Optimization of actively controlled structures using goal programming techniques

The problem of design of actively controlled structures subject to restrictions on the damping parameters of the closed-loop system is formulated and solved as a multiobjective optimization problem. The purpose of control is to effectively suppress structural vibrations due to initial excitation. The cross-sectional areas of the members are treated as design variables. The structural weight and the controlled system energy are considered as objective functions for minimization. The goal programming approach is used for the solution of the multiobjective optimization problems. The procedure is illustrated through numerical simulations using two-bar and twelve-bar truss structures.     

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.     

Reliability and redundancy apportionment using crisp and fuzzy multiobjective optimization approaches

The reliability of a multistage system with several components in each stage can be improved either by using more reliable components, or by adding redundant components in parallel in any stage. In many practical situations where reliability enhancement is involved, the decision making is complicated because of the presence of several mutually conflicting goals. For example, in the reliability based design of a system, the designer may be required to maximize the reliability and minimize the cost, weight or volume. This work considers the problem of reliability allocation for multistage systems with components having time-dependent reliability. Two multiobjective optimization techniques are presented, coupled with heuristic procedures, to solve the mixed integer nonlinear programming problems. A generalization of the problem in the presence of vague information results in an ill-structured reliability apportionment problem. The solution of such multiobjective problems is also presented in the present work using the techniques of fuzzy optimization.     

Shape, sizing optimization and material selection based on mixed variables and genetic algorithm

In this work, we explore simultaneous designs of materials selection and structural optimization. As the material selection turns out to be a discrete process that finds the optimal distribution of materials over the design domain, it cannot be performed with common gradient-based optimization methods. In this paper, material selection is considered together with the shape and sizing optimization in a framework of multiobjective optimization of tracking the Pareto curve. The idea of mixed variables is often introduced in the case of mono-objective optimization. However, in the case of multi-objective optimization, we still face some hard key points related to the convexity and the continuity of the Pareto domain, which underline the originality of this work. In addition to the above aspect, there is a lack in the literature concerning the industrial applications that consider the mixed parameters. Continuous variables refer to structural parameters such as thickness, diameter and spring elastic constants while material ID is defined as binary design variable for each material. Both mechanical and thermal loads are considered in this work with the aim of minimizing the maximum stress and structural weight simultaneously. The efficiency of the design procedure is demonstrated through various numerical examples.     

Enhanced multiobjective particle swarm optimization in combination with adaptive weighted gradient-based searching

This article proposes a new multiobjective optimization method for structural problems based on multiobjective particle swarm optimization (MOPSO). A gradient-based optimization method is combined with MOPSO to alleviate constraint-handling difficulties. In this method, a group of particles is divided into two groups—a dominated solution group and a non-dominated solution group. The gradient-based method, utilizing a weighting coefficient method, is applied to the latter to conduct local searching that yields superior non-dominated solutions. In order to enhance the efficiency of exploration in a multiple objective function space, the weighting coefficients are adaptively assigned considering the distribution of non-dominated solutions. A linear optimization problem is solved to determine the optimal weighting coefficients for each non-dominated solution at each iteration. Finally, numerical and structural optimization problems are solved by the proposed method to verify the optimization efficiency.     

An optimization design of vehicle axle system based on multiobjective cooperative optimization algorithm

ABSTRACT

To address multiobjective, multi constraint and time-consuming structural optimization problems in a vehicle axle system, a multiobjective cooperative optimization model of a vehicle axle structure is established. In light of the difficulty in the nondominated sorting of the NSGA-II algorithm caused by inconsistent effects of the uniformity objective function and physical objective function, this paper combines a multiobjective genetic algorithm with cooperative optimization and presents a strategy for handling the optimization of a vehicle axle structure. The uniformity objective function of the sub discipline is transformed to its self-constraint. Taking the multiobjective optimization of a vehicle axle system as an example, a multiobjective cooperative optimization design for the system is carried out in ISIGHT. The results show that the multiobjective cooperative optimization strategy can simplify the complexity of optimization problems and that the multiobjective cooperative optimization method based on an approximate model is favorable for accuracy and efficiency, thereby providing a theoretical basis for the optimization of similar complex structures in practical engineering.     

