Generalized Benders’ Decomposition for topology optimization problems

This article considers the non-linear mixed 0–1 optimization problems that appear in topology optimization of load carrying structures. The main objective is to present a Generalized Benders’ Decomposition (GBD) method for solving single and multiple load minimum compliance (maximum stiffness) problems with discrete design variables to global optimality. We present the theoretical aspects of the method, including a proof of finite convergence and conditions for obtaining global optimal solutions. The method is also linked to, and compared with, an Outer-Approximation approach and a mixed 0–1 semi definite programming formulation of the considered problem. Several ways to accelerate the method are suggested and an implementation is described. Finally, a set of truss topology optimization problems are numerically solved to global optimality.     

Optimization of topology and shape by combining phase field modeling and discrete stochastic algorithms

We propose a concept for the comprehensive optimization of load bearing structures. The main idea is to decompose the optimization task. First, we use a phase field model to evolve the topology as basic design. The parameters of this process are the filling level of the design domain and the tendency to nucleate more or less holes. Second, we use an interface to set up a subsequent structural model, where cross-sections and positions of nodes are simultaneous optimization parameters of a metaheuristic optimization method. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Topology optimization of periodic Mindlin plates

Periodic structures exhibit unique dynamic characteristics that make them act as tunable mechanical filters for wave propagation. As a result, waves can propagate along the periodic structures only within specific frequency bands called the ‘pass bands’ and wave propagation is completely blocked within other frequency bands called the ‘stop bands’ or ‘band gaps’. The spectral width of these bands can be optimized using topology optimization. In this paper, topology optimization is used to maximize the fundamental natural frequency of Mindlin plates while enforcing periodicity. A finite element model for Mindlin plates is presented and used along with an optimization algorithm that accounts for the periodicity constraint in order to determine the optimal topologies of plates with various periodic configurations. The obtained results demonstrate the effectiveness of the proposed design optimization approach in generating periodic plates with optimal natural frequency and wide stop bands. The presented approach can be invaluable design tool for many structures in order to control the wave propagation in an attempt to stop/confine the propagation of undesirable disturbances.     

应力和位移约束下连续体结构拓扑优化

同时考滤应力和位移约束的连续体结构拓扑优化问题,很难用现有的均匀方法或变密度方法等求解。主要困难在于难以建立应力和位移约束与拓扑设计变量间显式关系式;即使建立了这种关系,也由于优化问题规模过大,利用常规的数学规划方法难以求解。隋允康、杨德庆曾提出了基于独立连续拓扑变量及映射变换（ＩＣＭ）的桁架结构拓扑优化模型。本文在此基础上,建立了以重量为目标,考虑应力和位移约束的连续体结构拓扑优化模型,并推导出     

基于LiToSim平台的海上风机过渡段优化软件开发

针对海上风机过渡段结构,考虑风机多尺度优化模型和所受环境荷载采取极端情况下,引入双向渐进结构拓扑优化方法,以全局应力最小化为目标、体积为约束,对风机过渡段进行优化设计;并在自主研发的LiToSim平台基础上,嵌入风机优化数值计算程序,最终形成一款关于海上风机过渡段拓扑优化的定制化软件TUR/TOPT.借助定制化软件TUR/TOPT平台,对比过渡段传统柔度优化与应力优化结果,突显出应力优化在减材设计过程中结构应力明显降低且能有效避免应力集中等方面的优势;TUR/TOPT软件的生成在风机建设选型过程中具有重要指导价值.     

基于参数化水平集法的材料非线性子结构拓扑优化

针对利用传统水平集法进行非线性结构拓扑优化计算过程复杂及计算效率低等问题,将参数化水平集方法引入材料非线性结构拓扑优化中。通过全局径向基函数插值初始水平集函数,建立了以插值系数为设计变量、结构的应变能最小为目标函数、材料用量为约束条件的材料非线性结构拓扑优化模型,利用有限元分析对材料非线性结构建立平衡方程,并用迭代法求解。同时,采用子结构法划分设计区域为若干个子区域,将全自由度平衡方程的求解分解为缩减的平衡方程和多个子结构内部位移的求解,减小了计算成本。算例表明,这种处理非线性关系的方法可以在保证数值稳定的同时提高计算效率,得到边界清晰、结构合理的拓扑优化构形。     

Weight‐Reduction of Sandwich Structures by an Efficient Topology Optimization Technique

