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 Hybrid Parallel Strategy for Isogeometric Topology Optimization via CPU/GPU Heterogeneous Computing

This paper aims to solve large-scale and complex isogeometric topology optimization problems that consumesignificant computational resources. A novel isogeometric topology optimization method with a hybrid parallelstrategy of CPU/GPU is proposed, while the hybrid parallel strategies for stiffness matrix assembly, equationsolving, sensitivity analysis, and design variable update are discussed in detail. To ensure the high efficiency ofCPU/GPU computing, a workload balancing strategy is presented for optimally distributing the workload betweenCPU and GPU. To illustrate the advantages of the proposedmethod, three benchmark examples are tested to verifythe hybrid parallel strategy in this paper. The results show that the efficiency of the hybrid method is faster thanserial CPU and parallel GPU, while the speedups can be up to two orders of magnitude.

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.

Topology Optimization of Two Fluid Heat Transfer Problems for Heat Exchanger Design

Topology optimization of thermal-fluid coupling problems has received widespread attention.This article proposes a novel topology optimization method for laminar two-fluid heat exchanger design.The proposed method utilizes an artificial density field to create two permeability interpolation functions that exhibit opposing trends,ensuring separation between the two fluid domains.Additionally,a Gaussian function is employed to construct an interpolation function for the thermal conductivity coefficient.Furthermore,a computational program has been developed on the OpenFOAM platform for the topology optimization of two-fluid heat exchangers.This program leverages parallel computing,significantly reducing the time required for the topology optimization process.To enhance computational speed and reduce the number of constraint conditions,we replaced the conventional pressure drop constraint condition in the optimization problem with a pressure inlet/outlet boundary condition.The 3D optimization results demonstrate the characteristic features of a surface structure,providing valuable guidance for designing heat exchangers that achieve high heat exchange efficiency while minimizing excessive pressure loss.At the same time,a new structure appears in large-scale topology optimization,which proves the effectiveness and stability of the topology optimization program written in this paper in large-scale calculation.

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.

Multi-Material Topology Optimization of 2D Structures Using Convolutional Neural Networks

In recent years,there has been significant research on the application of deep learning(DL)in topology optimization(TO)to accelerate structural design.However,these methods have primarily focused on solving binary TO problems,and effective solutions for multi-material topology optimization(MMTO)which requires a lot of computing resources are still lacking.Therefore,this paper proposes the framework of multiphase topology optimization using deep learning to accelerate MMTO design.The framework employs convolutional neural network(CNN)to construct a surrogate model for solving MMTO,and the obtained surrogate model can rapidly generate multi-material structure topologies in negligible time without any iterations.The performance evaluation results show that the proposed method not only outputs multi-material topologies with clear material boundary but also reduces the calculation cost with high prediction accuracy.Additionally,in order to find a more reasonable modeling method for MMTO,this paper studies the characteristics of surrogate modeling as regression task and classification task.Through the training of 297 models,our findings show that the regression task yields slightly better results than the classification task in most cases.Furthermore,The results indicate that the prediction accuracy is primarily influenced by factors such as the TO problem,material category,and data scale.Conversely,factors such as the domain size and the material property have minimal impact on the accuracy.

A Subdivision-Based Combined Shape and Topology Optimization in Acoustics

We propose a combined shape and topology optimization approach in this research for 3D acoustics by using the isogeometric boundary element method with subdivision surfaces.The existing structural optimization methods mainly contain shape and topology schemes,with the former changing the surface geometric profile of the structure and the latter changing thematerial distribution topology or hole topology of the structure.In the present acoustic performance optimization,the coordinates of the control points in the subdivision surfaces fine mesh are selected as the shape design parameters of the structure,the artificial density of the sound absorbing material covered on the structure surface is set as the topology design parameter,and the combined topology and shape optimization approach is established through the sound field analysis of the subdivision surfaces boundary element method as a bridge.The topology and shape sensitivities of the approach are calculated using the adjoint variable method,which ensures the efficiency of the optimization.The geometric jaggedness and material distribution discontinuities that appear in the optimization process are overcome to a certain degree by the multiresolution method and solid isotropic material with penalization.Numerical examples are given to validate the effectiveness of the presented optimization approach.

Ballistic performance of additive manufacturing 316l stainless steel projectiles based on topology optimization method

Material and structure made by additive manufacturing(AM)have received much attention lately due to their flexibility and ability to customize complex structures.This study first implements multiple objective topology optimization simulations based on a projectile perforation model,and a new topologic projectile is obtained.Then two types of 316L stainless steel projectiles(the solid and the topology)are printed in a selective laser melt(SLM)machine to evaluate the penetration performance of the projectiles by the ballistic test.The experiment results show that the dimensionless specific kinetic energy value of topologic projectiles is higher than that of solid projectiles,indicating the better penetration ability of the topologic projectiles.Finally,microscopic studies(scanning electron microscope and X-ray micro-CT)are performed on the remaining projectiles to investigate the failure mechanism of the internal structure of the topologic projectiles.An explicit dynamics simulation was also performed,and the failure locations of the residual topologic projectiles were in good agreement with the experimental results,which can better guide the design of new projectiles combining AM and topology optimization in the future.

