Truss topology optimization with discrete design variables—Guaranteed global optimality and benchmark examples

This paper considers the problem of optimal truss topology design subject to multiple loading conditions. We minimize a weighted average of the compliances subject to a volume constraint. Based on the ground structure approach, the cross-sectional areas are chosen as the design variables. While this problem is well-studied for continuous bar areas, we consider in this study the case of discrete areas. This problem is of major practical relevance if the truss must be built from pre-produced bars with given areas. As a special case, we consider the design problem for a single available bar area, i.e., a 0/1 problem. In contrast to the heuristic methods considered in many other approaches, our goal is to compute guaranteed globally optimal structures. This is done by a branch-and-bound method for which convergence can be proven. In this branch-and-bound framework, lower bounds of the optimal objective function values are calculated by treating a sequence of continuous but non-convex relaxations of the original mixed-integer problem. The main effect of using this approach lies in the fact that these relaxed problems can be equivalently reformulated as convex problems and, thus, can be solved to global optimality. In addition, these convex problems can be further relaxed to quadratic programs for which very efficient numerical solution procedures exist. By exploiting this special problem structure, much larger problem instances can be solved to global optimality compared to similar mixed-integer problems. The main intention of this paper is to provide optimal solutions for single and multiple load benchmark examples, which can be used for testing and validating other methods or heuristics for the treatment of this discrete topology design problem.     

Structural optimization with several discrete design variables per part by outer approximation

The article proposes an optimal design approach to minimize the mass of load carrying structures with discrete design variables. The design variables are chosen from catalogues, and several variables are assigned to each part of the structure. This allows for more design freedom than only choosing parts from a catalogue. The problems are modelled as mixed 0–1 nonlinear problems with nonconvex continuous relaxations. An algorithm based on outer approximation is proposed to find optimized designs. The capabilities of the approach are demonstrated by optimal design of a space frame (jacket) structure for offshore wind turbines, with requirements on natural frequencies, strength, and fatigue lifetime.     

An application of relative difference quotient algorithm to topology optimization of truss structures with discrete variables

In this paper, the definitions of cross-sectional variable and topological variable are advanced, and a mathematical model of topology optimization of truss structures with discrete variables including two kinds of variables is developed. The model has considered the coupling relations between cross-sectional variables and topological variables, so that is reflects the innate characteristics of topology optimization as a combinatorial optimization problem. Moreover, problems such as limit stress and singular solution of structural optimization can be overcome by using this model. The model of topology optimization of truss structures with discrete variables including two kinds of variables is solved directly by using the relative difference quotient algorithm. The computational results are satisfactory and some new topologies and better solutions are obtained.     

Structural topology optimization with design-dependent pressure loads

One way to solve topology optimization of continuum structures under design-dependent pressure loads is to recover the loading surface at each step of the minimization process. During the topology evolution, the intermediate topologies obtained by using the SIMP (Solid Isotropic Material with Penalization) method actually can be regarded as gray scale images, for which the paper proposes a new material boundary identification scheme based on image segmentation technique. The Distance Regularized Level Set Evolution (DRLSE) method proposed by Li et al., IEEE Trans Image Process 19(12):3243–3254 (2010) is utilized to detect the image edge. Then the pressure boundary is represented as the zero level contour of a level set function (LSF). Inheriting the merits of the level set method, the current scheme can handle the topological change of the pressure boundary efficiently and be easily extended to the three-dimensional problems. In addition, the scheme is more stable compared with the conventional loading surface searching methods since it works well for the intermediate topologies with local scattered densities. A new optimization framework is also proposed to avoid the load sensitivity analysis. Four numerical examples are presented to show the validity and advantages of the proposed scheme.     

Suspension system performance optimization with discrete design variables

Suspension systems on commercial vehicles have become an important feature meeting the requirements from costumers and legislation. The performance of the suspension system is often limited by available catalogue components. Additionally the suspension performance is restricted by the travel speed which highly influences the ride comfort. In this article a suspension system for an articulated dump truck is optimized in sense of reducing elapsed time for two specified duty cycles without violating a certain comfort threshold level. The comfort threshold level is here defined as a whole-body vibration level calculated by ISO 2631-1. A three-dimensional multibody dynamics simulation model is applied to evaluate the suspension performance. A non-gradient optimization routine is used to find the best possible combination of continuous and discrete design variables including the optimum operational speed without violating a set of side constraints. The result shows that the comfort level converges to the comfort threshold level. Thus it is shown that the operational speed and hence the operator input influences the ride comfort level. Three catalogue components are identified by the optimization routine together with a set of continuous design variables and two operational speeds one for each load case. Thus the work demonstrates handling of human factors in optimization of a mechanical system with discrete and continuous design variables.     

