Multi-objective Structural Optimization Research Articles

Adjoint Variable Method for Multi-Objective Sizing and Shape Optimization

With smooth objective functions and constraint conditions, gradient-based methods can be used to solve multi-objective optimization problems efficiently. However, when applied to structural sizing optimization problems, using the Finite Element Method (FEM) and a finite difference scheme to calculate sensitivities can be computationally expensive. The adjoint variable method can be used to reduce computational cost. In order to solve multi-objective structural sizing and shape optimization problems efficiently, this paper proposes using the adjoint variable method. The adjoint variable method efficiently calculates multiple sensitivities for objectives that involve structural responses and cuts down computational cost by reducing the number of sensitivity calculations required per design variable.

Fuzzy multiobjective optimization of mechanical structures

This paper proposes a method that takes imperfections in multiobjective optimization problems into account. The main objective is to determine a feasible design space for fuzzy design variables that respects fuzzy restriction rules and optimizes the defuzzified objective functions. The proposed method uses the Fuzzy Finite Element Method (FFEM) to calculate fuzzy solutions (namely displacements and frequencies) and genetic algorithms for exploring the design space of fuzzy design variables. These elements are explained in detail and then applied to design a drop-tower impactor sled defined with imperfections.

Artificial immune system for multi-objective design optimization of composite structures

We present a new, generic method/model for multi-objective design optimization of laminated composite components using a novel multi-objective optimization algorithm developed on the basis of the artificial immune system (AIS) paradigm. A co-variant of the popular clonal selection principle called as the Objective Switching Clonal Selection Algorithm (OSCSA) has been developed and implemented successfully for the multi-objective design optimization of composites. The problem is formulated with multiple objectives of minimizing weight and the total cost of the composite component to achieve a specified strength. The primary optimization variables are—the number of layers, its stacking sequence (the orientation of the layers) and thickness of each layer. The classical lamination theory is utilized to determine the stresses in the component and the design is evaluated based on three failure criteria; failure-mechanism-based failure criteria, maximum stress failure criteria and the Tsai-Wu failure criteria. The optimization method is validated for a number of different loading configurations—uniaxial, biaxial and bending loads. The design optimization has been carried for both variable stacking sequences as well as fixed standard stacking schemes and a comparative study of the different design configurations evolved has been presented.

LOCAL SEARCH FOR MULTIOBJECTIVE OPTIMIZATION OF STEEL FRAMES

A single-point local search method is presented as a simplification of the multiobjective tabu search. Some improvements are made to reach the Pareto front within small number of function evaluations. The performance of the proposed method is first verified by a small mathematical problem. It is shown that accurate Pareto solutions with good diversity are obtained by using the proposed method. Pareto optimal solutions are next found for a four-span five-story steel building frame. The objective functions are the total structural volume and the compliance under the specified set of loads. It is shown that about 30 Pareto solutions with good accuracy can be found by carrying out structural analysis 200 times. The proposed method has very few problem-dependent parameters. Therefore, the method can be applied to multiobjective optimization of large structures, for which the population-based method is difficult to be applied.

Vector evaluated particle swarm optimization (VEPSO) for multi-objective design optimization of composite structures

We present a generic method/model for multi-objective design optimization of laminated composite components, based on vector evaluated particle swarm optimization (VEPSO) algorithm. VEPSO is a novel, co-evolutionary multi-objective variant of the popular particle swarm optimization algorithm (PSO). In the current work a modified version of VEPSO algorithm for discrete variables has been developed and implemented successfully for the, multi-objective design optimization of composites. The problem is formulated with multiple objectives of minimizing weight and the total cost of the composite component to achieve a specified strength. The primary optimization variables are – the number of layers, its stacking sequence (the orientation of the layers) and thickness of each layer. The classical lamination theory is utilized to determine the stresses in the component and the design is evaluated based on three failure criteria; failure mechanism based failure criteria, Maximum stress failure criteria and the Tsai–Wu failure criteria. The optimization method is validated for a number of different loading configurations – uniaxial, biaxial and bending loads. The design optimization has been carried for both variable stacking sequences, as well fixed standard stacking schemes and a comparative study of the different design configurations evolved has been presented.

