Web Layout Design of Large Cavity Structures Based on Topology Optimization

Large cavity structures are widely employed in aerospace engineering, such as thin-walled cylinders, blades andwings. Enhancing performance of aerial vehicles while reducing manufacturing costs and fuel consumptionhas become a focal point for contemporary researchers. Therefore, this paper aims to investigate the topologyoptimization of large cavity structures as a means to enhance their performance, safety, and efficiency. By usingthe variable density method, lightweight design is achieved without compromising structural strength. Theoptimization model considers both concentrated and distributed loads, and utilizes techniques like sensitivityfiltering and projection to obtain a robust optimized configuration. The mechanical properties are checked bycomparing the stress distribution and displacement of the unoptimized and optimized structures under the sameload. The results confirm that the optimized structures exhibit improved mechanical properties, thus offering keyinsights for engineering lightweight, high-strength large cavity structures.

Multi-Material Topology Optimization for Spatial-Varying Porous Structures

This paper aims to propose a topology optimization method on generating porous structures comprising multiple materials.The mathematical optimization formulation is established under the constraints of individual volume fraction of constituent phase or total mass,as well as the local volume fraction of all phases.The original optimization problem with numerous constraints is converted into a box-constrained optimization problem by incorporating all constraints to the augmented Lagrangian function,avoiding the parameter dependence in the conventional aggregation process.Furthermore,the local volume percentage can be precisely satisfied.The effects including the globalmass bound,the influence radius and local volume percentage on final designs are exploited through numerical examples.The numerical results also reveal that porous structures keep a balance between the bulk design and periodic design in terms of the resulting compliance.All results,including those for irregular structures andmultiple volume fraction constraints,demonstrate that the proposedmethod can provide an efficient solution for multiple material infill structures.

A non-probabilistic reliability topology optimization method based on aggregation function and matrix multiplication considering buckling response constraints

A non-probabilistic reliability topology optimization method is proposed based on the aggregation function and matrix multiplication.The expression of the geometric stiffness matrix is derived,the finite element linear buckling analysis is conducted,and the sensitivity solution of the linear buckling factor is achieved.For a specific problem in linear buckling topology optimization,a Heaviside projection function based on the exponential smooth growth is developed to eliminate the gray cells.The aggregation function method is used to consider the high-order eigenvalues,so as to obtain continuous sensitivity information and refined structural design.With cyclic matrix programming,a fast topology optimization method that can be used to efficiently obtain the unit assembly and sensitivity solution is conducted.To maximize the buckling load,under the constraint of the given buckling load,two types of topological optimization columns are constructed.The variable density method is used to achieve the topology optimization solution along with the moving asymptote optimization algorithm.The vertex method and the matching point method are used to carry out an uncertainty propagation analysis,and the non-probability reliability topology optimization method considering buckling responses is developed based on the transformation of non-probability reliability indices based on the characteristic distance.Finally,the differences in the structural topology optimization under different reliability degrees are illustrated by examples.

Probabilistic-Ellipsoid Hybrid Reliability Multi-Material Topology Optimization Method Based on Stress Constraint

This paper proposes a multi-material topology optimization method based on the hybrid reliability of the probability-ellipsoid model with stress constraint for the stochastic uncertainty and epistemic uncertainty of mechanical loads in optimization design.The probabilistic model is combined with the ellipsoidal model to describe the uncertainty of mechanical loads.The topology optimization formula is combined with the ordered solid isotropic material with penalization(ordered-SIMP)multi-material interpolation model.The stresses of all elements are integrated into a global stress measurement that approximates the maximum stress using the normalized p-norm function.Furthermore,the sequential optimization and reliability assessment(SORA)is applied to transform the original uncertainty optimization problem into an equivalent deterministic topology optimization(DTO)problem.Stochastic response surface and sparse grid technique are combined with SORA to get accurate information on the most probable failure point(MPP).In each cycle,the equivalent topology optimization formula is updated according to the MPP information obtained in the previous cycle.The adjoint variable method is used for deriving the sensitivity of the stress constraint and the moving asymptote method(MMA)is used to update design variables.Finally,the validity and feasibility of the method are verified by the numerical example of L-shape beam design,T-shape structure design,steering knuckle,and 3D T-shaped beam.

