3 Location of critical failure surface, convergence and other problems
3.4 Verification of the global minimization algorithm
The majority of the modern global optimization schemes have not been used in slope stability analysis in the past. The SA, SHM, MHM, PSO, Tabu and ant-colony methods were first used by Cheng (2003), Cheng et al. (2007b,e,f) and Cheng (2007) with various modifications to suit the slope stability problems. For the first demonstration of the applicability of these modern optimization methods, eight test problems are used to illustrate the effective-ness of Cheng’s (2003) proposal on the modified SA algorithm, and problems 4 and 8 are shown in Figures 3.16 and 3.17. Problems 1–3 are similar to problem 4 except for the external load. For problems 1–4 which are shown in Figure 3.16, in total there are two types of soils with a water table. In lem 1, there is no external load while the horizontal load is applied in prob-lem 2. Vertical load is applied in probprob-lem 3 while both vertical load and horizontal loads are applied in problem 4. Problems 5–7 are also similar to problem 8 except for the external load. For problems 5–8 which are shown in Figure 3.17, in total there are three types of soils, a water table and a perched water table. In problem 5, there is no external load while the horizontal load is applied in problem 6. Vertical load is applied in problem 7 while both verti-cal and horizontal loads are applied in problem 8. The cohesive strengths of Figure 3.15 Flowchart for the ant-colony algorithm.
Initialize the parameters: m, Na, Ni
The optimization problem is transformed into a graph; the equivalent pheromone is distributed on circles (shown in Figure 3.14)
Each ant starts from the initial point, chooses the values of control variables and goes back to the initial point after obtaining a solution
Updating the pheromone deposited on each circle
The above procedure is called one iteration, and altogether Ni iterations need implementation
The maximum length of subdivisions of all variables is lower than the pre-specified value: if yes, the algorithm stops; if not, the solid circles of each variable containing the maximum pheromone are equally divided into m elements
soils for problems 1–4 are 5 and 2 kPa, respectively, for soil 1 and soil 2 while the corresponding cohesive strengths for problems 5–8 are 5, 2 and 5 kPa.
The friction angles of soils for problems 1–4 are 35° and 32°, respectively, for soil 1 and soil 2 while the corresponding friction angles for problems 5–8 are 32°, 30° and 35°. The unit weight of soil is kept constant at 19 kNm−3in all these cases. It is not easy to minimize the factor of safety for these problems by a manual trial and error approach as the precise location of the failure sur-face will greatly influence the factor of safety. The minimum reference factors of safety are determined by an inefficient but robust pattern search approach.
To limit the amount of computer time used, the number of slices is limited to 5 in these studies and the slices are divided evenly.
The critical solution from the present study is shown in Figures 3.16 and 3.17 by ABCDEF. The x-ordinates of the left exit end A (4.0 for problems 1–
4 and 5.0 for problems 5–8) and right exit end F (14.0 for all problems) of the failure surfaces are fixed so that only the y-ordinates of B, C, D, E are variables (x-ordinates of B, C, D, E are obtained by even division). There are hence four control variables in the present study. Based on the critical result BCDE obtained from the minimization analysis (round up to two decimal
0 8 10 12
14 A
B C
D E
F soil 1
soil 2
16 18 20 22 24
2 4 6 8 10 12 14 16 18
Figure 3.16 Problem 4 with horizontal and vertical load (critical failure surface is shown by ABCDEF).
Source: Reproduced with permission of Taylor & Francis.
places), a grid is set up 0.5 m directly above and below B, C, D, E as obtained by the simulated annealing analysis. The spacings between the upper and lower bounds are hence 1.0 m for all the four control variables. The grid spac-ing for each control variable is 0.01 m so that each control variable can take 101 possible locations. The present grid spacing is fine enough for pattern search minimization and all the possible combinations of failure surfaces are tried, which are 101 × 101 × 101 × 101 or 10,406,041 combinations.
The factors of safety shown in Table 3.4 have clearly illustrated that the combined use of the failure surface generation and the simulated annealing method is able to minimize the factors of safety with high precision, and the results are similar to those obtained by a pattern search based on 10,406,041 trials. The location of the critical failure surface obtained from the simulated annealing analysis for problem 4 shown in Table 3.5 is very close to that obtained by the pattern search and similar results are also obtained for all the other problems. The results in Tables 3.4 and 3.5 have clearly illustrated the capability of the proposed modified SA algorithm in minimizing the factors of safety, so that the burden of engineers can be relieved by the adoption of mod-ern global optimization techniques. Besides the simulated annealing method, the other global optimization techniques as modified by Cheng’s methods (2007b) can also be worked with satisfaction for all these eight problems.
To illustrate the advantages of the present dynamic bound technique as compared with the classical static bounds to the control variables, the same problems are considered with static bounds analysis. The static bounds are
0
Figure 3.17 Problem 8 with horizontal and vertical load (critical failure surface is shown by ABCDEF).
Source: Reproduced with permission of Taylor & Francis.
defined as 0.5 m above and below the critical failure surface BCDE and the results are shown in Table 3.6 (the same minimum values are obtained from the two analyses). It is clear that the present proposal can greatly reduce the time of computation as compared with the classical simulated annealing tech-nique which is highly beneficial for real problems. This advantage is particu-larly important when the number of control variables is great.