Extended capability for automated design of frame-stiffened,submersible, cylindrical shells

This paper presents a procedure and computer program for the minimum weight design of circular, cylindrical, ‘T’ frame (ring) reinforced, submersible shells where all metal thicknesses may be confined to specified gage thickness values. Using the designer specified parameters defining shell radius shell length, eccentricity, operating depth, design factors of safety, construction materials properties and when used, the specified gage thickness values, the program will generate those values of skin thickness stifiener web and flange thicknesses, stiffener web depth and flange width, and if desired, stiffener spacing that will produce the smallest shell weight to liquid weight displaced ratio.Experience with the program has demonstrated that there is usually little weight penalty associated with the use of discrete metal thickness values when the stiffener spacing can be optimized. This weight penalty can, however, be significant where the number of stiffeners is held fixed.     

Mixed-integer second-order cone programming for global optimization of compliance of frame structure with discrete design variables

Frame structures are extensively used in mechanical, civil, and aerospace engineering. Besides generating reasonable designs of frame structures themselves, frame topology optimization may serve as a tool providing us with conceptual designs of diverse engineering structures. Due to its nonconvexity, however, most of existing approaches to frame topology optimization are local optimization methods based on nonlinear programming with continuous design variables or (meta)heuristics allowing some discrete design variables. Presented in this paper is a new global optimization approach to the frame topology optimization with discrete design variables. It is shown that the compliance minimization problem with predetermined candidate cross-sections can be formulated as a mixed-integer second-order cone programming problem. The global optimal solution is then computed with an existing solver based on a branch-and-cut algorithm. Numerical experiments are performed to examine computational efficiency of the proposed approach.     

Optimal design of steel structures using standard sections

The problem of optimal structural design having linked discrete variables is addressed. For such applications, when a discrete value for a variable is selected, values for other variables linked to it must also be selected from a table. The design of steel structures using available sections is a major application area of such problems. Three strategies that combine a continuous variable optimization method with a genetic algorithm, simulated annealing, and branch and bound method are presented and implemented into a computer program for their numerical evaluation. Three structural design problems are solved to study the performance of the proposed methods. CPU times for solution of the problems with discrete variables are large. Strategies are suggested to reduce these times.     

On the (non-)optimality of Michell structures

Optimal analytical Michell frame structures have been extensively used as benchmark examples in topology optimization, including truss, frame, homogenization, density and level-set based approaches. However, as we will point out, partly the interpretation of Michell’s structural continua as discrete frame structures is not accurate and partly, it turns out that limiting structural topology to frame-like structures is a rather severe design restriction and results in structures that are quite far from being stiffness optimal. The paper discusses the interpretation of Michell’s theory in the context of numerical topology optimization and compares various topology optimization results obtained with the frame restriction to cases with no design restrictions. For all examples considered, the true stiffness optimal structures are composed of sheets (2D) or closed-walled shell structures (3D) with variable thickness. For optimization problems with one load case, numerical results in two and three dimensions indicate that stiffness can be increased by up to 80 % when dropping the frame restriction. For simple loading situations, studies based on optimal microstructures reveal theoretical gains of +200 %. It is also demonstrated how too coarse design discretizations in 3D can result in unintended restrictions on the design freedom and achievable compliance.     

Discrete variable optimization of plate structures using dual methods

This study presents an efficient method for optimum design of plate and shell structures, when the design variables are continuous or discrete. Both sizing and shape design variables are considered. First the structural responses, such as element forces, are approximated in terms of some intermediate variables. By substituting these approximate relations into the original design problem, an explicit nonlinear approximate design task with high quality approximation is achieved. This problem with continuous variables can be solved very efficiently by means of numerical optimization techniques, the results of which are then used for discrete variable optimization. Now, the approximate problem is converted into a sequence of second level approximation problems of separable form, each of which is solved by a dual strategy with discrete design variables. The approach is efficient in terms of the number of required structural analyses, as well as the overall computational cost of optimization. Examples are offered and compared with other methods to demonstrate the features of the proposed method.     