Use of Metamodels in the Multi-Objective Optimization of Mechanical Structures with Uncertainties

In this paper, we propose a method which takes into account the propagation of uncertainties in the finite element models in a multi-objective optimization procedure. This method is based on the coupling of the Stochastic Response Surface Method (SRSM) and a genetic algorithm of NSGA type (Non-dominated Sorting Genetic Algorithm). The SRSM is based on the use of Stochastic Finite Element Method (SFEM) via the use of the perturbation method. Thus, we can avoid the use of Monte Carlo simulation, whose cost is prohibitive in the optimization problems, especially when the finite element models are large and with a considerable number of design parameters. The objective of this study is, on the one hand, to quantify efficaciously the effects of these uncertainties on the variability of responses which we wish to optimize, and on the other hand, to calculate solutions which are both optimal and robust resulting from the numerical simulation. At the end of a multi-objective optimization procedure, the space of optimal solutions is generally of a large dimension. The solutions obtained are practically non-exploitable by the designer. To facilitate this interpretation, a study of sensitivity a posteriori can be exploited in order to eliminate the non-significant design parameters. The use of the clusters resulting from the Self-Organizing Maps of Kohonen (SOM) is also suggested for a rational management of the design space. The importance of the methodology that we have used along with suggestions for its performances are highlighted by two numerical examples. The criterion of quality selected consists in obtaining the best compromise: the minimal computing time versus the maximum precision of results.

Optimization of Radar Absorbing Structure Using the Genetic Algorithm

In this study, multi-objective optimization of radar absorbing structure with circuit analog structure using the genetic algorithm was investigated, at the basis of the study on the influence of the size of the circuit analog, the electromagnetic parameter and the thickness of the medium ply on the properties of the microwave-absorbing composite. Based on the concept of Pareto optimality, Sharing and Niche technology was applied in the algorithm(NSGA), and the calculating results converged at the Pareto front dividedly. The study have been showed that the calculating values fit well with that of the experiment, which indicate that this algorithm is proper and has extensive adaptability. The results also showed that introducing circuit analog structure(CAS)into the radar absorbing structure composite design can improve its wave-absorbing properties. So, the radar absorbing structure composite with CAS is a promising radar absorbing structure composite form.

Target-matching test problem for multiobjective topology optimization using genetic algorithms

This paper describes the multiobjective topology optimization of continuum structures solved as a discrete optimization problem using a multiobjective genetic algorithm (GA) with proficient constraint handling. Crucial to the effectiveness of the methodology is the use of a morphological geometry representation that defines valid structural geometries that are inherently free from checkerboard patterns, disconnected segments, or poor connectivity. A graph- theoretic chromosome encoding, together with compatible reproduction operators, helps facilitate the transmission of topological/shape characteristics across generations in the evolutionary process. A multicriterion target-matching problem developed here is a novel test problem, where a predefined target geometry is the known optimum solution, and the good results obtained in solving this problem provide a convincing demonstration and a quantitative measure of how close to the true optimum the solutions achieved by GA methods can be. The methodology is then used to successfully design a path-generating compliant mechanism by solving a multicriterion structural topology optimization problem.

Multi-objective optimization of wind-excited structures

A procedure for the optimization of wind-excited structures is proposed. It is based on the simulated annealing algorithm combined with the dynamic analysis of the response either in the frequency or in the time domain. When the step-by-step dynamic analysis is used, it can handle the case of flexible structures, like masts and lattice towers, whose geometric non-linear effects cannot be neglected. The procedure allows for multiple variables and objectives. The proposed method is used to optimize the configuration of a cable-stayed mast subjected to turbulent wind loading. The results showed that the algorithm is reasonably independent of the first guess configuration and is effective in avoiding local minima. Its wide field of applicability and ease of implementation make the proposed algorithm a powerful design tool for structural engineers.

Multiobjective topology optimization of structures using genetic algorithms with chromosome repairing

In this work, a genetic algorithm (GA) for multiobjective topology optimization of linear elastic structures is developed. Its purpose is to evolve an evenly distributed group of solutions to determine the optimum Pareto set for a given problem. The GA determines a set of solutions to be sorted by its domination properties and a filter is defined to retain the Pareto solutions. As an equality constraint on volume has to be enforced, all chromosomes used in the genetic GA must generate individuals with the same volume value; in the coding adopted, this means that they must preserve the same number of “ones” and, implicitly, the same number of “zeros” along the evolutionary process. It is thus necessary: (1) to define chromosomes satisfying this propriety and (2) to create corresponding crossover and mutation operators which preserve volume. Optimal solutions of each of the single-objective problems are introduced in the initial population to reduce computational effort and a repairing mechanism is developed to increase the number of admissible structures in the populations. Also, as the work of the external loads can be calculated independently for each individual, parallel processing was used in its evaluation. Numerical applications involving two and three objective functions in 2D and two objective functions in 3D are employed as tests for the computational model developed. Moreover, results obtained with and without chromosome repairing are compared.

