Study On Efficient Algorithm Of Probabilistic Structural Design Optimization

Due to increasing competition between enterprises and shortage of energy and resources, optimum design has received more and more attention in recent decades. Since uncertainties are unavoidable in practical engineering, it becomes a general understanding that uncertainty widespread in material properties, loads and structural modeling must be taken into account in structural optimization formulation. Probabilistic structural design optimization (PSDO) is one of the most important formulations to consider uncertainty in structural optimization. Since the calculation of probabilistic constraints is in essence an optimization problem with numerous iterations, the straight approach to PSDO is to form a two-level optimization and the computation cost may be prohibitive, which prevents the application of PSDO in engineering. An efficient approach for PSDO is essential in this area.The present thesis adopts the recently proposed performance measure approach (PMA) to evaluate probabilistic constraints. Based on the approximate probabilistic performance measure (PPM) and its sensitivities at the current design, a novel sequential approximate programming for PSDO is constructed. In implementation of the algorithm, it is found that PPM should always be obtained by minimizing the performance function no matter performance function value is positive or negative at the mean of the random variables. This corrects the error in literatures. Furthermore, chaotic control strategy is introduced to deal with convergence failure such as oscillation, chaotic motion in iteration for PPM evaluation which leads to the failure of optimization iteration. Through these improvements, the new sequential approximate programming realizes concurrent convergence of both design optimization and PPM calculation. It also has high efficiency and less dependence on the probabilistic distribution of random variables and the target reliability of probability constraints. In comparison with various algorithms in literatures, the present algorithm is more efficient, stable and reliable. The research work carried out in the present thesis could be summarized as follows.1. The iterative formulae to compute reliability index in reliability index approach (RIA) and PPM in PMA are described in detail. The consistence of PMA and RIA in the sense of evaluation of probabilistic constraints is discussed and moreover the mathematical optimal model of PPM calculation is clarified.Generally one gets positive reliability index in RIA. However, when the performance measure at the origin in the standard normal space, namely the mean value point in the original random space, is negative, the failure probability exceeds 50% and one gets negative reli- ability index. In some literatures it was suggested that the performance measure should be minimized or maximized to compute PPM in PMA according to positive or negative reliability index respectively in RIA. Through computational results and figures analysis, it is found that the violated probabilistic constraint may be misjudged to be satisfied when using maximizing the performance measure for negative reliability index. If still using minimizing, the right judgment consistent with RIA can be gotten. This argument is further proved by reduction to absurdity. So through this argument combined with the previous attestation in literatures, one can easily draw the conclusion: whether the performance measure at the mean of the random variables is positive or negative, one should always minimize the performance measure to compute PPM in PMA and this would make PMA consistent with RIA for evaluation of probabilistic constraints. This conclusion clarifies the mathematical optimal model of probabilistic constraints calculation in PMA and makes it possible to produce effective and efficient approach for PSDO. (In Chapter 2)2. The straightforward approach to the performance-measure-based PSDO is to form a two-level optimization by simply connecting algorithms of PPM calculation and design optimization. It is well known that the computation cost can be prohibitive when the associated function evaluation is time consuming. To overcome this difficulty, the sequential approximate programming(SAP) strategy, which is successfully used in deterministic structural optimization, is extended. In PMA with SAP, a sequence of approximate programming of PSDO are formulated and solved before the final optimum is located. In each sub programming, rather than relying on direct linear Taylor expansion of PPM, the author develops a formulation for approximate PPM and its sensitivity at the current design point and uses its linearization instead. The approximate PPM and its sensitivity are obtained by approximating the optimality conditions in the vicinity of the minimum performance target point (MPTP). The error analysis shows that in theε-vicinity of optimum design and the corresponding MPTP, the difference between the normal linear Taylor expansion of PPM and the linear expansion of approximate PPM is of higher order ofε. Finally, the algorithm is connected with the commercial structural analysis and optimization software in order to optimize complex practical structures. Through the application of this algorithm to the most relevant examples frequently cited in the similar studies, it is proved that concurrent convergence of both design optimization and PPM calculation is realized, which agrees well with the error analysis. The performance of PMA with SAP is quite effective and has good adaptability and the probabilistic distribution of random variables and the target reliability of probability constraints have little effect on the algorithm performance. (In Chapter 3)3. The advanced mean-value (AMV) method is well suitable for PMA due to its simplicity and efficiency. However, when the AMV iterative scheme is applied for some nonlinear performance functions, the iterative sequences could fail to converge. Then both PMA two-level and PMA with SAP,. which are based on performance measure of probabilistic constraints, fail for these problems. In this thesis, it is firstly illustrated by numerical examples that the iterative sequences generated by AMV iterative scheme could fall into the periodic oscillation, bifurcation and even chaos. And the chaos dynamics analysis on this iterative procedure is carried out. Then, the stability transformation method of chaos feedback control is employed to the convergence control of iterative failure of AMV. The unstable fixed points embedded in the periodic and chaotic orbit are stabilized and the expected stable convergent solutions are obtained. Once the evaluation of probabilistic constraints can be carried out successfully, the design optimization is performed by PMA two-level or PMA with SAP. The numerical results demonstrate that the convergence control using stability transformation method is effective and PMA with sAP is slightly more efficient. Moreover, the stability transformation method of chaos feedback control provides a relatively effective approach for the probabilistic optimization problem with uniform random variables, which is considered to be a conundrum. (In Chapter 4)4. The possibility of using the probabilistic reliability analysis to tackle the bounded-but-unknown uncertainties is exploded. Firstly, a non-probabilistic measure of reliability based on interval variables proposed in literature is discussed and it is pointed out that the essence of this measure is that a structure is reliable if there is no failure even in the worst case and would lead to severe conservatism. Then a similar non-probabilistic measure of reliability based on convex model is proposed and it omits some off-chance occurrences and is not so much conservative as that based on interval model. Because in the sense of maximum information entropy, the uniform distribution is the most conservative probabilistic distribution of random variable for which only the variable interval is known, and a relatively effective approach for the probabilistic optimization problem with uniform random variables has been developed, the author models the bounded-but-unknown variables as uniformly distributed random variables and use the probabilistic reliability analysis to deal with the bounded-but-unknown uncertainties. The computational results show that the probabilistic reliability measure is comparatively close to the non-probabilistic measure of reliability based on convex model. At the end, optimum design of structures with bounded-but-unknown uncertainties is performed using these three reliability measures. Example results also illustrate that the optimum result using the probabilistic reliability measure is comparatively close to that using the non-probabilistic measure of reliability based on convex model and less conservative than that of interval model. (In Chapter 5) The research of this dissertation is supported by Major Program of Natural Science Foundation of China (No. 10332010, 10672030), National Key Basic Research Development Program of China (Grant no. 2006CB601205) and National Creative Research Team Program of China (No. 10421002), which is gratefully acknowledged by the author.