A heuristic algorithm for the weight minimization of sandwich structures by a specific kind of topology optimization is presented. The method employs a preexisting algorithm for the layerwise topology optimization of symmetric laminates under in‐plane loads and expands this method for the case of bending. During the optimization procedure the algorithm adds or subtracts material from the layers of the face sheets and the core of the sandwich plate in regions of high or low stresses respectively. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An efficient sensitivity computation strategy for the evolutionary structural optimization (ESO) of continuum structures subjected to self-weight loads

This work presents a modified version of the evolutionary structural optimization procedure for topology optimization of continuum structures subjected to self-weight forces. Here we present an extension of this procedure to deal with maximum stiffness topology optimization of structures when different combinations of body forces and fixed loads are applied. Body forces depend on the density distribution over the design domain. Therefore, the value and direction of the loading are coupled to the shape of the structure and they change as the material layout of the structure is modified in the course of the optimization process. It will be shown that the traditional calculation of the sensitivity number used in the ESO procedure does not lead to the optimum solution. Therefore, it is necessary to correct the computation of the element sensitivity numbers in order to achieve the optimum design. This paper proposes an original correction factor to compute the sensitivities and enhance the convergence of the algorithm. The procedure has been implemented into a general optimization software and tested in several numerical applications and benchmark examples to illustrate and validate the approach, and satisfactorily applied to the solution of 2D, 3D and shell structures, considering self-weight load conditions. Solutions obtained with this method compare favourably with the results derived using the SIMP interpolation scheme.     

Topology Optimization of Flexible Multibody Systems Using Equivalent Static Loads and Displacement Fields

Topology optimization techniques are applied in most cases for static applications. However, recently topology optimization procedures for structures under dynamic loads have been the focus of several studies. In this work, a topology optimization scheme for flexible multibody systems using equivalent static loads and displacement fields is investigated. The optimization problem is formulated using a homogenization method, more precisely, the solid isotropic material with penalization (SIMP) approach. The objective function in the optimization problem is the compliance and the method of moving asymptotes is used as optimizer. The objective function and the sensitivities are computed directly from the displacement field computed in the dynamic simulation. The examples of a 2-arm manipulator and a slider-crank mechanism are presented and the results are discussed to verify the improved dynamical behavior through this optimization method. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiple-Load Truss Topology and Sizing Optimization: Some Properties of Minimax Compliance

This paper considers the mathematical properties of discrete or discretized mechanical structures under multiple loadings which are optimal w.r.t. maximal stiffness. We state a topology and/or sizing problem of maximum stiffness design in terms of element volumes and displacements. Multiple loads are handled by minimizing the maximum of compliance of all load cases, i.e., minimizing the maximal sum of displacements along an applied force. Generally, the problem considered may contain constraints on the design variables. This optimization problem is first reformulated in terms of only design variables. Elastic equilibrium is hidden in potential energy terms. It is shown that this transformed objective function is convex and continuous, including infinite values. We deduce that maximum stiffness structures are dependent continuously on the bounds of the element volumes as parameters. Consequently, solutions to sizing problems with small positive lower bounds on the design variables can be considered as good approximations of solutions to topology problems with zero lower bounds. This justifies heuristic approaches such as the well-known stress-rationing method for solving truss topology problems.     

Efficient topology optimization of thermo-elasticity problems using coupled field adjoint sensitivity analysis method

We develop a unified and efficient adjoint design sensitivity analysis (DSA) method for weakly coupled thermo-elasticity problems. Design sensitivity expressions with respect to thermal conductivity and Young's modulus are derived. Besides the temperature and displacement adjoint equations, a coupled field adjoint equation is defined regarding the obtained adjoint displacement field as the adjoint load in the temperature field. Thus, the computing cost is significantly reduced compared to other sensitivity analysis methods. The developed DSA method is further extended to a topology design optimization method. For the topology design optimization, the design variables are parameterized using a bulk material density function. Numerical examples show that the DSA method developed is extremely efficient and the optimal topology varies significantly depending on the ratio of mechanical and thermal loadings.     

A level-set method for shape optimization

We study a level-set method for numerical shape optimization of elastic structures. Our approach combines the level-set algorithm of Osher and Sethian with the classical shape gradient. Although this method is not specifically designed for topology optimization, it can easily handle topology changes for a very large class of objective functions. Its cost is moderate since the shape is captured on a fixed Eulerian mesh. To cite this article: G. Allaire et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1125–1130.     