On the validity of using small positive lower bounds on design variables in discrete topology optimization

It is proved that an optimal {ε, 1} ⁿ solution to a “ε-perturbed” discrete minimum weight problem with constraints on compliance, von Mises stresses and strain energy densities, is optimal, after rounding to {0, 1} ⁿ , to the corresponding “unperturbed” discrete problem, provided that the constraints in the perturbed problem are carefully defined and ε > 0 is sufficiently small.     

Structural topology optimization of multibody systems

Flexible multibody dynamics (FMD) has found many applications in control, analysis and design of mechanical systems. FMD together with the theory of structural optimization can be used for designing multibody systems with bodies which are lighter, but stronger. Topology optimization of static structures is an active research topic in structural mechanics. However, the extension to the dynamic case is less investigated as one has to face serious numerical difficulties. One way of extending static structural topology optimization to topology optimization of dynamic flexible multibody system with large rotational and transitional motion is investigated in this paper. The optimization can be performed simultaneously on all flexible bodies. The simulation part of optimization is based on an FEM approach together with modal reduction. The resulting nonlinear differential-algebraic systems are solved with the error controlled integrator IDA (Sundials) wrapped into Python environment by Assimulo (Andersson et al. in Math. Comput. Simul. 116(0):26–43, 2015). A modified formulation of solid isotropic material with penalization (SIMP) method is suggested to avoid numerical instabilities and convergence failures of the optimizer. Sensitivity analysis is central in structural optimization. The sensitivities are approximated to circumvent the expensive calculations. The provided examples show that the method is indeed suitable for optimizing a wide range of multibody systems. Standard SIMP method in structural topology optimization suggests stiffness penalization. To overcome the problem of instabilities and mesh distortion in the dynamic case we consider here additionally element mass penalization.     

Topology optimization using a dual method with discrete variables

This paper deals with topology optimization of continuous structures in static linear elasticity. The problem consists in distributing a given amount of material in a specified domain modelled by a fixed finite element mesh in order to minimize the compliance. As the design variables can only take two values indicating the presence or absence of material (1 and 0), this problem is intrinsicallydiscrete. Here, it is solved by a mathematical programming method working in the dual space and specially designed to handle discrete variables. This method is very wellsuited to topology optimization, because it is particularly efficient for problems with a large number of variables and a small number of constraints. To ensure the existence of a solution, the perimeter of the solid parts is bounded. A computer program including analysis and optimization has been developed. As it is specialized for regular meshes, the computational time is drastically reduced. Some classical 2-D and new 3-D problems are solved, with up to 30,000 design variables. Extensions to multiple load cases and to gravity loads are also examined.     

Structural topology optimization using ant colony optimization algorithm

The ant colony optimization (ACO) algorithm, a relatively recent bio-inspired approach to solve combinatorial optimization problems mimicking the behavior of real ant colonies, is applied to problems of continuum structural topology design. An overview of the ACO algorithm is first described. A discretized topology design representation and the method for mapping ant's trail into this representation are then detailed. Subsequently, a modified ACO algorithm with elitist ants, niche strategy and memory of multiple colonies is illustrated. Several well-studied examples from structural topology optimization problems of minimum weight and minimum compliance are used to demonstrate its efficiency and versatility. The results indicate the effectiveness of the proposed algorithm and its ability to find families of multi-modal optimal design.     

Structural topology optimization under limit analysis

Structural and Multidisciplinary Optimization - The objective of this paper is to look for structural designs arising from topological optimization procedures that aim at maximizing the loading...     

Structural topology optimization with constraints on multi-fastener joint loads

This paper addresses an important problem of design constraints on fastener joint loads that are well recognized in the design of assembled aircraft structures. To avoid the failure of fastener joints, standard topology optimization is extended not only to minimize the structural compliance but also to control shear loads intensities over fasteners. It is shown that the underlying design scheme is to ameliorate the stiffness distribution over the structure in accordance with the control of load distributions over fastener joints. Typical examples are studied by means of topology optimization with joint load constraints and the standard compliance design. The effects of joint load constraints are highlighted by comparing numerical optimization results obtained by both methods. Meanwhile, resin models of optimized designs are fabricated by rapid prototyping process for loading test experiments to make sure the effectiveness of the proposed method.     

高压巡线机器人穿越耐张杆塔的控制算法

通过对高压输电线路中耐张杆塔的电磁场进行理论和试验分析,根据电磁场的分布规律对导航电磁传感器的布置进行试验研究,提出了采用传感器补偿算法的控制对策。试验表明:采用传感器补偿算法的控制对策,巡线机器人自主穿越耐张杆塔障碍是可行的。     

声发射技术在高压输电塔锚杆腐蚀检测中的应用

在高压输电塔的锚杆顶端施加一瞬态激振力,由布设在锚杆顶端的声发射传感器接收反射的信号.对接收的声发射信号进行了时频分析和小波分析.通过对比可以看出,声发射信号的特征频率、能量与腐蚀的状况有很好的相关性.因此,声发射技术是一种可靠有效的检测高压输电塔锚杆腐蚀的方法.     

