A two-level delimitative and combinatorial algorithm for discrete optimization of structures

In this paper, a two-level delimitative and combinatorial algorithm for a kind of (0,1,2) programming is proposed and applied to discrete optimization of structures. The algorithm generates all combinations in a certain order of magnitude of the function of the objective by using a two-level generating method and eliminates the majority of infeasible or nonoptimum combinations by using a two-level delimitative algorithm, so that computational efficiency is greater. Additionally, a (0,1,2) programming model of discrete structural optimization is established and the local optimum solution can be obtained by using this algorithm, thus it provides a method to judge whether or not the approximate optimum solution obtained by the heuristic algorithm is a local optimum solution.     

An application of relative difference quotient algorithm to topology optimization of truss structures with discrete variables

In this paper, the definitions of cross-sectional variable and topological variable are advanced, and a mathematical model of topology optimization of truss structures with discrete variables including two kinds of variables is developed. The model has considered the coupling relations between cross-sectional variables and topological variables, so that is reflects the innate characteristics of topology optimization as a combinatorial optimization problem. Moreover, problems such as limit stress and singular solution of structural optimization can be overcome by using this model. The model of topology optimization of truss structures with discrete variables including two kinds of variables is solved directly by using the relative difference quotient algorithm. The computational results are satisfactory and some new topologies and better solutions are obtained.     

Chaotic imperialist competitive algorithm for optimum design of truss structures

The imperialist competitive algorithm is a new socio-politically motivated optimization algorithm which recently is applied for structural problems. This paper utilizes the idea of using chaotic systems instead of random processes in the imperialist competitive algorithm. The resulting method is called chaotic imperialist competitive algorithm (CICA) in which chaotic maps are utilized to improve the movement step of the algorithm. Some well-studied truss structures are chosen to evaluate the efficiency of the new algorithm.     

Discrete optimum design of truss structures using artificial bee colony algorithm

Over the past few years, swarm intelligence based optimization techniques such as ant colony optimization and particle swarm optimization have received considerable attention from engineering researchers and practitioners. These algorithms have been used in the solution of various engineering problems. Recently, a relatively new swarm based optimization algorithm called the Artificial Bee Colony (ABC) algorithm has begun to attract interest from researchers to solve optimization problems. The aim of this study is to present an optimization algorithm based on the ABC algorithm for the discrete optimum design of truss structures. The ABC algorithm is a meta-heuristic optimization technique that mimics the process of food foraging of honeybees. Originally the ABC algorithm was developed for continuous function optimization problems. This paper describes the modifications made to the ABC algorithm in order to solve discrete optimization problems and to improve the algorithm’s performance. In order to demonstrate the effectiveness of the modified algorithm, four structural problems with up to 582 truss members and 29 design variables were solved and the results were compared with those obtained using other well-known meta-heuristic search techniques. The results demonstrate that the ABC algorithm is very effective and robust for the discrete optimization designs of truss structural problems.     

A gradient-based approach for discrete optimum design

In this paper, structural optimization with discrete variables in engineering design is modeled and investigated as a zero-one programming problem. The zero-one programming problem is first reformulated as an equivalent continuous problem through replacing the zero-one constraints by complementarity constraints, then as an equivalent ordinary nonlinear programming problem with the help of the NCP(nonlinear complementarity problem) function. Furthermore, an aggregate function method is introduced with the aim to simplify computation in the following augmented Lagrangian method. Numerical experiments on discrete optimum design showed the proposed method is promising.     

Analysis and optimum design of fibre-reinforced composite structures

The optimal design of a carbon-fibre-reinforced plastic (CFRP) sandwich-like structure with aluminium (Al) webs is addressed. The material parameters are determined using tensile tests, whereafter the results of an analytical model, a numerical model and an experimental setup are compared. The analytical and numerical approximations are then used to optimize the structure in a multi-algorithm approach for minimum cost and maximum stiffness. The selected algorithm and approximation are motivated by their accuracy and computational efficiency. The CFRP plates are optimized with respect to ply arrangement, while the complete sandwich-like structure is optimized with respect to the combination of manufacturing and material cost. Design constraints on maximum deflection of the total structure, buckling of the CFRP composite plates, buckling of the Al webs, stress in the composite plates and stress in the Al stiffeners are included in the formulation. For the different phases in the optimization process, we use the recently proposed particle swarm optimization algorithm, a dynamic search technique and a continuous-discrete optimization technique .     

An approach to solving discrete vector optimization problems over a combinatorial set of permutations

Complex discrete multicriteria problems over a combinatorial set of permutations are analyzed. Some properties of an admissible domain for a combinatorial multicriteria problem embedded into an arithmetic Euclidian space are considered. Optimality conditions are obtained for different types of effective solutions. A new approach to solving the problems formulated is constructed and substantiated. This work was supported by the Fundamental Research Fund of Ukraine (project Φ251/094). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 158–172, May–June 2008.     

