Topology optimization of planar linkage mechanisms

In spite of increasing interest in gradient‐based topology optimization of linkage mechanisms, it is still difficult to solve practical, realistic problems. Besides the apparent difficulty resulting from high nonlinearity, the optimization problem faces other major difficulties: difficulty to satisfy the discrete DOF condition with continuous design variables and lack of intrinsic mechanisms to generate distinct black‐and‐white layouts. To deal with the DOF issue, we propose a new formulation, which maximizes a single objective function, the energy transmittance efficiency. It is shown that the efficiency function maximization handles DOF redundancy and deficiency simultaneously. To obtain distinct linkage layouts, a common practice is to introduce an artificial mass constraint and/or to remove unnecessary links during optimization. However, we do not use any artificial mass constraint but post‐process the optimized result to obtain the final layout by a special post‐processing algorithm. In this study, the linkage design model consists of nonlinear ground bars and zero‐length springs. The springs are used to fix bar‐connecting nodes to the ground, generating pinned joints. After verifying the effectiveness of the proposed approach for four‐bar linkage synthesis, we synthesize an automobile steering mechanism satisfying the Ackermann condition. The steering mechanism problem is solved here for the first time. Copyright © 2014 John Wiley & Sons, Ltd.     

Reliability‐based topology optimization of structures under stress constraints

In this paper, we propose an approach for reliability‐based design optimization where a structure of minimum weight subject to reliability constraints on the effective stresses is sought. The reliability‐based topology optimization problem is formulated by using the performance measure approach, and the sequential optimization and reliability assessment method is employed. This strategy allows for decoupling the reliability‐based topology optimization problem into 2 steps, namely, deterministic topology optimization and reliability analysis. In particular, the deterministic structural optimization problem subject to stress constraints is addressed with an efficient methodology based on the topological derivative concept together with a level‐set domain representation method. The resulting algorithm is applied to some benchmark problems, showing the effectiveness of the proposed approach.     

基于小波的二维转动装置优化设计与摩擦试验研究

针对地面对接仿真试验台对二维转动装置的要求,对多种方案分析后,设计了一种抗弯件式结构形式的二维转动装置,它能满足质量轻、刚度强度大和摩擦力矩小的要求.利用基于小波的约束随机方向法对该装置进行了优化设计,这种优化方法不仅提高了优化速度,而且克服了传统约束随机方向法不易求得全局最优解的缺点,优化后其质量降低了30%.随后对该装置的摩擦力矩进行了测试,主动仿真试验台二维转动装置最大摩擦力矩5.N·m,被动仿真试验台二维转动装置最大摩擦力矩1.325N·m.说明抗弯件式二维转动装置是一种比较合理的结构,为仿真试验台真实的模拟对接过程奠定了基础.     

桁架结构智能布局优化设计

结构的布局优化由于涉及尺寸、形状和拓扑三个层次的综合设计而成为优化问题中的难点,结合桁架结构提出了一个基于多个初始基结构的布局优化方法。以智能生成的、型式多样合理的基结构代替传统模型中的单一基结构,然后从不同基结构下的拓扑优化结果中找出最优设计。在克服传统基结构法有可能限制求解空间而丢失最优解这一局限性的同时,将形状和拓扑优化设计有效分离,降低了求解的难度,并且结合拓扑变化法,实现了桁架结构从选型生成、分析计算到优化设计的一体化智能设计过程。算例表明:利用该文提出的方法进行桁架结构的最优布局设计是可靠有效的。     

A new method to generate the ground structure in truss topology optimization

The aim of this article is to provide an effective method to generate the ground structure in truss topology optimization. The core of this method is to place nodal points for the ground structure at the intersection of the first and third principal stress trajectories, which are obtained by solving the equivalent static problem in the design domain with a homogeneous isotropic material property. It is applicable to generate the ground structure for arbitrary regular and irregular geometric design domains. The proposed method is tested on some benchmark examples in truss topology optimization. The optimization model is a standard linear programming problem based on plastic design and solved by the interior point algorithm. Compared with other methods, the proposed method may use a well-defined ground structure with fewer nodes and bars, resulting in faster solution convergence, which shows it to be efficient.     

