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.     

Seleção de topologias ótimas de estruturas elásticas 2D com restrição de tensão – via Smooth Evolutionary Structural Optimization

This paper approaches the topology optimization problems in plane linear elasticity considering the minimization of the volume with restriction of the stress employing an index of performance for monitoring the meeting of the optimum region. It is used for this purpose the classical evolutionary structural optimization, or ESO ‐ evolutionary structural optimization. This procedure is based on systematic and gradual removal of the elements with lower stress compared with the maximum stress of the structure. This procedure also known as a process “hard‐kill”. It is proposed a variant of the ESO method, called SESO ‐ Smoothing ESO, which is based on the philosophy that if an element is not really necessary for the structure, its contribution to the structural stiffness will gradually diminish until it has no longer influence in the structure, so its removal is performed smoothly. That is, their removal is done smoothly, reducing the values of the constitutive matrix of the element as if it were in the process of damage. A new performance index for the monitoring of this evolutionary process smoothed is proposed herein. The applications of ESO and SESO are made with the finite element method, but considering a high order triangular element based on the free formulation. Finally, it is implemented a spatial filter in terms of stress control, which was associated with SESO technique proved to be very stable and efficient in eliminating the formation of the checkerboard.     

n-sided polygonal hybrid finite elements with unified fundamental solution kernels for topology optimization

In topology optimization, the optimized design can be obtained based on spatial discretization of design domain using natural polygonal finite elements to reduce the influence of mesh geometry on topology optimization solutions. However, the natural polygonal finite elements require separate interpolants for each type of elements and involve troublesome domain integrals. In this study, an alternative n-sided polygonal hybrid finite element possessing multiple-node connection is formulated in a unified form to compress the checkerboard patterns caused by numerical instability in topology optimization. Different from the natural polygonal finite elements, the present polygonal hybrid finite elements involve two sets of independent displacement fields. The intra-element displacement field defined inside the element is approximated by the linear combination of the fundamental solution of the problem to achieve the purpose of the local satisfaction of the governing equations of the problem, but not the specific boundary conditions and the inter-element continuity conditions. To overcome such drawback, the inter-element displacement field defined over the entire element boundary is independently approximated by means of the conventional shape function interpolation. As a result, only line integrals along the element boundary are involved in the computation, whose dimension is reduced by one compared to the domain integrals in the natural polygonal finite elements, and more importantly, allowing us to flexibly construct any polygons from Voronoi tessellations in discretizing complex design domains using same fundamental solution kernels. Numerical results obtained indicate that the present n-sided polygonal hybrid finite elements can produce more accurate displacement solutions and smaller mean compliance, compared to the standard finite elements and the natural polygonal finite elements.     

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.     

Formulation and Execution of Structural Topology Optimization for Practical Design Solutions

Topology optimization has gained prime significance due to increasing demands of lightweight components. In this paper, a general mathematical formulation of topology optimization is presented with some imperative manufacturing constraints for maximizing the stiffness of a structure with mixed boundary conditions. A methodology is implemented to determine the optimal configuration of operative structural components by executing TOSCA in batch-process mode with ANSYS software. CAD viable design is attained by smoothing the topological optimized surfaces. The geometry at the maximum stressed areas is also optimized. Analysis of the customized reduced weight configuration reveals that it comprises the harmonized stress distribution and improved structural performance.     

Una formulación de mínimo peso con restricciones en tensión en optimización topológica de estructuras

Optimum design of structures has been traditionally focused on the analysis of shape and dimensions optimization problems. However, more recently a new discipline has emerged: the topology optimization of the structures. This discipline states innovative models that allow to obtain optimal solutions without a previous definition of the type of structure being considered. These formulations obtain the optimal topology and the optimal shape and size of the resulting elements. The most usual formulations of the topology optimization problem try to obtain the structure of maximum stiffness. These approaches maximize the stiffness for a given amount of material to be used. These formulations have been widely analyzed and applied in engineering but they present considerable drawbacks from a numerical and from a practical point of view. In this paper the author propose a different formulation, as an alternative to maximum stiffness approaches, that minimizes the weight and includes stress constraints. The advantages of this kind of formulations are crucial since the cost of the structure is minimized, which is the most frequent objective in engineering, and they guarantee the structural feasibility since stresses are constrained. In addition, this approach allows to avoid some of the drawbacks and numerical instabilities related to maximum stiffness approaches. Finally, some practical examples have been solved in order to verify the validity of the results obtained and the advantages of the proposed formulation.     

