A mixed integer programming for robust truss topology optimization with stress constraints

This paper presents a mixed integer programming (MIP) formulation for robust topology optimization of trusses subjected to the stress constraints under the uncertain load. A design‐dependent uncertainty model of the external load is proposed for dealing with the variation of truss topology in the course of optimization. For a truss with the discrete member cross‐sectional areas, it is shown that the robust topology optimization problem can be reduced to an MIP problem, which is solved globally. Numerical examples illustrate that the robust optimal topology of a truss depends on the magnitude of uncertainty. Copyright © 2010 John Wiley & Sons, Ltd.     

A hierarchical method for discrete structural topology design problems with local stress and displacement constraints

In this paper, we present a hierarchical optimization method for finding feasible true 0–1 solutions to finite‐element‐based topology design problems. The topology design problems are initially modelled as non‐convex mixed 0–1 programs. The hierarchical optimization method is applied to the problem of minimizing the weight of a structure subject to displacement and local design‐dependent stress constraints. The method iteratively treats a sequence of problems of increasing size of the same type as the original problem. The problems are defined on a design mesh which is initially coarse and then successively refined as needed. At each level of design mesh refinement, a neighbourhood optimization method is used to treat the problem considered. The non‐convex topology design problems are equivalently reformulated as convex all‐quadratic mixed 0–1 programs. This reformulation enables the use of methods from global optimization, which have only recently become available, for solving the problems in the sequence. Numerical examples of topology design problems of continuum structures with local stress and displacement constraints are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

A level set approach for topology optimization with local stress constraints

The purpose of this work is to present a level set‐based approach for the structural topology optimization problem of mass minimization submitted to local stress constraints. The main contributions are threefold. First, the inclusion of local stress constraints by means of an augmented Lagrangian approach within the level set context. Second, the proposition of a constraint procedure that accounts for a continuous activation/deactivation of a finite number of local stress constraints during the optimization sequence. Finally, the proposition of a logarithmic scaling of the level set normal velocity as an additional regularization technique in order to improve the minimization sequence. A set of benchmark tests in two dimensions achieving successful numerical results assesses the good behavior of the proposed method. In these examples, it is verified that the algorithm is able to identify stress concentrations and drive the design to a feasible local minimum. Copyright © 2014 John Wiley & Sons, Ltd.     

频率双向渐进优化方法中新的插值技术

研究了拓扑结构优化中的插值技术。针对传统的频率双向渐进优化方法所采取的代数外插方式只能用于规则的矩形单元或长方体单元的情形,根据当前结构的主振型,建立静力平衡方程组,从而计算出边界单元新增加结点的模态位移。这种插值技术可以将双向渐进优化方法推广运用于任意形状的单元。仿真算例表明,该方法有效可行。     

A mass constraint formulation for structural topology optimization with multiphase materials

This work is focused on the topology optimization of lightweight structures consisting of multiphase materials. Instead of adopting the common idea of using volume constraint, a new problem formulation with mass constraint is proposed. Meanwhile, recursive multiphase materials interpolation (RMMI) and uniform multiphase materials interpolation (UMMI) schemes are discussed and compared based on numerical tests and theoretical analysis. It is indicated that the nonlinearity of the mass constraint introduced by RMMI brings numerical difficulties to attain the global optimum of the optimization problem. On the contrary, the UMMI‐2 scheme makes it possible to formulate the mass constraint in a linear form with separable design variables. One such formulation favors very much the problem resolution by means of mathematical programming approaches, especially the convex programming methods. Moreover, numerical analysis indicates that fully uniform initial weighting is beneficial to seek the global optimum when UMMI‐2 scheme is used. Besides, the relationship between the volume constraint and mass constraint is theoretically revealed. The filtering technique is adapted to avoid the checkerboard pattern related to the problem with multiphase materials. Numerical examples show that the UMMI‐2 scheme with fully uniform initial weighting is reliable and efficient to deal with the structural topology optimization with multiphase materials and mass constraint. Meanwhile, the mass constraint formulation is evidently more significant than the volume constraint formulation. Copyright © 2011 John Wiley & Sons, Ltd.     