正交-遗传试验算法的分析与应用

为了解决并联机器人机构的优化设计问题,提出一种基于正交试验设计法和遗传算法的优化方法。在简要讨论正交试验设计法和遗传算法的基本原理基础上,对两种方法的寻优算法、各个参数的对应关系作了比较分析,探讨了用正交表构造遗传算法中初始种群的方法。提出一种适用于设计变量多且适应度函数难求的“一代”正交-遗传试验法的思路和方法。将该方法应用于一种新型四自由度并联机器人机构的结构优化设计,得出以机构全域条件数为目标函数的机构结构优化尺寸方案。实例证明这种优化方法行之有效。     

An enhanced genetic algorithm for structural topology optimization

Genetic algorithms (GAs) have become a popular optimization tool for many areas of research and topology optimization an effective design tool for obtaining efficient and lighter structures. In this paper, a versatile, robust and enhanced GA is proposed for structural topology optimization by using problem‐specific knowledge. The original discrete black‐and‐white (0–1) problem is directly solved by using a bit‐array representation method. To address the related pronounced connectivity issue effectively, the four‐neighbourhood connectivity is used to suppress the occurrence of checkerboard patterns. A simpler version of the perimeter control approach is developed to obtain a well‐posed problem and the total number of hinges of each individual is explicitly penalized to achieve a hinge‐free design. To handle the problem of representation degeneracy effectively, a recessive gene technique is applied to viable topologies while unusable topologies are penalized in a hierarchical manner. An efficient FEM‐based function evaluation method is developed to reduce the computational cost. A dynamic penalty method is presented for the GA to convert the constrained optimization problem into an unconstrained problem without the possible degeneracy. With all these enhancements and appropriate choice of the GA operators, the present GA can achieve significant improvements in evolving into near‐optimum solutions and viable topologies with checkerboard free, mesh independent and hinge‐free characteristics. Numerical results show that the present GA can be more efficient and robust than the conventional GAs in solving the structural topology optimization problems of minimum compliance design, minimum weight design and optimal compliant mechanisms design. It is suggested that the present enhanced GA using problem‐specific knowledge can be a powerful global search tool for structural topology optimization. Copyright © 2005 John Wiley & Sons, Ltd.     

Calibration of a class of non‐linear viscoelasticity models with sensitivity assessment based on duality

The calibration of constitutive models is based on the solution of an optimization problem, whereby the sought parameter values minimize an objective function that measures the discrepancy between experimental observations and the corresponding simulated response. By the introduction of an appropriate adjoint problem, the resulting formulation becomes well suited for a gradient‐based optimization scheme. A class of viscoelastic models is studied, where a discontinuous Galerkin method is used to integrate the governing evolution equation in time. A practical solution algorithm, which utilizes the time‐flow structure of the underlying evolution equation, is presented. Based on the proposed formulation it is convenient to estimate the sensitivity of the calibrated parameters with respect to measurement noise. The sensitivity is computed using a dual method, which compares favourably with the conventional primal method. The strategy is applied to a viscoelasticity model using experimental data from a uniaxial compression test. Copyright © 2006 John Wiley & Sons, Ltd.     

Optimal design of multi-response experiments using semi-definite programming

Optimal design of multi-response experiments for estimating the parameters of multi-response linear models is a challenging problem. The main drawback of the existing algorithms is that they require the solution of many optimization problems in the process of generating an optimal design that involve cumbersome manual operations. Furthermore, all the existing methods generate approximate design and no method for multi-response n-exact design has been cited in the literature. This paper presents a unified formulation for multi-response optimal design problem using Semi-Definite Programming (SDP) that can generate D-, A- and E-optimal designs. The proposed method alleviates the difficulties associated with the existing methods. It solves a one-shot optimization model whose solution selects the optimal design points among all possible points in the design space. We generate both approximate and n-exact designs for multi-response models by solving SDP models with integer variables. Another advantage of the proposed method lies in the amount of computation time taken to generate an optimal design for multi-response models. Several test problems have been solved using an existing interior-point based SDP solver. Numerical results show the potentials and efficiency of the proposed formulation as compared with those of other existing methods. The robustness of the generated designs with respect to the variance-covariance matrix is also investigated.     

