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考虑有界场的几何不确定性非概率可靠性拓扑优化

NON-PROBABILITY RELIABILITY-BASED TOPOLOGY OPTIMIZATION AGAINST GEOMETRIC UNCERTAINTY WITH A BOUNDED FIELD MODEL

  • 摘要: 在结构的加工制造过程中, 由于加工误差等原因不可避免会在结构上产生一定的几何缺陷, 比如结构长度误差及厚度分布不均等问题. 这些几何不确定性会使结构产生一定的性能波动, 影响结构的安全性. 文章研究的是考虑厚度分布不均的几何不确定性, 因此属于“场不确定性”问题. 考虑实际工程中样本数量有限, 无法准确地获得不确定性概率分布信息, 随机场概率可靠性理论不再适用. 文章基于非概率有界场模型提出了一种考虑结构几何不确定性的非概率可靠性拓扑优化模型. 在模型中几何不确定性通过不确定阈值场函数来表征, 而不确定阈值场则通过一个非概率有界场模型来描述. 该非概率可靠性优化模型为嵌套优化问题, 内层是进行结构的非概率可靠性评估, 外层是采用基于材料场级数展开(material-field series-expansion)的拓扑优化方法来确定结构的最优布局. 优化模型的灵敏度信息是通过伴随法灵敏度分析求得, 并采用了基于移动渐近线法(the method of moving asymptotes)的梯度优化算法来进行该优化问题的求解. 通过数值算例验证了所提出的基于有界场模型的几何不确定性非概率可靠性优化模型的有效性.

     

    Abstract: In the process of structural processing and manufacturing, most products inevitably exhibit certain degree of geometric uncertainties due to processing errors and other reasons, including the structural length error and uneven thickness distribution. These geometric uncertainties will cause certain performance fluctuations in the structure, leading to structural failure and posing certain safety hazards. In this paper, the geometric uncertainty of uneven thickness distribution is considered. Because the thickness distribution of the structure changes with space, it belongs to the "field uncertainty" problem. Considering the limited number of samples in the practical engineering, it is impossible to obtain the information of uncertainty probability distribution accurately, and the traditional random field method based on the probability theory is no longer applicable. Therefore, this paper proposes a non-probabilistic reliability-based topology optimization model considering the geometric uncertainty of the structure with spatial distribution characteristics based on the non-probabilistic bounded field model. In the non-probabilistic reliability-based topology optimization model, the geometric uncertainty is represented by an uncertain threshold field function, and the uncertain threshold field is described by a non-probabilistic bounded field model. The reliability-based topology optimization model is a nested optimization problem, in which the inner-loop optimization problem is used to conduct the non-probability reliability assessment of the structure, and the outer-loop optimization problem is expressed as determining the optimum topology layout of the structure based on the series expansion of material fields topology optimization method. The sensitivity information of the optimization model is obtained by the adjoint sensitivity analysis, and the gradient-based optimization algorithm based on method of moving asymptotes is used to solve the optimization problem. The differential sensitivity analysis method is used to verify the correctness of the analytical sensitivity analysis in this paper. Numerical examples are also presented to illustrate the effectiveness of the proposed non-probabilistic reliability-based topology optimization against geometric uncertainty with the bounded field model.

     

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