A novel surrogate-based approach for optimal design of electromagnetic-based circuits

A new geometric design centring approach for optimal design of central processing unit-intensive electromagnetic (EM)-based circuits is introduced. The approach uses norms related to the probability distribution of the circuit parameters to find distances from a point to the feasible region boundaries by solving nonlinear optimization problems. Based on these normed distances, the design centring problem is formulated as a max–min optimization problem. A convergent iterative boundary search technique is exploited to find the normed distances. To alleviate the computation cost associated with the EM-based circuits design cycle, space-mapping (SM) surrogates are used to create a sequence of iteratively updated feasible region approximations. In each SM feasible region approximation, the centring process using normed distances is implemented, leading to a better centre point. The process is repeated until a final design centre is attained. Practical examples are given to show the effectiveness of the new design centring method for EM-based circuits.     

Sequential algorithms for structural design optimization under tolerance conditions

A deterministic optimization usually ignores the effects of uncertainties in design variables or design parameters on the constraints. In practical applications, it is required that the optimum solution can endure some tolerance so that the constraints are still satisfied when the solution undergoes variations within the tolerance range. An optimization problem under tolerance conditions is formulated in this article. It is a kind of robust design and a special case of a generalized semi-infinite programming (GSIP) problem. To overcome the deficiency of directly solving the double loop optimization, two sequential algorithms are then proposed for obtaining the solution, i.e. the double loop optimization is solved by a sequence of cycles. In each cycle a deterministic optimization and a worst case analysis are performed in succession. In sequential algorithm 1 (SA1), a shifting factor is introduced to adjust the feasible region in the next cycle, while in sequential algorithm 2 (SA2), the shifting factor is replaced by a shifting vector. Several examples are presented to demonstrate the efficiency of the proposed methods. An optimal design result based on the presented method can endure certain variation of design variables without violating the constraints. For GSIP, it is shown that SA1 can obtain a solution with equivalent accuracy and efficiency to a local reduction method (LRM). Nevertheless, the LRM is not applicable to the tolerance design problem studied in this article.     

MULTIOBJECTIVE DESIGN OF A MARINE VEHICLE WITH AID OF DESIGN KNOWLEDGE

To generate the Pareto optimal set efficiently in multiobjective optimization, a hybrid optimizer is developed by coupling the genetic algorithm and the direct search method. This method determines a candidate region around the global optimum point by using the genetic algorithm, then searches the global optimum point by the direct search method concentrating in this region, thus reducing calculation time and increasing search efficiency. Although the hybrid optimizer provides cost-effectiveness, the design optimization process involves a number of tasks which require human expertise and experience. Therefore, methods of optimization and associated programs have been used mostly by experts in the real design world. Hence, this hybrid optimizer incorporates a knowledge-based system with heuristic and analytic knowledge, thereby narrowing the feasible space of the objective function. Some domain knowledge is retrieved from database and design experts. The obtained knowledge is stored in the knowledge base. The results of this paper, through application to marine vehicle design with multiobjective optimization, show that the hybrid optimizer with aid of design knowledge can be a useful tool for multiobjective optimum design. © 1997 John Wiley & Sons, Ltd.     

A new hybrid method for optimal circuit design using semi-definite programming

In this article a new method for yield optimization (design centring) is introduced. The method has a statistical-geometrical nature, hence it is called hybrid. The method exploits the semi-definite programming applications in approximating the feasible region with two bounding ellipsoids. These ellipsoids are obtained using a two phase algorithm. In the first phase, the minimum volume ellipsoid enclosing the feasible region is obtained. The largest ellipsoid that can be inscribed inside the feasible region is obtained in the second phase. The centres of these bounding ellipsoids are used as design centres. In the second phase, an additional polytopic region approximation is constructed. A comparison between the obtained region approximations is given. Saving in the number of circuit simulations needed for yield optimization is also considered. Practical examples are given to show the effectiveness of the new method.     