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)     

An efficient optimization method for mechanical shells considering cut-outs

In this contribution an optimization method for shell structures is presented. This method was developed in order to perform a simultaneous optimization of the shape and position of the mid surface and a topology optimization to introduce cut-outs. A topology optimization method for continuum structures is combined with a manufacturing constraint for deep drawable sheet metals. It is shown, how more than a million design variables can be handled efficiently using a mathematical optimization algorithm for the design update and the finite element method for the structural simulation. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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。     

On Class Size of p-Singular Elements in Finite Groups

The object of this article is to study the connection between the structure of a finite group and the conjugacy class sizes of the p-singular elements of a finite group. The finite groups G in which every conjugacy class size of the p-singular elements satisfies some given arithmetic conditions are classified.     

Finite element simulation of a layout optimisation technique by photoelastic stress minimisation

Layout optimisation to minimise maximum Tresca stress by photoelastic stress minimisation technique is simulated by finite element method: elements in the design domain that are lowly stressed are slowly removed resulting in a structure having minimum Tresca stress. The FEM simulation consists of analysing-monitoring the Tresca stress of elements in the design domain and “removing” material by declaring the element stiffness matrix of those possessing small stress values as of negligible stiffness in the subsequent step of the optimisation process. The lower bounds and upper bounds of stress limits for the “removal” criterion have to be appropriately chosen and effects of sharp notches introduced by removing finite elements should be properly taken into account for successful optimisation. The FEM simulation can be made fully automatic and can be extended to cases of complex geometry, loading material properties as well as to other objective functions of the optimisation problem.     

Stochastic Material and Topology Design

In order to find a robust optimal topology or material design with respect to stochastic variations of the model parameters of a mechanical structure, the basic optimization problem under stochastic uncertainty must be replaced by an appropriate deterministic substitute problem. Starting from the equilibrium equation and the yield/strength conditions, the problem can be formulated as a stochastic (linear) program “with recourse”. Hence, by discretization the design space by finite elements, linearizing the yield conditions, in case of discrete probability distributions the resulting deterministic substitute problems are linear programs with a dual decomposition data structure.     

基于离散变量的拓扑优化方法

单元敏度的不准确估计是离散拓扑优化算法数值不稳定的原因之一,特别是添加材料时,传统的敏度计算公式给出的估计误差较大,甚至有时估计符号都是错误的．为了克服这一问题,通过对弹性平衡增量方程的摄动分析构造了新的增量敏度估计公式．这一新的公式无论是添加材料还是删除材料都能较准确地估计出目标函数增量,它可以看作是通过非局部单元刚度阵对传统敏度分析公式的修正．以此为基础构建了一种基于离散变量的拓扑优化算法,它可以从任意单元上添加或删除材料以使目标函数减小,同时为避免优化过程中重新划分网格,采用了单元软杀策略以小刚度材料模拟空单元．这一方法的主要优点是简单,不需要太多的数学计算,特别有利于工程实际的应用．     

