Presented at the 7th European Conference on Numerical Methods in Geotechnical Engineering, Trondheim, 2010
1 INTRODUCTION
Several computational programs are available, with which the stability of a soil body can be calculated with a limit equilibrium method. In such a program, a slip surface is analyzed with a certain methodology, for example, the method “Bishop” (Bishop 1995) to determine its stability. The user enters an area in which the program needs to find the circle with the minimal stability factor.
Searching such a space usually happens by calculat-ing all possible slip circles with correspondcalculat-ing tan-gent lines and reporting the one with the minimal safety. This algorithm has several disadvantages: • It is sequential and therefore time consuming • There is no guarantee the (global) minimal safety factor will be found.
• A small displacement or change in boundary condi-tions of the grid can lead to fundamentally different answers.
• A small change in boundary conditions can lead to fundamentally different answers.
• Much experience and understanding of the method is required, even though this is not obvious.
Other search routines (for example hill climbing) have great disadvantages as well. In the recent past Genetic Algorithms (Barricelli, Nils Aall 1957) are used more frequently as a search procedure and it seems to be a well-suited method to find the repre-sentative slip plane with the minimal safety factor. For “Flood Control 2015”, a genetic algorithm (GA) has been implemented in the stability program MStab.
Genetic algorithms process a mathematical represen-tation of a solution of an analyzed problem. For Bishop’s method, this representation is a vector con-taining the X and Y value of the centre of the circle, and the radius of the circle. This representation can be seen as an individual and a sum of individuals for a population. An individual can be tested for its fitness, for example with Bishop’s method.
The genetic algorithm improves the quality of a population is a similar way as nature does. Two indi-viduals cross their DNA, there is a chance for muta-tions and a new individual is created. Two new indi-viduals fight, and the fittest one continues to the next generation.
The algorithm seems to be faster and better at finding a global minimum. A disadvantage is that the results are not always reproducible. On top of that, there will be a very strong tendency to find the global minimum, while sometimes, a local minimum is inter-esting as well. This can be overcome using penalties steering the result in the desired direction. Because of its high speed, a genetic algorithm makes it possi-ble to find a free slip surface with Janbu’s or Spencer’s method.
This paper will present in the second section that the GA fundamentally works using an analytical simplifi-cation of Bishop’s formula. The next section shows the efficiency of the GA by comparing calculation time and accuracy of a grid based method to the gorithm. Thereafter, the efficiency of the genetic al-gorithm is explained. Finally, it is shown that a GA can perform a free surface search using Spencer’s method.
A Genetic Algorithm for Solving Slope Stability Problems:
from Bishop to a Free Slip Plane
R. van der Meij & J. B. Sellmeijer
Deltares, Delft, Netherlands
ABSTRACT: Finding the safety factor of an embankment using a limit equilibrium method requires a search algorithm to find the representative slip circle. Because of the complex solution space, a grid based method is most often preferred. This paper presents a genetic algorithm as an alternative. This genetic algorithm gives ac-curate results faster then a traditional grid based method. Because of its efficiency, the genetic algorithm is even able to find a free slip surface using Spencer’s method with the lowest safety factor.
An analytical formulation of Bishop’s method is de-rived for a simplified embankment in order to have an analytical safety factor to test the genetic algorithm. Only the crest, slope and surface level of an em-bankment will be considered, as can be seen in Figure 1. H is the height of the embankment, L is the length of the slope. Different angles defining the slip circle are defined with 0 through 3. The location of the
centre of the slip circle is defined with X and Y using the outer crest as a reference point. The radius of the circle is defined as R. The subsoil is divided in three area’s. Area I is underneath the crest, area II is un-derneath the slope and area III is unun-derneath the ground level.
Figure 1: Slip circle entering in zone I and exiting in zone III
For the purpose of simplicity, no water pressures are considered. The soil is cohesive and homogenous without internal friction. The explicit result depends on the zone (I, II or III) where the circle enters and exits the soil body. In total, four types or circles can be distinguished. A circle that enters through the crest and exits on the surface level, as shown in fig-ure 1, has the safety factor of which the result of the derivation is shown in equation (1).
The safety factor of the circle that enters through the crest and exits in the slope of the embankment is given in equation (2) and the safety factor of a circle that enters through the slope and exits on the surface level is given in equation (3). Finally, the safety fac-tor of a circle that enters and exits in the slope of the embankment is given in equation (4).
To calculate the safety of the embankment, one first needs to check which case is relevant, and the safety factor can be calculated directly. These formula’s are programmed in Matlab to compare Matlab’s genetic algorithm with the genetic algorithm we wish to im-plement in the stability program MStab.
