A Chaotic Response Surface Method for Non-Gaussian Stochastic Analysis of Structural Responses
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摘要: 传统响应面法应用于非Gauss随机结构时影响计算效率和精度. 为此，提出了非Gauss响应量分析的混沌响应面法. 首先根据随机变量的概率分布类型构造了混合型广义混沌多项式，据此建立了非Gauss响应量的随机展开式；利用高阶一维广义混沌多项式的根构造了非Gauss基本随机变量空间的概率配点，并基于系数矩阵行满秩原则遴选非Gauss随机变量空间的最优概率配点；进而利用最小二乘法确定了响应面的待定系数，据此建立了非Gauss响应面的广义混沌表达式. 最后，通过对比分析，验证了混沌响应面法能够以较少的配点、较低的展开阶次取得更高的计算精度和效率.Abstract: A chaotic response surface method was proposed to improve the computational efficiency and accuracy of the traditional response surface method for stochastic analysis of structural responses involving non-Gaussian random variables. Firstly, the non-Gaussian response variable was expanded by a hybrid generalized polynomial chaos constructed according to the probability distribution function types of the basic random variables. Secondly, the candidate probability collocation points in the non-Gaussian probability space were determined through the combination of the roots of the 1D generalized polynomial chaos with the next higher order, then the probability optimal collocation points in the non-Gaussian probability space were picked out based on the full row rank principle of the coefficient matrix. Finally, the unknown coefficients of the proposed response surface were determined by means of the least squares method. Comparison of examples shows that, the proposed method requires fewer collocation points and lower expansion orders, and achieves higher calculation accuracy and efficiency than those of the traditional response surface methods.
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图 3 F3的均值和标准差
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 3. Means and standard deviations of F3
图 9 不同变异系数下的顶层梁水平位移的统计特征值
Figure 9. Statistical parameters of the horizontal displacement at the top floor with different coefficients of variation
表 1 对应不同概率分布类型的广义混沌多项式
Table 1. Generalized polynomial chaos for different probabilistic distributions
distribution generalized polynomial chaos weight function domain Gauss Hermite $(1 / \sqrt{2 {\rm{\mathsf{π}}}}) \mathrm{e}^{-\xi^2 / 2}$ (-∞，+∞) uniform Legendre 1 [－1, 1] beta Jacobi (1+ξ)β(1－ξ)α [－1, 1] exponential Laguerre e－ξ [0，+∞) Gamma generalized Laguerre ξαe－ξ [0，+∞) 
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表 2 2维3阶广义混沌多项式
Table 2. The mixed generalized polynomial chaos (m=2, p=3)
Ψ3(ξ1, ξ2) α=(α1, α2) H3(ξ1)L0(ξ2)=(ξ13-3ξ1)×1 (3, 0) H2(ξ1)L1(ξ2)=(ξ12－1)×ξ2 (2, 1) $H_1\left(\xi_1\right) L_2\left(\xi_2\right)=\xi_1 \times \frac{1}{2}\left(3 \xi_2^2-1\right)$ (1, 2) $H_0\left(\xi_1\right) L_3\left(\xi_2\right)=1 \times \frac{1}{2}\left(5 \xi_2^3-3 \xi_2\right)$ (0, 3)
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表 3 CRSM与HRSM计算功能函数F2的结果
Table 3. Results of the CRSM and the HRSM for F2
method total probability collocation point optimal probability collocation point mean standard
deviationcollocation
point numbermean standard
deviationcollocation
point numberMCS 3.444 5×103 2.598 7×103 103 - - - HRSM2 3.211 2×103 2.656 3×103 9 3.195 5×103 2.648 8×103 6 HRSM3 3.548 6×103 2.501 9×103 16 3.534 0×103 2.488 9×103 10 HRSM4 3.411 9×103 2.682 1×103 25 3.410 7×103 2.676 2×103 15 HRSM5 3.457 0×103 2.539 5×103 36 3.454 1×103 2.538 7×103 21 HRSM6 3.445 4×103 2.639 2×103 49 3.445 3×103 2.639 6×103 28 CRSM2 3.445 1×103 2.588 8×103 9 3.445 1×103 2.588 8×103 6 CRSM3 3.444 3×103 2.592 6×103 16 3.444 3×103 2.592 6×103 10
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表 4 随机变量的统计特性
Table 4. Statistical characteristics of random variables
random variable distribution mean coefficient of variation Eb Gauss 20 GPa 0.15 Ec Gauss 22 GPa 0.15 P1 beta 10 kN 0.15 P2 beta 100 kN 0.15 q uniform 8.0 N/m 0.15
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表 5 顶层水平位移u的统计特征值
Table 5. Statistical parameters of horizontal displacement at top floor
method total probability collocation point optimal probability collocation point mean/m standard
deviation/mcollocation
point numbermean/m standard
deviation/mcollocation
point numberMCS 5.472 5×10-3 1.040 9×10-3 105 - - - HRSM2 5.473 3×10-3 9.000 4×10-4 243 5.480 8×10-3 8.878 8×10-4 21 HRSM3 5.474 7×10-3 1.144 3×10-3 1 024 5.459 4×10-3 1.134 1×10-3 56 HRSM4 5.478 5×10-3 9.934 5×10-4 3 125 5.477 6×10-3 9.783 1×10-4 126 HRSM5 5.476 2×10-3 1.082 1×10-3 7 776 5.477 8×10-3 1.083 2×10-3 252 CRSM2 5.471 9×10-3 1.052 4×10-3 243 5.471 9×10-3 1.024 0×10-3 21 CRSM3 5.472 2×10-3 1.040 7×10-3 1 024 5.473 9×10-3 1.041 5×10-3 56
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