Full-Scale Isogeometric Topology Optimization of Cellular Structures Based on Kirchhoff-Love Shells

Cellular thin-shell structures are widely applied in ultralightweight designs due to their high bearing capacity and strength-to-weight ratio.In this paper,a full-scale isogeometric topology optimization(ITO)method based on Kirchhoff-Love shells for designing cellular tshin-shell structures with excellent damage tolerance ability is proposed.This method utilizes high-order continuous nonuniform rational B-splines(NURBS)as basis functions for Kirchhoff-Love shell elements.The geometric and analysis models of thin shells are unified by isogeometric analysis(IGA)to avoid geometric approximation error and improve computational accuracy.The topological configurations of thin-shell structures are described by constructing the effective density field on the controlmesh.Local volume constraints are imposed in the proximity of each control point to obtain bone-like cellular structures.To facilitate numerical implementation,the p-norm function is used to aggregate local volume constraints into an equivalent global constraint.Several numerical examples are provided to demonstrate the effectiveness of the proposed method.After simulation and comparative analysis,the results indicate that the cellular thin-shell structures optimized by the proposed method exhibit great load-carrying behavior and high damage robustness.

Topological optimization of ballistic protective structures through genetic algorithms in a vulnerability-driven environment

Reducing the vulnerability of a platform,i.e.,the risk of being affected by hostile objects,is of paramount importance in the design process of vehicles,especially aircraft.A simple and effective way to decrease vulnerability is to introduce protective structures to intercept and possibly stop threats.However,this type of solution can lead to a significant increase in weight,affecting the performance of the aircraft.For this reason,it is crucial to study possible solutions that allow reducing the vulnerability of the aircraft while containing the increase in structural weight.One possible strategy is to optimize the topology of protective solutions to find the optimal balance between vulnerability and the weight of the added structures.Among the many optimization techniques available in the literature for this purpose,multiobjective genetic algorithms stand out as promising tools.In this context,this work proposes the use of a in-house software for vulnerability calculation to guide the process of topology optimization through multi-objective genetic algorithms,aiming to simultaneously minimize the weight of protective structures and vulnerability.In addition to the use of the in-house software,which itself represents a novelty in the field of topology optimization of structures,the method incorporates a custom mutation function within the genetic algorithm,specifically developed using a graph-based approach to ensure the continuity of the generated structures.The tool developed for this work is capable of generating protections with optimized layouts considering two different types of impacting objects,namely bullets and fragments from detonating objects.The software outputs a set of non-dominated solutions describing different topologies that the user can choose from.

Explicit Topology Optimization Design of Stiffened Plate Structures Based on the Moving Morphable Component(MMC)Method

This paper proposes an explicit method for topology optimization of stiffened plate structures.The present work is devoted to simultaneously optimizing stiffeners’shape,size and layout by seeking the optimal geometry parameters of a series of moving morphable components(MMC).The stiffeners with straight skeletons and the stiffeners with curved skeletons are considered to enhance the modeling and optimization capability of the current approach.All the stiffeners are represented under the Lagrangian-description framework in a fully explicit way,and the adaptive ground structure method,as well as dynamically updated plate/shell elements,is used to obtain optimized designs with more accurate analysis results.Compared with existing works,the proposed approach provides an explicit description of the structure.Thus,a stiffened plate structure with clear stiffener distribution and smooth geometric boundary can be obtained.Several numerical examples provided,including straight and curved stiffeners,hierarchical stiffeners,and a stiffened plate with a cutout,validate the effectiveness and applicability of the proposed approach.

DYNAMIC RESPONSE OPTIMIZATION DESIGN FOR ENGINEERING STRUCTURES BASED ON RELIABILITY

In many practical structures, physical parameters of material and applied loads have random property.To optimize this kind of structures,an optimum mathematical model was built.This model has reliability constraints on dynamic stress and displacement and upper & lower limits of the design variables. The numerical characteristic of dynamic response and sensitivity of dynamic response based on probability of structure were deduced respectively. By equivalent disposing, the reliability constraints were changed into conventional forms. The SUMT method was used in the optimization process.Two examples illustrate the correctness and practicability of the optimum model and solving approach.

AWK-TIS:An Improved AK-IS Based on Whale Optimization Algorithm and Truncated Importance Sampling for Reliability Analysis

In this work,an improved active kriging method based on the AK-IS and truncated importance sampling(TIS)method is proposed to efficiently evaluate structural reliability.The novel method called AWK-TIS is inspired by AK-IS and RBF-GA previously published in the literature.The innovation of the AWK-TIS is that TIS is adopted to lessen the sample pool size significantly,and the whale optimization algorithm(WOA)is employed to acquire the optimal Krigingmodel and themost probable point(MPP).To verify the performance of theAWK-TISmethod for structural reliability,four numerical cases which are utilized as benchmarks in literature and one real engineering problem about a jet van manipulate mechanism are tested.The results indicate the accuracy and efficiency of the proposed method.

