Single and multiobjective structural optimization in discrete-continuous variables using simulated annealing

A multivariable optimization technique based on the Monte-Carlo method used in statistical mechanics studies of condensed systems is adapted for solving single and multiobjective structural optimization problems. This procedure, known as simulated annealing, draws an analogy between energy minimization in physical systems and objective function minimization in structural systems. The search for a minimum is simulated by a relaxation of the statistical mechanical system where a probabilistic acceptance criterion is used to accept or reject candidate designs. To model the multiple objective functions in the problem formulation, a cooperative game theoretic approach is used. Numerical results obtained using three different annealing strategies for the single and multiobjective design of structures with discrete-continuous variables are presented. The influence of cooling schedule parameters on the optimum solutions obtained is discussed. Simulation results indicate that, in several instances, the optimum solutions obtained using simulated annealing outperform the optimum solutions obtained using some gradient-based and discrete optimization techniques. The results also indicate that simulated annealing has substantial potential for additional applications in optimization, especially for problems with mixed discrete-continuous variables.     

平面桁架结构拓扑优化设计的改进智能算法

结构拓扑及形状退火算法(STSA)用于桁架结构拓扑优化设计,其优化特点为注重结构构型的改变而较少考虑结构的力学性能,而针对既定几何构型的桁架结构截面优化,满应力准则法(FSD)具有明显优势,因此,将其引入退火历程改进STSA。提出结构几何构型状态相对稳定判别方法,并以结构构型状态相对稳定作为引入FSD的最佳时机形成杂交算法。算例分析表明:该改进智能算法使寻优搜索过程更为稳定,其表现为搜索效率、鲁棒性和最优解均得以改善。     

A Genetic Algorithm Hybrid for Constructing Optimal Response Surface Designs

Hybrid heuristic optimization methods can discover efficient experiment designs in situations where traditional designs cannot be applied, exchange methods are ineffective, and simple heuristics like simulated annealing fail to find good solutions. One such heuristic hybrid is GASA (genetic algorithm–simulated annealing), developed to take advantage of the exploratory power of the genetic algorithm, while utilizing the local optimum exploitive properties of simulated annealing. The successful application of this method is demonstrated in a difficult design problem with multiple optimization criteria in an irregularly shaped design region. Copyright © 2004 John Wiley & Sons, Ltd.     

Shape optimal design of elastic space frames with non-linear response

This paper describes an approach to shape optimal design of elastic space frames with kinematically non-linear response. A space frame structure is treated as to be assembled from several frame design elements each of them being defined as a skeleton lying on a rational Bézier patch. The design variables may influence the control points of each patch and the cross-sectional quantities of beam elements. Highly accurate beam finite elements are employed based on a modified formulation of the beam element proposed by Jelenić and Saje. The modified element is able to account for arbitrary initial curvature and it fits nicely into the context of both the proposed design element technique and the optimization process. The formulation of the shape optimal design problem in form of non-linear mathematical programming problem and its solution by employing gradient-based methods of mathematical programming are discussed briefly. The theory is illustrated in detail with three numerical examples. © 1998 John Wiley & Sons, Ltd.     

On structural shape optimization using an embedding domain discretization technique

This contribution presents a novel approach to structural shape optimization that relies on an embedding domain discretization technique. The evolving shape design is embedded within a uniform finite element background mesh which is then used for the solution of the physical state problem throughout the course of the optimization. We consider a boundary tracking procedure based on adaptive mesh refinement to separate between interior elements, exterior elements, and elements intersected by the physical domain boundary. A selective domain integration procedure is employed to account for the geometric mismatch between the uniform embedding domain discretization and the evolving structural component. Thereby, we avoid the need to provide a finite element mesh that conforms to the structural component for every design iteration, as it is the case for a standard Lagrangian approach to structural shape optimization. Still, we adopt an explicit shape parametrization that allows for a direct manipulation of boundary vertices for the design evolution process. In order to avoid irregular and impracticable design updates, we consider a geometric regularization technique to render feasible descent directions for the course of the optimization. Copyright © 2016 John Wiley & Sons, Ltd.     