Application of manufacturing constraints to structural optimization of thin-walled structures

Topology optimization can be a very useful tool for creating conceptual designs for vehicles. Structures suggested by topology optimization often turn out to be difficult to implement in manufacturing processes. Presently, rail vehicle structures are made by welding sheet metal parts. This leads to many complications and increased weight of the vehicle. This article presents a new design concept for modern rail vehicle structures made of standardized, thin-walled, closed, steel profiles that fulfil the stress and manufacturing requirements. For this purpose, standard software for topology optimization was used with a new way of preprocessing the design space. The design methodology is illustrated by an example of the topology optimization of a freight railcar. It is shown that the methodology turns out to be a useful tool for obtaining optimal structure design that fulfils the assumed manufacturing constraints.     

Topology optimization of continuum structures with local stress constraints

We introduce an extension of current technologies for topology optimization of continuum structures which allows for treating local stress criteria. We first consider relevant stress criteria for porous composite materials, initially by studying the stress states of the so-called rank 2 layered materials. Then, on the basis of the theoretical study of the rank 2 microstructures, we propose an empirical model that extends the power penalized stiffness model (also called SIMP for Solid Isotropic Microstructure with Penalization for inter-mediate densities). In a second part, solution aspects of topology problems are considered. To deal with the so-called ‘singularity’ phenomenon of stress constraints in topology design, an ϵ-constraint relaxation of the stress constraints is used. We describe the mathematical programming approach that is used to solve the numerical optimization problems, and show results for a number of example applications. © 1998 John Wiley & Sons, Ltd.     

A fixed‐grid bidirectional evolutionary structural optimization method and its applications in tunnelling engineering

Topology optimization has exhibited an exceptional capability of improving structural design. However, several typical topology optimization algorithms are finite element (FE) based, where mesh‐dependent zigzag representation of boundaries is barely avoidable in both intermediate and final results. To tackle the problem, this paper proposes a new fixed‐grid‐based bidirectional evolutionary structural optimization method, namely FG BESO. The adoption of an FG FE framework enables a continuous boundary change in the course of topology optimization, which provides a means of dealing with not only the non‐smooth boundary of the final design but also the interpretation of intermediate densities. As a class of important practical application, it is interesting to make use of the new FG BESO method to the reinforcement design for underground tunnels. To accommodate the FG BESO technique to geological engineering applications, a nodal sensitivity is derived for a two‐phase material model comprising the artificial reinforcement and original rock. In this paper, some innovative topological designs of tunnel reinforcements are presented for minimizing the floor and sidewall heaves under different geological loading conditions. Copyright © 2007 John Wiley & Sons, Ltd.     

基于演化算法的带频率约束的桁架结构形状和尺寸优化

具有频率约束的桁架结构形状和尺寸优化设计是一个难度大的非线性动力优化问题。形状和尺寸变量的耦合通常导致收敛困难,而频率约束则使得动力灵敏度分析困难,传统的优化准则法和数学规划法难于求解。将单纯形算法、子空间变维技术、均匀变异有机融入郭涛算法,提出一种混合演化算法,避开繁琐的动力灵敏度分析,简单、有效地求解这类桁架形状和尺寸优化问题。典型的桁架算例验证了算法的有效性和可靠性。     

Modelling topology optimization problems as linear mixed 0–1 programs

This paper deals with topology optimization of discretized continuum structures. It is shown that a large class of non‐linear 0–1 topology optimization problems, including stress‐ and displacement‐constrained minimum weight problems, can equivalently be modelled as linear mixed 0–1 programs. The modelling approach is applied to some test problems which are solved to global optimality. Copyright © 2003 John Wiley & Sons, Ltd.     

Multimaterial topology optimization with multiple volume constraints: Combining the ZPR update with a ground‐structure algorithm to select a single material per overlapping set

Multimaterial topology optimization often leads to members containing composite materials. However, in some instances, designers might be interested in using only one material for each member. Therefore, we propose an algorithm that selects a single preferred material from multiple materials per overlapping set. We develop the algorithm, based on the evaluation of both the strain energy and the cross‐sectional area of each member, to control the material profile (ie, the number of materials) in each subdomain of the final design. This algorithm actively and iteratively selects materials to ensure that a single material is used for each member. In this work, we adopt a multimaterial formulation that handles an arbitrary number of volume constraints and candidate materials. To efficiently handle such volume constraints, we employ the ZPR (Zhang‐Paulino‐Ramos) design variable update scheme for multimaterial optimization, which is based upon the separability of the dual objective function of the convex subproblem with respect to Lagrange multipliers. We provide an alternative derivation of this update scheme based on the Karush‐Kuhn‐Tucker conditions. Through numerical examples, we demonstrate that the proposed material selection algorithm, which can be readily implemented in multimaterial optimization, along with the ZPR update scheme, is robust and effective for selecting a single preferred material among multiple materials.     