A Hopfield Neural Network Approach to the Dual Response Problem

The application of neural networks to optimization problems has been an active research area since the early 1980s. Unconstrained optimization, constrained optimization and combinatorial optimization problems have been solved using neural networks. This study presents a new approach using Hopfield neural networks (HNNs) for solving the dual response system (DRS) problems. The major aim of the proposed method is to produce a string of solutions, rather than a ‘one‐shot’ optimum solution, to make the trade‐offs available between the mean and standard deviation responses. This gives more flexibility to the decision‐maker in exploring alternative solutions. The proposed method has been tested on two examples. The HNN results are very close to those obtained by using the NIMBUS (Nondifferentiable Interactive Multiobjective Bundle‐based Optimization System) algorithm. Choosing an appropriate solution method for a certain multi‐objective optimization problem is not easy, as has been made abundantly clear. Unlike the NIMBUS method, the HNN approach does not set any specific assumptions on the behaviour or the preference structure of the decision maker. As a result, the proposed method will still work and generate alternative solutions whether or not the decision maker has enough time and capabilities for co‐operation. Copyright © 2005 John Wiley & Sons, Ltd.     

Path‐generation of articulated mechanisms by shape and topology variations in non‐linear truss representation

This paper presents studies on an optimization‐based method for path‐generation of articulated mechanisms. An extended truss ground‐structure approach is taken in which both the shape and topology of the truss are designed using cross‐sectional areas and nodal positions as design variables. This leads to a technique for simultaneous type and dimensional synthesis of articulated mechanisms. For the analysis part it is essential to control the mechanism configuration so that the mechanism remains within a given configuration space, thus stabilizing the optimization process and resulting in realistic solutions. This can be achieved by using the Levenberg–Marquardt method. The design method is illustrated by a number of design cases for both closed and open input and output paths. Copyright © 2005 John Wiley & Sons, Ltd.     

A Heaviside function-based density representation algorithm for truss-like buckling-induced mechanism design

Motivated by key advances in manufacturing techniques, the tailoring of materials to achieve novel properties such as energy dissipation properties has been the focus of active research in engineering and materials science over the past decade. The goal of material design is to determine the optimal spatial layout to achieve a desired macroscopic constitutive response. However, the manufacturing abilities are the key factors to constrain the feasible design space, eg, minimum length and geometry complexity. Traditional density-based method, where each element works as a variable, always results in complicated geometry with large number of small intricate features. To address the aforementioned challenges, a new density field representation technique, named, Heaviside function-based geometric representation algorithm, is proposed in this paper, where density field is represented by truss-like components. Truss-like components have less control parameters and easier to handle for sensitivities derivation, especially for distance sensitivities. Using bar components to explicitly represent density field can explore design space effectively and generate simple structures without any intricate small features at borders. Furthermore, this density representation method is mesh independent and design variables are reduced significantly so that optimization problem can be solved efficiently using small-scale optimization algorithm, eg, sequential quadratic programming. However, finding a reasonable initial component distribution is critical to avoid optimization failure. To overcome this difficulty, a jump-start method is proposed by solving inverse subproblem. The overall optimization progress is divided into three stages, ie, the first stage is obtaining coarse snap-through material configuration based on traditional density-based method; the second stage is an inverse optimization problem to fit the geometry component to the solution obtained in stage I; and the stage III is maximizing the energy dissipation capacity. To demonstrate the powerful ability in design buckling-induced mechanism of the proposed density representation algorithm, buckling-induced energy dissipation mechanism with snap-through behavior to achieve the desired energy dissipation capacity considering failure constraint is demonstrated through four numerical examples.     

