Topology optimization of truss-like continua with three families of members model under stress constraints

A method to minimize structural volume under stress constraints subject to multiple load cases is presented. A new material model is developed and employed to simulate the constitutive relation of the truss-like continuum. It is assumed that there are infinite numbers of members with infinitesimal spaces along three orientations at any position. The densities and orientations of members at all nodes are taken as design variables. An iterative optimization method is presented. In one iteration step, design variables are optimized separately and independently. The orientations of three families of members at every node are optimized by mathematics program method. Fully stressed criterion is adapted to optimize member densities. In the every iteration step, member densities are adjusted to make their strains smaller than the permissible values while the stress states is assumed to keep unchanged. The densities and orientations of members at any point inside an element are obtained by their interpolating the corresponding values (i.e., the densities and orientations) at the nodes of the element. These values vary continuously inside an element and the intermediate values are not suppressed. By using this technique the optimal truss-like continuum is formed, which represents for member distribution field. Once parts of members are chosen, discrete truss can be constructed according to the continuous member distributive field. This discrete structure is a nearly optimal structure. In above process, there are no numerical instabilities such as checkerboard and mesh-dependency. Numerical examples are presented to demonstrate the effectiveness of the present method.     

Topology optimization of truss-like continuum structures for natural frequencies

A method to maximize the natural frequencies of vibration of truss-like continua with the constraint of material volume is presented. Truss-like is a kind of particular anisotropic continuum, in which there are finite numbers of members with infinitesimal spaces. Structures are analyzed by finite element method. The densities and orientations of members at nodes are taken as design variables. The densities and orientations of members in elements are interpolated by these values at nodes; therefore they vary continuously in design domain. For no intermediate densities being suppressed, there is no numerical instability, such as checkerboard patterns and one-node connected hinges. The natural frequency and its sensitivities of truss-like continuum are derived. Optimization is achieved by the techniques of moving asymptotes and steepest descent. Several numerical examples are provided to demonstrate this optimization method.     

最小位移类桁架连续体拓扑设计优化

为将拓扑优化中的柔度最小化问题拓展到一般位移最小化问题,用有限元划分设计域,采用类桁架连续体材料模型,并假设杆件在设计域内连续分布.将杆件在节点位置的密度和方向作为设计变量,将指定位置和方向的位移作为目标函数,采用基于目标函数梯度的优化准则法,通过优化杆件的连续分布场形成拓扑优化的类桁架连续体.该方法可结合结构力学的基本概念,选择部分杆件形成拓扑优化刚架.     

Optimization of least-weight grillages by finite element method

A numerical method is presented for the minimization of the volume of grillages with a stress constraint. The material distribution in the design domain is optimized by a fully-stressed criterion using a finite element method. The densities and orientations of the beams at nodes in grillages are taken as design variables, which vary in the design domain continuously. As intermediate densities are not suppressed in the optimization procedure, numerical instabilities are completely avoided. As a result, the optimal distribution fields of moments, deformation and material are obtained simultaneously. Subsequently the discrete structures are determined from the optimal distribution fields. The optimization procedure is accomplished by the computer program automatically. The capability of the proposed procedure is demonstrated on several classical benchmark problems.     

Force density method for simultaneous optimization of geometry and topology of trusses

A new method of simultaneous optimization of geometry and topology is presented for plane and spatial trusses. Compliance under single loading condition is minimized for specified structural volume. The difficulties due to existence of melting nodes are successfully avoided by considering force density, which is the ratio of axial force to the member length, as design variable. By using the fact that the optimal truss is statically determinate with the same absolute value of stress in existing members, the compliance and structural volume are expressed as explicit functions of force density only. After obtaining optimal cross-sectional area, nodal locations, and topology, the cross-sectional areas and nodal coordinates are further optimized using a conventional method of nonlinear programming. Accuracy of the optimal solution is verified through examples of plane trusses and a spatial truss. It is shown that various nearly optimal solutions can be found using the proposed method.     