A new hole insertion method for level set based structural topology optimization

Structural shape and topology optimization using level set functions is becoming increasingly popular. However, traditional methods do not naturally allow for new hole creation and solutions can be dependent on the initial design. Various methods have been proposed that enable new hole insertion; however, the link between hole insertion and boundary optimization can be unclear. The new method presented in this paper utilizes a secondary level set function that represents a pseudo third dimension in two‐dimensional problems to facilitate new hole insertion. The update of the secondary function is connected to the primary level set function forming a meaningful link between boundary optimization and hole creation. The performance of the method is investigated to identify suitable parameters that produce good solutions for a range of problems. Copyright © 2012 John Wiley & Sons, Ltd.     

Topology optimization of continuum structures with stress constraints and uncertainties in loading

Topology optimization using stress constraints and considering uncertainties is a serious challenge, since a reliability problem has to be solved for each stress constraint, for each element in the mesh. In this paper, an alternative way of solving this problem is used, where uncertainty quantification is performed through the first‐order perturbation approach, with proper validation by Monte Carlo simulation. Uncertainties are considered in the loading magnitude and direction. The minimum volume problem subjected to local stress constraints is formulated as a robust problem, where the stress constraints are written as a weighted average between their expected value and standard deviation. The augmented Lagrangian method is used for handling the large set of local stress constraints, whereas a gradient‐based algorithm is used for handling the bounding constraints. It is shown that even in the presence of small uncertainties in loading direction, different topologies are obtained when compared to a deterministic approach. The effect of correlation between uncertainties in loading magnitude and direction on optimal topologies is also studied, where the main observed result is loss of symmetry in optimal topologies.     

A multiple surrogate assisted evolutionary algorithm for optimization involving iterative solvers

In many real-world optimization problems, the underlying objective and constraint function(s) are evaluated using computationally expensive iterative simulations such as the solvers for computational electro-magnetics, computational fluid dynamics, the finite element method, etc. The default practice is to run such simulations until convergence using termination criteria, such as maximum number of iterations, residual error thresholds or limits on computational time, to estimate the performance of a given design. This information is used to build computationally cheap approximations/surrogates which are subsequently used during the course of optimization in lieu of the actual simulations. However, it is possible to exploit information on pre-converged solutions if one has the control to abort simulations at various stages of convergence. This would mean access to various performance estimates in lower fidelities. Surrogate assisted optimization methods have rarely been used to deal with such classes of problem, where estimates at various levels of fidelity are available. In this article, a multiple surrogate assisted optimization approach is presented, where solutions are evaluated at various levels of fidelity during the course of the search. For any solution under consideration, the choice to evaluate it at an appropriate fidelity level is derived from neighbourhood information, i.e. rank correlations between performance at different fidelity levels and the highest fidelity level of the neighbouring solutions. Moreover, multiple types of surrogates are used to gain a competitive edge. The performance of the approach is illustrated using a simple 1D unconstrained analytical test function. Thereafter, the performance is further assessed using three 10D and three 20D test problems, and finally a practical design problem involving drag minimization of an unmanned underwater vehicle. The numerical experiments clearly demonstrate the benefits of the proposed approach for such classes of problem.     

A velocity field level set method for shape and topology optimization

In this paper, we propose a new implementation of the level set shape and topology optimization, the velocity field level set method. Therein, the normal velocity field is constructed with specified basis functions and velocity design variables defined on a given set of points that are independent of the finite element mesh. A general mathematical programming algorithm can be employed to find the optimal normal velocities on the basis of the sensitivity analysis. As compared with conventional level set methods, mapping the variational boundary shape optimization problem into a finite‐dimensional design space and the use of a general optimizer makes it more efficient and straightforward to handle multiple constraints and additional design variables. Moreover, the level set function is updated by the Hamilton‐Jacobi equation using the normal velocity field; thus, the inherent merits of the implicit representation is retained. Therefore, this method combines the merits of both the general mathematical programming and conventional level set methods. Integrated topology optimization of structures with embedded components of designable geometries is considered to show the capability of this method to deal with general design variables. Several numerical examples in 2D or 3D design domains illustrate the robustness and efficiency of the method using different basis functions.     

