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 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.     

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.     

An efficient and practical approach to obtain a better optimum solution for structural optimization

For many structural optimization problems, it is hard or even impossible to find the global optimum solution owing to unaffordable computational cost. An alternative and practical way of thinking is thus proposed in this research to obtain an optimum design which may not be global but is better than most local optimum solutions that can be found by gradient-based search methods. The way to reach this goal is to find a smaller search space for gradient-based search methods. It is found in this research that data mining can accomplish this goal easily. The activities of classification, association and clustering in data mining are employed to reduce the original design space. For unconstrained optimization problems, the data mining activities are used to find a smaller search region which contains the global or better local solutions. For constrained optimization problems, it is used to find the feasible region or the feasible region with better objective values. Numerical examples show that the optimum solutions found in the reduced design space by sequential quadratic programming (SQP) are indeed much better than those found by SQP in the original design space. The optimum solutions found in a reduced space by SQP sometimes are even better than the solution found using a hybrid global search method with approximate structural analyses.     

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.     

Simultaneous optimal design of structural topology, actuator locations and control parameters for a plate structure

 Simultaneous optimization with respect to the structural topology, actuator locations and control parameters of an actively controlled plate structure is investigated in this paper. The system consists of a clamped-free plate, a H ₂ controller and four surface-bonded piezoelectric actuators utilized for suppressing the bending and torsional vibrations induced by external disturbances. The plate is represented by a rectangular design domain which is discretized by a regular finite element mesh and for each element the parameter indicating the presence or absence of material is used as a topology design variable. Due to the unavailability of large-scale 0–1 optimization algorithms, the binary variables of the original topology design problem are relaxed so that they can take all values between 0 and 1. The popular techniques in the topology optimization area including penalization, filtering and perimeter restriction are also used to suppress numerical problems such as intermediate thickness, checkerboards, and mesh dependence. Moreover, since it is not efficient to treat the structural and control design variables equally within the same framework, a nested solving approach is adopted in which the controller syntheses are considered as sub processes included in the main optimization process dealing with the structural topology and actuator locations. The structural and actuator variables are solved in the main optimization by the method of moving asymptotes, while the control parameters are designed in the sub optimization processes by solving the Ricatti equations. Numerical examples show that the approach used in this paper can produce systems with clear structural topology and high control performance. Received 16 November 2001 / Accepted 26 February 2002     

HYBRID MULTIMODAL OPTIMIZATION WITH CLUSTERING GENETIC STRATEGIES

Abstract

Two-stage hybrid multimodal optimization approaches that combine cluster identification techniques in genetic algorithms with sharing and gradient-based local search methods are proposed. The multimodal optimization comprises the use of a sharing function implementation in genetic searches to pursue multiple local optima and subsequent executions of local searches to locate each local optimum when an extreme-containing region is identified. A new cluster identification technique is proposed for automatic and adaptive identification of the locations and sizes of design clusters in genetic algorithms with sharing. The first stage of the hybrid multimodal optimization is to use sharing-enhanced genetic algorithms for the identification of the near-optimum designs inside extreme-containing regions. The second stage simply involves consecutive employment of efficient gradient-based local searches by using the near-optimum designs as initial designs. Two strategies defining the coupling of the genetic search and local searches are proposed. The proposed hybrid optimization strategies are tested in a number of illustrative multimodal optimization problems.     

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.     

Optimal placement of actuators and sensors in control-augmented structural optimization

A control-augmented structural synthesis methodology is presented in which actuator and sensor placement is treated in terms of (0, 1) variables. Structural member sizes and control variables are treated simultaneously as design variables. A multiobjective utopian approach is used to obtain a compromise solution for inherently conflicting objective functions such as structural mass, control effort and number of actuators. Constraints are imposed on natural frequencies, peak transient displacements and accelerations, peak actuator forces and dynamic stability as well as controllability and observability of the system. The combinatorial aspects of the mixed (0, 1)-continuous variable design optimization problem are made tractable by combining approximation concepts with branch and bound techniques. Some numerical results for example problems are presented to illustrate the efficacy of the design procedure set forth.     