(1)
(2)
(3)
(4)
Figure 2: Solution space of Bishop’s equation above a slope
Figure 2 shows the solution space for a fixed radius. Matlab has a complex GA tool. Because it is difficult to understand and reproduce, a simple GA specifi-cally built to minimize the above equation is pro-grammed as well. This GA is called the “MStab GA” as it will be used in MStab in the future. Because of the chaotic convergence procedure of a GA, 10000 runs have been performed to analyze the precision. “Pop” stands for the size of the population, “Gen” stands for the number of generations. The average value of the optimum and its standard deviation are presented in Table 1. 0 1 2 3 X Y R H L I II III 0 1 2 3 X Y R H L I II III 2 3 0 reaction 2 2 2 1 2 2 1 1 soil g 12 2 2 2 R M c F M H R H L H Y L X 2 2 0 2 2 2 1 2 2 2 1 1 g 12 2 2 2 1 R c F H F R F H L H F Y L F X 2 2 0 2 2 2 1 2 2 2 1 1 g 12 2 2 2 1 R c F H F R F H L H Y H F L X L F 2 2 0 2 2 2 1 g 6 2 1 R c F H F F f H L
Presented at the 7th European Conference on Numerical Methods in Geotechnical Engineering, Trondheim, 2010
Simulation Matlab GA MStab GA
Pop Gen average St. dev. Avg. St. dev.
50 50 2.7828 4.30E-3 2.7790 1.6192e-4
100 100 2.7792 5.26E-4 2.7789 3.1659e-5 Table 1: Average and standard deviation of the results of the Matlab GA and the MStab GA.
A population of 50 individuals running 50 genera-tions seems to be sufficient to get an answer with less then 1% error. The precision of the methods is alike although the deviation of the MStab GA is an order of magnitude lower.
3 IMPLEMENTATION OF BISHOP’S AND VAN’S METHOD IN MSTAB
The “MStab GA” as mentioned previously is imple-mented in the stability program MStab in order to find the representative slip circle. The grid and GA are compared with the limit equilibrium methods Bishop and Van (Van 2001).
Figure 3: Representative slip circle using grid and tangent lines
Figure 3 on the previous page shows the representa-tive slip circle found with the grid search algorithm, figure 4 below shows the representative slip plane found with the GA. Figure 5 shows the slip plane found with the Van’s method.
Figure 4: Representative slip circle found with a GA
Table 2 compares the calculation time of the different search algorithms with Bishop’s method. The repre-sentative circle is found each time because it is al-ready contained in the initial small search area. The calculation time of the grid method is directly pro-portional to the size of the grid. The calculation time of the GA only depends on the population size and the number of generations, so it does not vary.
Figure 5: Representative slip plane method Van
BHP Grid Small GA small Large grid Large GA full GA Calc. time[s] 2,5 5,0 31 5,0 5,0 f [-] 1,10 1,10 1,10 1,10 1,10
Table 2: Calculation time grid versus GA with increasing search area.
One can see that for a small search area the grid method is the quickest. As the search area increases, the grid becomes relatively slower. This phenomenon is amplified with Van’s method as the search space is more complex. Van Small grid Small GA Larger grid Larger GA Large grid Large GA Full GA Calc. t [s] 4,8 17 19,6 16 263,4 13,5 10,5 f [-] 1,11 1,12 1,09 1,08 1,08 1,08 1,09
Table 3: Calculation time grid versus GA with increasing search area.
The grid method is only faster if the user specifies the location of the slip plane very well. If the search area increases, the GA becomes relatively faster. Abso-lutely, the calculation time also decreases. This is be-cause more geometrically impossible slip planes are in the population and therefore not analyzed. The faster calculation leads to less precision but Table 3 shows it is still sufficient. Searching the entire area is impossible with a grid method and can be performed rapidly with the GA.
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The calculation time of a grid based method is a function of the calculation time of a single analysis times a * b * c (see figure 5)
Figure 5: combination of calculations for bishop analysis
The optimization procedure of a GA is fundamentally different. In each dimension, a near value needs to be selected and through a number of generations (n) the right combination will be found. For bishop’s method, n * (a+b+c) calculations need to be per-formed for the optimization. Earlier in this paper, it has been shown that 50 is a good value for n.
Van’s analysis (Figure 6) uses 5 parameters to de-scribe the slip circle. A grid based method uses a*b*c*d*e calculations. A GA based method uses n * (a+b+c+d+e) calculations. With an increasing search area and more search dimensions, the GA be-comes a more efficient alternative.