Design space optimization using a numerical design continuation method

A generalized optimization problem in which design space is also a design to be found is defined and a numerical implementation method is proposed. In conventional optimization, only a portion of structural parameters is designated as design variables while the remaining set of other parameters related to the design space are often taken for granted. A design space is specified by the number of design variables, and the layout or configuration. To solve this type of design space problems, a simple initial design space is selected and gradually improved while the usual design variables are being optimized. To make the design space evolve into a better one, one may increase the number of design variables, but, in this transition, there are discontinuities in the objective and constraint functions. Accordingly, the sensitivity analysis methods based on continuity will not apply to this discontinuous stage. To overcome the difficulties, a numerical continuation scheme is proposed based on a new concept of a pivot phase design space. Two new categories of structural optimization problems are formulated and concrete examples shown. Copyright © 2001 John Wiley & Sons, Ltd.     

Particle-swarm harmony search for water network design

The optimal design of water distribution networks is a non-linear, multi-modal, and constrained problem classified as an NP-hard combinatorial problem. Because of the drawbacks of calculus-based algorithms, the problem has been tackled by assorted stochastic algorithms, such as the genetic algorithm, simulated annealing, tabu search, shuffled frog-leaping algorithm, ant colony optimization algorithm, harmony search, cross entropy, and scatter search. This study proposes a modified harmony search algorithm incorporating particle swarm concept. This algorithm was applied to the design of four bench-mark networks (two-loop, Hanoi, Balerma, and New York City networks), with good results.     

Optimal design of material properties and material distribution for multiple loading conditions

An extension of recent work¹ on the simultaneous optimization of material and structure to address the design of structures under multiple loading conditions is presented. Material properties are represented in the most general form possible, namely, as elements of the unrestricted set of positive-semi-definite constitutive tensors of a linearly elastic continuum. Existence of solutions can be shown when the objective is a weighted average of compliances and a resource constraint measured as the 2-norm or the trace of the constitutive tensors is included. The optimized material properties can be derived analytically. The optimization of the layout of the material leads to a sizing problem of structural optimization involving a non-linear, non-smooth elasticity analysis. The computational solution of this problem is discussed and illustrated with examples.     

Buckling optimization of composite laminated adaptive structures

This paper deals with the optimal design of laminated composite plates with integrated piezoelectric actuators. Refined finite element models based on equivalent single layer high-order shear deformation theories are used. These models are combined with simulated annealing, a stochastic global optimization technique, in order to find the optimal location of piezoelectric actuators and also to find the optimal fiber reinforcement angles in both cases having the objective of maximizing the buckling load of the composite adaptive plate structure. To show the performance of the proposed optimization models, two illustrative and simple examples are presented and discussed. In one of these examples a comparison between the simulated annealing technique and a gradient based optimization scheme, is carried out.     

基于序贯层次Kriging模型的微型飞行器机身结构设计优化

在基于仿真模型的工程设计优化中,采用高精度、高成本的分析模型会导致计算量大,采用低精度、低成本的分析模型会导致设计优化结果的可信度低,难以满足实际工程的要求。为了有效平衡高精度与低成本之间的矛盾关系,通过建立序贯层次Kriging模型融合高/低精度数据,采用大量低成本、低精度的样本点反映高精度分析模型的变化趋势,并采用少量高成本、高精度的样本点对低精度分析模型进行校正,以实现对优化目标的高精度预测。为了避免层次Kriging模型误差对优化结果的影响,将层次Kriging模型与遗传算法相结合,根据6σ设计准则计算每一代最优解的预测区间,具有较大预测区间的当前最优解即为新的高精度样本点。同时,在优化过程中序贯更新层次Kriging模型,提高最优解附近的层次Kriging模型的预测精度,从而保证设计结果的可靠性。将所提出的方法应用于微型飞行器机身结构的设计优化中,以验证该方法的有效性和优越性。采用具有不同单元数的网格模型分别作为低精度分析模型和高精度分析模型,利用最优拉丁超立方设计分别选取60个低精度样本点和20个高精度样本点建立初始层次Kriging模型,采用本文方法求解并与直接采用高精度仿真模型求解的结果进行比较。结果表明,所提出的方法能够有效利用高/低精度样本点处的信息,建立高精度的层次Kriging模型;本文方法仅需要少量的计算成本就能求得近似最优解,有效提高了设计效率,为类似的结构设计优化问题提供了参考。     

Coupling ant colony optimization and the extended great deluge algorithm for the discrete facility layout problem

