Locating critical sliding surface of slopes by estimation of distribution algorithm
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摘要:
在求解边坡临界滑动面问题的优化算法中，大多都存在结构复杂、参数取值困难或寻优效果差的缺点。为此，将基于高斯分布模型的分布估计算法与基于简化Bishop法的滑面安全系数计算模型相结合，建立具有简单生物协同和竞争思想的临界滑动面搜索新方法；然后，针对3自由度问题，设计了一种局部搜索方法，以弥补分布估计算法局部搜索性能差的劣势。将标准方法和改进方法分别应用于边坡断面复杂度依次递增的3道算例中，完成对标准方法正交试验结果的极差分析和多因素方差分析，并实现标准算法与改进算法计算结果的对比分析。结果表明：（1）标准分布估计算法可以计算出边坡的临界滑动面；（2）当算例简单时，各控制因素都不会对计算结果产生显著性差异；当算例复杂时，只有种群规模呈现出显著性；（3）改进算法比标准算法的计算结果更佳、搜索速度更快，且有效降低了种群规模大小对计算结果的影响。探索分布估计算法在边坡临界滑动面搜索问题中的应用，为解决该类问题开辟出了一条新的研究途径。初步校验表明，该模型鲁棒性更好，具有广阔的应用前景。
Abstract:Most of the optimization algorithms to solve the slope critical sliding surface have the disadvantages of complex structure, difficult to determine the parameter value, and poor optimization effect. This study introduced the estimation of the distribution algorithm based on the Gaussian distribution model, and combined with the sliding surface calculation and analysis model using the simplified Bishop method, to establish a new critical sliding surface search method with simple biological collaboration and competition ideas; secondly, a local search method for the 3-degrees of freedom was designed to compensate for the poor local search performance of the estimation of distribution algorithm. The standard and improvement methods were applied to the three calculation examples of increasing slope section complexity, respectively. The orthogonal experimental results from the standard method were validated by range analysis and multivariate analysis of variance, and the comparative analysis of the calculation of the standard algorithm and the improved algorithm was conducted. The results show that the standard estimation of distribution algorithm can be used to calculate the critical sliding surface of slopes. When the calculated case is simple, the control factors have limited influence on the calculated results; when it is complex, the population size has a significant influence. Compared to the standard algorithm, the improvement algorithm has better calculation and faster speed, and can effectively reduce the impact of the population size on the calculation. The preliminary verification shows that the model is more robust and has a broad application prospect. This study provides a new insight to explore the application of the distribution estimation algorithm in the slope critical sliding surface.
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表 1 正交试验结果
Table 1. Orthogonal experimental results
试验号 控制因子 安全系数 n m/n α β 算例一 算例二 算例三 1 60 1/2 0.6 0.1 0.985 3 1.395 7 1.755 6 2 60 1/3 0.7 0.2 0.985 3 1.395 5 1.756 6 3 60 1/4 0.8 0.3 0.985 5 1.395 5 1.757 2 4 60 1/5 0.9 0.4 0.985 3 1.396 0 1.758 9 5 120 1/2 0.7 0.3 0.985 4 1.395 9 1.757 2 6 120 1/3 0.6 0.4 0.985 5 1.395 6 1.756 6 7 120 1/4 0.9 0.1 0.985 3 1.395 5 1.755 6 8 120 1/5 0.8 0.2 0.985 5 1.395 7 1.756 4 9 240 1/2 0.8 0.4 0.985 3 1.395 6 1.756 8 10 240 1/3 0.9 0.3 0.985 4 1.395 7 1.756 1 11 240 1/4 0.6 0.2 0.985 3 1.395 7 1.755 9 12 240 1/5 0.7 0.1 0.985 3 1.395 5 1.755 8 13 480 1/2 0.9 0.2 0.985 3 1.395 3 1.756 1 14 480 1/3 0.8 0.1 0.985 3 1.395 4 1.755 1 15 480 1/4 0.7 0.4 0.985 3 1.395 6 1.755 9 16 480 1/5 0.6 0.3 0.985 3 1.395 5 1.756 2 
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表 2 各算例临界滑面统计
Table 2. Critical slip surfaces for each calculated case
算例 分类 FS Xs/m Xe/m Yo/m 算例一 maxFS 0.985 5 −0.000 4 21.314 4 28.119 2 minFS 0.985 3 0.000 3 21.265 4 28.506 8 均值 0.985 4 0.001 0 21.298 2 28.585 5 标准差 8.165 0e-05 0.007 5 0.031 0 0.269 1 算例二 maxFS 1.396 0 −0.025 2 20.961 3 17.855 1 minFS 1.395 3 −0.002 7 21.037 3 17.655 7 均值 1.395 6 −0.036 0 21.040 7 17.718 0 标准差 1.769 0e-04 0.022 3 0.048 8 0.145 5 算例三 maxFS 1.758 9 −2.305 7 20.740 1 12.127 4 minFS 1.755 1 −1.948 0 20.273 1 11.805 2 均值 1.756 4 −2.108 2 20.418 2 11.847 2 标准差 0.000 8 0.281 3 0.242 5 0.371 4
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表 3 多因素方差分析结果
Table 3. Results of multivariate analysis of variance
算例 因子 平方和 df 均方 F p 显著性排序 算例一 种群规模 3.50×10−8 3 1.17×10−8 1.400 0 0.394 4 1 精英比例 5.00×10−9 3 1.70×10−9 0.200 0 0.890 4 4 学习因子 1.50×10−8 3 5.00×10−9 0.600 0 0.657 5 3 变异因子 2.00×10−8 3 6.70×10−9 0.800 0 0.570 6 2 残差 2.50×10−8 3 8.30×10−9 算例二 种群规模 1.37×10−7 3 4.56×10−8 0.695 2 0.613 8 1 精英比例 3.69×10−8 3 1.23×10−8 0.187 3 0.898 8 3 学习因子 1.69×10−8 3 5.60×10−9 0.085 7 0.963 2 4 变异因子 8.19×10−8 3 2.73×10−8 0.415 9 0.755 0 2 残差 1.97×10−7 3 6.56×10−8 算例三 种群规模 1.06×10−5 3 3.53×10−6 8.572 3 0.037 9 1 精英比例 1.33×10−6 3 4.42×10−7 1.072 9 0.477 6 3 学习因子 7.20×10−7 3 2.40×10−7 0.583 0 0.665 7 4 变异因子 5.14×10−6 3 1.71×10−6 4.158 0 0.136 1 2 残差 1.24×10−6 3 4.12×10−7
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表 4 临界滑面的统计表
Table 4. Statistical values of the critical slip surface
计算方法 n Xs/m Xe/m Yo/m FS 标准分布
估计算法9 −2.614 1 20.169 8 12.037 5 1.759 9 18 −2.541 3 20.657 4 12.461 4 1.759 7 36 −2.133 8 20.614 9 12.001 3 1.756 6 72 −2.162 4 20.450 5 11.863 8 1.755 7 改进分布
估计算法9 −2.198 2 20.406 6 11.805 4 1.754 7 18 −2.243 2 20.368 3 11.8322 1.754 8 36 −2.138 1 20.510 6 11.7997 1.755 0 72 −2.264 2 20.289 6 11.804 1 1.754 6
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