A GA‐based technique for layout optimization of truss with stress and displacement constraints

Various developments of increasing complexity involved in layout optimization are discussed. The use of conventional GA in layout optimization is briefly mentioned with emphasis on its limitations and conditions imposed in finding the optimal design. The proposed new technique is applied to the benchmark example of Michell's truss for verification. The approach has also been applied to new examples of bridge truss and crane truss problems in order to demonstrate the generality and robustness for topology optimization. The approach is extended to include dual stress‐displacements constraints since many practical problems involve these two constraints simultaneously. Two‐bar and 10‐bar trusses are solved as examples for layout optimization with both stress and displacement constraints with satisfactory results. The effect of mutation on the final topology is also discussed. The major drawbacks of the ground structure approach are overcome in this proposed new method. The optimal designs obtained demonstrate the ability, robustness and generality of using the proposed new technique in layout optimization problems. Copyright © 2001 John Wiley & Sons, Ltd.     

Stochastic inverse heat conduction using a spectral approach

An adjoint‐based functional optimization technique in conjunction with the spectral stochastic finite element method is proposed for the solution of an inverse heat conduction problem in the presence of uncertainties in material data, process conditions and measurement noise. The ill‐posed stochastic inverse problem is restated as a conditionally well‐posed L₂ optimization problem. The gradient of the objective function is obtained in a distributional sense by defining an appropriate stochastic adjoint field. The L₂ optimization problem is solved using a conjugate‐gradient approach. Accuracy and effectiveness of the proposed approach is appraised with the solution of several stochastic inverse heat conduction problems. Copyright © 2004 John Wiley & Sons, Ltd.     

智能制造系统AGV小车及缓冲区容量仿真优化

具有MHS(material handling system)的智能制造系统AGV(automated guided vehicle)小车及缓冲区最大容量配置优化,属于典型的非线性整数规划问题。由于约束无法用封闭形式表达,因此较难获得问题的精确解。为此,本文提出了仿真优化的方法以获得问题的近似解。首先,对AGV小车及缓冲区最大容量配置优化问题进行了描述;其次,基于Em plant平台建立了具有MHS的智能制造系统仿真模型;然后,基于不同的优化目标,提出了3种仿真优化方法;最后,通过仿真试验对上述3种方法进行了分析与比较。分析表明,本文提出的方法及优化结果,可为企业配置AGV小车及缓冲区最大容量提供决策支持。     

Robust compliance topology optimization based on the topological derivative concept

In this paper, we present an approach for robust compliance topology optimization under volume constraint. The compliance is evaluated considering a point‐wise worst‐case scenario. Analogously to sequential optimization and reliability assessment, the resulting robust optimization problem can be decoupled into a deterministic topology optimization step and a reliability analysis step. This procedure allows us to use topology optimization algorithms already developed with only small modifications. Here, the deterministic topology optimization problem is addressed with an efficient algorithm based on the topological derivative concept and a level‐set domain representation method. The reliability analysis step is handled as in the performance measure approach. Several numerical examples are presented showing the effectiveness of the proposed approach. Copyright © 2015 John Wiley & Sons, Ltd.     

Worst‐case design in head impact crashworthiness optimization

This paper describes the solution of a worst‐case design optimization problem of head impact in automotive design. The worst‐case design process uses an optimization algorithm that can locate saddlepoints: points in the design space where the objective function is maximized with respect to some design variables (worst case) while it is minimized with respect to other design variables simultaneously. The worst‐case design methodology is first tested using two analytic functions. Both functions contain saddlepoints, while the second one also has a random analytic noise component and an integer variable. Thereafter, the methodology is applied to the worst‐case design of a crashworthiness head impact problem. The head impact problem contains both numerical noise and an integer variable. For the first analytical case, the effect of separability of the maximization and minimization variables is investigated by rotating the design variable axes. For the second analytical case, analytical noise in the form of a modified Griewank function and an integer variable is added. For the head impact problem, cases are presented where maximization and minimization are first performed separately, and then in a combined fashion to locate the saddle point. The case studies illustrate the power of this approach in the automotive occupant safety design field. Copyright © 2003 John Wiley & Sons, Ltd.     

