Adaptive Structural Optimization Methods

Abstract

Although the field of structural optimization has undergone substantial growth the last decades, the efficiency of the programs used to generate structurally optimal designs for industrial applications remains questionable. To make matters worse, as structural problems become more complex, e.g. by adding multi-disciplinary optimization targets to the problem formulation, the slow convergence speed associated with the inefficiency of these programs turns into a significant issue. Thus, one can argue that there is a need for improvements in the optimization methods used to address such problems. One of the most prominent methods to tackle structural optimization problems is SAO. This method is comprised of 2 steps: approximating locally the non-analytical functions that describe the studied phenomenon and solving the resultant nonlinear optimization problem. This report is a thorough investigation of the function approximation part on a fundamental level, and an attempt to improve the quality of the generated approximation. We start by conducting a literature survey of the state-of-the-art function approximation methods and evaluating their performance - in terms of convergence speed - under various scenarios. Then, we focus our attention to the most widely applied family of approximations (i.e. the MMA) by locating and understanding their inadequacies. These inadequacies are then addressed by proposing a solution that includes the generation of novel mixed approximation schemes that can outperform their constituent members for a set of small-scale, albeit challenging, test problems. Apart from implementing, validating and evaluating fairly different approximation schemes, we also generate approximation selection criteria, whose output is the 'optimal' approximation method for any given function at an arbitrary point. The outcome of this process is an 'Optimization Laboratory' that offers an in-depth understanding of the behaviour of any function approximation method under examination on a fundamental level. It is effectively an object-oriented software library that can be used to compare different approximation schemes on a set of problems, while offering a detailed insight on the quality of any approximation at any given point. Numerical examples that support our assertions are presented, building a strong case in favor of the possible superiority of a mixed scheme compared to its component members, as far as adaptiveness, robustness and total computational cost of SAO algorithms are concerned. More specifically, all approximations used herein are analyzed with a benchmark profiling method. Different sets of initial points, multiple convergence criteria and several tolerances are used in order to obtain an objective comparison between the considered methods. The results of this thorough comparison are summarized in performance profile plots and discussed thereafter. The results of the present research suggest that the generated mixed schemes are superior to their constituent members for the considered problem set. A significant reduction in the number of iterations required to converge to a local optimum was achieved and overall convergence was improved. Even in the most conservative scenario, the best mixed scheme outperformed all others for 46% of the implemented problems (compared to MMA that did so for just above 15% of them). What is more, improved robustness was also achieved by successfully solving 95% of the considered problems (compared to 72% for MMA). However, this improved performance is accompanied by an increase in the computational cost of the approximation process. To this end, an analysis of the relative computational cost of mixed schemes compared to the state-of-the-art approximation methods - in terms of time spent per iteration on the approximation - follows the performance results and a method to estimate the viability of approximation enhancement is proposed. Finally, conclusions on the relative performance of all methods are drawn from the aforementioned comparison and recommendations for further research on improving the performance and lowering the computational cost of mixed schemes are given.