An interactive decision framework for multiple objective production planning

The formation of realistic implementable medium-range production plans requires explicit recognition of the multiple conflicting objectives of production planning. However, suggested applications of multiobjective optimization to production planning have been limited to goal programming procedures which fail to capitalize on the intrinsic flexibility of a multiobjective model. Alternatively, interactive multiobjective solution techniques could be used to allow planners to enhance decision making without excessive computational effort. This study describes an interactive multiple objective decision framework and evaluates its effectiveness via a multiobjective capacitated lot sizing model based on a real manufacturing facility. The results suggest that this approach is an effective solution strategy and useful decision aid for complex production planning problems.     

A genetic algorithm approach to single and multiobjective structural optimization with discrete–continuous variables

A search procedure with a philosophical basis in molecular biology is adapted for solving single and multiobjective structural optimization problems. This procedure, known as a genetic algorithm (GA). utilizes a blending of the principles of natural genetics and natural selection. A lack of dependence on the gradient information makes GAs less susceptible to pitfalls of convergence to a local optimum. To model the multiple objective functions in the problem formulation, a co-operative game theoretic approach is proposed. Examples dealing with single and multiobjective geometrical design of structures with discrete–continuous design variables, and using artificial genetic search are presented. Simulation results indicate that GAs converge to optimum solutions by searching only a small fraction of the solution space. The optimum solutions obtained using GAs compare favourably with optimum solutions obtained using gradient-based search techniques. The results indicate that the efficiency and power of GAs can be effectively utilized to solve a broad spectrum of design optimization problems with discrete and continuous variables with similar efficiency.     

Multiobjective design of actively controlled structures using a hybrid optimization method

A multiobjective approach to the combined structure and control optimization problem for flexible space structures is presented. The proposed formulation addresses robustness considerations for controller design, as well as a simultaneous determination of optimum actuator locations. The structural weight, controlled system energy, stability robustness index and damping augmentation provided by the active controller are considered as objective functions of the multiobjective problem which is solved using a cooperative game-theoretic approach. The actuator locations and the cross-sectional areas of structural members are treated as design variables. Since the actuator locations are spatially discrete, whereas the cross-sectional areas are continuous, the optimization problem has mixed discrete-continuous design variables. A solution approach to this problem based on a hybrid optimization scheme is presented. The hybrid optimizer is a synergetic blend of artificial genetic search and gradient-based search techniques. The computational procedure is demonstrated through the design of an ACOSS-FOUR space structure. The optimum solutions obtained using the hybrid optimizer are shown to outperform the optimum results obtained using gradient-based search techniques.     

Nonlinear programming on a microcomputer

The Davidon-Fletcher-Powell (DFP) algorithm for multivariable unconstrained nonlinear function minimization is implemented on a desk microcomputer—the Hindustan Computers MICRO-2200—with a floppy disk memory extension peripheral. The program is developed using the programming keys of the system. The program is closely based on FMFP, a well-known FORTRAN subroutine of the algorithm. Among the exercises performed are two standard nonlinear programming (NLP) test problems, namely the Rosenbrock and Wood function minimization and two SUMT (Sequential Unconstrained Minimization Technique) illustration problems in structural optimization, i.e. two-bar and three-bar truss weight minimization examples. Experience bears out the potential of desk calculating systems as tools in nonlinear programming algorithm research. Also, the small-scale structural optimization capability afforded by such systems is expected to be of significance in teaching and preliminary design contexts.     

Multi-stage optimization method for air-intake system of hovercraft based on autonomous optimization

Abstract

In this study, an optimal structural design program was designed and developed for Computational Fluid Dynamics based on self-optimization, effectively reducing the time required for structural optimization. Through experimental design using this program, the effects of various design variables on the optimization objectives were evaluated, and an adaptive simulated annealing algorithm was used for global optimization. Furthermore, response surface methodology and a nonlinear quadratic programming algorithm were utilized to obtain a global optimum solution after repeated iterations. Moreover, using a hovercraft air-intake system as the optimized object, the total pressure loss of the system was completely optimized by using a porous medium model and Matlab analysis program, and the accuracy of the structural design optimization program was validated. After the global optimization, the total pressure loss of the air-intake system was reduced by 20.5% compared to the original model. An average nonuniformity of 4.36% of engine inlet speed and 5% local nonuniformity of 11.19% satisfy the design requirements of the hovercraft engine. This method can be directly applied to engineering optimization problems as well as multiobjective optimization tasks after improving the relevant methodologies.     