A method for shape and topology optimization of truss-like structure

We present a method for the shape and topology optimization of truss-like structure. First, an initial design of a truss-like structure is constructed by a mesh generator of the finite element method because a truss-like structure can be described by a finite element mesh. Then, the shape and topology of the initial structure is optimized. In order to ensure a truss-like structure can be easily manufactured via intended techniques, we assume the beams have the same size of cross-section, and a method based on the concept of the SIMP method is used for the topology optimization. In addition, in order to prevent intersection of beams and zero-length beams, a geometric constraint based on the signed area of triangle is introduced to the shape optimization. The optimization method is verified by several 2D examples. Influence on compliance of the representative length of beams is investigated.     

Concurrent material and structural optimization of hollow plate with truss-like material

In this paper, optimum stress distribution for hollow plates composed of linear cellular materials (LCMs), a kind of truss-like material, is investigated. To reduce the computational cost, we model the material as micropolar continua representation. Two classes of design variables, relative density, and cell-size distribution of truss-like materials are to be determined by optimization under given total material volume constraint. And the concurrent designs of material and structure are obtained for three different optimization formulations. For the first formulation, we aim at the minimization of the maximum stress that appears at the initial uniform design; for the second formulation, we minimize the highest stress within the specified point set. As the yield strength of truss-like material is dependent on the relative material density, we minimize the ratio of stress over the corresponding yield strength along the hole boundary in our third formulation, which maximizes the strength reserve and seems more rational. The numerical results for the three objectives validate the concurrent optimization method proposed in this paper. And the influence of ply angle (angle between the principle direction of material and the axes of the system’s coordinates) on the optimum result is discussed. The dependence of optimum design on finite element meshes is also investigated. An approximate discrete model is established to verify the method proposed in this paper, and the stress concentration near a hole is reduced significantly.     

Achieving stress-constrained topological design via length scale control

A new suite of computational procedures for stress-constrained continuum topology optimization is presented. In contrast to common approaches for imposing stress constraints, herein it is proposed to limit the maximum stress by controlling the length scale of the optimized design. Several procedures are formulated based on the treatment of the filter radius as a design variable. This enables to automatically manipulate the minimum length scale such that stresses are constrained to the allowable value, while the optimization is driven to minimizing compliance under a volume constraint – without any direct constraints on stresses. Numerical experiments are presented that incorporate the following : 1) Global control over the filter radius that leads to a uniform minimum length scale throughout the design; 2) Spatial variation of the filter radius that leads to local manipulation of the minimum length according to stress concentrations; and 3) Combinations of the two above. The optimized designs provide high-quality trade-offs between compliance, stress and volume. From a computational perspective, the proposed procedures are efficient and simple to implement: essentially, stress-constrained topology optimization is posed as a minimum compliance problem with additional treatment of the length scale.     

Simultaneous material selection and geometry design of statically determinate trusses using continuous optimization

In this work, we explore simultaneous geometry design and material selection for statically determinate trusses by posing it as a continuous optimization problem. The underlying principles of our approach are structural optimization and Ashby’s procedure for material selection from a database. For simplicity and ease of initial implementation, only static loads are considered in this work with the intent of maximum stiffness, minimum weight/cost, and safety against failure. Safety of tensile and compression members in the truss is treated differently to prevent yield and buckling failures, respectively. Geometry variables such as lengths and orientations of members are taken to be the design variables in an assumed layout. Areas of cross-section of the members are determined to satisfy the failure constraints in each member. Along the lines of Ashby’s material indices, a new design index is derived for trusses. The design index helps in choosing the most suitable material for any geometry of the truss. Using the design index, both the design space and the material database are searched simultaneously using gradient-based optimization algorithms. The important feature of our approach is that the formulated optimization problem is continuous, although the material selection from a database is an inherently discrete problem. A few illustrative examples are included. It is observed that the method is capable of determining the optimal topology in addition to optimal geometry when the assumed layout contains more links than are necessary for optimality.     

