Improvement in Numerical Solution of Probability Density Evolution Equation and Slope Stability Analysis
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摘要:
为提升概率密度演化方法在高维结构随机响应分析中的计算精度与效率问题，提出基于总变差不增（TVD）格式的Van Leer通量限制器及超球体均匀选点方法. 首先，针对差分求解的TVD格式，分析Roe-Sweby通量限制器的不足，选取Van Leer通量限制器对数值求解精度进行优化，利用数值算例验证该方法的有效性；其次，基于现有数学证明结论，提出
N 维标准正态分布在超球体内与超立方体内的概率计算和量化证明超球体高维空间选点方法，进一步提出超球体-好格子选点法，实现超球体内生成均匀点集及概率计算；最后，以某在建高边坡工程为依托，建立边坡稳定性概率密度演化方程，采用本文改进方法对边坡稳定系数进行计算. 研究结果表明：在正常服役工况和降雨工况下，与10 000次蒙特卡洛模拟结果相比，边坡稳定系数的概率积分点最小误差分别为0.4%和0.2%，最大误差分别为2.9%和4.2%；在解决复杂高维结构随机问题时，本文提出的概率密度演化方程数值求解改进方法具有良好的计算精度和计算效率.Abstract:To improve the computational accuracy and efficiency of the probability density evolution method in analyzing high-dimensional stochastic responses of complex structures, a total variation diminishing (TVD) scheme using Van Leer’s flux limiter and a hypersphere-based uniform point selection method are proposed. First, for the TVD scheme in difference solving, the limitations of the Roe-Sweby flux limiter are analyzed. Van Leer’s flux limiter is selected to optimize numerical solution accuracy, and its effectiveness is verified through numerical examples. Second, based on existing mathematical findings, the probability calculation formulas for the n-dimensional standard normal distribution within hyperspheres and hypercubes are derived, by which the hypersphere-based high-dimensional point selection method is quantitatively validated. Furthermore, a hypersphere-based good lattice point method is developed to generate uniform point sets and perform probability calculations within hyperspheres. Finally, taking a high slope project under construction as a case study, a probability density evolution equation for slope stability is established, and the proposed improved method is applied to calculate the slope stability coefficient. The results show that under normal service conditions and rainfall conditions, compared with 10 000 Monte Carlo simulations, the minimum errors of the probability integral for the slope stability coefficient are 0.4% and 0.2%, and the maximum errors are 2.9% and 4.2%, respectively. When addressing high-dimensional stochastic structural problems, the proposed numerical solution method demonstrates satisfactory computational accuracy and efficiency.
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表 1 不同风化层性能参数
Table 1. Material properties of various weathered layers
边坡结构 γ/kN c/kPa $\varphi $/（°） 风化情况 平均值 变异
系数平均值 变异
系数平均值 变异
系数土状强风化 19.5 0.22 30 0.21 35 0.09 中等风化 27.0 0.01 2000 0.10 40 0.05 微风化 27.0 0.01 3500 0.06 42 0.03 
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表 2 正常工况下各概率积分点
Table 2. Probability integral points for normal conditions
方法 0.01 0.05 0.10 0.20 0.50 0.80 0.90 0.95 0.99 MCS 1.709 1.869 1.954 2.057 2.254 2.452 2.555 2.640 2.801 Van Leer 1.685 1.832 1.923 2.038 2.245 2.464 2.592 2.693 2.887 相差率/% 1.404 1.984 1.596 0.927 0.408 0.493 1.417 1.984 2.986
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表 3 降雨工况下各概率积分点
Table 3. Probability integral points for rainfall conditions
积分值 0.0001 0.001 0.01 0.02 0.05 0.1 0.5 0.90 0.99 MCS 1.246 1.379 1.611 1.667 1.755 1.835 2.144 2.487 2.832 Van Leer 1.248 1.375 1.542 1.605 1.705 1.799 2.139 2.520 2.865 相差率/% 0.246 0.266 4.277 3.676 2.841 1.958 0.243 1.335 1.157
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