This article uses a hybrid optimization approach to solve the discrete facility layout problem (FLP), modelled as a quadratic assignment problem (QAP). The idea of this approach design is inspired by the ant colony meta-heuristic optimization method, combined with the extended great deluge (EGD) local search technique. Comparative computational experiments are carried out on benchmarks taken from the QAP-library and from real life problems. The performance of the proposed algorithm is compared to construction and improvement heuristics such as H63, HC63-66, CRAFT and Bubble Search, as well as other existing meta-heuristics developed in the literature based on simulated annealing (SA), tabu search and genetic algorithms (GAs). This algorithm is compared also to other ant colony implementations for QAP. The experimental results show that the proposed ant colony optimization/extended great deluge (ACO/EGD) performs significantly better than the existing construction and improvement algorithms. The experimental results indicate also that the ACO/EGD heuristic methodology offers advantages over other algorithms based on meta-heuristics in terms of solution quality.     

Optimization of truss topology using tabu search

A design procedure for integrating topological considerations in the framework of structural optimization is presented. The proposed approach is capable of considering multiple load conditions, stress, displacement and local/global buckling constraints, and multiple objective functions in the problem formulation. Further, since the proposed method permits members to be added to or deleted from an existing topology and the topology is not defined by member areas, the difficulty of not being able to reach singular optima is also avoided. These objectives are accomplished using a discrete optimization procedure which uses 0–1 topological variables to optimize alternate designs. Since the topological variables are discrete in nature and the member cross-sections are assumed to be continuous, the topological optimization problem has mixed discrete-continuous variables. This non-linear programming problem is solved using a memory-based combinatorial optimization technique known as tabu search. Numerical results obtained using tabu search for single and multiobjective topological optimization of truss structures are presented. To model the multiple objective functions in the problem formulation, a cooperative game theoretic approach is used. The results indicate that the optimum topologies obtained using tabu search compare favourably, and in some instances, outperform the results obtained using the ground–structure approach. However, this improvement occurs at the expense of a significant increase in computational burden owing to the fact that the proposed approach necessitates that the geometry of each trial topology be optimized.     

Shape optimal design of elastic planar frames with non-linear response

This paper describes an approach to shape optimal design of elastic planar frames with non-linear response. The foundation of the proposed approach forms an appropriate strategy of shape representation of the structure, based on the design element technique. A frame structure is treated as to be assembled from several frame design elements, which in turn may consist of several appropriately joined beam finite elements. The shape of each frame design element is defined by convenient functions involving Bezier blending polynomials. The original formulation of the beam finite element, proposed by Saje, is modified in order to fit nicely into the context of the frame design element technique. The formulation of the shape optimal design problem in a form of a problem of non-linear mathematical programming and its solution by employing gradient-based methods of mathematical programming are discussed briefly. The theory is illustrated with two numerical examples.     

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.     

SIMULATED ANNEALING AND PARALLEL PROCESSING: AN IMPLEMENTATION FOR CONSTRAINED GLOBAL DESIGN OPTIMIZATION

Global optimization becomes important as more and more complex designs are evaluated and optimized for superior performance. Often parametric designs are highly constrained, adding complexity to the design problem. In this work simulated annealing (SA), a stochastic global optimization technique, is implemented by augmenting it with a feasibility improvement scheme (FIS) that makes it possible to formulate and solve a constrained optimization problem without resorting to artificially modifying the objective function. The FIS is also found to help recover from the infeasible design space rapidly. The effectiveness of the improved algorithm is demonstrated by solving a welded beam design problem and a two part stamping optimization problem. Large scale practical design problems may prohibit the efficient use of computationally intensive iterative algorithms such as SA. Hence the FIS augmented SA algorithm is implemented on an Intel iPSC/860 parallel super-computer using a data parallel structure of the algorithm for the solution of large scale optimization problems. The numerical results demonstrate the effectiveness of the FIS as well as the parallel version of the SA algorithm. Expressions are developed for the estimation of the speedup of iterative algorithms running on a parallel computer with hyper-cube interconnection topology. Computational speedup in excess of 8 is achieved using 16 processors. The timing results given for the example problems provide guidelines to designers in the use of parallel computers for iterative processes.     