Multiobjective Evolutionary Structural Optimization Using Combined Static/Dynamic Control Parameters

A thermal protection system (TPS) is responsible for protecting a spacecraft’s components from melting due to high reentry temperatures. In the design of TPS, both maximum thermal stress and minimum natural frequency must be considered due to the combined thermoacoustic environment inherent in high-speed vehicle applications. A multiobjective structural optimization method for the three-dimensional acreage TPS design is developed using an evolutionary structural optimization (ESO) algorithm. The static control parameter used to find the optimum in minimum thermal stress design is modified to address an irregular mode-switching phenomenon, as well as to improve the modal stiffness in dynamic analysis. Two objectives are optimized simultaneously, namely, the maximization of fundamental natural frequency and the minimization of maximum thermal stress. The proposed modified control parameter is demonstrated in the design of a metallic TPS using the method of weighted objectives. The results are then compared with the conventional ESO sensitivity approach. This work concludes by applying the methodology which makes use of both topology and shape optimization in the design of an acreage TPS.

Applications of multi-objective structure optimization

We present applications of multi-objective evolutionary optimization of feed-forward neural networks (NN) to two real world problems, car and face classification. The possibly conflicting requirements on the NNs are speed and classification accuracy, both of which can enhance the embedding systems as a whole. We compare the results to the outcome of a greedy optimization heuristic (magnitude-based pruning) coupled with a multi-objective performance evaluation. For the car classification problem, magnitude-based pruning yields competitive results, whereas for the more difficult face classification, we find that the evolutionary approach to NN design is clearly preferable.

A new hybrid fuzzy-goal programming scheme for multi-objective topological optimization of static and dynamic structures under multiple loading conditions

This work presents a hybrid fuzzy-goal multi-objective programming scheme for topological optimization of continuum structures, in which both static and dynamic loadings are considered. The proposed methodology fortopological optimization first employs a fuzzy-goal programming scheme at the top level for multi-objective problems with static and dynamic objectives. For the static objective with multi-stiffness cases in the fuzzy-goal formulation, a hybrid approach, involving a hierarchical sequence approach or a hierarchical sequence approach coupled with a compromise programming method, is especially suggested for the statically loaded multi-stiffness structure at the sublevel. Concerning dynamic optimization problems of freevibration cases, nonstructural mass, oscillation of the objective function, and repeated eigenvalues are also discussed. Solid Isotropic Material with Penalization density–stiffness interpolation scheme is used to indicate the dependence ofmaterial modulus upon regularized element densities. The globally convergent version of the method of moving asymptotes and the sequential linear programming method areboth employed as optimizers. Several applications have been applied to demonstrate the validation of the presented methodologies.

Multi-objective optimization of tire carcass contours using a systematic aspiration-level adjustment procedure

While forming a basic tire configuration and supporting most static and dynamic loads of automobiles, tire carcass influences major tire performances according to its contour. Among significant tire performances, we in this study intend to improve the automobile maneuverability and the tire durability by optimizing the sidewall carcass contour. In order to effectively maximize these multi-objectives, we refine the conventional satisficing trade-off methods (STOM) which were proposed originally for the multi-objective structural optimization, by introducing a systematic aspiration-level adjustment procedure. According to the systematic procedure, we perform the sidewall contour optimization that ideally distributes the sidewall carcass tension and minimizes strain-energy density at the belt edge. Since the tire analysis is highly nonlinear problem we employ an incremental analysis scheme, together with the finite-difference sensitivity scheme. Through the numerical experiment, we confirmed that the refined multi-objective optimization technique systematically leads to a final optimum sidewall contour, together with the stable and rapid convergence.

