Interval uncertainty analysis method based on adaptive radial basis function model
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摘要:
针对区间不确定性分析问题，提出一种基于径向基函数模型的自适应不确定性分析算法。首先提出一种适用于径向基函数模型的采集函数-潜在最值函数，并针对最大/小值问题特性将其细分为潜在最大/小值函数。针对区间不确定性分析问题，建立基于潜在最大/小值函数的序列优化框架，完成区间不确定性分析问题的高效高精度求解。通过3个算例的分析结果表明：相比于粒子群优化（PSO）算法和顶点法等算法，所提算法能够在保证计算精度的同时，有效提高区间不确定性问题的分析效率；相对于采用拉丁超立方法一次抽样构建径向基函数模型，之后使用粒子群优化算法计算响应边界（LHS+PSO）的方法，所提算法通过采集函数序列更新模型，提高了模型在局部区域的模型逼近效果，保证了区间不确定性问题的分析求解精度。
Abstract:Considering the problem of interval uncertainty analysis, an adaptive uncertainty analysis method based on radial basis function model was proposed. Firstly, an acquisition function, also called the potential maximum function, which can be combined with the radial basis function model, was presented, and subdivided into potential maximum/minimum functions according to the characteristics of the maximum/minimum problem. Then, for the interval uncertainty analysis problem, a sequential optimization framework based on the potential maximum/minimum function was established to complete the efficient and high-precision solution of the interval uncertainty analysis problem. Three examples showed that, the proposed method can improve the computational efficiency of particle swarm optimization (PSO) and vertex method with accurate solution; also, the method refined the model sequentially through the proposed acquisition function, so compared with the method in which the Latin hypercube sampling is used to perform the “one-shot” sampling for radial basis function model constructing, and the response bounds is estimated through particle swarm optimization (LHS+PSO), it can guarantee the accuracy of the predicted bounds by improving approximate accuracy of the model in local regions.
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图 4 25杆桁架示意图
注：图中编号1～12为节点编号；编号（1）～（25）为杆编号。
Figure 4. Schematic diagram of a 25-bar truss
图 5 航空发动机涡轮叶片的有限元模型和应力分布云图
Figure 5. Finite element model and stress distribution of turbine blade
表 1 算例1的所有分析结果
Table 1. Evaluation results of Example 1
算法类型 $g （{{\boldsymbol{x}}}） $上边界 $g （{{\boldsymbol{x}}}） $下边界 调用次数 取值 相对误差/% 取值 相对误差/% 精确解 59.95 −8.10 PSO 59.95 0 −8.10 0 7600 顶点法 51.25 14.51 −4 51.25 4 LHS+PSO 55.37 7.64 −5.97 26.83 80 N-PGBO 59.94 0.017 −8.10 0 80 ARBF 59.94 0.017 −8.10 0 66 
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表 2 算例2中区间变量参数
Table 2. Interval variables for Example 2
变量 区间 ${A_1}$/${\text{m}}{{\text{m}}^2}$ $[630,770]$ ${A_2}$/${\text{m}}{{\text{m}}^2}$ $[5\;580,6\;820]$ ${A_3}$/${\text{m}}{{\text{m}}^2}$ $[4\;770,5\;830]$ ${A_4}$/${\text{m}}{{\text{m}}^2}$ $[7\;920,9\;680]$ $L$/${\text{m}}$ $[13.5,16.5]$ ${F_1}$/${\text{kN}}$ $[1\;601.2,1\;957.1]$ ${F_2}$/${\text{kN}}$ $[2\;001.6,2\;446.4]$ ${F_3}$/${\text{kN}}$ $[1\;601.2,1\;957.1]$ ${F_4}$/${\text{kN}}$ $[1\;201,1\;467.8]$
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表 3 算例2的所有分析结果
Table 3. Evaluation results of Example 2
算法类型 节点7垂直位移上边界 节点7垂直位移下边界 调用次数 取值/mm 相对误差/% 取值/mm 相对误差/% PSO 54.62 28.04 4600 顶点法 54.62 0 28.04 0 512 LHS+PSO 59.74 9.37 21.00 25.11 70 N-PBGO 54.62 0 28.04 0 36 ARBF 54.69 0.13 27.95 0.32 57
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表 4 算例3中区间变量相关参数
Table 4. Interval variables for Example 3
变量 区间 $E$/109 Pa [200, 250] $ {C}_{{\mathrm{CTE}}} $/10−6 K−1 [11, 14] $\mu $ $[0.24,0.3]$ $\lambda $/（W/（m·K）） $[10.3,12.7]$ ${p_1}$/105 Pa [7.2, 8.8] ${p_2}$/105 Pa [5.4, 6.6]
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表 5 算例3的所有分析结果
Table 5. Evaluation results of Example 3
算法类型 涡轮叶片最大变形量上边界 涡轮叶片最大变形量下边界 调用次数 取值/mm 相对误差/% 取值/mm 相对误差/% PSO 2.08 1.31 4600 顶点法 2.08 0 1.31 0 64 LHS+PSO 2.23 7.21 1.17 10.69 50 N-PBGO 2.08 0 1.31 0 41 ARBF 2.08 0 1.31 0 40
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