The optimum shape of cooling towers

Numerous computer optimization techniques have been developed and applied primarily to the design of structures composed of discrete elements. Continuous surface structures have been optimized primarily by methods based upon the differential or integral calculus (e.g. the calculus of variations). However, the determination of the optimal shape of continuous surface structures can also be approached by algebraic methods more suitable for digital computation. If the coordinates of the middle surface of a shell are expressed by a finite polynomial series, an optimization problem in a finite set of discrete variables results. In the present work, this method is applied to a particular example of a shell of revolution: a natural draft cooling tower. A simple preliminary design model is formulated in order to evaluate the potential savings due to numerical optimization, and the resulting nonlinear programming problem is solved by iterated linear programming. The results indicate that the method is feasible and that significant savings might be attainable by computerized shape optimization.     

A pseudo-discrete rounding method for structural optimization

A new heuristic method aimed at efficiently solving the mixed-discrete nonlinear programming (MDNLP) problem in structural optimization, and denotedselective dynamic rounding, is presented. The method is based on the sequential rounding of a continuous solution and is in its current form used for the optimal discrete sizing design of truss structures. A simple criterion based on discrete variable proximity is proposed for selecting the sequence in which variables are to be rounded, and allowance is made for both upward and downward rounding. While efficient in terms of the required number of function evaluations, the method is also effective in obtaining a low discrete approximation to the global optimum. Numerical results are presented to illustrate the effectiveness and efficiency of the method.     

Structural optimization with several discrete design variables per part by outer approximation

The article proposes an optimal design approach to minimize the mass of load carrying structures with discrete design variables. The design variables are chosen from catalogues, and several variables are assigned to each part of the structure. This allows for more design freedom than only choosing parts from a catalogue. The problems are modelled as mixed 0–1 nonlinear problems with nonconvex continuous relaxations. An algorithm based on outer approximation is proposed to find optimized designs. The capabilities of the approach are demonstrated by optimal design of a space frame (jacket) structure for offshore wind turbines, with requirements on natural frequencies, strength, and fatigue lifetime.     

Mixed-integer linear programming approach for global discrete sizing optimization of frame structures

This paper focuses on discrete sizing optimization of frame structures using commercial profile catalogs. The optimization problem is formulated as a mixed-integer linear programming (MILP) problem by including the equations of structural analysis as constraints. The internal forces of the members are taken as continuous state variables. Binary variables are used for choosing the member profiles from a catalog. Both the displacement and stress constraints are formulated such that for each member limit values can be imposed at predefined locations along the member. A valuable feature of the formulation, lacking in most contemporary approaches, is that global optimality of the solution is guaranteed by solving the MILP using branch-and-bound techniques. The method is applied to three design problems: a portal frame, a two-story frame with three load cases and a multiple-bay multiple-story frame. Performance profiles are determined to compare the MILP reformulation method with a genetic algorithm.     

A genetic algorithm with memory for mixed discrete-continuous design optimization

This paper describes a new approach for reducing the number of the fitness function evaluations required by a genetic algorithm (GA) for optimization problems with mixed continuous and discrete design variables. The proposed additions to the GA make the search more effective and rapidly improve the fitness value from generation to generation. The additions involve memory as a function of both discrete and continuous design variables, multivariate approximation of the fitness function in terms of several continuous design variables, and localized search based on the multivariate approximation. The approximation is demonstrated for the minimum weight design of a composite cylindrical shell with grid stiffeners.     