Genetic Optimization via Diverse Crossover Intelligence

An intelligent crossover methodology within the genetic algorithm (GA) is explored within both mathematical and finite element arenas improving both design and solution convergence time. This improved intelligent crossover outperforms the traditional genetic algorithm combined with a rule-based approach utilizing domain specific knowledge developed by Webb, et al. [1]. The encoding of the improved crossover consists of two chromosome strings within the genetic algorithm where the first string represents the design or solution string, and the second string represents chromosome crossover string intelligence. This improved crossover methodology saves the best population members or designs evaluated from each generation and applies crossover chromosome intelligence to the best saved population members paired with globally selected parents. Enhanced features of this crossover methodology employ the random selection of the best designs from the prior generation as a potential parent coupled with alternating intelligence pairing methods. In addition to this approach, two globally selected parents possess the ability to mate utilizing crossover chromosome string intelligence maintaining the integrity of a global GA search. Overall, the final population following crossover employs both global and best generation design chromosome strings to maximize creativity while enhancing the solution search. This is a modification to a conventional GA that can be translated into GA encoding. This technique is explored initially through a Base 10 mathematical application followed by the examination of plate structural optimization considering stress and displacement constraints. Results from crossover intelligence are compared with the conventional genetic algorithm and from Webb, et al. [1] which illustrates the outcome of a two phase genetic optimization algorithm.

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.

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 Simple and Efficient Structural Topology Optimization Implementation Using Open-Source Software for All Steps of the Algorithm:Modeling, Sensitivity Analysis and Optimization

This work analyzes the implementation of a continuous method of structural topology optimization(STO)using open-source software for all stages of the topology optimization problem:modeling,sensitivity analysis and optimization.Its implementation involves three main components:numerical analysis using the Finite Element Method(FEM),sensitivity analysis using an Adjoint method and an optimization solver.In order to allow the automated numerical solution of Partial Differential Equations(PDEs)and perform a sensitivity analysis,FEniCS and Dolfin Adjoint software are used as tools,which are open-source code.For the optimization process,Ipopt(Interior Point OPTimizer)is used,which is a software package for nonlinear optimization scale designed to find(local)solutions of mathematical optimization problems.The topological optimization method used is based on the SIMP-Solid IsotropicMaterial with Penalization interpolation.The considered problem is the minimization of compliance/maximization of stiffness,considering the examples of recurrent structures in the literature in 2D and 3D.A density filtering algorithm based on Helmholtz formulation is used.The complete code involves 51 lines of programming and is presented and commented in detail in this article.

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.

ICM method for topology optimization of multimaterial continuum structure with displacement constraint

A new topology optimization method is formulated for lightweight design of multimaterial structures, using the independent continuous mapping (ICM) method to minimize the weight with a prescribed nodal displacement constraint. Two types of independent topological variable are used to identify the presence of elements and select the material for each phase, to realize the interpolations of the element stiffness matrix and total weight. Furthermore, an explicit expression for the optimized formulation is derived, using approximations of the displacement and weight given by first- and second-order Taylor expansions. The optimization problem is thereby transformed into a standard quadratic programming problem that can be solved using a sequential quadratic programming approach. The feasibility and effectiveness of the proposed multimaterial topology optimization method are demonstrated by determining the best load transfer path for four numerical examples. The results reveal that the topologically optimized configuration of the multimaterial structure varies with the material properties, load conditions, and constraint. Firstly, the weight of the optimized multimaterial structure is found to be lower than that composed of a single material. Secondly, under the precondition of a displacement constraint, the weight of the topologically optimized multimaterial structure decreases as the displacement constraint value is increased. Finally, the topologically optimized multimaterial structures differ depending on the elastic modulus of the materials. Besides, the established optimization formulation is more reliable and suitable for use in practical engineering applications with structural performance parameters as constraint.

