فرمولاسیون برنامه ریزی بهبودیافته متوالی خطی برای به حداقل رساندن وزن ساختاری

This paper presents an optimization algorithm for weight minimization of structures. The algorithm––denoted as LSTRLP (line search trust region linear programming)––combines sequential linear programming (SLP) and Trust region methods (TRM). LSTRLP solves a linearized sub-problem in each design cycle and accepts or rejects intermediate designs based on a line-search strategy which detects if the eventual improvement in cost is actually the largest possible. It is to be noticed that the present work is the closure to several studies carried out by the present authors in order to improve the overall efficiency and robustness of the sequential linear programming method. The LSTRLP algorithm is implemented by an optimization code written in Fortran 90. The optimization code is tested in eight cases of weight minimization of bar truss and frame structures. The test cases include examples of large-scale and configuration optimization. The results obtained here are compared to those presented in literature. The optimizations are run also with sequential quadratic programming (SQP) routines implemented in commercial software. The results indicate that LSTRLP is competitive with recently published algorithms and commercial software.

Sequential linear programming (SLP) is very popular in practical engineering since the linear solvers utilized to solve the linearized sub-problems are easily available to designers. In optimum design of structures, SLP is more attractive than other optimization methods because it requires structural analysis only for computing gradients of the cost function and constraints. Trust region methods (TRM) are universally acknowledged robust and versatile optimization algorithms because of their excellent global convergence properties granted by the fact that TRM adaptively manage the amount of movement allowed in design space when approximation models are used in the optimization process. Over the years, much work has been developed on optimization algorithms based on SLP [1], [2], [3], [4], [5] and [6] and TRM [7], [8], [9], [10], [11] and [12] methods. In the last few years, Lamberti and Pappalettere [13], [14] and [15] made considerable effort to enhance the overall efficiency of the SLP method. In particular, they proposed a novel approach to move limit definition based on computing and, then, limiting the approximation error introduced in the optimization process when the original non-linear optimization problem is replaced by the linearized sub-problem. An optimization code based on this rationale was presented in Ref. [13]. The code, denoted as linearization error amplitude move limits (LEAML), resulted very efficient in sizing optimization problems of bar truss structures with up to 200 elements, 96 design variables and 3500 constraints. However, LEAML did not include any strategy for choosing the points at which non-linear constraints should be evaluated after that individual move limits are determined for each design variable. This weakness suggested not using LEAML in highly non-linear problems like large-scale or shape optimization problems. The formulation of LEAML was significantly improved by the linearization error sequential linear programming (LESLP) code described in Refs. [14] and [15]. LESLP included a trust region model for accepting or rejecting move limits. This fact allowed to increase the design freedom and to reduce, thus, the number of design cycles. The results presented in Ref. [14] indicated clearly that LESLP outperformed LEAML. In fact, LESLP solved successfully large-scale problems (with up to 720 structural elements and 200 design variables) and performed well also in configuration optimization problems of bar truss structures. Further investigations reported in Ref. [15] show that LESLP was efficient also in configuration optimization of frame structures and, remarkably, it was insensitive to even very different starting designs. Finally, LESLP was competitive with commercial optimizers implementing state-of-art optimization algorithms such as sequential quadratic programming. However, in spite of the excellent numerical behavior exhibited by LESLP, the formulation presented in Refs. [14] and [15] has clearly a weak point in that there is no search for optimal step along the direction found by solving the linearized sub-problem. This fact leads to carry out additional design cycles because the improvement in cost function is less significant than that could actually be. For instance, if the linearization point is pretty far away from the constraint domain boundaries, the optimizer should explore a much longer step than the step obtained by solving the linearized sub-problem (the and vectors respectively denote the linearization point and the intermediate solution). The new optimization algorithm presented in this paper overcomes the aforementioned limitation. In fact, it includes line search along the direction found in the current iteration also when the iteration might be considered successful since it resulted in design improvements. In addition, the new formulation includes a strategy for checking on constraint domain non-convexity. The new algorithm has been implemented by a numerical code denoted in the rest of this paper as LSTRLP where the acronym stands for line search trust region linear programming. The LSTRLP code was successfully applied on structures such as a 444 element dome, a 45 element cantilevered bar truss, a 47 element power line, a frame comprised of 45 I-beam elements and to pure configuration optimization problems such as the classical Michell’s semi-circular arch, a 25 element transmission tower, a 52 element dome and a bridge comprised of 37 elements (10 beams and 27 trusses). Hence, the set of design problems is certainly exhaustive because it includes examples of large-scale and configuration optimization. The present paper is divided as follows. After the Section 1, the description of LSTRLP and a pseudo code of the algorithm are contained in Section 2. Section 3 describes the optimization problems chosen as test cases. The results of the optimization runs are presented and discussed in Section 4. Finally, Section 5 summarizes the main findings of the research and presents the conclusions drawn from this study.

This paper described an improved formulation for structural optimization based on the combination of sequential linear programming (SLP) and trust region (TR) methods. The new algorithm, denoted as LSTRLP, enhanced the formulation implemented in the LESLP algorithm previously developed by the present authors. The main difference between LSTRLP and the previous formulation lies in the fact that the present algorithm performed line search along the direction found in a given iteration also when that iteration might be considered successful since it resulted in design improvements. In addition, LSTRLP included a strategy for checking on constraint domain non-convexity. LSTRLP was tested in eight optimization problems of truss and frame structures: a 444 element dome, a 45 element cantilevered bar truss, a 47 element power line and a frame comprised of 45 I-beam elements, the classical Michell’s semi-circular arch (with 13 elements), a 25 element transmission tower, a 52 element dome and a bridge comprised of 37 elements (10 beams and 27 trusses). The set of design problems was certainly indicative because it included examples of large-scale and configuration optimization of structures under multiple loading conditions. In order to draw general conclusions, LSTRLP was compared to commercial optimizers and also to formulations developed ad hoc for design optimization of skeletal structures. The results of the optimization runs showed that LSTRLP was more efficient than LESLP in non-convex optimization problems (cantilevered bar truss structure and frame structure). In large-scale problems (444 element dome), using LSTRLP allowed to reduce the number of design cycles and hence the overall computational cost. In general, LSTRLP was competitive with commercial optimization packages and its performance resulted comparable to that of tailored algorithms (in the case of the power line) or even better (in the case of the structures optimized with configuration variables only). In particular, the convergence speed of LSTRLP resulted insensitive to the initial design while the performance of commercial optimizers and recently published algorithms was significantly influenced by the difference between the initial configuration and the optimum design. The arguments developed and the results presented in this research support the conclusion that the LSTRLP algorithm is an efficient and robust tool for structural weight minimization.