An interactive approach based on a discrete differential evolution algorithm for a class of integer bilevel programming problems

This paper presents an interactive approach based on a discrete differential evolution algorithm to solve a class of integer bilevel programming problems, in which integer decision variables are controlled by an upper-level decision maker and real-value or continuous decision variables are controlled by a lower-level decision maker. Using the Karush--Kuhn–Tucker optimality conditions in the lower-level programming, the original discrete bilevel formulation can be converted into a discrete single-level nonlinear programming problem with the complementarity constraints, and then the smoothing technique is applied to deal with the complementarity constraints. Finally, a discrete single-level nonlinear programming problem is obtained, and solved by an interactive approach. In each iteration, for each given upper-level discrete variable, a system of nonlinear equations including the lower-level variables and Lagrange multipliers is solved first, and then a discrete nonlinear programming problem only with inequality constraints is handled by using a discrete differential evolution algorithm. Simulation results show the effectiveness of the proposed approach.     

基于MSC Nastran的离散变量优化算法的实现

实际工程中存在大量的离散变量优化问题,基于MSC Nastran优化框架实现新的离散变量算法,有利于新算法本身的推广应用和解决大规模的实际复杂工程问题.通过修改MSC Nastran输入文件的方法实现离散变量的优化算法——GSFP算法.GSFP是基于广义形函数的离散变量优化算法,它将离散变量优化问题转化成连续变量优化问题,通过惩罚等措施使得最优设计结果最终收敛到离散解,该方法能够解决大规模的实际离散变量优化问题.最后以桁架截面选型优化为应用背景,给出GSFP算法实现的基本原理和方法.     

Embedding linear programming in multi objective genetic algorithms for reducing the size of the search space with application to leakage minimization in water distribution networks

This paper shows how embedding a local search algorithm, such as the iterated linear programming (LP), in the multi-objective genetic algorithms (MOGAs) can lead to a reduction in the search space and then to the improvement of the computational efficiency of the MOGAs. In fact, when the optimization problem features both continuous real variables and discrete integer variables, the search space can be subdivided into two sub-spaces, related to the two kinds of variables respectively. The problem can then be structured in such a way that MOGAs can be used for the search within the sub-space of the discrete integer variables. For each solution proposed by the MOGAs, the iterated LP can be used for the search within the sub-space of the continuous real variables. An example of this hybrid algorithm is provided herein as far as water distribution networks are concerned. In particular, the problem of the optimal location of control valves for leakage attenuation is considered. In this framework, the MOGA NSGAII is used to search for the optimal valve locations and for the identification of the isolation valves which have to be closed in the network in order to improve the effectiveness of the control valves whereas the iterated linear programming is used to search for the optimal settings of the control valves. The application to two case studies clearly proves the reduction in the MOGA search space size to render the hybrid algorithm more efficient than the MOGA without iterated linear programming embedded.     

Minimum cost design of a welded orthogonally stiffened cylindrical shell

In this study the optimal design of a cylindrical orthogonally stiffened shell member of an offshore fixed platform truss, loaded by axial compression and external pressure, is investigated. Ring stiffeners of welded box section and stringers of halved rolled I-section are used. The design variables considered in the optimization are the shell thickness as well as the dimensions and numbers of stiffeners. The design constraints relate to the shell, panel ring and panel stringer buckling, as well as manufacturing limitations. The cost function includes the cost of material, forming of plate elements into cylindrical shape, welding and painting. In the optimization a number of relatively new mathematical optimization methods (leap-frog - LFOPC, Dynamic-Q, ETOPC, and particle swarm - PSO) are used, in order to ensure confidence that the finally computed optimum design is accurately determined, and indeed corresponds to a global minimum. The continuous optimization procedures are adapted to allow for discrete values of the design variables to be used in the final manufacturing of the truss member. A comparison of the computed optimum costs of the stiffened and un-stiffened assemblies, shows that significant cost savings can be achieved by orthogonal stiffening, since the latter allows for considerable reduction of the shell thickness, which results in large material and manufacturing cost savings.     

