Topology optimization of beam cross-section considering warping deformation

The beam cross-section optimization problems have been very important as beams are widely used as efficient load-carrying structural components. Most of the earlier investigations focus on the dimension and shape optimization or on the topology optimization along the axial direction. An important problem in beam section design is to find the location and direction of stiffeners, for the introduction of a stiffener in a closed beam section may result in a topologically different configuration from the original; the existing section shape optimization theory cannot be used. The purpose of this paper is to formulate a section topology optimization technique based on an anisotropic beam theory considering warping of sections and coupling among deformations. The formulation and corresponding solving method for the topology optimization of beam cross-sections are proposed. In formulating the topology optimization problem, the minimum averaged compliance of the beam is taken as objective, and the material density of every element is used as design variable. The schemes to determine the rigidity matrix of the cross-sections and the sensitivity analysis are presented. Several kinds of topologies of the cross-section under different load conditions are given, and the effect of load condition on the optimum topology is analyzed.     

On the (non-)optimality of Michell structures

Optimal analytical Michell frame structures have been extensively used as benchmark examples in topology optimization, including truss, frame, homogenization, density and level-set based approaches. However, as we will point out, partly the interpretation of Michell’s structural continua as discrete frame structures is not accurate and partly, it turns out that limiting structural topology to frame-like structures is a rather severe design restriction and results in structures that are quite far from being stiffness optimal. The paper discusses the interpretation of Michell’s theory in the context of numerical topology optimization and compares various topology optimization results obtained with the frame restriction to cases with no design restrictions. For all examples considered, the true stiffness optimal structures are composed of sheets (2D) or closed-walled shell structures (3D) with variable thickness. For optimization problems with one load case, numerical results in two and three dimensions indicate that stiffness can be increased by up to 80 % when dropping the frame restriction. For simple loading situations, studies based on optimal microstructures reveal theoretical gains of +200 %. It is also demonstrated how too coarse design discretizations in 3D can result in unintended restrictions on the design freedom and achievable compliance.     

连续体结构的参数化水平集统一优化

针对传统分步式结构优化设计的不足,提出一种同时进行结构拓扑、形状和尺寸统一优化的设计方法.首先采用水平集函数描述统一的结构优化模型和几何尺寸边界,通过引入紧支径向插值基函数将结构拓扑优化变量、形状优化变量和尺寸优化变量变换为基函数的扩展系数;然后取该扩展系数为设计变量,借助一种参数的变化表达3种优化要素对结构性能的影响,将复杂的多变量优化问题变换为相对简单的参数优化问题,有利于与相对成熟的优化算法相结合提高求解效率;进一步用R函数将其融合为一个整体,构造出统一优化模型,并用最优化准则法进行求解.最后通过数值案例证明了该方法的有效性和精确性.     

An integrated approach to topology,sizing, and shape optimization

Topology optimization has become very popular in industrial applications, and most FEM codes have implemented certain capabilities of topology optimization. However, most codes do not allow simultaneous treatment of sizing and shape optimization during the topology optimization phase. This poses a limitation on the design space and therefore prevents finding possible better designs since the interaction of sizing and shape variables with topology modification is excluded. In this paper, an integrated approach is developed to provide the user with the freedom of combining sizing, shape, and topology optimization in a single process.     

Shape optimization of buckling sensitive structures

The optimal design of structures with distinct geometrically non-linear behavior has attracted a great deal of interest in the last years mainly with respect to sizing for prescribed external loads. In the present contribution a method is proposed to maximize the critical load under certain constraints, e.g. for a given volume, allowing varying shape as well as cross-sections. The combination of direct computation of the critical load and path-following methods is integrated into a general optimization procedure consisting of mathematical programming techniques, sensitivity analysis and computer aided geometric design methods. The formulation includes imperfection sensitivity as an important part within the optimization process.     

Uniform crashworthiness optimization of car body for high-speed trains

The paper deals with a new uniform crashworthiness concept of car bodies optimization of high-speed trains. The design optimization was done from the point of view of structural protection of occupants’ survival space. For the reason that it is impossible to find a highly probable scenario for the derailment, the authors decided to find the solution in the form of rigid frame structure (survival cells), which will provide safety space for the passengers. In the optimization example a typical passenger car body was divided into cells of approximately equal dimensions. The optimization problem was to minimize the mass of the structure with stress constraints. The survival cell was subjected to a sequence of high value loads. The loads are acting in an asynchronous way in three load directions what gives the optimized structure uniform crashworthiness. The optimization strategy consists of three stages. In the first step, the constant criterion surface algorithm (CCSA) of topology optimization is applied to find a preliminary solutions. For improving the manufacture properties of this solution, a new concept of design space constraints was proposed. The sizing optimization with evolutionary algorithms was used to define a thin-walled structure in the second step. For evolutionary optimization a standard procedure was employed. Finally, CCSA optimization algorithm was applied again to remove excessive material from a car body structure. As the optimization result a new design proposition of a car body with multiple survival cells of high uniform stiffness was obtained. By maintaining passengers’ survival space, the passive safety of a high-speed car body was significantly increased.     