A study of mathematical programming methods for structural optimization. Part I: Theory

A comprehensive study of various mathematical programming methods for structural optimization is presented. In recent years, many modern optimization techniques and convergence results have been developed in the field of mathematical programming. The aim of this paper is twofold: (a) to discuss the applicability of modern optimization techniques to structural design problems, and (b) to present mathematical programming methods from a unified and design engineers' viewpoint. Theoretical aspects are considered here, while numerical results of test problems are discussed in a companion paper. Special features possessed by structural optimization problems, together with recent developments in mathematical programming (recursive quadratic programming methods, global convergence theory), have formed a basis for conducting the study. Some improvements of existing methods are noted and areas for future investigation are discussed.     

A general methodological analysis for optimum design

Generalized applications of modern numerical analysis methods—while digital computers experienced a fast development—produced a first revolution in design techniques, allowing one to perform computations considered unapproachable until that time. Introduction of Computer Aided Design (CAD) techniques—while high-performance graphic peripherals experience a fast development—is actually producing a second revolution, by making easy and fast most routine design tasks. However, the introduction of Computer Aided OPTIMUM Design techniques has not yet produced the expected third revolution, in spite of the big amount of research and the interest of its potential applications. The authors think that this fact is due mainly to the dispersion of the optimum design research, and to the lack of a well established doctrine. In this paper we approach the design process from a general methodological perspective, suitable to be applied to a wide range of problems. The design process is organized in several related levels. This approach leads naturally to the concept of optimum design and to the statement of a general mathematical programming problem. The practical application of this methodology to any particular problem takes an efficient and modular form. First and second order sensitivity analysis techniques are introduced from the general formulation, and alternative techniques (adjoint state) of the direct differentiation method are discussed. DAO², a powerful and versatile computer aided optimum design system by the Finite Element Method, has been developed by the authors¹ according to this general methodology. The system can solve efficiently 2D and 3D structural fixed-geometry and shape optimization problems. The power and viability of this methodology is illustrated by the solution to a structural optimization problem. The shape of the central section of an arch dam is optimized. A linear elastic structural FEM analysis is simultaneously performed for plane stress and for radial symmetry—while constraints are imposed for several load cases—taking into account the construction and loading stages. It should be emphasized that the same optimum design is reached in a small number of iterations starting from two significantly different initial designs.     

Level Diagrams analysis of Pareto Front for multiobjective system redundancy allocation

Reliability-based and risk-informed design, operation, maintenance and regulation lead to multiobjective (multicriteria) optimization problems. In this context, the Pareto Front and Set found in a multiobjective optimality search provide a family of solutions among which the decision maker has to look for the best choice according to his or her preferences. Efficient visualization techniques for Pareto Front and Set analyses are needed for helping decision makers in the selection task.In this paper, we consider the multiobjective optimization of system redundancy allocation and use the recently introduced Level Diagrams technique for graphically representing the resulting Pareto Front and Set. Each objective and decision variable is represented on separate diagrams where the points of the Pareto Front and Set are positioned according to their proximity to ideally optimal points, as measured by a metric of normalized objective values. All diagrams are synchronized across all objectives and decision variables. On the basis of the analysis of the Level Diagrams, we introduce a procedure for reducing the number of solutions in the Pareto Front; from the reduced set of solutions, the decision maker can more easily identify his or her preferred solution.     

Analysis and optimization of parameter combinations in wire electrical discharge machining

This paper presents a formulation and solution of a multi-objective optimization problem for the selection of the best control settings on a wire electrical discharge machine. The measures of performance for the model are taken to be metal removal rate and surface finish quality. A factorial design model is used to predict the measures of performance as a function of a variety of control settings. To aid in selecting the best combination of settings the concept of a non-dominated point is introduced. The non-dominated points are related to two important complementary optimization problems. Two techniques for the computation of non-dominated points are presented—one using explicit enumeration and the other based on the principles of dynamic programming.     

Multiobjective robust optimization method for drawbead design in sheet metal forming

It is recognized that fracture and wrinkling in sheet metal forming can be eliminated via an appropriate drawbead design. Although deterministic multiobjective optimization algorithms and finite element analysis (FEA) have been applied in this respect to improve formability and shorten design cycle, the design could become less meaningful or even unacceptable when considering practical variation in design variables and noises of system parameters. To tackle this problem, we present a multiobjective robust optimization methodology to address the effects of parametric uncertainties on drawbead design, where the six sigma principle is adopted to measure the variations, a dual response surface method is used to construct surrogate model and a multiobjective particle swarm optimization is developed to generate robust Pareto solutions. In this paper, the procedure of drawbead design is divided into two stages: firstly, equivalent drawbead restraining forces (DBRF) are obtained by developing a multiobjective robust particle swarm optimization, and secondly the DBRF model is integrated into a single-objective particle swarm optimization (PSO) to optimize geometric parameters of drawbead. The optimal design showed a good agreement with the physical drawbead geometry and remarkably improve the formability and robust. Thus, the presented method provides an effective solution to geometric design of drawbead for improving product quality.     