Multi-objective feasibility enhanced particle swarm optimization

This article introduces a new method entitled multi-objective feasibility enhanced partical swarm optimization (MOFEPSO), to handle highly-constrained multi-objective optimization problems. MOFEPSO, which is based on the particle swarm optimization technique, employs repositories of non-dominated and feasible positions (or solutions) to guide feasible particle flight. Unlike its counterparts, MOFEPSO does not require any feasible solutions in the initialized swarm. Additionally, objective functions are not assessed for infeasible particles. Such particles can only fly along sensitive directions, and particles are not allowed to move to a position where any previously satisfied constraints become violated. These unique features help MOFEPSO gradually increase the overall feasibility of the swarm and to finally attain the optimal solution. In this study, multi-objective versions of a classical gear-train optimization problem are also described. For the given problems, the article comparatively evaluates the performance of MOFEPSO against several popular optimization algorithms found in the literature.     

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.     

An improvement of Kriging based sequential approximate optimization method via extended use of design of experiments

When Kriging is used as a meta-model for an inequality constrained function, approximate optimal solutions are sometimes infeasible in the case where they are active at the constraint boundary. This article explores the development of a Kriging-based meta-model that enhances the constraint feasibility of an approximate optimal solution. The trust region management scheme is used to ensure the convergence of the approximate optimal solution. The present study proposes a method of enhancing the constraint feasibility in which the currently infeasible design is replaced by the most feasible-usable design during the sequential approximate optimization process. An additional convergence condition is also included to reinforce the design accuracy and feasibility. Latin hypercube design and (2n+1) design are used as tools for design of experiments. The proposed approach is verified through a constrained mathematical function problem and a number of engineering optimization problems to support the proposed strategies.     

A deterministic global approach for mixed-discrete structural optimization

This study proposes a novel approach for finding the exact global optimum of a mixed-discrete structural optimization problem. Although many approaches have been developed to solve the mixed-discrete structural optimization problem, they cannot guarantee finding a global solution or they adopt too many extra binary variables and constraints in reformulating the problem. The proposed deterministic method uses convexification strategies and linearization techniques to convert a structural optimization problem into a convex mixed-integer nonlinear programming problem solvable to obtain a global optimum. To enhance the computational efficiency in treating complicated problems, the range reduction technique is also applied to tighten variable bounds. Several numerical experiments drawn from practical structural design problems are presented to demonstrate the effectiveness of the proposed method.     

有限元线法求解非线性模型问题——Ⅱ.扭转杆的最优截面

作为有限元线法(FEMOL)求解非线性问题的系列工作之二,本文将该法应用于形状优化问题,对扭转杆的截面优化这一模型问题作了分析求解。文中首先对双连域截面的扭转问题作了FEMOL推导,然后允许结线的长度改变以描述不同的截面形状,再利用若干变换技巧将形状变量及优化条件引入常微分方程(ODE)体系中,从而将问题转换成标准的非线性ODE问题,并由ODE求解器进行求解。文中算例显示了本法对形状优化问题的求解具有方法简洁、实施方便、效率显著等优点。     

A surrogate assisted parallel multiobjective evolutionary algorithm for robust engineering design

A number of multi-objective evolutionary algorithms have been proposed in recent years and many of them have been used to solve engineering design optimization problems. However, designs need to be robust for real-life implementation, i.e. performance should not degrade substantially under expected variations in the variable values or operating conditions. Solutions of constrained robust design optimization problems should not be too close to the constraint boundaries so that they remain feasible under expected variations. A robust design optimization problem is far more computationally expensive than a design optimization problem as neighbourhood assessments of every solution are required to compute the performance variance and to ensure neighbourhood feasibility. A framework for robust design optimization using a surrogate model for neighbourhood assessments is introduced in this article. The robust design optimization problem is modelled as a multi-objective optimization problem with the aim of simultaneously maximizing performance and minimizing performance variance. A modified constraint-handling scheme is implemented to deal with neighbourhood feasibility. A radial basis function (RBF) network is used as a surrogate model and the accuracy of this model is maintained via periodic retraining. In addition to using surrogates to reduce computational time, the algorithm has been implemented on multiple processors using a master–slave topology. The preliminary results of two constrained robust design optimization problems indicate that substantial savings in the actual number of function evaluations are possible while maintaining an acceptable level of solution quality.     