Metamaterial topology optimization of nonpneumatic tires with stress and buckling constraints

This article presents the design of a metamaterial for the shear layer of a nonpneumatic tire using topology optimization, under stress and buckling constraints. These constraints are implemented for a smooth maximum function using global aggregation. A linear elastic finite element model is used, implementing solid isotropic material with penalization. Design sensitivities are determined by the adjoint method. The method of moving asymptotes is used to solve the numerical optimization problem. Two different optimization statements are used. Each requires a compliance limit and some aspect of continuation. The buckling analysis is linear, considering the generalized eigenvalue problem of the conventional and stress stiffness matrices. Various symmetries, base materials, and starting geometries are considered. This leads to novel topologies that all achieve the target effective shear modulus of 10 MPa, while staying within the stress constraint. The stress-only designs generally were susceptible to buckling failure. A family of designs (columnar, noninterconnected representative unit cells) that emerge in this study appears to exhibit favorable properties for this application.     

A fast method for binary programming using first‐order derivatives,with application to topology optimization with buckling constraints

We present a method for finding solutions of large‐scale binary programming problems where the calculation of derivatives is very expensive. We then apply this method to a topology optimization problem of weight minimization subject to compliance and buckling constraints. We derive an analytic expression for the derivative of the stress stiffness matrix with respect to the density of an element in the finite‐element setting. Results are presented for a number of two‐dimensional test problems.Copyright © 2012 John Wiley & Sons, Ltd.     

Bidirectional evolutionary structural optimization for nonlinear structures under dynamic loads

This study aims to develop efficient numerical optimization methods for finding the optimal topology of nonlinear structures under dynamic loads. The numerical models are developed using the bidirectional evolutionary structural optimization method for stiffness maximization problems with mass constraints. The mathematical formulation of topology optimization approach is developed based on the element virtual strain energy as the design variable and minimization of compliance as the objective function. The suitability of the proposed method for topology optimization of nonlinear structures is demonstrated through a series of two- and three-dimensional benchmark designs. Several issues relating to the nonlinear structures subjected to dynamic loads such as material, geometric, and contact nonlinearities are addressed in the examples. It is shown that the proposed approach generates more reliable designs for nonlinear structures.     

A multiobjective topology optimization approach for cost and time minimization in additive manufacturing

The ever-present drive for increasingly high-performance designs realized on shorter timelines has fostered the need for computational design generation tools such as topology optimization. However, topology optimization has always posed the challenge of generating difficult, if not impossible to manufacture designs. The recent proliferation of additive manufacturing technologies provides a solution to this challenge. The integration of these technologies undoubtedly has the potential for significant impact in the world of mechanical design and engineering. This work presents a new methodology which mathematically considers additive manufacturing cost and build time alongside the structural performance of a component during the topology optimization procedure. Two geometric factors, namely, the surface area and support volume required for the design, are found to correlate to cost and build time and are controlled through the topology optimization procedure. A novel methodology to consider each of these factors dynamically during the topology optimization procedure is presented. The methodology, based largely on the use of the spatial gradient of the density field, is developed in such a way that it does not leverage the finite element discretization scheme. This work investigates a problem that has not yet been explored in the literature: direct minimization of support material volume in density-based topology optimization. The entire methodology is formulated in a smooth and differentiable manner, and the sensitivity expressions required by gradient based optimization solvers are presented. A series of example problems are provided to demonstrate the efficacy of the proposed methodology.     

Applying multiple objective tabu search to continuous optimization problems with a simple neighbourhood strategy

One of the first multiple objective versions of the tabu search (TS) algorithm is proposed by the author. The idea of applying TS to multiple objective optimization is inspired from its solution structure. TS works with more than one solution (neighbourhood solutions) at a time and this situation gives the opportunity to evaluate multiple objectives simultaneously in one run. The selection and updating stages are modified to enable the original TS algorithm to work with more than one objective. In this paper, the multiple objective tabu search (MOTS) algorithm is applied to multiple objective non‐linear optimization problems with continuous variables using a simple neighbourhood strategy. The algorithm is applied to four mechanical components design problems. The results are compared with several other solution techniques including multiple objective genetic algorithms. It is observed that MOTS is able to find better and much wider spread of solutions than the reported ones. Copyright © 2005 John Wiley & Sons, Ltd.