A level-set-based topology and shape optimization method for continuum structure under geometric constraints

Recent advances in level-set-based shape and topology optimization rely on free-form implicit representations to support boundary deformations and topological changes. In practice, a continuum structure is usually designed to meet parametric shape optimization, which is formulated directly in terms of meaningful geometric design variables, but usually does not support free-form boundary and topological changes. In order to solve the disadvantage of traditional step-type structural optimization, a unified optimization method which can fulfill the structural topology, shape, and sizing optimization at the same time is presented. The unified structural optimization model is described by a parameterized level set function that applies compactly supported radial basis functions (CS-RBFs) with favorable smoothness and accuracy for interpolation. The expansion coefficients of the interpolation function are treated as the design variables, which reflect the structural performance impacts of the topology, shape, and geometric constraints. Accordingly, the original topological shape optimization problem under geometric constraint is fully transformed into a simple parameter optimization problem; in other words, the optimization contains the expansion coefficients of the interpolation function in terms of limited design variables. This parameterization transforms the difficult shape and topology optimization problems with geometric constraints into a relatively straightforward parameterized problem to which many gradient-based optimization techniques can be applied. More specifically, the extended finite element method (XFEM) is adopted to improve the accuracy of boundary resolution. At last, combined with the optimality criteria method, several numerical examples are presented to demonstrate the applicability and potential of the presented method.     

A revised discrete Lagrangian-based search algorithm for the optimal design of skeletal structures using available sections

Issues relating to the application of the discrete Lagrangian method (DLM) to the discrete sizing optimal design of skeletal structures are addressed. The resultant structure, whether truss or rigid frame, is subjected to stress and displacement constraints under multiple load cases. The members’ sections are selected from an available set of profiles. A table that contains sectional properties for all the available profiles is used directly in structural optimization. Each profile in the table is assigned by a unique profile number, which is used as the integer design variable for each of the structural members. It is proposed that we use a revised DLM search algorithm with static weighting to design trusses and rigid frames for minimum weight. Five examples are used to demonstrate the feasibility of the method. It is shown that, for monotonic as well as nonmonotonic constraint functions, the DLM is effective and robust for the discrete sizing design of skeletal structures.     

Shape and discrete sizing optimization of timber trusses by considering of joint flexibility

The paper presents the shape and discrete sizing optimization of timber trusses with the consideration of joint flexibility. The optimization was performed by the Mixed-Integer Non-linear Programming (MINLP) approach. In the optimization model an economic objective function for minimizing the structure’s self-manufacturing costs was defined. The design conditions in accordance with Eurocode 5 were considered as optimization constraints. The internal forces and deflections were calculated by finite element analysis. The structural stiffness matrix was composed by considering fictiously decreased cross-sectional areas of all the flexibly connected elements. The cross-section dimensions and the number of fasteners were defined as discrete sizing variables, while the joint coordinates were considered as shape variables. The applicability of the proposed approach is demonstrated through some numerical examples, presented at the end of the paper.     

Research on Optimization Strategy of Missile Preliminary Design

Two system optimization architectures are proposed for missile system preliminary design, taking into account aerodynamics, weights and sizing, propulsion and trajectory. Approximation methods are investigated in order to reduce problem dimensionality and to improve the efficiency of optimization process.     

Weight/shape structural optimization exploiting rigid movement

The paper explores the possibility of performing limited shape/size optimization to obtain a minimum weight design using rigid body movement. A two step approach is adopted where changes in the shape parameters are made separately from changes in the size parameters. The sizing step uses a conventional optimization, STARS, which is available commercially. For simplicity the method is demonstrated using planar arches subject to a variety of load and boundary conditions.     

基于ANSYS Workbench的T形结构优化设计

针对T形结构传统设计周期长、材料利用率低、设计成本高等问题,使用SolidWorks建立数字模型,将其转换成ANSYS Workbench可读的格式文件,进行拓扑优化设计。对T形结构在载荷作用下进行最优化设计,建立以单元材料密度为设计变量,以结构最小柔顺度为目标函数,以质量减少百分比为约束函数的数学模型。采用ANSYS Workbench的Topology Optimization模块进行拓扑优化设计,对比优化前、后结构的应力和变形,可知运用拓扑优化技术实现T形结构的轻量化设计合理有效。     

Simultaneous shape and topology optimization of shell structures

In this research, Method of Moving Asymptotes (MMA) is utilized for simultaneous shape and topology optimization of shell structures. It is shown that this approach is well matched with the large number of topology and shape design variables. The currently practiced technology for optimization is to find the topology first and then to refine the shape of structure. In this paper, the design parameters of shape and topology are optimized simultaneously in one go. In order to model and control the shape of free form shells, the NURBS (Non Uniform Rational B-Spline) technology is used. The optimization problem is considered as the minimization of mean compliance with the total material volume as active constraint and taking the shape and topology parameters as design variables. The material model employed for topology optimization is assumed to be the Solid Isotropic Material with Penalization (SIMP). Since the MMA optimization method requires derivatives of the objective function and the volume constraint with respect to the design variables, a sensitivity analysis is performed. Also, for alleviation of the instabilities such as mesh dependency and checkerboarding the convolution noise cleaning technique is employed. Finally, few examples taken from literature are presented to demonstrate the performance of the method and to study the effect of the proposed concurrent approach on the optimal design in comparison to the sequential topology and shape optimization methods.     

PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes

We present an efficient Matlab code for structural topology optimization that includes a general finite element routine based on isoparametric polygonal elements which can be viewed as the extension of linear triangles and bilinear quads. The code also features a modular structure in which the analysis routine and the optimization algorithm are separated from the specific choice of topology optimization formulation. Within this framework, the finite element and sensitivity analysis routines contain no information related to the formulation and thus can be extended, developed and modified independently. We address issues pertaining to the use of unstructured meshes and arbitrary design domains in topology optimization that have received little attention in the literature. Also, as part of our examination of the topology optimization problem, we review the various steps taken in casting the optimal shape problem as a sizing optimization problem. This endeavor allows us to isolate the finite element and geometric analysis parameters and how they are related to the design variables of the discrete optimization problem. The Matlab code is explained in detail and numerical examples are presented to illustrate the capabilities of the code.     

Topology optimization of multicomponent optomechanical systems for improved optical performance

The stringent and conflicting requirements imposed on optomechanical instruments and the ever-increasing need for higher resolution and quality imagery demands a tightly integrated system design approach. Our aim is to drive the thermomechanical design of multiple components through the optical performance of the complete system. To this end, we propose a new method combining structural-thermal-optical performance analysis and topology optimization while taking into account both component and system level constraints. A 2D two-mirror example demonstrates that the proposed approach is able to improve the system’s spot size error by 95% compared to uncoupled system optimization while satisfying equivalent constraints.     

Optimization of practical trusses with constraints on eigenfrequencies, displacements, stresses, and buckling

In this paper we consider the optimization of general 3D truss structures. The design variables are the cross-sections of the truss bars together with the joint coordinates, and are considered to be continuous variables. Using these design variables we simultaneously carry out size optimization (areas) and shape optimization (joint positions). Topology optimization (removal and introduction of bars) is only considered in the sense that bars of minimum cross-sectional area will have a negligible influence on the performance of the structure. The structures are subjected to multiple load cases and the objective of the optimizations is minimum mass with constraints on (possibly multiple) eigenfrequencies, displacements, and stresses. For the case of stress constraints, we deal differently with tensile and compressive stresses, for which we control buckling on the element level. The stress constraints are imposed in correlation with industrial standards, to make the optimized designs valuable from a practical point of view. The optimization problem is solved using SLP (Sequential Linear Programming).     

Topology optimization of trusses with local stability constraints and multiple loading conditions—a heuristic approach

In truss topology optimization against buckling constraints, the extension from considering a single load case to include multiple loading conditions remains an unsolved problem in the ground structure approach. The present paper suggests a heuristic method attempting to take the multiple load situation into account. A method by Pedersen (1993, 1994) considering only single loading conditions is generalized to include multiple load cases. Based on the ground structure approach the algorithm allows for variable ground structures allowing for, for instance, geometrical restrictions such as concave or even disconnected design domains (Smith 1995b).     

Preference-based topology optimization for vehicle concept design with concurrent static and crash load cases

In the simulation-based design process of automotive structures, an increasing amount of multi-disciplinary requirements have to be considered. Methods of topology optimization can be used to devise structural concepts early in the design process to obtain the best possible structural layout as starting point for further development steps. Especially relevant for the vehicle design process is the concurrent consideration of static load requirements representing normal operating conditions and energy absorption requirements targeting passive safety in crash events. When the disciplines are considered separately, the heuristic Hybrid Cellular Automaton topology optimization is a suitable method. However, in practical applications, both disciplines are usually addressed sequentially. This complicates the overall process and may reduce the quality of the final optimization result, since optimization objectives may be conflicting. We propose a preference-based Scaled Energy Weighting approach to address the topology optimization of both disciplines concurrently. The main idea is to decouple the user preference from the scaling of the different magnitudes of energies. This enables a multi-objective optimization and ultimately the selection of the desired trade-off solution. We first validate the capability of the method to provide structures optimized for stiffness and energy absorption objectives on beam examples. Finally, the method is applied to optimize a concept structure of an industrial vehicle body, demonstrating its practical feasibility.     

Design optimization as an engineering tool

A number of examples on design optimization from different disciplines such as structural mechanics, manufacturing costs, fluid flow, acoustics, topology, mechanical systems and environmental load are presented. All examples have Swedish or Danish origin. Techniques to solve certain special optimization problems are discussed. Some of those are multiple loading cases in shape optimization problems involving contact zones, production tolerance sensitivity design, volume dependence and fail safe design.     

Topological structural synthesis

While considerable attention has been given to the optimization of structures using sizing design variables, little consideration has been given to topological design variables. Topological design schemes can be divided into methods that remove members from a basic structure and methods that add members to a basic structure. Algorithms for both of these approaches are presented, and it is demonstrated with small examples thatthey will produce the minimum weight topology.