工程结构混合离散变量优化的模拟退火方法

本文针对工程结构优化设计中普遍存在的混合离散变量的现象，探索了应用模拟退火算法直接获得最优解的方法，从而不再对离散变量作“规格化”后处理。文中根据混合离散变量的特点，提出了几种邻域状态的产生函数和迭代方案，给出了适宜的模拟退火过程的冷却进度表。算例表明该方法能有效地解决工程结构混合离散变量的优化问题。     

Topology optimization for thermo-mechanical compliant actuators using mesh-free methods

This article presents an alternative topology optimization method for the design of compliant actuators using mesh-free methods, in which the thermo-mechanical multi-physics modelling and geometrically non-linear analysis are included. The relatively new mesh-free method rather than the standard finite element method (FEM) is used to discretize the design domain and interpolate the bulk density field, because the mesh-free method is in some cases more capable of modelling the large-displacement compliant mechanisms involving the geometrical non-linearity. An interpolation scheme is used to indicate the dependence of material properties on element pseudo densities which are distributed to the corresponding integration points, and the method for imposing essential boundary conditions in mesh-free methods is also discussed. Furthermore, the adjoint approach is incorporated into the mesh-free method to perform the design sensitivity analysis. The optimization problem is established mathematically as a non-linear programming problem to which a sequential convex programming method is applied. The effectiveness of the proposed method is demonstrated by using a widely studied example.     

Design of manufacturing systems by a hybrid approach with neural network metamodelling and stochastic local search

Metamodels are models of simulation models. Metamodels are able to estimate the simulation responses corresponding to a given combination of input variables. A simulation metamodel is easier to manage and provides more insights than simulation alone. Traditionally, the multiple regression analysis is utilized to develop the metamodel from a set of simulation experiments. Simulation can consequentially benefit from the metamodelling in post-simulation analysis. A backpropagation (BP) neural network is a proven tool in providing excellent response predictions in many application areas and it outperforms regression analysis for a wide array of applications. In this paper, a BP neural network is used to generate metamodels for simulated manufacturing systems. For the purpose of optimal manufacturing systems design, mathematical models can be formulated by using the mapping functions generated from the neural network metamodels. The optimization model is then solved by a stochastic local search approach, simulated annealing (SA), to obtain an optimal configuration with respect to the objective of the systems design. Instead of triggering the detailed simulation programs, the SA-based optimization procedure evaluates the simulation outputs by the neural network metamodels. By using the SA-based optimization algorithm, the solution space of the studied problem is extensively exploited to escape the entrapment of local optima while the number of time consuming simulation runs is reduced. The proposed methodology is illustrated to be both effective and efficient in solving a manufacturing systems design problem through an example.     

Multidisciplinary design and optimization of an air launched satellite launch vehicle using a hybrid heuristic search algorithm

A multidisciplinary design and optimization strategy for a multistage air launched satellite launch vehicle comprising of a solid propulsion system to low earth orbit with the implementation of a hybrid heuristic search algorithm is proposed in this article. The proposed approach integrated the search properties of a genetic algorithm and simulated annealing, thus achieving an optimal solution while satisfying the design objectives and performance constraints. The genetic algorithm identified the feasible region of solutions and simulated annealing exploited the identified feasible region in search of optimality. The proposed methodology coupled with design space reduction allows the designer to explore promising regions of optimality. Modules for mass properties, propulsion characteristics, aerodynamics, and flight dynamics are integrated to produce a high-fidelity model of the vehicle. The objective of this article is to develop a design strategy that more efficiently and effectively facilitates multidisciplinary design analysis and optimization for an air launched satellite launch vehicle.     

Metric-driven mesh optimization using a local simulated annealing algorithm

We report on results obtained with a metric-driven mesh optimization procedure for simplicial meshes based on the simulated annealing (SA) method. The use of SA improves the chances of removing pathological clusters of bad elements, that have the tendency to lock into frozen configurations in difficult regions of the model such as corners and complex face intersections, prejudicing the overall quality of the final grid. A local version of the algorithm is developed that significantly lowers the computational cost. Numerical examples illustrate the effectiveness of the proposed methodology, which is compared to a classical greedy Gauss–Seidel optimization. Substantial improvement in the quality of the worst elements of the grid is observed for the local simulated annealing optimization. Furthermore, the method appears to be robust to the choice of the algorithmic parameters. Copyright © 2006 John Wiley & Sons, Ltd.