Optimality criteria coupled adaptive mesh method for optimal shape design of Stokes flow

An adaptive mesh method combined with the optimality criteria algorithm is applied to optimal shape design problems of fluid dynamics. The shape sensitivity analysis of the cost functional is derived. The optimization problem is solved by a simple but robust optimality criteria algorithm, and an automatic local adaptive mesh refinement method is proposed. The mesh adaptation, with an indicator based on the material distribution information, is itself shown as a shape or topology optimization problem. Taking advantages of this algorithm, the optimal shape design problem concerning fluid flow can be solved with higher resolution of the interface and a minimum of additional expense. Details on the optimization procedure are provided. Numerical results for two benchmark topology optimization problems are provided and compared with those obtained by other methods. Copyright © 2016 John Wiley & Sons, Ltd.     

Topology Optimization Procedure Based on Structure Stress History

The structure design points stress analysis for succeeding optimization steps is the base of the topology optimization algorithm. Because the material density of the material is proportional to the strain energy we can predict how to change the density for each design point for the next optimization steps. Proposed procedure make the optimization process faster and final topology is finally more optimal than topology obtained using standard optimization procedures. This stress analyzing procedure, can be treated as a hardware of the sensors being the part of the smart structure for the real time structure reconstruction. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shape sensitivity analysis of a quasi‐electrostatic piezoelectric system in multilayered media

The optimization of shape and topology of piezo‐patches or layered piezo‐electrical material attached to structural parts, such as elastic bodies, plates and shells, plays a major role in the design of smart structures, as piezo‐mechanic‐acoustic devices in loudspeakers or energy harvesters. While the design for time‐harmonic motions is genuinely frequency‐dependent, as has been reported in the literature in the context of density optimization with the SIMP‐method, time‐varying piezoelectric material has not been investigated with respect to the optimal design so far. Therefore, shape sensitivities for layered piezoelectric material and time‐varying loads and charges are derived in this paper. In particular, we provide the shape‐derivatives for nested piezo‐layers associated with a class of shape functional. More general layers can be dealt with similar approach. Copyright © 2010 John Wiley & Sons, Ltd.     

Bi-directional evolutionary topology optimization of geometrically nonlinear continuum structures with stress constraints

This paper proposes a design method to maximize the stiffness of geometrically nonlinear continuum structures subject to volume fraction and maximum von Mises stress constraints. An extended bi-directional evolutionary structural optimization (BESO) method is adopted in this paper. BESO method based on discrete variables can effectively avoid the well-known singularity problem in density-based methods with low density elements. The maximum von Mises stress is approximated by the p-norm global stress. By introducing one Lagrange multiplier, the objective of the traditional stiffness design is augmented with p-norm stress. The stiffness and p-norm stress are considered simultaneously by the Lagrange multiplier method. A heuristic method for determining the Lagrange multiplier is proposed in order to effectively constrain the structural maximum von Mises stress. The sensitivity information for designing variable updates is derived in detail by adjoint method. As for the highly nonlinear stress behavior, the updated scheme takes advantages from two filters respectively of the sensitivity and topology variables to improve convergence. Moreover, the filtered sensitivity numbers are combined with their historical sensitivity information to further stabilize the optimization process. The effectiveness of the proposed method is demonstrated by several benchmark design problems.     

Evolutionary structural optimization for problems with stiffness constraints

This paper presents a simple evolutionary procedure based on finite element analysis to minimize the weight of structures while satisfying stiffness requirements. At the end of each finite element analysis, a sensitivity number, indicating the change in the stiffness due to removal of each element, is calculated and elements which make the least change in the stiffness; of a structure are subsequently removed from the structure. The final design of a structure may have its weight significantly reduced while the displacements at prescribed locations are kept within the given limits. The proposed method is capable of performing simultaneous shape and topology optimization. A wide range of problems including those with multiple displacement constraints, multiple load cases and moving loads are considered. It is shown that existing solutions of structural optimization with stiffness constraints can easily be reproduced by this proposed simple method. In addition some original shape and layout optimization results are presented.     

Topology Optimization Design of Heat Convection Problems With Variable-Density Cells北大核心CSCD

为获得优异的散热结构设计,发展了一种基于腐蚀-扩散算子的变密度胞元层级结构设计方法.通过腐蚀-扩散算子得到了一系列拓扑相似但体积分数不同的变密度微结构,计算并拟合得到变密度微结构等效热传导系数曲线.在此基础上,采用移动渐近线法更新宏观设计变量,将变密度微结构植入相应体积分数的宏观单元中完成装配.通过数值算例对不同优化方法下温度场的热柔顺度、平均温度、方差等参数进行了比较分析,结果表明,变密度胞元层级结构比传统单尺度胞元结构和周期胞元结构具有更好的散热性能.