﻿ 边坡稳定梯度法优化计算中新的差分格式与收敛准则 A New Difference Scheme and Convergence Criterion for Gradient Optimization Method for Slope Stability

Vol.04 No.04(2015), Article ID:16365,14 pages
10.12677/AAM.2015.44043

A New Difference Scheme and Convergence Criterion for Gradient Optimization Method for Slope Stability

Mengxi Wu1, Fanmin He1, Zhenggang Zhan2, Fuping Fan2

1Institute of Mechanics, Chinese Academy of Sciences, Beijing

2Powerchina Guiyang Engineering Limited, Guiyang Guizhou

Received: Oct. 29th, 2015; accepted: Nov. 14th, 2015; published: Nov. 19th, 2015

ABSTRACT

It is necessary to use optimization methods to find the most dangerous sliding surface for the safety factor of slope stability calculated by the limit equilibrium method. Gradient method is an accurate optimization method, however there may fail to find accurately the most dangerous sliding surface. An improved gradient optimization method with a descent difference scheme for the calculation of the direction vector is proposed. The descent difference scheme is superior to the central difference scheme both in accuracy and consuming time. The problem of wrong search direction occurring in the central difference scheme is dissolved in this scheme. The problem of convergence criterion used in the classical optimization method based on the gradient of the objective function is pointed out. A new convergence criterion for single variable optimization or for multivariable optimization along the gradient direction is proposed. The stability of three test examples is analyzed with a two-stage search method for circular slip surface. The gradient method combined with the descent difference scheme is an accurate and efficient method with an ability of avoiding to fall to a local minimum in a search process. The errors of search results in the test examples are less than the given convergence error. The proposed convergence criterion is appropriate.

Keywords:Slope Stability, Difference Scheme, Convergence Criterion, Optimization Method

1中国科学院力学研究所，北京

2中国电建集团贵阳勘测设计研究院，贵州 贵阳

1. 介绍

2. 改进的边坡稳定分析梯度优化方法

2.1. 基于目标函数梯度的改进优化方法

(1)

(2)

(3)

(4)

(a) (b)

Figure 1. Sketch of steepest descent search (a) traditional methods; (b) improved methods

(5)

(6)

2.2. 目标函数梯度方向计算的下降差分格式

(7)

(8)

(9)

(a) (b) (c)

Figure 2. Sketch of the descent difference scheme (a) descent forward; (b) descent backward; (c) descent neither forward nor backward

2.3. 收敛准则

(10)

Figure 3. Comparison of the true error with the difference of the two consecutive iterations

(11)

(12)

(13)

(14)

2.4. 圆弧滑动面优化变量

(15)

Figure 4. Variables defining a circular slip surface

(16)

(17)

2.5. 圆弧滑动面的两级搜索方法

(18)

3. 基于圆弧滑动算例的方法验证

Figure 5. Determination of single peak intervals

3.1. 经典算例介绍

3.2. 下降差分格式有效性验证

3.3. 两种差分格式搜索结果的比较

3个算例两种差分格式搜索结果的比较分别列于表5表6表7。下降差分格式不同搜索路径的终点和安全系数都比中心差分格式的结果更靠近。下降差分格式所得安全系数结果的最大差异在TE1、TE2

(a) (b)(c)

Figure 6. Configuration and mesh of the (a) TE1, (b) TE2 and (c) TE2

Figure 7. Contours of the vertical normal stress of TE1/kPa

Figure 8. Contours of the pore water pressure of TE3/kPa

Figure 9. Contours of the vertical normal stress of TE3/kPa

(a) (b)(c)

Figure 10. Search paths with several initial points for (a) TE1, (b) TE2 and TE3

Table 1. Geotechnical parameters for TE1

Table 2. Geotechnical parameters for TE2

Table 3. Geotechnical parameters for TE3

Table 4. Comparison of the factors of safety by this study and the reference answers

Table 5. Comparison of the search results with the two difference schemes in TE1

Table 6. Comparison of the search results with the two difference schemes in TE2

TE1和TE2中采用中心差分格式没有找到最小值的No. 2搜索路径终点的搜索信息如图11所示。两个例子中中心差分格式搜索落入一个等高线凹形区域O点，安全系数沿着中心差分格式计算出来的方向安全系数不下降。图中点A，B，C，D和O后面括号里的数字是这些点的安全系数。它们表明TE1中O点不是极值，TE2中O点靠近一个局部极值点。用下降差分格式的搜索路径在TE1中通过了O点附近区域，在TE2中在局部极值点跳出来了搜索得以继续。显然与下降差分格式结合的改进的梯度方法是精确的且有能力跳出局部极值，而与中心差分格式结合则不是精确的，甚至会陷入局部极值。

3.4. 收敛准则的验证

TE1、TE2、TE3几个收敛误差值下最小安全系数分别列于表8表9表10。不妨将等于0.0001的最小计算结果当作真实值，表中括号内的数值为大于0.0001时的计算误差。不同搜索路径所得结果的误差均小于给定误差标准。表明所给的收敛准则是合适的。

Table 7. Comparison of the search results with the two difference schemes in TE3

Table 8. Comparison of the minimum factors of safety under several convergence standards in TE1

Table 9. Comparison of the minimum factors of safety under several convergence standards in TE2

Table 10. Comparison of the minimum factors of safety under several convergence standards in TE3

(a) (b)

Figure 11. The search information in the dashed box in Figure 10 (a) and (b)

4. 总结和结论

A New Difference Scheme and Convergence Criterion for Gradient Optimization Method for Slope Stability[J]. 应用数学进展, 2015, 04(04): 343-356. http://dx.doi.org/10.12677/AAM.2015.44043

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