Figure 7 shows an approach for an analysis of a free slip plane. An upper and a lower bound of the slip
Figure 6: combination of calculations for Van’s analysis
plane is defined, and in between 13 straight lines are defined. Including the surface lines, 15 points on these lines have to be found that, together, have the lowest safety factor.
Figure 7: Approach for a free slip plane
Assuming we allow 10 points per line, with a grid based method, 1015 calculations have to be per-formed. With a GA based method, n * (10+10 + 10 …) = 150 * n calculations need to be performed. This makes a free surface search feasible. Most other search algorithms have the curse of dimension (Bell-man 1957) whereby the calculation time exponen-tially increases with the number of degrees of free-dom in the problem. Because the search time increases with the sum of the number of degrees of freedom, this curse is overcome.
5 FREE SLIP SURFACE SEARCH
Figure 7 shows an approach for a free surface search. As a limit equilibrium method, one can choose for example Janbu’s or Spencer’s method. In this case, Spencer’s method is chosen. An upper and lower boundary is defined with 15 points. The first point is connected through the surface line on the crest, the second through 14th point is connected with a straight line in between, and the last point is again connected by the surface line. The genetic algorithm must find the combination of points on the lines that has the lowest safety factor.
The optimization is by far not as straightforward as in Bishop’s method. Bishop will always be able to calculate a safety factor given a centre for the circle and a tangent line. Spencer is not able to produce a safety factor if a sudden increase of the slip surface slope comes across. There are two fundamental ways of addressing this issue. The unrealistically high pas-sive earth pressures can be cut off in such a case by the limit equilibrium method. This is common prac-tice in Bishop’s method. Alternatively, unrealistic slip planes can also be avoided when defining the ge-nome. This issue has not yet been addressed, but as the method is very robust, it already works.
Figures 8, 9 and 10 present the representative slip plane of respectively a Bishop, Van and Spencer analysis. One can see that as the shape of the slip plane becomes more complex, the safety factor de-creases.
Presented at the 7th European Conference on Numerical Methods in Geotechnical Engineering, Trondheim, 2010
Figure 8: slope stability calculated with Bishop’s Method, f=1,08
Figure 9: slope stability calculated with Van’s Method, f=1,06
tom sand layer, the slip plane tends to be deep and long. It is difficult to describe this surface with a cir-cle, and therefore Bishop’s method gives a relative high safety factor of 1,08. Van’s method is designed to analyze such problems and consequently gives a lower safety factor of 1,06.
The fact that Spencer’s method combined with the genetic algorithm gives a significant lower safety fac-tor of 0,97 is remarkable. Especially, if one takes into account that the passive shear force is cut off in Bishop’s and Van’s method, but not in Spencer’s method. If this cut off is also implemented in Spencer’s method, the safety factor will be lower and the passive wedge can exit more steeply.
CONCLUSIONS
A Genetic Algorithm is an optimization procedure to find the representative slip circle that has several ad-vantages above a grid based method. First, the ge-netic algorithm can find the correct minimum, even if the solution space is very complex. The method is good at finding the global minimum, even if there are several local minima.
Even though the algorithm does not converge di-rectly via the same path to the solution, the standard deviation of the solution is relatively small and there-fore reliable.
same amount of time. One can also choose to have a quick answer with a relative good precision in very little time. The time of an analysis is known in ad-vance as the number of generations are fixed. This makes it a good procedure when many automated calculations are performed.
The genetic algorithm theoretically works for all limit equilibrium methods. Its relative efficiency increases with a larger search space and also with a larger number of parameters to be optimized. With Van’s method, the genetic algorithm is in general faster then a grid based method. Finding a free slip plane using a grid based method is not possible whereas the efficiency of the genetic algorithm does make it fea-sible as the genetic algorithm overcomes the curse of dimension.
An analysis based on a free slip plane gives a signifi-cantly lower factor of safety with a better limit equi-librium model.
REFERENCES
Barricelli, Nils Aall (1957). "Symbiogenetic evolution proc-esses realized by artificial methods". Methodos: 143–182 Bellman, R.E. (1957). Dynamic Programming. Princeton
Uni-versity Press, Princeton, NJ.
Bishop, C. M. (1995). Neural Networks for Pattern Recogni-tion. Oxford University Press, ISBN 0-19-853864-2 Bishop, W. (1955). “The use of the slip circle in the stability
analysis of slopes”. Geotechnique, Vol 5, 7-17.
Van, M. A. (2001). “New approach for uplift induced slope failure”. XVth International Conference on Soil Mechanics and Geotechnical Engineering, Istanbul. 2285-2288
