تعریف محدودیت حرکت در بهینه سازی ساختاری با برنامه ریزی خطی پی در پی. قسمت دوم: نمونه های عددی

A variety of numerical methods have been proposed in literature in purpose to deal with the complexity and non-linearity of structural optimization problems. In practical design, sequential linear programming (SLP) is very popular because of its inherent simplicity and because linear solvers (e.g. Simplex) are easily available. However, SLP performance is sensitive to the definition of proper move limits for the design variables which task itself often involves considerable heuristics. This research presents a new SLP algorithm (LESLP) that implements an advanced technique for defining the move limits. The linearization error sequential linear programming (LESLP) algorithm is formulated so to overcome the traditional limitations of the SLP method. In a companion paper [Comput. Struct. 81 (2003) 197] the basics of the LESLP formulation along with a guide to programming are provided. The new algorithm is successfully tested in weight minimisation problems of truss structures with up to hundreds of design variables and thousands of constraints: sizing and configuration problems are considered. Optimization problems of non-truss structures are also presented. The numerical efficiency, advantages and drawbacks of LESLP are discussed and compared to those of other SLP algorithms recently published or implemented in commercial software packages.

In practical optimization, the sequential linear programming (SLP) method [1], [2], [3] and [4] is very popular because of its simple theoretical foundation and because reliable linear solvers are readily available (i.e. Simplex packages). Although SLP performs very well in convex programming problems with nearly linear objective and constraint inequality functions, the method is not globally convergent. Moreover, numerical problems like convergence to local or infeasible optima and objective function oscillations may occur if the method is not well controlled. Therefore, SLP is often considered a “poor” method by many theoreticians. However, SLP techniques may be enhanced by using move limits which are additional side constraints that define a region of the design space where the solution of the linearized sub-problem will lie. An efficient and robust approximate method for structural optimization should define the move limits so that the approximate sub-problem portrays the original non-linear problem well. Besides, high accuracy of the approximation will eliminate cost function oscillations and will avoid that the optimizer gets stuck in infeasible regions. Wujek and Renaud [5] stated that a proper choice of the move limits should ensure that the objective function improves monotonously, that each intermediate solution is feasible, that the design variable movement is controlled in order to maintain the approximation error at a reasonable level. The aforementioned requirements were fulfilled by the move limit definition strategy proposed by Lamberti and Pappalettere [6]. The resulting algorithm (LEAML) was superior over other SLP techniques but involved some heuristics. Hence, the original formulation of Ref. [6] has been substantially modified in this research. The new algorithm, called linearization error sequential linear programming (LESLP), is formulated so to avoid any uncertainties and guesswork in the SLP procedure. The move limit domain is built based on the linearization error εLIM introduced with the linear approximation: LESLP uses a combination of “low-fidelity” and “high-fidelity” models that, respectively, evaluate the cost function only or all the non-linear functions. Also, line searches are performed to enforce the move limit domain to lie in regions of the design space where the cost function is very likely to improve. Furthermore, the move limits are accepted or rejected and recalculated based on trust region models that enhance the robustness of the procedure. Finally, LESLP provides schemes to accept, improve or reject intermediate designs. Even infeasible designs can be handled because LESLP tries to find regions where the constraints are satisfied or violated the least. The basics of LESLP formulation and a basic guide to programming have been presented in a companion paper [7]. The present paper discusses in detail the numerical efficiency of LESLP in solving a large variety of design optimization problems.

This research work discussed the experiences done with different tools for structural optimization. A new SLP algorithm called LESLP was tested in 20 weight minimisation problems of truss structures and four non-truss problems ranging from mathematical programming examples to structures to be designed under multiple loading conditions. The set of design problems was very indicative as it included large scale and configuration problems along with the fact that the non-truss test cases included highly non-linear problems or cases usually solved with multi-level optimization schemes. The results obtained with LESLP were in excellent agreement with the literature. LESLP showed the same efficiency of more sophisticate algorithms that implement formulations even tailored to the kind of optimization problems considered. Furthermore, LESLP resulted very efficient compared to other SLP techniques or well-known commercial optimizers (DOT). In fact, the present code designed always the most efficient structures within very few optimization cycles. LESLP was also very competitive in terms of structural analyses required to converge to the optimum. The required CPU times resulted smaller than for other SLP algorithms but also strictly depending on the number of approximate sub-problems formulated and solved; in fact, the most significant fraction of CPU time was required by the linear solver. Although LESLP exhibited some failures in designing structures with larger amount of design freedom (more design variables), it behaved much better than the other SLP algorithms considered in this study. Optimum designs were found substantially insensitive to input parameters such as the initial value of the allowable linearization error and the starting design point. LESLP was also rather robust when noises and uncertainties were introduced in the optimization. Furthermore, the behaviour of LESLP was insensitive to the choice of hardware platform and compiling options. Finally, it was found that using fast compilers did not result in significant reductions of the CPU time. The results presented in this study support the conclusion that LESLP is a very powerful and versatile tool for structural optimization. The fact that LESLP and the other algorithms considered here were used as “black-boxes” emphasises the superiority of LESLP over the other codes. However, the arguments and the results presented in this paper are obviously not exhaustive. Although a large variety of optimization problems were solved very efficiently, the present authors advice that the findings of this investigation should be considered general only for bar truss structures.