A portfolio optimization model with three objectives and discrete variables

We formulate the portfolio selection as a tri-objective optimization problem so as to find tradeoffs between risk, return and the number of securities in the portfolio. Furthermore, quantity and class constraints are introduced into the model in order to limit the proportion of the portfolio invested in assets with common characteristics and to avoid very small holdings. Since the proposed portfolio selection model involves mixed integer decision variables and multiple objectives finding the exact efficient frontier may be very hard. Nevertheless, finding a good approximation of the efficient surface which provides the investor with a diverse set of portfolios capturing all possible tradeoffs between the objectives within limited computational time is usually acceptable. We experiment with the current state of the art evolutionary multiobjective optimization techniques, namely the Non-dominated Sorting Genetic Algorithm II (NSGA-II), Pareto Envelope-based Selection Algorithm (PESA) and Strength Pareto Evolutionary Algorithm 2 (SPEA2), for solving the mixed-integer multiobjective optimization problem and provide a performance comparison among them using metrics proposed by the community.     

Multidisciplinary design optimization with discrete and continuous variables of various uncertainties

As a powerful design tool, Reliability Based Multidisciplinary Design Optimization (RBMDO) has received increasing attention to satisfy the requirement for high reliability and safety in complex and coupled systems. In many practical engineering design problems, design variables may consist of both discrete and continuous variables. Moreover, both aleatory and epistemic uncertainties may exist. This paper proposes the formula of RFCDV (Random/Fuzzy Continuous/Discrete Variables) Multidisciplinary Design Optimization (RFCDV-MDO), uncertainty analysis for RFCDV-MDO, and a method of RFCDV-MDO within the framework of Sequential Optimization and Reliability Assessment (RFCDV-MDO-SORA) to solve RFCDV-MDO problems. A mathematical problem and an engineering design problem are used to demonstrate the efficiency of the proposed method.     

A discrete level-set topology optimization code written in Matlab

This paper presents a compact Matlab implementation of the level-set method for topology optimization. The code can be used to minimize the compliance of a statically loaded structure. Simple code modifications to extend the code for different and multiple load cases are given. The code is inspired by a Matlab implementation of the solid isotropic material with penalization (SIMP) method for topology optimization (Sigmund, Struct Multidiscipl Optim 21:120–127, 2001). Including the finite element solver and comments, the code is 129 lines long. The code is intended for educational purposes, and in particular it could be used alongside the Matlab implementation of the SIMP method for topology optimization to demonstrate the similarities and differences between the two approaches.     

Direct gradient projection method with transformation of variables technique for structural topology optimization

This paper proposes an efficient and reliable topology optimization method that can obtain a black and white solution with a low objective function value within a few tens of iterations. First of all, a transformation of variables technique is adopted to eliminate the constraints on the design variables. After that, the optimization problem is considered as aiming at the minimum compliance in the space of design variables which is supposed to be solved by iterative method. Based on the idea of the original gradient projection method, the direct gradient projection method (DGP) is proposed. By projecting the negative gradient of objective function directly onto the hypersurface of the constraint, the most promising search direction from the current position is obtained in the vector space spanned by the gradients of objective and constraint functions. In order to get a balance between efficiency and reliability, the step size is constrained in a rational range via a scheme for step size modification. Moreover, a grey elements suppression technique is proposed to lead the optimization to a black and white solution at the end of the process. Finally, the performance of the proposed method is demonstrated by three numerical examples including both 2D and 3D problems in comparison with the typical SIMP method using the optimality criteria algorithm.     

Methods for optimization of nonlinear problems with discrete variables: A review

The methods for discrete-integer-continuous variable nonlinear optimization are reviewed. They are classified into the following six categories: branch and bound, simulated annealing, sequential linearization, penalty functions, Lagrangian relaxation, and other methods. Basic ideas of each method are described and details of some of the algorithms are given. They are transcribed into a step-by-step format for easy implementation into a computer. Under other methods, rounding-off, heuristic, cutting-plane, pure discrete, and genetic algorithms are described. For nonlinear problems, none of the methods are guaranteed to produce the global minimizer; however, good practical solutions can be obtained.Notation BBM branch and bound method - D set of discrete values for all the discrete variables - D _i set of discrete values for thei-th variable - d _ij j-th discrete value for thei-th variable - f cost function to be minimized - f ^* upper bound for the cost function - g _i i-th constraint function - IP integer programming - ILP integer linear programming - L Lagrangian - LP linear programming - m total number of constraints - MDLP mixed-discrete linear programming - MDNLP mixed-discrete nonlinear programming - n _d number of discrete variables - NLP nonlinear programming - p number of equality constraints; acceptance probability used in simulated annealing - q _i number of discrete values for thei-th variable - SLP sequential linear programming - SQP sequential quadratic programming - x design variable vector of dimension n - x _iL smallest allowed value for thei-th variable - x _iU largest allowed value for thei-th variable - the gradient operator