An Overview of Sequential Approximation in Topology Optimization of Continuum Structure

This paper offers an extensive overview of the utilization of sequential approximate optimization approaches in the context of numerically simulated large-scale continuum structures.These structures,commonly encountered in engineering applications,often involve complex objective and constraint functions that cannot be readily expressed as explicit functions of the design variables.As a result,sequential approximation techniques have emerged as the preferred strategy for addressing a wide array of topology optimization challenges.Over the past several decades,topology optimization methods have been advanced remarkably and successfully applied to solve engineering problems incorporating diverse physical backgrounds.In comparison to the large-scale equation solution,sensitivity analysis,graphics post-processing,etc.,the progress of the sequential approximation functions and their corresponding optimizersmake sluggish progress.Researchers,particularly novices,pay special attention to their difficulties with a particular problem.Thus,this paper provides an overview of sequential approximation functions,related literature on topology optimization methods,and their applications.Starting from optimality criteria and sequential linear programming,the other sequential approximate optimizations are introduced by employing Taylor expansion and intervening variables.In addition,recent advancements have led to the emergence of approaches such as Augmented Lagrange,sequential approximate integer,and non-gradient approximation are also introduced.By highlighting real-world applications and case studies,the paper not only demonstrates the practical relevance of these methods but also underscores the need for continued exploration in this area.Furthermore,to provide a comprehensive overview,this paper offers several novel developments that aim to illuminate potential directions for future research.

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.

Topology Optimization of Metamaterial Microstructures for Negative Poisson’s Ratio under Large Deformation Using a Gradient-Free Method

Negative Poisson’s ratio(NPR)metamaterials are attractive for their unique mechanical behaviors and potential applications in deformation control and energy absorption.However,when subjected to significant stretching,NPR metamaterials designed under small strain assumption may experience a rapid degradation in NPR performance.To address this issue,this study aims to design metamaterials maintaining a targeted NPR under large deformation by taking advantage of the geometry nonlinearity mechanism.A representative periodic unit cell is modeled considering geometry nonlinearity,and its topology is designed using a gradient-free method.The unit cell microstructural topologies are described with the material-field series-expansion(MFSE)method.The MFSE method assumes spatial correlation of the material distribution,which greatly reduces the number of required design variables.To conveniently design metamaterials with desired NPR under large deformation,we propose a two-stage gradient-free metamaterial topology optimization method,which fully takes advantage of the dimension reduction benefits of the MFSE method and the Kriging surrogate model technique.Initially,we use homogenization to find a preliminary NPR design under a small deformation assumption.In the second stage,we begin with this preliminary design and minimize deviations in NPR from a targeted value under large deformation.Using this strategy and solution technique,we successfully obtain a group of NPR metamaterials that can sustain different desired NPRs in the range of[−0.8,−0.1]under uniaxial stretching up to 20% strain.Furthermore,typical microstructure designs are fabricated and tested through experiments.The experimental results show good consistency with our numerical results,demonstrating the effectiveness of the present gradientfree NPR metamaterial design strategy.

The Topology Optimization of Oil Cylinder Mounting

The method of the structural topology optimization is often used to design machine in the early stage of the mechanical design.And the mechanical structures use the topology design to produce a new still and lightweight part with the different loading.A new structure is created through overlapping these new optimized structure.

Systematical Development of NVH Engineering for Vehicle Electrical Powertrain Based on an Optimized V-Model

For systematical NVH development of vehicle (especially for mass-production passenger vehicles) electric powertrain, an optimized V-Model is designed and has been implemented in the entire component-vehicle development, which integrates three individual branches: simulation, validation and optimization. Compared to the V-models in the traditional sense, this optimized V-model is not only driven by requirement and task accomplishment but also maximum optimization of NVH system performance. In this case, developing procedures are capable to be efficiently iterated and the NVH engineering can be expanded into 3D with this V-model.

基于TLBO算法的不确定性条件下复杂产品协同设计的可靠性拓扑优化

复杂产品的拓扑优化设计可以显著节省材料和节能,有效地降低惯性力和机械振动。本研究以一种大吨位液压机作为典型的复杂产品,用于阐述该优化方法。本文提出了一种基于可靠性与优化解耦模型和基于教学学习的优化(TLBO)算法的可靠性拓扑优化方法。将由板结构形成的支撑物作为拓扑优化对象,重量轻、稳定性好。将不确定性下的可靠性优化和结构拓扑优化协同处理。首先,利用有限差分法将优化问题中的不确定性参数修正为确定性参数。然后,将不确定性可靠性分析和拓扑优化的复杂嵌套解耦。最后,利用TLBO算法求解解耦模型,该算法参数少,求解速度快。TLBO算法采用了自适应教学因子,在初始阶段实现了更快的收敛速度,并在后期进行了更精细的搜索。本文给出了一个液压机基板结构的数值实例,说明了该方法的有效性。

Open-Source Codes of Topology Optimization: A Summary for Beginners to Start Their Research

Topology optimization(TO),a numerical technique to find the optimalmaterial layoutwith a given design domain,has attracted interest from researchers in the field of structural optimization in recent years.For beginners,opensource codes are undoubtedly the best alternative to learning TO,which can elaborate the implementation of a method in detail and easily engage more people to employ and extend the method.In this paper,we present a summary of various open-source codes and related literature on TO methods,including solid isotropic material with penalization(SIMP),evolutionary method,level set method(LSM),moving morphable components/voids(MMC/MMV)methods,multiscale topology optimization method,etc.Simultaneously,we classify the codes into five levels,fromeasy to difficult,depending on their difficulty,so that beginners can get started and understand the form of code implementation more quickly.