An evolution equation based approach to topology optimization

The objective of topology optimization is to find a mechanical structure with maximum stiffness and minimal amount of used material for given boundary conditions [2]. There are different approaches. Either the structure mass is held constant and the structure stiffness is increased or the amount of used material is constantly reduced while specific conditions are fulfilled. In contrast, we focus on the growth of a optimal structure from a void model space and solve this problem by introducing a variational problem considering the spatial distribution of structure mass (or density field) as variable [3]. By minimizing the Gibbs free energy according to Hamilton's principle in dynamics for dissipative processes, we are able to find an evolution equation for the internal variable describing the density field. Hence, our approach belongs to the growth strategies used for topology optimization. We introduce a Lagrange multiplier to control the total mass within the model space [1]. Thus, the numerical solution can be provided in a single finite element environment as known from material modeling. A regularization with a discontinuous Galerkin approach for the density field enables us to suppress the well-known checkerboarding phenomena while evaluating the evolution equation within each finite element separately [4]. Therefore, the density field is no additional field unknown but a Gauß-point quantity and the calculation effort is strongly reduced. Finally, we present solutions of optimized structures for different boundary problems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A influência do peso próprio na otimização topológica de estruturas elásticas 2D – via técnica numérica Smooth Evolutionary Structural Optimization (SESO)

This paper deals with topology optimization in plane elastic‐linear problems considering the influence of the self weight in efforts in structural elements. For this purpose it is used a numerical technique called SESO (Smooth ESO), which is based on the procedure for progressive decrease of the inefficient stiffness element contribution at lower stresses until he has no more influence. The SESO is applied with the finite element method and is utilized a triangular finite element and high order. This paper extends the technique SESO for application its self weight where the program, in computing the volume and specific weight, automatically generates a concentrated equivalent force to each node of the element. The evaluation is finalized with the definition of a model of strut‐and‐tie resulting in regions of stress concentration. Examples are presented with optimum topology structures obtaining optimal settings.     

On the sobriety of the inverse topology

An algebraic frame L with the finite intersection property (FIP) on compact elements is said to be polarised if every minimal prime element in it is complemented. In this note, we give a necessary and sufficient condition for the inverse topology on the set of minimal prime elements of such a frame to be sober. We also establish some sufficient conditions for sobriety when the polarisation condition is relaxed.     

An isogeometric boundary element approach for topology optimization using the level set method

Among the various types of structural optimization, topology has been occupying a prominent place over the last decades. It is considered the most versatile because it allows structural geometry to be determined taking into account only loading and fixing constraints. This technique is extremely useful in the design phase, which requires increasingly complex computational modeling. Modern geometric modeling techniques are increasingly focused on the use of NURBS basis functions. Consequently, it seems natural that topology optimization techniques also use this basis in order to improve computational performance. In this paper, we propose a way to integrate the isogeometric boundary techniques to topology optimization through the level set function. The proposed coupling occurs by describing the normal velocity field from the level set equation as a function of the normal shape sensitivity. This process is not well behaved in general, so some regularization technique needs to be specified. Limiting to plane linear elasticity cases, the numerical investigations proposed in this study indicate that this type of coupling allows to obtain results congruent with the current literature. Moreover, the additional computational costs are small compared to classical techniques, which makes their advantage for optimization purposes evident, particularly for boundary element method practitioners.     

Large scale structural synthesis

A general purpose optimization program is coupled to a large scale finite element program to provide an efficient tool for structural synthesis. The resulting interface program may be used to design structures for minimum weight, subject to constraints on stress, displacement, and vibration frequencies. A variety of state-of-the-art techniques are employed, including design variable linking, constraint deletion, reciprocal variables, and formal approximations. The capability is demonstrated with the design of a gear housing using 30 design variables and over 5000 nonlinear inequality constraints. The finite element model consists of over 1600 elements and 7000 displacement degrees of freedom. The design required six detailed finite element analyses and approximately one hour on a Cray-1s supercomputer. It is concluded that structures of practical size and complexity can be efficiently designed using numerical optimization.     

Topology optimization based on the first optimization steps strain energy analysis.

The paper presents the algorithm for obtaining “the near optimal” topology very fast, based on the first optimization steps strain energy analysis. The objective function is the compliance of the structure. It is equal to the strain energy and it is minimized under the body mass constraints. Because the material density is proportional to the strain energy accumulated in discrete material point, the identification of more and less effort domains especially for the first optimization steps let analyse the structure from the topology point of view. This identification is the base of relative faster finding out material domains and void domains within the design domain which leads to the topology, which is very similar to the optimal topology even from the strain energy level point of view. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)