Soft computing approach for reliability optimization: State-of-the-art survey

In the broadest sense, reliability is a measure of performance of systems. As systems have grown more complex, the consequences of their unreliable behavior have become severe in terms of cost, effort, lives, etc., and the interest in assessing system reliability and the need for improving the reliability of products and systems have become very important. Most solution methods for reliability optimization assume that systems have redundancy components in series and/or parallel systems and alternative designs are available. Reliability optimization problems concentrate on optimal allocation of redundancy components and optimal selection of alternative designs to meet system requirement. In the past two decades, numerous reliability optimization techniques have been proposed. Generally, these techniques can be classified as linear programming, dynamic programming, integer programming, geometric programming, heuristic method, Lagrangean multiplier method and so on. A Genetic Algorithm (GA), as a soft computing approach, is a powerful tool for solving various reliability optimization problems. In this paper, we briefly survey GA-based approach for various reliability optimization problems, such as reliability optimization of redundant system, reliability optimization with alternative design, reliability optimization with time-dependent reliability, reliability optimization with interval coefficients, bicriteria reliability optimization, and reliability optimization with fuzzy goals. We also introduce the hybrid approaches for combining GA with fuzzy logic, neural network and other conventional search techniques. Finally, we have some experiments with an example of various reliability optimization problems using hybrid GA approach.     

Flower pollination algorithm: A novel approach for multiobjective optimization

Multiobjective design optimization problems require multiobjective optimization techniques to solve, and it is often very challenging to obtain high-quality Pareto fronts accurately. In this article, the recently developed flower pollination algorithm (FPA) is extended to solve multiobjective optimization problems. The proposed method is used to solve a set of multiobjective test functions and two bi-objective design benchmarks, and a comparison of the proposed algorithm with other algorithms has been made, which shows that the FPA is efficient with a good convergence rate. Finally, the importance for further parametric studies and theoretical analysis is highlighted and discussed.     

Polymorphic uncertain nonlinear programming approach for maximizing the capacity of V-belt driving

In this paper, a polymorphic uncertain nonlinear programming (PUNP) approach is developed to formulate the problem of maximizing the capacity in a system of V-belt driving with uncertainties. The constructed optimization model is found to consist of a nonlinear objective function and some nonlinear constraints with some parameters which are of uncertain nature. These uncertain parameters are interval parameters, random interval parameters, fuzzy parameters or fuzzy interval parameters. To find a robust solution of the problem, a deterministic equivalent formulation (DEF) is established for the polymorphic uncertain nonlinear programming model. For a given satisfaction level, this DEF turns out to be a nonlinear programming involving only interval parameters. A solution method, called a sampling based interactive method, is developed such that a robust solution of the original model with polymorphic uncertainties is obtained by using standard smooth optimization techniques. The proposed method is applied into a real-world design of V-belt driving, and the results indicate that both the PUNP approach and the developed algorithm are useful to the optimization problem with polymorphic uncertainty.     

Reliability optimization using multiobjective ant colony system approaches

The multiobjective ant colony system (ACS) meta-heuristic has been developed to provide solutions for the reliability optimization problem of series-parallel systems. This type of problems involves selection of components with multiple choices and redundancy levels that produce maximum benefits, and is subject to the cost and weight constraints at the system level. These are very common and realistic problems encountered in conceptual design of many engineering systems. It is becoming increasingly important to develop efficient solutions to these problems because many mechanical and electrical systems are becoming more complex, even as development schedules get shorter and reliability requirements become very stringent. The multiobjective ACS algorithm offers distinct advantages to these problems compared with alternative optimization methods, and can be applied to a more diverse problem domain with respect to the type or size of the problems. Through the combination of probabilistic search, multiobjective formulation of local moves and the dynamic penalty method, the multiobjective ACSRAP, allows us to obtain an optimal design solution very frequently and more quickly than with some other heuristic approaches. The proposed algorithm was successfully applied to an engineering design problem of gearbox with multiple stages.