Design of optimum structures

Design sensitivity analysis plays a central role in structural optimization, since virtually all optimization methods require computation of derivatives of structural response quantities with respect to design variables. Three fundamentally different approaches to design sensitivity analysis are presented. These have been used extensively in the structural optimization literature. They are the virtual load method, the state space method, and the design space method. An analysis of these methods indicates that the state space and design space methods are more general than the virtual load method. Moreover, the virtual load method, when applicable, is a special case of the state space method. Any one of these procedures may be incorporated into an optimality criterion or a mathematical programming method for structural optimization.

Methods of Design Sensitivity Analysis in Structural Optimization

Approximation concepts and dual method algorithms are combined to create a method for minimum weight design of structural systems. Approximation concepts convert the basic mathematical programming statement of the structural synthesis problem into a sequence of explicit primal problems of separable form. These problems are solved by constructing explicit dual functions, which are maximized subject to nonnegativity constraints on the dual variables. It is shown that the joining together of approximation concepts and dual methods can be viewed as a generalized optimality criteria approach. The dual method is successfully extended to deal with pure discrete and mixed continuous-discrete design variable problems. The power of the method presented is illustrated with numerical results for example problems, including a metallic swept wing and a thin delta wing with fiber composite skins.

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Dual methods and approximation concepts in structural synthesis

The type of mathematical modeling selected for the optimum design problems of steel skeletal frames affects the size and mathematical complexity of the programming problem obtained. Survey on the structural optimization literature reveals that there are basically two types of design optimization formulation. In the first type only cross sectional properties of frame members are taken as design variables. In such formulation when the values of design variables change during design cycles, it becomes necessary to analyze the structure and update the response of steel frame to the external loading. Structural analysis in this type is a complementary part of the design process. In the second type joint coordinates are also treated as design variables in addition to the cross sectional properties of members. Such formulation eliminates the necessity of carrying out structural analysis in every design cycle. The values of the joint displacements are determined by the optimization techniques in addition to cross sectional properties. The structural optimization literature contains structural design algorithms that make use of both type of formulation. In this study a review is carried out on mathematical and metaheuristic algorithms where the effect of the mathematical modeling on the efficiency of these algorithms is discussed.

https://downloads.hindawi.com/journals/mpe/2013/271031.pdf

Mathematical and Metaheuristic Applications in Design Optimization of Steel Frame Structures: An Extensive Review

The purpose of this paper is to present refinements in a steepest descent algorithm for optimal design of structures and numerical experience that demonstrates its numerical efficiency. The algorithm is based on a state space optimization technique that was initially developed and applied to optimal control problems. Design constraints are divided into four distinct subsets, the special characteristics of which are exploited to improve computational efficiency of the algorithm. Some further improvements in structural analysis, design sensitivity analysis, and constrained steepest descent programming calculations are presented. Optimum designs for three test problems are obtained and convergence rates are compared with results presented in the literature.

Efficient optimal design of structures by generalized steepest descent programming

Optimum structural design is formulated as a problem in nonlinear mathematical programming, i.e., a problem of selecting n design variables subject to m > n constraints such that an objective function is minimized. The design variables correspond to the geometrical or mechanical properties of the structural system, the constraints correspond to safety requirements such as limitations on stresses and deflections, and the objective function corresponds to weight or cost. The gradient projection method of nonlinear programming is described and an algorithm is presented for high-speed computation. Examples are presented in which the method is used to obtain least-weight designs of typical indeterminate rigid frames made of standard steel wide flange shapes. The examples illustrate that the programming approach affords a sound mathematical basis on which to develop rational methods of structural design.

Alfredo H.-S. Aug

Papers

Structural Optimization by Nonlinear Programming