On the prediction of material properties and topology for optimal continuum structures

A new formulation is presented for mathematical modelling to predict material properties for the optimal design of continuum structures. The method is based on an extended form of an already established characterization for continuum design, where the material properties tensor for an arbitrary structural continuum appears as the design variable. The extension is comprised of means to represent an independently specifiedunit relative cost factor, which appears simply as a weighting function in the argument of the isoperimetric (cost) constraint of the original model. A procedure is demonstrated where optimal black/white topology is predicted out of a sequence of solutions to material properties design problems having thisgeneralized cost formulation form. A systematic adjustment is made in the unit relative cost field for each subsequent solution step in the sequence, and at the stage identified with final topology, no more than a small fraction of a percent of the total element area in the system has material property density off the bounding black or white levels. This technique is effective for the prediction of optimal black/white topology design for design around obstacles of arbitrary shape, as well as the more unusual topology design problems. Results are presented for 2D examples of both types of problem. In addition to the treatment for (the usual) minimum compliance design, an alternate formulation of the design problem is presented as well, one that provides for the prediction of optimum topology with a generalized measure of compliance as the objective.     

A stress–based approach to the optimal design of structures with unilateral behavior of material or supports

The paper presents a stress-based approach that copes with the optimal design of truss-like elastic structures in case of unilateral behavior of material or ground supports. The conventional volume-constrained minimization of compliance is coupled with a set of local stress constraints that are enforced, all over the domain or along prescribed boundaries, to control the arising of members with tension-only or compression-only strength. A Drucker–Prager failure criterion is formulated to provide a smooth approximation of the no-tension or no-compression conditions governing the stress field. A selection strategy is implemented to handle efficiently the multi-constrained formulation that is solved through mathematical programming. Benchmark examples are investigated to discuss the features of the achieved optimal designs, as compared with problems involving material and ground supports with equal behavior in tension and compression. Numerical simulations show that a limited set of constraints is needed in the first iterations to steer the solution of the energy-driven optimization towards designs accounting for the prescribed assumption of unilateral strength.     

On objective functions of minimizing the vibration response of continuum structures subjected to external harmonic excitation

This work deals with the topological design of vibrating continuum structures. The vibration of continuum structure is excited by time-harmonic external mechanical loading with prescribed frequency and amplitude. In comparison with well-known compliance minimization in static topology optimization, various objective functions are proposed in literature to minimize the response of vibrating structures, such as power flow, vibration transmission, and dynamic compliances, etc. Even for the dynamic compliance, different definitions are found in literature, which have quite different formulations and influences on the optimization results. The aim of this paper is to provide a comparison of these different objective functions and propose reference forms of objective functions for design optimization of vibration problems. Analytical solutions for two degrees of freedom system and topological design of plane structures in numerical examples are compared using different optimization formulations for given various excitation frequencies. The results are obtained by the finite element method and gradient based optimization using analytical sensitivity analysis. The optimized topologies and vibration response of the optimized structures are presented. The influence of excitation frequencies and the eigenfrequencies of the structure are discussed in the numerical examples.     

A design model to predict optimal two-material composite structures

A method is presented for the prediction of optimal configurations for two-material composite continuum structures. In the model for this method, both local properties and topology for the stiffer of the two materials are to be predicted. The properties of the second, less stiff material are specified and remain fixed. At the start of the procedure for computational solution, material composition of the structure is represented as a pure mixture of the two materials. This design becomes modified in subsequent steps into a form comprised of a skeleton of concentrated stiffer material, together with a nonoverlapping distribution of the second material to fill the original domain. Computational solutions are presented for two example design problems. A comparison among solutions for different ratios of stiffness between the two materials gives an indication of how the distribution of concentrated stiffer material varies with this factor. An example is presented as well to show how the method can be used to predict an efficient layout for rib-reinforcement of a stamped sheet metal panel.     

