reliable basis for the stable operation and ultra-precision positioning of ultra-high acceleration macro-motion platforms.

Structure Improvement and Optimization of Gantry Milling System for Complex Boring and Milling Machining Center

To enhance the efficiency and machining precision of the TX1600G complex boring and milling machining center,a study was conducted on the structure of its gantry milling system.This study aimed to mitigate the influence of factors such as structural quality,natural frequency,and stiffness.The approach employed for this investigation involved mechanism topology optimization.To initiate this process,a finite element model of the gantry milling system structure was established.Subsequently,an objective function,comprising strain energy and modal eigenvalues,was synthesized.This objective function was optimized through multi-objective topology optimization,taking into account certain mass fraction constraints and considering various factors,including processing technology.The ultimate goal of this optimization was to create a gantry milling structure that exhibited high levels of dynamic and static stiffness,a superior natural frequency,and reduced mass.To validate the effectiveness of these topology optimization results,a comparison was made between the new and previous structures.The findings of this study serve as a valuable reference for optimizing the structure of other components within the machining center.

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.

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.

Hydraulic Self Servo Swing Cylinder Structure Optimization and Dynamic Characteristics Analysis Based on Genetic Algorithm

The dynamic characteristics of hydraulic self servo swing cylinder were analyzed according to the hydraulic system natural frequency formula. Based on that,a method of the hydraulic self servo swing cylinder structure optimization based on genetic algorithm was proposed in this paper. By analyzing the four parameters that affect the dynamic characteristics, we had to optimize the structure to obtain as larger the Dm( displacement) as possible under the condition with the purpose of improving the dynamic characteristics of hydraulic self servo swing cylinder. So three state equations were established in this paper. The paper analyzed the effect of the four parameters in hydraulic self servo swing cylinder natural frequency equation and used the genetic algorithm to obtain the optimal solution of structure parameters. The model was simulated by substituting the parameters and initial value to the simulink model. Simulation results show that: using self servo hydraulic swing cylinder natural frequency equation to study its dynamic response characteristics is very effective.Compared with no optimization,the overall system dynamic response speed is significantly improved.

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.

Integrated design optimization of composite frames and materials for maximum fundamental frequency with continuous fiber winding angles

Fiber reinforced composite frame structure is an ideal lightweight and large-span structure in the fields of aerospace,satellite and wind turbine.Natural fundamental frequency is one of key indicators in the design requirement of the composite frame since structural resonance can be effectively avoided with the increase of the fundamental frequency.Inspired by the concept of integrated design optmization of composite frame structures and materials,the design optimization for the maximum structural fundamental frequency of fiber reinforced frame structures is proposed.An optimization model oriented at the maximum structural fundamental frequency under a composite material volume constraint is established.Two kinds of independent design variables are optimized,in which one is variables represented structural topology,the other is variables of continuous fiber winding angles.Sensitivity analysis of the frequency with respect to the two kinds of independent design variables is implemented with the semi-analytical sensitivity method.Some representative examples in the manuscript demonstrate that the integrated design optimization of composite structures can effectively explore coupled effects between structural configurations and material properties to increase the structural fundamental frequency.The proposed integrated optimization model has great potential to improve composite frames structural dynamic performance in aerospace industries.

Geometry and size optimization of stiffener layout for three-dimensional box structures with maximization of natural frequencies

Based on the growth mechanism of natural biological branching systems and inspiration from the morphology of plant root tips,a bionic design method called Improved Adaptive Growth Method(IAGM)has been proposed in the authors’previous research and successfully applied to the reinforcement optimization of three-dimensional box structures with respect to natural frequencies.However,as a kind of ground structure methods,the final layout patterns of stiffeners obtained by using the IAGM are highly subjected to their ground structures,which restricts the optimization effect and freedom to further improve the dynamic performance of structures.To solve this problem,a novel post-processing geometry and size optimization approach is proposed in this article.This method takes the former layout optimization result as start,and iteratively finds the optimal layout angles,locations,and lengths of stiffeners with a few design variables by optimizing the positions of some specific node lines called active node lines.At the same time,thick-nesses of stiffeners are also optimized to further improve natural frequencies of three-dimensional box structures.Using this method,stiffeners can be successfully separated from their ground structures and further effectively improve natural frequencies of three-dimensional box structures with less material consumption.Typical numerical examples are illustrated to validate the effectiveness and advantages of the suggested 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 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.

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.

Oxidization-induced structural optimization of Ni_(3)Fe-N-C derived from 3D covalent organic framework for high-efficiency and durable oxygen evolution reaction

NiFe composites have been regarded as promising candidates to replace commercial noble-based electrocatalysts for the oxygen evolution reaction(OER).However,their practical applications still suffer from poor conductivity,limited activity,durability.To address these issues,herein,by utilizing three-dimensional covalent organic framework(3D-COF)with porous confined structures and abundant coordinate N sites as the precursor,the partially oxidized Ni_(3)Fe nanoalloys wrapped by Ndoped carbon(N-C)layers are constructed via simple pyrolysis and subsequent oxidization.Benefiting from the 3D curved hierarchical structure,high-conductivity of Ni_(3)Fe and N-C layers,well-distributed active sites,the as-synthesized O-Ni_(3)Fe-NC catalyst demonstrates excellent activity and durability for catalyzing OER.Experimental and theoretical analyses disclose that both high-temperature oxidization and the OER process greatly promote the formation and exposure of the Ni(Fe)OOH active species as well as lower charge transfer resistance,inducing its optimized OER activity.The robust graphitized N-C layers with superior conductivity and their couplings with oxidized Ni_(3)Fe nanoalloys are beneficial for stabilizing catalytic centers,thereby imparting O-Ni_(3)Fe-N-C with such outstanding stability.This work not only provides a rational guidance for enriching and stabilizing high-activity catalytic sites towards OER but also offers more insights into the structural evolution of NiFe-based OER catalysts.

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.