Firefly-inspired algorithm for discrete optimization problems: An application to manufacturing cell formation

The canonical firefly algorithm is basically developed for continuous optimization problems. However, lots of practical problems are formulated as discrete optimization problems. The main purpose of this paper is to present the discrete firefly algorithm (DFA) to solve discrete optimization problems. In the DFA, we define a firefly's position in terms of changes of probabilities that will be in one state or the other. Then by using this metaheuristic algorithm, the manufacturing cell formation problem is solved. To illustrate the behavior of the proposed model and verify the performance of the algorithm, we introduce a number of numerical examples to illustrate the use of the foregoing algorithm. The performance evaluation shows the effectiveness of the DFA. The proposed metaheuristic algorithm should thus be useful to both researchers and practitioners.     

An improved algorithm for the prediction of optimum and suboptimum folding structures of long single-stranded RNA

We have previously reported an algorithm (Yamamoto and Yoshikura, 1985) for the prediction of optimum and suboptimum RNA folding structures. The calculation data was presented as an 'information map'. However, the result was affected by the starting point of calculation. In this paper, we have improved the method so that the result will not be affected by the starting point. In addition, we present a method of converting the information map into a set of pictures of optimal and suboptimal molecular structures.     

Fabrication aspects in the optimum design of welded structures

A brief survey of selected literature shows that fabrication aspects play an important role in the economic design of welded structures. Fabrication costs and production requirements should be considered in the optimum design procedure.In a detailed numerical example some optimal solutions are determined for a simple box beam having transverse diaphragms and carrying a fluctuating load. In the objective function the material and fabrication costs are taken into account. The cost function formula enables the investigation of structural solutions for wide ranges of cost factors.Constraints on fatigue strength, local buckling of plate elements, shear stress and deflection are considered. Based on the active constraints, the cost function may be expressed as a function of the beam height as the main variable. The minimum cross-sectional area, volume and cost designs are calculated.Concerning the fillet welds of the transverse diaphragms two solutions are considered: (1) without post treatment with a lower fatigue strength; (2) with post toe burr grinding with improved fatigue strength. In the second case the cost of post treatment is also included in the cost function.A comparison of the above solutions shows that the post treatment of transverse fillet welds may be economic despite of the additional cost.Notation A cross-sectional area - A _f cross-sectional area of a flange - a _w fillet weld size - b beam width - c ₁ , c₂ distances of diaphragms from the support - E modulus of elasticity - G = V mass - h web height - I _x moment of inertia - K total cost - K _m , K _f material and fabrication cost, respectively - k _m , k _f material and fabrication cost factor, respectively - L span length - L _w weld length - M bending moment - N number of cycles - n number of diaphragms - n ₁ number of diaphragms with post treated welds - p intensity of normal load - Q shear force - S _x statical moment - T time - t _f , t _w /2 flange and web thickness, respectively - V volume - W _x section modulus - W ₀ required section modulus - deflection - limiting plate slenderness of webs - difficulty factor - _f limiting plate slenderness of the compression flange - = k _f /k_m ratio of cost factors - number of structural elements - v Poisson's ratio - density - normal stress - _f fatigue strength - shear stress - _f fatigue strength in shear - _w shear stress in welds     

An algorithm for the prediction of search trajectory in nonlinear programming problems and optimum design

A simple modification of some methods of mathematical (nonlinear) programming is suggested (Newton's method and the steepest descent method are taken as examples). The modification is made in order to reduce the number of steps for some sequential search methods. The reduction is achieved by extrapolation of the results obtained by the previous search steps. The computation of prognosis points is proposed instead of the large amount of calculation necessary to do thek-th step of the approach process. The extrapolation formulae are obtained by using elements of the random numbers theory. Results of computational tests and solutions of optimum design problems for a frame and a shell demonstrate the efficiency of the proposed method.     

Decoupled approach to integrated optimum design of structures and robust control systems

Integrated optimum design of structures and control systems is studied by using H ₂ and robust control formulations. It is derived that conventional simultaneous optimization approach by using these robust control laws can be approximated by a decoupled optimization approach in which the structures are optimized by shaping the structural singular values and then the controllers can be designed, namely, decoupled, sequential or successive design approach. It is shown that the proposed decoupled optimization approach can be used to design optimum robust structures and has certain advantages over the conventional simultaneous optimization procedures such as it avoids the drawbacks of pure robust control laws and faster, especially if the number of degrees of freedom (DOF) of the associated structure is large. The bounds for achievable robustness measures are also obtained. Following, simultaneous and decoupled optimization approaches are applied to active control of two structures. The optimization results are presented, and it is concluded that the proposed decoupled optimization approach yields the achieved global minimum much faster than the simultaneous optimization approach.     