Linear and nonlinear topology optimization design with projection-based ground structure method (P-GSM)

A new topology optimization scheme called the projection-based ground structure method (P-GSM) is proposed for linear and nonlinear topology optimization designs. For linear design, compared to traditional GSM which are limited to designing slender members, the P-GSM can effectively resolve this limitation and generate functionally graded lattice structures. For additive manufacturing-oriented design, the manufacturing abilities are the key factors to constrain the feasible design space, for example, minimum length and geometry complexity. Conventional density-based method, where each element works as a variable, always results in complex geometry with large number of small intricate features, while these small features are often not manufacturable even by 3D printing and lose its geometric accuracy after postprocessing. The proposed P-GSM is an effective method for controlling geometric complexity and minimum length for optimal design, while it is capable of designing self-supporting structures naturally. In optimization progress, some bars may be disconnected from each other (floating in the air). For buckling-induced design, this issue becomes critical due to severe mesh distortion in the void space caused by disconnection between members, while P-GSM has ability to overcome this issue. To demonstrate the effectiveness of proposed method, three different design problems ranging from compliance optimization to buckling-induced mechanism design are presented and discussed in details.     

Design optimization using multiple dominance relations

A challenge in engineering design is to choose suitable objectives and constraints from many quantities of interest, while ensuring an optimization is both meaningful and computationally tractable. We propose an optimization formulation that can take account of more quantities of interest than existing formulations, without reducing the tractability of the problem. This formulation searches for designs that are optimal with respect to a binary relation within the set of designs that are optimal with respect to another binary relation. We then propose a method of finding such designs in a single optimization by defining an overall ranking function to use in optimizers, reducing the cost required to solve this formulation. In a design under uncertainty problem, our method obtains the most robust design that is not stochastically dominated faster than a multiobjective optimization. In a car suspension design problem, our method obtains superior designs according to a k-optimality condition than previously suggested multiobjective approaches to this problem. In an airfoil design problem, our method obtains designs closer to the true lift/drag Pareto front using the same computational budget as a multiobjective optimization.     

基于变频率区间约束的结构材料优化设计

针对频率约束的结构材料优化问题,基于结构拓扑优化思想,提出变频率区间约束的结构材料优化方法。借鉴均匀化及ICM(独立、连续、映射)方法,以微观单元拓扑变量倒数为设计变量,导出宏观单元等效质量矩阵及导数,进而获得频率一阶近似展开式。结合变频率区间约束思想,获得以结构质量为目标函数、频率为约束条件的连续体微结构拓扑优化近似模型;采用对偶方法求解。通过算例验证该方法的有效性及可行性,表明考虑质量矩阵变化影响所得优化结果更合理。     

Root cause analysis strategy for robust design domain recognition

Design domain identification with desirable attributes (e.g. feasibility, robustness and reliability) provides advantages when tackling large-scale engineering optimization problems. For the purpose of dealing with feasibility robustness design problems, this article proposes a root cause analysis (RCA) strategy to identify desirable design domains by investigating the root causes of performance indicator variation for the starting sampling initiation of evolutionary algorithms. The iterative dichotomizer 3 method using a decision tree technique is applied to identify reduced feasible design domain sets. The robustness of candidate domains is then evaluated through a probabilistic principal component analysis-based criterion. The identified robust design domains enable optimal designs to be obtained that are relatively insensitive to input variations. An analytical example and an automotive structural optimization problem are demonstrated to show the validity of the proposed RCA strategy.     