Reliability-Based Topology Optimization of Fail-Safe Structures Using Moving Morphable Bars

This paper proposes an effective reliability design optimizationmethod for fail-safe topology optimization(FSTO)considering uncertainty based on the moving morphable bars method to establish the ideal balance between cost and robustness,reliability and structural safety.To this end,a performancemeasure approach(PMA)-based doubleloop optimization algorithmis developed tominimize the relative volume percentage while achieving the reliability criterion.To ensure the compliance value of the worst failure case can better approximate the quantified design requirement,a p-norm constraint approach with correction parameter is introduced.Finally,the significance of accounting for uncertainty in the fail-safe design is illustrated by contrasting the findings of the proposed reliabilitybased topology optimization(RBTO)method with those of the deterministic design method in three typical examples.Monte Carlo simulation shows that the relative error of the reliability index of the optimized structure does not exceed 3%.

Explicit Isogeometric Topology Optimization Method with Suitably Graded Truncated Hierarchical B-Spline

This work puts forward an explicit isogeometric topology optimization(ITO)method using moving morphable components(MMC),which takes the suitably graded truncated hierarchical B-Spline based isogeometric analysis as the solver of physical unknown(SGTHB-ITO-MMC).By applying properly basis graded constraints to the hierarchical mesh of truncated hierarchical B-splines(THB),the convergence and robustness of the SGTHB-ITOMMC are simultaneously improved and the tiny holes occurred in optimized structure are eliminated,due to the improved accuracy around the explicit structural boundaries.Moreover,an efficient computational method is developed for the topological description functions(TDF)ofMMC under the admissible hierarchicalmesh,which consists of reducing the dimensionality strategy for design space and the locally computing strategy for hierarchical mesh.We apply the above SGTHB-ITO-MMC with improved efficiency to a series of 2D and 3Dcompliance design problems.The numerical results show that the proposed SGTHB-ITO-MMC method outperforms the traditional THB-ITO-MMCmethod in terms of convergence rate and efficiency.Therefore,the proposed SGTHB-ITO-MMC is an effective way of solving topology optimization(TO)problems.

Topology Optimization for Steady-State Navier-Stokes Flow Based on Parameterized Level Set Based Method

In this paper,we consider solving the topology optimization for steady-state incompressibleNavier-Stokes problems via a new topology optimization method called parameterized level set method,which can maintain a relatively smooth level set function with a local optimality condition.The objective of topology optimization is to􀀀nd an optimal con􀀀guration of theuid and solid materials that minimizes power dissipation under a prescribeduid volume fraction constraint.An arti􀀀cial friction force is added to the Navier-Stokes equations to apply the no-slip boundary condition.Although a great deal of work has been carried out for topology optimization ofuidow in recent years,there are few researches on the topology optimization ofuidow with physical body forces.To simulate theuidow in reality,the constant body force(e.g.,gravity)is considered in this paper.Several 2D numerical examples are presented to discuss the relationships between the proposed method with Reynolds number and initial design,and demonstrate the feasibility and superiority of the proposed method in dealing with unstructuredmesh problems.Three 3D numerical examples demonstrate the proposedmethod is feasible in three-dimensional.

Topology Optimization for Harmonic Excitation Structures with Minimum Length Scale Control Using the Discrete Variable Method

Continuumtopology optimization considering the vibration response is of great value in the engineering structure design.The aimof this studyis toaddress the topological designoptimizationof harmonic excitationstructureswith minimumlength scale control to facilitate structuralmanufacturing.Astructural topology design based on discrete variables is proposed to avoid localized vibration modes,gray regions and fuzzy boundaries in harmonic excitation topology optimization.The topological design model and sensitivity formulation are derived.The requirement of minimum size control is transformed into a geometric constraint using the discrete variables.Consequently,thin bars,small holes,and sharp corners,which are not conducive to the manufacturing process,can be eliminated from the design results.The present optimization design can efficiently achieve a 0–1 topology configuration with a significantly improved resonance frequency in a wide range of excitation frequencies.Additionally,the optimal solution for harmonic excitation topology optimization is not necessarily symmetric when the load and support are symmetric,which is a distinct difference fromthe static optimization design.Hence,one-half of the design domain cannot be selected according to the load and support symmetry.Numerical examples are presented to demonstrate the effectiveness of the discrete variable design for excitation frequency topology optimization,and to improve the design manufacturability.