Reconstruction of polycrystalline structures: a new application of combinatorial optimization

Many known materials possess polycrystalline structure. The images produced by plane cuts through such structures are polygonal complexes. The problem of finding the edges, when only the vertices of a given polygonal complex are known, is considered. A combinatorial optimization model is proposed whose solution yields an approximation of the complex. The problem itself is solved using simulated annealing. Encouraging first experiments are presented.     

间歇精馏常规设计和优化设计的模型及算法

建立了板式间歇精馏塔在恒馏出液组成操作状况下常规设计及优化设计的数学模型。常规设计模型用数值方法编程求解,对二元理想及非理想溶液均适用。优化设计模型以间歇精馏系统年效益最大为优化目标,用菲波那契法求解单变量优化问题,用复合形法求解多变量优化问题。模型同时考虑对整个间歇精馏系统(包括塔主体、塔顶冷凝器及塔底再沸器)进行优化,更符合工程实际情况。求解模型可得到间歇精馏过程最优的一系列设计和操作参数(如理论板数,塔径,操作回流比,塔釜蒸发量,釜残液组成,冷凝器传热面积及冷却水出口温度,再沸器传热面积及加热蒸汽温度等)。算例表明,对恒馏出液组成间歇精馏单变量及多变量优化设计比常规设计分别提高年效益2．6％和18．9％。     

Efficient optimum design of structures—Program DDDU

An efficient optimization algorithm is developed for engineering structures subject to multiple constraints. This highly non-linear and implicit problem is reduced to a combination of a sequence of quasi-linear constraints and explicit problems of the statically determined structures. The method is based on the Kuhn—Tucker necessary conditions for optimality associated with a simple quadratic programme designed simultaneously to determine the Lagrange multipliers and to delete non-active constraints. A number of examples including trusses and wing structures show that the method is efficient when compared with other competing techniques.     

Applied soft computing for optimum design of structures

In this study a critical assessment of three metaheuristic optimization algorithms, namely differential evolution, harmony search and particle swarm optimization, is performed with reference to their efficiency and robustness for the optimum design of real-world structures. Furthermore, a neural network based prediction scheme of the structural response, required to assess the quality of each candidate design during the optimization procedure, is proposed. The proposed methodology is applied to an overhead crane structure using different finite element simulations corresponding to a solid discretization as well as mixed discretizations with shell-solid and beam-solid elements. The number of degrees of freedom (dof) resulted for the simulation of the structural response varies in the range of 60,000 to 1,400,000 dof leading to highly computational intensive problems.     

A revised discrete Lagrangian-based search algorithm for the optimal design of skeletal structures using available sections

Issues relating to the application of the discrete Lagrangian method (DLM) to the discrete sizing optimal design of skeletal structures are addressed. The resultant structure, whether truss or rigid frame, is subjected to stress and displacement constraints under multiple load cases. The members’ sections are selected from an available set of profiles. A table that contains sectional properties for all the available profiles is used directly in structural optimization. Each profile in the table is assigned by a unique profile number, which is used as the integer design variable for each of the structural members. It is proposed that we use a revised DLM search algorithm with static weighting to design trusses and rigid frames for minimum weight. Five examples are used to demonstrate the feasibility of the method. It is shown that, for monotonic as well as nonmonotonic constraint functions, the DLM is effective and robust for the discrete sizing design of skeletal structures.     

An optimum time scale for discrete Laguerre network

Any discrete-time stable transfer function can be expressed by a discrete-time Laguerre series with a chosen time scale. An optimum time scale such that an index is minimized is derived. This index ensures that the coefficients of higher-order Laguerre functions go toward zero quickly. The solution derived requires the knowledge of the impulse response of the discrete plant. Cases of first-order plants, second-order underdamped plants, and plants with multiunit delay are also discussed     

Harmony search based algorithm for the optimum design of grillage systems to LRFD-AISC

Harmony search based optimum design method is presented for the grillage systems. This numerical technique imitates the musical performance process that takes place when a musician searches for a better state of harmony. Jazz improvisation seeks to find musically pleasing harmony similar to the optimum design process which seeks to find the optimum solution. The design algorithm considers the serviceability and ultimate strength constraints which are implemented from Load and Resistance Factor Design—American Institute of Steel Construction (LRFD-AISC). It selects the appropriate W-sections for the transverse and longitudinal beams of the grillage system out of 272 discrete W-section designations given in LRFD-AISC. This selection is carried out such that the design limitations described in LRFD-AISC are satisfied and the weight of the system is the minimum. Many design examples are considered to demonstrate the efficiency of the algorithm presented.