Plate/shell structure topology optimization of orthotropic material for buckling problem based on independent continuous topological variables

The purpose of the present work is to study the buckling problem with plate/shell topology optimization of orthotropic material. A model of buckling topology optimization is established based on the independent, continuous, and mapping method, which considers structural mass as objective and buckling critical loads as constraints. Firstly, composite exponential function (CEF) and power function (PF) as filter functions are introduced to recognize the element mass, the element stiffness matrix, and the element geometric stiffness matrix. The filter functions of the orthotropic material stiffness are deduced. Then these filter functions are put into buckling topology optimization of a differential equation to analyze the design sensitivity. Furthermore, the buckling constraints are approximately expressed as explicit functions with respect to the design variables based on the first-order Taylor expansion. The objective function is standardized based on the second-order Taylor expansion. Therefore, the optimization model is translated into a quadratic program. Finally, the dual sequence quadratic programming (DSQP) algorithm and the global convergence method of moving asymptotes algorithm with two different filter functions (CEF and PF) are applied to solve the optimal model. Three numerical results show that DSQP&CEF has the best performance in the view of structural mass and discretion.

Topology Optimization of Compliant Mechanisms with Geometrical Nonlinearities Using the Ground Structure Approach

The majority of topology optimization of compliant mechanisms uses linear finite element models to find the structure responses.Because the displacements of compliant mechanisms are intrinsically large,the topological design can not provide quantitatively accurate result.Thus,topological design of these mechanisms considering geometrical nonlinearities is essential.A new methodology for geometrical nonlinear topology optimization of compliant mechanisms under displacement loading is presented.Frame elements are chosen to represent the design domain because they are capable of capturing the bending modes.Geometrically nonlinear structural response is obtained by using the co-rotational total Lagrange finite element formulation,and the equilibrium is solved by using the incremental scheme combined with Newton-Raphson iteration.The multi-objective function is developed by the minimum strain energy and maximum geometric advantage to design the mechanism which meets both stiffness and flexibility requirements, respectively.The adjoint method and the direct differentiation method are applied to obtain the sensitivities of the objective functions. The method of moving asymptotes（MMA） is employed as optimizer.The numerical example is simulated to show that the optimal mechanism based on geometrically nonlinear formulation not only has more flexibility and stiffness than that based on linear formulation,but also has better stress distribution than the one.It is necessary to design compliant mechanisms using geometrically nonlinear topology optimization.Compared with linear formulation,the formulation for geometrically nonlinear topology optimization of compliant mechanisms can give the compliant mechanism that has better mechanical performance.A new method is provided for topological design of large displacement compliant mechanisms.

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.

Power Flow Response Based Dynamic Topology Optimization of Bi-material Plate Structures

Work on dynamic topology optimization of engineering structures for vibration suppression has mainly addressed the maximization of eigenfrequencies and gaps between consecutive eigenfrequencies of free vibration, minimization of the dynamic compliance subject to forced vibration, and minimization of the structural frequency response. A dynamic topology optimization method of bi-material plate structures is presented based on power flow analysis. Topology optimization problems formulated directly with the design objective of minimizing the power flow response are dealt with. In comparison to the displacement or velocity response, the power flow response takes not only the amplitude of force and velocity into account, but also the phase relationship of the two vector quantities. The complex expression of power flow response is derived based on time-harmonic external mechanical loading and Rayleigh damping. The mathematical formulation of topology optimization is established based on power flow response and bi-material solid isotropic material with penalization(SIMP) model. Computational optimization procedure is developed by using adjoint design sensitivity analysis and the method of moving asymptotes(MMA). Several numerical examples are presented for bi-material plate structures with different loading frequencies, which verify the feasibility and effectiveness of this method. Additionally, optimum results between topological design of minimum power flow response and minimum dynamic compliance are compared, showing that the present method has strong adaptability for structural dynamic topology optimization problems. The proposed research provides a more accurate and effective approach for dynamic topology optimization of vibrating structures.

THE SATELLITE STRUCTURE TOPOLOGY OPTIMIZATION BASED ON HOMOGENIZATION METHOD AND ITS SIZE SENSITIVITY ANALYSIS

With the development of satellite structure technology, more and more design parameters will affect its structural performance. It is desirable to obtain an optimal structure design with a minimum weight, including optimal configuration and sizes. The present paper aims to describe an optimization analysis for a satellite structure, including topology optimization and size optimization. Based on the homogenization method, the topology optimization is carried out for the main supporting frame of service module under given constraints and load conditions, and then the sensitivity analysis is made of 15 structural size parameters of the whole satellite and the optimal sizes are obtained. The numerical result shows that the present optimization design method is very effective.