Optimum design of trusses with discrete sizing and shape variables

The objective here is to present a method for optimizing truss structures with discrete design variables. The design variables are considered to be sizing variables as well as coordinates of joints. Both types of variables can be discrete simultaneously. Mixed continuous-discrete variables can also be considered. To increase the efficiency of the method, the structural responses, such as forces and displacements are approximated in each design cycle. The approximation of responses is carried out with respect to the design variables or their reciprocals. By substituting these approximate functions of the responses into the original design problem, an explicit high quality approximation is achieved, the solution of which does not require the detailed finite element analysis of the structure in each sub-optimization iteration. First it is assumed that all the design variables are continuous and a continuous variable optimization is performed. With the results of this step, the branch and bound method is employed on the same approximate problem to achieve a discrete solution. The numerical results indicate that the method is efficient and robust in terms of the required number of structural analyses. Several examples are presented to show the efficiency of the method.     

离散非线性规划问题的改进遗传算法

针对实际离散非线性规划问题,分析了离散与连续变量优化问题和求解方法的不同及特性．根据离散变量与遗传算法的特点,将单纯形搜索与算术交叉思想相结合,提出离散单纯形交叉算子以提高遗传算法的局部寻优能力,将种群逐步向离散极值点进行引导,实现算法的快速离散寻优．同时,设计了离散变异算子,使遗传算子真正在离散空同中进行搜索．基于梯度下降思想提出离散修复算子,提高算法对非线性约束的处理能力．实际离散非线性规划问题的应用研究验证了方法的有效性．     

Application of mathematical programming in the design of optimal control systems†

The problem of applying the various computational methods of mathematical programming in the design of an optimal control system is discussed. A general case of non-linear, non-autonomous, state equations, subject to inequality constraints on both state and control variables, is considered. Both continuous and discrete time systems are investigated. In case of discrete time systems, the sampling intervals are assumed generally unequal and aperiodic, with inequality constraints imposed upon them.

Systems like these impose considerable computational difficulties when treated by the maximum principle or dynamie programming. Using mathematical programming, one may simplify a wide class of those computational problems.

Several examples of applying mathematical programming to particular control problems are presented.     

Multi-database exploration of large design spaces in the framework of cascade evolutionary structural sizing optimization

In discrete sizing optimization of truss and frame structures the design variables take values from databases, which are usually populated with a relatively small number of cross-section types and sizes. The aim of this work is to allow the use of large-size databases in discrete structural sizing optimization problems, in order to enrich the set of design variable options and increase the potential of achieving high-quality optimal designs. For this purpose, the concept of coarse database is introduced, according to which smaller-size versions of an appropriately ordered large database can be constructed. This concept is combined with the idea of cascading, which allows a single optimization problem to be tackled with a number of autonomous optimization stages. Under this context, several coarse versions of the same full-size database are formed, in order to utilize a different database in each cascade stage executed with an evolutionary optimization algorithm. The first optimization stages of the resulting multi-database cascade procedure make use of the coarsest database versions available and serve the purpose of basic design space exploration. The last stages exploit finer databases (including the original full-size database) and aim in fine tuning the achieved optimal solution. Based on the reported numerical results, multi-database cascading proves to be an effective tool for the handling of large databases and corresponding extensive design spaces in the framework of discrete structural sizing optimization applications.     

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.     

‘Truncate,replicate, sample’: A method for creating integer weights for spatial microsimulation

Iterative proportional fitting (IPF) is a widely used method for spatial microsimulation. The technique results in non-integer weights for individual rows of data. This is problematic for certain applications and has led many researchers to favour combinatorial optimisation approaches such as simulated annealing. An alternative to this is ‘integerisation’ of IPF weights: the translation of the continuous weight variable into a discrete number of unique or ‘cloned’ individuals. We describe four existing methods of integerisation and present a new one. Our method – ‘truncate, replicate, sample’ (TRS) – recognises that IPF weights consist of both ‘replication weights’ and ‘conventional weights’, the effects of which need to be separated. The procedure consists of three steps: (1) separate replication and conventional weights by truncation; (2) replication of individuals with positive integer weights; and (3) probabilistic sampling. The results, which are reproducible using supplementary code and data published alongside this paper, show that TRS is fast, and more accurate than alternative approaches to integerisation.