Normative design of organizations. II. Organizational structure

For pt.I. see ibid., p. 346-59. This paper presents a multiobjective structural optimization process of designing an organization to execute a specific mission. We provide mathematical formulations for optimization problems arising in Phases II and III of our organizational design process and polynomial algorithms to solve the corresponding problems. Our organizational design methodology applies specific optimization techniques at different phases of the design, efficiently matching the structure of a mission (in particular, the one defined by the courses of action obtained from mission planning) to that of an organization. It allows an analyst to obtain an acceptable tradeoff among multiple mission and design objectives, as well as between computational complexity and solution efficiency (desired degree of suboptimality).

Multi-objective optimization of skeletal structures under static and seismic loading conditions

Almost every real world problem involves simultaneous optimization of several incommensurable and often competing objectives which constitutes a multi-objective optimization problem. In multi-objective optimization problems the optimal solution is not unique as in single-objective optimization problems. This paper is concerned with large-scale structural optimization of skeletal structures such as space frames and trusses, under static and/or seismic loading conditions with multiple objectives. Combinatorial optimization methods and in particular algorithms based on evolution strategies are implemented for the solution of this type of problems. In treating seismic loading conditions a number of accelerograms are produced from the elastic design response spectrum of the region. These accelerograms constitute the multiple loading conditions under which the structures are optimally designed. This approach for treating seismic loading is compared with an approximate design approach, based on simplifications adopted by the seismic codes, in the framework of multi-objective optimization.

Genetic Algorithm Development for Multiobjective Optimization of Structures

Genetic algorithms (GAs) have the characteristic of maintaining a population of solutions and can search in a parallel manner for many nondominated solutions. These features coincide with the requirement of seeking a Pareto optimal set in a multiobjective (multicriteria, vector) optimization problem. The rationale for multiobjective optimization via GAs is that at each generation, the fitness of each individual is defined according to its nondominated property. Because nondominated individuals are assigned the highest fitness values, the convergence of a population will go to the nondominated zone: the Pareto optimal set. Based on this concept, a Pareto GA, whose goal is to locate the Pareto optimal set of a multiobjective optimization problem, is developed. In this GA, to avoid missing Pareto optimal points during evolutionary processes, a new concept called Pareto-set filter is adopted. At each generation, the points of rank 1 are put into the filter and undergo a nondominated check. In addition, a niche technique is provided to prevent genetic drift in population evolution. This technique sets a replacement rule for reproduction procedures. For a constrained optimization problem, a revised penalty function method is introduced to transfer a constrained problem into a nonconstralned one. The transferred function of a point contains information on a point's status (feasible or infeasible), position in a search region, and distance to the Pareto optimal set Two multiobjective optimization examples, a 25-bar space truss optimal design (objectives: structural weight and virtual work, constraints: stresses) and a four-bar pyramid truss with control system (objectives: structural weight and control effort, constraints: closed-loop frequencies) are provided to demonstrate analysis procedures of the proposed Pareto GA.

Genetic algorithm development for multiobjective optimization of structures

Genetic algorithm development for multiobjective optimization of structures

96/05383 Waste disposal and reprocessing by the Orleans District Authority

A multi-objective structural optimization of integrated thermal desalination and thermal compression systems is performed, whereby the performance ratio of the structures is maximized while the specific area requirements are minimized. With the aid of the superstructure developed in part I, herein the thermal compression of vapor streams produced in intermediate multi-effect distillation (MED) effects as opposed to the common practice of compressing vapors produced in the last effect, is examined. The study concludes that intermediate vapor compression results in significant reductions in area requirements, as well as significant increases in maximum distillate production capacities. Moreover, the optimal location of vapor extraction is heavily dependent on the exact distillate production requirement in question. Two novel configuration forms are informed by the optimization. The first is an integrated MED–TVC + MED + MSF system, while the second is an integrated MED–TVC + MSF system.

Multiobjective Shape Optimization of Linear Elastic Structures Considering Multiple Loading Conditions : Dealing with Mean Compliance Minimization problems

We describe numerical analysis methods for multiobjective shape optimization of linear elastic structures. As an example, we consider a multiloading mean compliance minimization problem with a volume constraint. The methods presented here are based on the traction method, in which the speed field representing the domain variation is analyzed. A weighted lp norm method with four types of norm is employed to scalarize the multiobjective functionals. The shape gradient functions for each scalarized objective functional are obtained using the Lagrange multiplier method. A general-purpose finite element code is used to perform the numerical analyses. Numerical analysis results for a multiply connected plate problem and a solid structure problem under multiloading conditions are presented to demonstrate the validity of the traction method in obtaining Pareto optimal solutions.