LAGRANGEAN RELAXATION AND SUBGRADIENT OPTIMIZATION APPLIED TO OPTIMAL DESIGN WITH DISCRETE SIZING

The discrete sizing problem in optimal design is adressed. Lagrangean dual approaches earlier published are briefly reviewed and it is noted that quite sophisticated procedures have been used to solve the dual problems. The simple concept of Lagrangean relaxation combined with subgradient optimization and Lagrangean heuristics has, however, not been applied to the discrete sizing problem. In this paper a scheme based on this concept is described and tested on some small problems. The results indicate that subgradient optimization is completely capable of solving the dual problem. Moreover it is possible to devise heuristics that construct feasible solutions to the original problem, using the Lagrangean subproblem solution.     

Topology optimization of geometrically nonlinear structures using an evolutionary optimization method

The topology optimization using isolines/isosurfaces and extended finite element method (Iso-XFEM) is an evolutionary optimization method developed in previous studies to enable the generation of high-resolution topology optimized designs suitable for additive manufacture. Conventional approaches for topology optimization require additional post-processing after optimization to generate a manufacturable topology with clearly defined smooth boundaries. Iso-XFEM aims to eliminate this time-consuming post-processing stage by defining the boundaries using isovalues of a structural performance criterion and an extended finite element method (XFEM) scheme. In this article, the Iso-XFEM method is further developed to enable the topology optimization of geometrically nonlinear structures undergoing large deformations. This is achieved by implementing a total Lagrangian finite element formulation and defining a structural performance criterion appropriate for the objective function of the optimization problem. The Iso-XFEM solutions for geometrically nonlinear test cases implementing linear and nonlinear modelling are compared, and the suitability of nonlinear modelling for the topology optimization of geometrically nonlinear structures is investigated.     

Quantized hopfield networks for reliability optimization

The use of neural networks in the reliability optimization field is rare. This paper presents an application of a recent kind of neural networks in a reliability optimization problem for a series system with multiple-choice constraints incorporated at each subsystem, to maximize the system reliability subject to the system budget. The problem is formulated as a nonlinear binary integer programming problem and characterized as an NP-hard problem. Our design of neural network to solve efficiently this problem is based on a quantized Hopfield network. This network allows us to obtain optimal design solutions very frequently and much more quickly than others Hopfield networks.     

Particle swarm optimization approach for multi-objective composite box-beam design

This paper presents a multi-agent search technique to design an optimal composite box-beam helicopter rotor blade. The search technique is called particle swarm optimization (‘inspired by the choreography of a bird flock’). The continuous geometry parameters (cross-sectional dimensions) and discrete ply angles of the box-beams are considered as design variables. The objective of the design problem is to achieve (a) specified stiffness value and (b) maximum elastic coupling. The presence of maximum elastic coupling in the composite box-beam increases the aero-elastic stability of the helicopter rotor blade. The multi-objective design problem is formulated as a combinatorial optimization problem and solved collectively using particle swarm optimization technique. The optimal geometry and ply angles are obtained for a composite box-beam design with ply angle discretizations of 10°, 15° and 45°. The performance and computational efficiency of the proposed particle swarm optimization approach is compared with various genetic algorithm based design approaches. The simulation results clearly show that the particle swarm optimization algorithm provides better solutions in terms of performance and computational time than the genetic algorithm based approaches.     

Gearbox design for uncertain load requirements using active robust optimization

Design and optimization of gear transmissions have been intensively studied, but surprisingly the robustness of the resulting optimal design to uncertain loads has never been considered. Active Robust (AR) optimization is a methodology to design products that attain robustness to uncertain or changing environmental conditions through adaptation. In this study the AR methodology is utilized to optimize the number of transmissions, as well as their gearing ratios, for an uncertain load demand. The problem is formulated as a bi-objective optimization problem where the objectives are to satisfy the load demand in the most energy efficient manner and to minimize production cost. The results show that this approach can find a set of robust designs, revealing a trade-off between energy efficiency and production cost. This can serve as a useful decision-making tool for the gearbox design process, as well as for other applications.