Enhanced damping of lightweight structures by semi-active joints

Summary Lightweight structures typically have low inherent structural damping. Effective vibration suppression is required, for example, in certain applications involving precision positioning. The present approach is based on friction damping in semi-active joints which allow relative sliding between the connected parts. The energy dissipation due to interfacial slip in the friction joints can be controlled by varying the normal pressure in the contact area using a piezo-stack actuator. This paper focuses on the optimal placement of semi-active joints for vibration suppression. The proposed method uses optimality criteria for actuator and sensor locations based on eigenvalues of the controllability and observability gramians. Optimal sensor/actuator placement is stated as a nonlinear multicriteria optimization problem with discrete variables and is solved by a stochastic search algorithm. At optimal locations, conventional rigid connections of a large truss structure are replaced by semi-active friction joints. Two different concepts for the control of the normal forces in the friction interfaces are implemented. In the first approach, each semi-active joint has its own local feedback controller, whereas the second concept uses a global, clipped-optimal controller. Simulation results for a 10-bay truss structure show the potential of the proposed semi-active concept. Dedicated to Professor Franz Ziegler on the occasion of his 70th birthday     

A hybrid global–local approach for optimization of injection gate locations in liquid composite molding process simulations

The optimization of injection gate locations in liquid composite molding processes by trial and error based methods is time consuming and requires an elevated level of intuition, even when high fidelity physics-based numerical models are available. Optimization based on continuous sensitivity equations (CSE) and gradient search algorithms focused towards minimizing the mold infusion time gives a robust approach that will converge to local optima based on the initial solution. Optimization via genetic algorithms (GA) utilizes natural selection as a means of finding the optimal solution in the global domain; the computed solution is at best, close to the global optimum with further refinement still possible. In this paper, we present a hybrid global–local search approach that combines evolutionary GAs with gradient-based searches via the CSE. The hybrid approach provides a global search with the GA for a predetermined amount of time and is subsequently further refined with a gradient-based search via the CSE. In our hybrid method, we utilize the efficiency of gradient searches combined with the robustness of the GA. The resulting combination has been demonstrated to provide better and more physically correct results than either method alone. The hybrid method provides optimal solutions more quickly than GA alone and more robustly than CSE based searches alone. A resin infusion quality parameter that measures the deviation from a near uniform mold volume infusion rate is defined. The effectiveness of the hybrid method with a modified objective function that includes both the infusion time and the defined mold infusion quality parameter is demonstrated.     

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.     

The harmony search heuristic algorithm for discrete structural optimization

Many methods have been developed and are in use for structural size optimization problems, in which the cross-sectional areas or sizing variables are usually assumed to be continuous. In most practical structural engineering design problems, however, the design variables are discrete. This paper proposes an efficient optimization method for structures with discrete-sized variables based on the harmony search (HS) heuristic algorithm. The recently developed HS algorithm was conceptualized using the musical process of searching for a perfect state of harmony. It uses a stochastic random search instead of a gradient search so that derivative information is unnecessary. In this article, a discrete search strategy using the HS algorithm is presented in detail and its effectiveness and robustness, as compared to current discrete optimization methods, are demonstrated through several standard truss examples. The numerical results reveal that the proposed method is a powerful search and design optimization tool for structures with discrete-sized members, and may yield better solutions than those obtained using current methods.     

Genetic approach to optimal topology/controller design of adaptive structures

Integrated optimization of structural topology and control system is considered. The problem is formulated as mixed discrete-continuous multi-objective programming with a linear quadratic regulator cost index, and measures of robustness and controllability as objectives. The Genetic Algorithm (GA), a guided random search technique, is adopted for the problem-solving. A member elimination strategy that allows deleted members to be recovered is suggested in the search procedure. As verification for the proposed method, optimum layout and actuator placement for a 45-bar truss is illustrated. Numerical results indicate that the genetic algorithm can converge to optimum solutions by searching only a minor fraction of the solution space. Discussions on the algorithm are presented. © 1998 John Wiley & Sons, Ltd.     

Shape and size optimization of trusses with multiple frequency constraints using harmony search and ray optimizer for enhancing the particle swarm optimization algorithm

In this paper, size and shape optimization of truss structures is performed using an efficient hybrid method. This algorithm uses a particle swarm strategy and ray optimizer, and utilizes additional harmony search for a better exploitation. Here, multiple frequency constraints are considered making the optimization a highly nonlinear problem. Some basic benchmark problems are solved by this hybrid method, and the numerical results demonstrate the efficiency and robustness of this method compared to other mathematical and heuristic algorithms.     