Topology optimization for the seismic design of truss-like structures

A practical optimization method is applied to design nonlinear truss-like structures subjected to seismic excitation. To achieve minimum weight design, inefficient material is gradually shifted from strong parts to weak parts of a structure until a state of uniform deformation prevails. By considering different truss structures, effects of seismic excitation, target ductility and buckling of the compression members on optimum topology are investigated. It is shown that the proposed method could lead to 60% less structural weight compared to optimization methods based on elastic behavior and equivalent static loads, and is efficient at controlling performance parameters under a design earthquake.     

Non-parametric free-form optimization method for frame structures

In this paper, we propose a parameter-free shape optimization method based on the variational method for designing the smooth optimal free-form of a spatial frame structure. A stiffness design problem where the compliance is minimized under a volume constraint is solved as an example of shape design problems of frame structures. The optimum design problem is formulated as a distributed-parameter shape optimization problem under the assumptions that each member is varied in the out-of-plane direction to the centroidal axis and that the cross section is prismatic. The shape gradient function and the optimality conditions are then theoretically derived. The optimal curvature distribution is determined by applying the derived shape gradient function to each member as a fictitious distributed force both to vary the member in the optimum direction and to minimize the objective functional without shape parametrization, while maintaining the members’ smoothness. The validity and practical utility of this method were verified through several design examples. It was confirmed that axial-force-carrying structures were obtained by this method.     

Simultaneous design of structural layout and discrete fiber orientation using bi-value coding parameterization and volume constraint

The so-called bi-value coding parameterization (BCP) method is developed for the simultaneous optimization of layout design and discrete fiber orientations of laminated structures related to the compliance minimization and natural frequency maximization. Both kinds of problems are transformed into a discrete material selection problem that is then solved as a continuous topology optimization problem with multiphase materials. A new form of the volume constraint is introduced in accordance with the BCP to control the material usage and material removal in the corresponding problem formulation. The BCP scheme assigning the integer value of +1 or -1 to each design variable for the unique “coding” is efficiently used to interpolate discrete fiber orientations and to identify the presence and removal of materials. Meanwhile, a general set-up strategy is proposed by assigning “uniform” weight values in BCP to ensure the feasibility of the initial starting point. Numerical tests illustrate that the BCP is efficient in dealing with both kinds of design problems including the volume constraint.     

空间桁架结构优化设计

为获得空间桁架结构的合理构型,以某空间设备支撑结构为例,分析结构材料在设计空间的分布形式和桁架结构的传力路径。在已知载荷约束和设计空间大小的条件下,基于连续体拓扑优化方法,以静态多工况刚度和动态固有频率为多目标函数进行优化分析。依据设计要求确定计算模型的结点数和结点位置,获得满足要求的空间桁架结构并进行优化设计。优化结果比原模型质量减少36.7%,一阶模态提高3.6%。     

Non-uniqueness and symmetry of optimal topology of a shell for minimum compliance

Uniqueness and symmetry of solution are investigated for topology optimization of a symmetric continuum structure subjected to symmetrically distributed loads. The structure is discretized into finite elements, and the compliance is minimized under constraint on the structural volume. The design variables are the densities of materials of elements, and intermediate densities are penalized to prevent convergence to a gray solution. A path of solution satisfying conditions for local optimality is traced using the continuation method with respect to the penalization parameter. It is shown that the rate form of the solution path can be formulated from the optimality conditions, and the uniqueness and bifurcation of the path are related to eigenvalues and eigenvectors of the Jacobian of the governing equations. This way, local uniqueness and symmetry breaking process of the solution are rigorously investigated through the bifurcation of a solution path.     

Evolutionary topology optimization of continuum structures with an additional displacement constraint

Evolutionary structural optimization (ESO) and its later version Bi-directional ESO (BESO) have been successfully applied to optimum material distribution problems for continuum structures. However, the existing ESO/BESO methods are limited to the topology optimization of an objective function such as mean compliance with a single constraint e.g. structural volume. The present work extends the BESO method to the stiffness optimization with a material volume constraint and a local displacement constraint. As a result, one will obtain a structure with the highest stiffness for a given volume while the displacement of a certain node does not exceed a prescribed limit. Several examples are presented to demonstrate the effectiveness of the proposed method.