【作者】 张文博；

【导师】 亢战；

【作者基本信息】 大连理工大学 ， 计算力学， 2016， 硕士

【摘要】 结构中的不确定性需要根据其特征及来源做有效的描述、分析和优化。如果样本数量不足以构建足够精确的概率模型且其本身仅有有界性质的情况下,适于采用非概率凸模型来描述。然而不合适的凸模型建立方法将导致可靠度计算及其优化的结果可信性降低。本文发展了一种在给定样本集合的情况下基于半定规划的最小椭球凸模型的建立方法。该方法将有界不确定性根据来源分组,并对每一组采用可保证全局最优解的半定规划列式求解其最小包络椭球。该方法还被应用于已知样本集合的非概率可靠性计算及基于可靠度的优化设计问题。该模型建立方法及分析和优化方法的正确性和效率在数值算例中得到了检验。而从不确定性的来源考虑,几何不确定性问题区别于载荷不确定性和材料不确定性问题。因为如光刻蚀和增材制造的制造误差有可能引起拓扑改变,这为不确定性问题提出了新的挑战——如何同时在随机分析以及鲁棒拓扑优化中考虑这样的不确定性。本文发展了一种隐式的随机水平集描述方法,并进行Karhunen-Loeve展开以有效描述不确定性。在此基础上发展了基于多项式混沌展开的分析方法,其中关于柔顺性的数值算例得到了指数级收敛。在基于水平集方法的鲁棒拓扑优化设计中,由于采用隐式的随机水平集描述,与特征尺寸相当的制造误差和其引起的拓扑改变得到了有效处理。在鲁棒拓扑优化的算例中,这种考虑几何不确定性的方法还呈现出了尺寸控制效应。更多还原

【Abstract】 The inevitable uncertainties in structure need to be treated according to their features and sources. When they are bounded without distribution information and cannot be described with probability model with sufficient samples, the non-probabilistic convex model is suitable due to the fact that it provide a smooth and convex bound reference for such uncertainties. However, improper modeling of the uncertainties may give rise to misleading non-probabilistic reliability analysis, thus result in either unsafe or over-conservative designs. This paper presents a construction method for minimum-volume ellipsoidal convex model based on semi-definite programming (SDP). In this method, the uncertain parameters are first divided into groups according to their sources. For each individual group of uncertainties, the minimum-volume ellipsoid problem is reformulated into a semi-definite programming (SDP) problem and thus can be efficiently solved to its global optimum. It is also applied in non-probabilistic reliability analysis and design optimization of structures with bounded uncertainties. The effectiveness and efficiency of the present techniques for convex model construction and the corresponding reliability analysis are demonstrated with numerical examples of structural topology optimization problems with bounded variations arising from different sources.Under the classification of uncertainty sources, the reason why geometric uncertainties are different from load uncertainties and material uncertainties, especially the possible topological changes arising from photolithography or additive manufacturing, is that they present a new challenge:How to account for such uncertainties in modelling, analysis and optimization. This thesis address this issue by developing an implicit characterization method with stochastic level set perturbation based on Karhunen-Loeve expansion. The analysis is treated by a pseudo spectral version of polynomial chaos expansion (PCE). In the robust topology optimization stage, an adjoint-variable shape sensitivity scheme is developed to account for geometric variations and topological changes. The numerical example concerning with compliance shows an exponential convergence of PCE. The effectiveness of the optimization method is demonstrated in compliance minimization problems. This robust topology optimization method also shows length-scale control effects.更多还原