Practical design optimization of truss structures using the genetic algorithms

Truss structures are widely employed in the industrialized world. They appear as bridges, towers, pylons, roof supports, building exoskeletons or high technology light space structures. This paper investigates the simultaneous size, geometry and topology optimization of real life large truss structures using genetic algorithms (GAs) as optimizer and finite element method as analyzer. In general, the large truss structures are constructed for practical reasons from the duplication of some basic structures called bays. Thus, the final optimum design may be reached by optimizing the characteristics of the basic bays instead of optimizing the whole structure. Both single and multiobjective functions based on the mass of the structure and the maximum nodal displacement have been considered as the cost functions. In order to obtain realistic optimal designs, the cross-sectional areas have been extracted from the standard profiles according to AISC codes and practical conditions are imposed on the bays. The design optimization problem is also constrained by the maximum stress, maximum slenderness ratio and the maximum and minimum cross-sectional area of the truss members. To accommodate all these constraints, two different penalty functions are considered. The first penalty function considers the normalization of violated constraints with respect to the allowable stress or slenderness ratio. The second penalty function is a constant function which is used to penalize the violations of the slenderness ratio. Three illustrative examples of realistic planar and space truss structures have been optimized to demonstrate the effectiveness of the proposed methodology. However, other criteria such as cost and/or manufacturability could be quantified and included in the optimization formulation.     

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.     

Optimal design of a variable-twist proprotor incorporating shape memory alloy hybrid composites

The tiltrotor blades, or proprotor, act as a rotor in the helicopter mode and a propeller in the airplane mode. The helicopter mode generally requires relatively a low built-in twist angle, whereas in the airplane mode, a high built-in twist is desired. Meeting these rather conflicting requirements make the tiltrotor design a challenging task. This paper explores an optimal design of a variable-twist proprotor that changes the built-in twist in an adaptive manner by using the shape memory alloy hybrid composite (SMAHC). The optimum design problem attempts to find the cross-section internal layout that maximizes the twist actuation of the variable-twist proprotor while satisfying a series of design constraints. An optimum design framework is constructed in the current work by combining various analysis and design tools, such as an active composite cross-sectional analysis, a nonlinear flexible multibody dynamics analysis, a 3-D strain analysis, and a gradient-based optimizer. The MATLAB is used to integrate and synthesize the individual tools. A static tip twist is chosen as an objective function that should be maximized for the best performance. The optimum results exhibit that the twist actuation of the variable-twist proprotor can be maximized while satisfying all the prescribed design constraints.     

DESIGN OF STRUCTURE CONTROL SYSTEM USING BOUNDED LQG

An approach for designing a structure and its control system for vibration suppression is presented. The control system is based on the Linear Quadratic Gaussian (LQG) and is modified to allow bounds on the actuators forces to simulate real actuators. The simultaneous design of the structure and control problem is formulated as a nonlinear optimization problem. The system is designed for minimum weight where the weight includes both the weight of the structure and the weight of the actuators. The weight of an actuator is assumed to be proportional to the bound on the maximum force that it can supply. The design variables include the cross-sectional areas of the structural members and the bounds on the actuator forces. The constraints are imposed on the closed loop frequency distribution and the time to reduce the energy of vibration to a small portion of the initial vibrational energy of the system. The structure is analyzed using a finite element approach. For illustration of the design approach, a truss structure idealized with rod elements is used.     

OPTIMAL STRUCTURAL DESIGN BY ANT COLONY OPTIMIZATION

Ant colony optimization (ACO) is a relatively new heuristic combinatorial optimization algorithm in which the search process is a stochastic procedure that incorporates positive feedback of accumulated information. The positive feedback (;i.e., autocatalysis) facility is a feature of ACO which gives an emergent search procedure such that the (common) problem of algorithm termination at local optima may be avoided and search for a global optimum is possible.

The ACO algorithm is motivated by analogy with natural phenomena, in particular, the ability of a colony of ants to ‘optimize’ their collective endeavours. In this paper the biological background for ACO is explained and its computational implementation is presented in a structural design context. The particular implementation of ACO makes use of a tabu search (TS) local improvement phase to give a computationally enhanced algorithm (ACOTS).

In this paper ACOTS is applied to the optimal structural design, in terms of weight minimization, of a 25-bar space truss. The design variables are the cross-sectional areas of the bars, which take discrete values. Numerical investigation of the 25-bar space truss gave the best (i.e., lowest to-date) minimum weight value. This example provides evidence that ACOTS is a useful and technically viable optimization technique for discrete-variable optimal structural design.