A ‘best points’ interpolation method for efficient approximation of parametrized functions

We present an interpolation method for efficient approximation of parametrized functions. The method recognizes and exploits the low‐dimensional manifold structure of the parametrized functions to provide good approximation. Basic ingredients include a specific problem‐dependent basis set defining a low‐dimensional representation of the parametrized functions, and a set of ‘best interpolation points’ capturing the spatial‐parameter variation of the parametrized functions. The best interpolation points are defined as solution of a least‐squares minimization problem which can be solved efficiently using standard optimization algorithms. The approximation is then determined from the basis set and the best interpolation points through an inexpensive and stable interpolation procedure. In addition, an a posteriori error estimator is introduced to quantify the approximation error and requires little additional cost. Numerical results are presented to demonstrate the accuracy and efficiency of the method. Copyright © 2007 John Wiley & Sons, Ltd.     

Multi-objective genetic algorithm for solving N-version program design problem

N-version programming (NVP) is a programming approach for constructing fault tolerant software systems. Generally, an optimization model utilized in NVP selects the optimal set of versions for each module to maximize the system reliability and to constrain the total cost to remain within a given budget. In such a model, while the number of versions included in the obtained solution is generally reduced, the budget restriction may be so rigid that it may fail to find the optimal solution. In order to ameliorate this problem, this paper proposes a novel bi-objective optimization model that maximizes the system reliability and minimizes the system total cost for designing N-version software systems. When solving multi-objective optimization problem, it is crucial to find Pareto solutions. It is, however, not easy to obtain them. In this paper, we propose a novel bi-objective optimization model that obtains many Pareto solutions efficiently.We formulate the optimal design problem of NVP as a bi-objective 0–1 nonlinear integer programming problem. In order to overcome this problem, we propose a Multi-objective genetic algorithm (MOGA), which is a powerful, though time-consuming, method to solve multi-objective optimization problems. When implementing genetic algorithm (GA), the use of an appropriate genetic representation scheme is one of the most important issues to obtain good performance. We employ random-key representation in our MOGA to find many Pareto solutions spaced as evenly as possible along the Pareto frontier. To pursue improve further performance, we introduce elitism, the Pareto-insertion and the Pareto-deletion operations based on distance between Pareto solutions in the selection process.The proposed MOGA obtains many Pareto solutions along the Pareto frontier evenly. The user of the MOGA can select the best compromise solution among the candidates by controlling the balance between the system reliability and the total cost.     

Topology optimization of acoustic–structure interaction problems using a mixed finite element formulation

The paper presents a gradient‐based topology optimization formulation that allows to solve acoustic–structure (vibro‐acoustic) interaction problems without explicit boundary interface representation. In acoustic–structure interaction problems, the pressure and displacement fields are governed by Helmholtz equation and the elasticity equation, respectively. Normally, the two separate fields are coupled by surface‐coupling integrals, however, such a formulation does not allow for free material re‐distribution in connection with topology optimization schemes since the boundaries are not explicitly given during the optimization process. In this paper we circumvent the explicit boundary representation by using a mixed finite element formulation with displacements and pressure as primary variables (a u /p‐formulation). The Helmholtz equation is obtained as a special case of the mixed formulation for the elastic shear modulus equating to zero. Hence, by spatial variation of the mass density, shear and bulk moduli we are able to solve the coupled problem by the mixed formulation. Using this modelling approach, the topology optimization procedure is simply implemented as a standard density approach. Several two‐dimensional acoustic–structure problems are optimized in order to verify the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.