Static strength reliability and sensitivity analysis of large composite structures based on surrogate models
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摘要: 基于代理模型的结构可靠性分析方法目前已有多种代理模型和多种失效概率计算方法，参数灵敏度分析也存在多种灵敏度指标，为探讨大尺寸复合材料结构可靠性分析方法，并研究不同代理模型和计算方法对民机复合材料结构静强度预测结果的差异性，以某型飞机碳纤维复合材料升降舵结构为研究对象，考虑单层板力学性能、厚度、密度和不同部位铺层角度等17种输入变量的不确定性，利用Matlab和Optistruct仿真软件，构建Kriging、PC-Kriging和支持向量回归机(SVR)三种代理模型，结合MCS和Subset两种方法求解结构失效概率，根据代理模型验证误差选取最准确的代理模型计算Morris基本效应筛选指标和Sobol指标，从而获得关键设计参数排序，为民机复合材料结构可靠性设计提供参考。Abstract: Multiple surrogate models and multiple calculation methods of structural probability of failure were proposed for the structural reliability analysis based on surrogate model. Multiple sensitivity indices were also proposed for the sensitivity analysis of input variables. For the purpose of studying the reliability analysis methods for large composite structures, and studying the differences among different surrogate models and methods to predict the static strength of composite structures of civil aircrafts, a composite elevator of a certain aircraft was taken as an example. The uncertainty of 17 input variables were considered, including mechanical properties, thickness, density and ply angles. Matlab and Optistruct softwares were used to obtain three surrogate models, including Kriging, PC-Kriging and SVR models. MCS and Subset simulate methods were applied to calculate the structural probability of failure. The best surrogate model can be obtained by comparing validation errors. The critical design parameters can be obtained by the importance ranking based on Morris elementary effects and Sobol indices, which is useful for the design of civil aircraft composite structures.
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Key words:
- composite structure /
- surrogate modelling /
- Kriging model /
- reliability analysis /
- sensitivity analysis
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图 1 某型飞机平尾结构示意图(含水平安定面和升降舵)
Figure 1. FEM model of the horizontal tail structure, including horizontal stabilizer and elevators
图 2 基于代理模型的可靠性及灵敏度分析流程
Figure 2. Flowchart of the procedure used for reliability and sensitivity analysis based on surrogate modelling
图 4 复合材料升降舵结构复合材料板最小强度比(RS)
Figure 4. Minimum strength ratio (RS) of the composite elevator structure
图 5 基于不同代理模型的复合材料升降舵结构最小Tsai-Wu强度比预测结果与有限元计算结果对比：(a) Kriging；(b) PC-Kriging；(c) SVR
Figure 5. Comparison between the lowest Tsai-Wu RS FEM result and prediction based on different metamodels: (a) Kriging; (b) PC-Kriging; (c) SVR of the composite elevator structure
图 6 基于不同代理模型的复合材料升降舵结构失效概率收敛性：(a) Kriging；(b) PC-Kriging；(c) SVR(失效概率计算方法均为MCS)
Figure 6. Convergence of failure probability of the composite elevator structure based on different metamodels: (a) Kriging; (b) PC-Kriging; (c) SVR (MCS has been applied for the evaluation of Pf)
图 7 复合材料升降舵结构输入变量灵敏度分析及排序：(a) Morris指标；(b) Sobol指标
Figure 7. Sensitivity analysis and ranking of input variables of the composite elevator structure: (a) Morris index; (b) Sobol index
图 8 复合材料升降舵结构Morris和Sobol灵敏度指标排序比较
Figure 8. Morris and Sobol sensitivity index ranking comparison of the composite elevator structure
表 1 复合材料升降舵结构随机输入变量正态分布
Table 1. Normal distribution of random input variables of the composite elevator structure
Variable Mean Standard deviation Description E11 /GPa 60 6.0 Elastic modulus in 11 direction E22 /GPa 54 5.4 Elastic modulus in 22 direction G12 /GPa 3.7 0.37 Elastic modulus in 12 direction G13 /GPa 3.0 0.30 Elastic modulus in 13 direction G23 /GPa 3.0 0.30 Elastic modulus in 23 direction ρ /(kg·m−3) 1440 144.0 Density Xt /MPa 830 83.0 Longitudinal tensile strength Xc /MPa 650 65.0 Longitudinal compressive strength Yt /MPa 250 25.0 Transverse tensile strength Yc /MPa 230 23.0 Transverse compressive strength S12 /MPa 100 10.0 In-plane shear strength t /mm 0.216 0.0216 Thickness of a lamina α1/(°) 0 4.5 (0°,90°) ply angle for skin panel α2/(°) 45 4.5 (±45°) ply angle for skin panel α3/(°) 0 4.5 (0°,90°) ply angle for spar α4/(°) 45 4.5 (±45°) ply angle for spar α5/(°) 45 4.5 (±45°) ply angle for ribs 
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表 2 基于不同代理模型的模型误差和失效概率结果对比
Table 2. Comparison of errors and failure probability results based on different surrogate models
Metamodel $ {e_{{\text{LOO}}}} $ $ {e_{{\text{val }}}} $ $P_{\text{f}}^{{{\mathrm{MCS}}} }$(Evaluations) $P_{\text{f}}^{{\mathrm{Subset}}}$(Evaluations) Kriging 4.56×10−2 8.00×10−3 7.02×10−4 (106) 7.66×10−4 (25900) PC-Kriging 3.61×10−2 2.04×10−2 6.78×10−4 (106) 5.27×10−4 (25900) SVR 1.22×10−1 5.08×10−2 1.60×10−4 (106) 1.18×10−4 (25900) Notes: $ {e_{{\text{LOO}}}} $ and $ {e_{{\text{val }}}} $are leave-one-out cross-validation error and validation error, respectively; $P_{\text{f}}^{{{\mathrm{MCS}}} }$ and $P_{\text{f}}^{{\mathrm{Subset}}}$ are failure probability based on MCS and Subset methods, respectively.
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