Presented at the 7th European Conference on Numerical Methods in Geotechnical Engineering, Trondheim, 2010
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 accurate 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.
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 calculating all possible slip circles with corresponding tangent 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 conditions 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 representative 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 representation of a solution of an analyzed problem. For
Bishop’s method, this representation is a vector containing 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 individuals cross their DNA, there is a chance for mutations and a new individual is created. Two new individuals 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 interesting 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 possible 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 simplification 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 algorithm. Thereafter, the efficiency of the genetic algorithm is explained. Finally, it is shown that a GA
can perform a free surface search using Spencer’s
method.
2 TESTING THE GA IN MATLAB
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 figure 1, has the safety factor of which the result of the
derivation is shown in equation (1).
An analytical formulation of Bishop’s method is derived 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 embankment 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 underneath the slope and area III is underneath the
ground level.
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 factor of a circle that enters and exits in the slope of the
embankment is given in equation (4).
X
0
1
2
Y
3
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 implement in the stability program MStab.
L
I
H
R
II
III
Figure 1: Slip circle entering in zone I and exiting in zone III
F
F
F
M reaction
M soil
c 1
2
gH F R
c 1
2
gH F R
2
c
2
gH R
1
12
1
12
F2 H2
H2
L2
2
2
L2
2
1
12
F
F2 H2
L2
c 1
gH F
1
6
3
1
2
2
2
2
F
f
HF
0
H Y
2
H Y
2
1
2
L X
(1)
2
R2
0
1
2
R2
0
2
Y
1
2
LF
X
2
(2)
2
(3)
R2
1
2
HF
0
R2
H2
2
L2
L X
1
2
LF
(4)
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 specifically built to minimize the above equation is programmed 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.
Figure 2: Solution space of Bishop’s equation above a slope
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 generations 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 implemented 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).
Table 2 compares the calculation time of the different
search algorithms with Bishop’s method. The representative circle is found each time because it is already contained in the initial small search area. The
calculation time of the grid method is directly proportional 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.
Materials
zanddijk
kleidijk
zandklei
humeklei
zand
zanddijk
kleidijk
T1
humeklei
zandklei
humeklei
humeklei
Figure 5: Representative slip plane method Van
Materials
zanddijk
kleidijk
zandklei
humeklei
zand
BHP
Calc.
time[s]
f [-]
Grid
Small
GA
small
Large
grid
Large
GA
2,5
5,0
31
5,0
5,0
1,10
1,10
1,10
1,10
1,10
full GA
Table 2: Calculation time grid versus GA with increasing
search area.
zanddijk
kleidijk T1
humeklei
zandklei
humeklei
humeklei
zand
Figure 3: Representative slip circle using grid and tangent
lines
Figure 3 on the previous page shows the representative 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.
Materials
zanddijk
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
Calc.
t [s]
f [-]
Small
grid
Small
GA
Larger
grid
Larger
GA
Large
grid
Large
GA
Full
GA
4,8
17
19,6
16
263,4
13,5
10,5
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.
kleidijk
zandklei
humeklei
zand
zanddijk
kleidijk T1
humeklei
zandklei
humeklei
humeklei
zand
MStab 9.10 : HELL1Amstab82bhp.sti
Figure 4: Representative slip circle found with a GA
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. Absolutely, the calculation time also decreases. This is because 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.
4 CHOICE OF GA VERSUS GRID METHOD
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 7: Approach for a free slip plane
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 performed 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 describe 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 becomes 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
Assuming we allow 10 points per line, with a grid
based method, 1015 calculations have to be performed. 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 (Bellman 1957) whereby the calculation time exponentially increases with the number of degrees of freedom 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 passive earth pressures can be cut off in such a case by
the limit equilibrium method. This is common practice in Bishop’s method. Alternatively, unrealistic slip
planes can also be avoided when defining the genome. This issue has not yet been addressed, but as
the method is very robust, it already works.
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.
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 decreases.
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
Figure 10: slope stability calculated with the genetic algorithm and Spencer’s Method, f=0,97
Because of the high pore water pressures in the bottom sand layer, the slip plane tends to be deep and
long. It is difficult to describe this surface with a circle, 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.
A much larger search space can be investigated in the
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 advance as the number of generations are fixed. This
makes it a good procedure when many automated
calculations are performed.
The fact that Spencer’s method combined with the
genetic algorithm gives a significant lower safety factor 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.
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 feasible as the genetic algorithm overcomes the curse of
dimension.
CONCLUSIONS
A Genetic Algorithm is an optimization procedure to
find the representative slip circle that has several advantages above a grid based method. First, the genetic 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 directly via the same path to the solution, the standard
deviation of the solution is relatively small and therefore reliable.
An analysis based on a free slip plane gives a significantly lower factor of safety with a better limit equilibrium model.
REFERENCES
Barricelli, Nils Aall (1957). "Symbiogenetic evolution processes realized by artificial methods". Methodos: 143–182
Bellman, R.E. (1957). Dynamic Programming. Princeton University Press, Princeton, NJ.
Bishop, C. M. (1995). Neural Networks for Pattern Recognition. 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