According to the upper bound theory, any prescribed failure surface will be an upper bound to the true solution. For the critical failure surface which cor-responds to the global minimum, some of the difficulties and interesting phe-nomena in locating the critical failure surface will be discussed. Consider a one-dimensional function y=f(x) defined over a solution domain AB shown in Figure 3.1. The local minima where the gradients of the function are equal to 0 (f′(x) = 0) are given by points C and D, while the global minimum is defined by point E. If the y-ordinate of B is lower than the y-ordinate of E, point B will then be the global minimum, but the gradient of the function is not equal to 0 at B. Cheng (2003) has demonstrated that this situation can happen for a slope stability problem using an example from the ACADS (1989) study. For the multi-variable optimization analysis required by the slope stability problem, the factor of safety objective function is highly com-plicated, and the problem will be a complicated N-P hard type, which has attracted the attention of many researchers.
Another special feature about the critical failure surface for a simple slope is shown in Figure 3.2. There are only minor changes in the factors of safety if the trial failure surfaces fall within the shaded region as shown in Figure 3.2. In this respect, there is no strong need to determine the precise location of the critical failure surface if the geometry and ground conditions for a slope are simple. For complicated slopes or slopes with a soft band which will be illustrated in this chapter, it is however possible that a minor change in the location of the failure surface can induce a major change in the factor of safety. Under this case, the robustness of the optimization algorithm will be important for the success in locating the critical solution.
Failure surfaces can be divided into the circular and the non-circular failure surfaces. A circular failure surface is actually a sub-set of the non-circular
failure surface, but it is useful because: (1) some stability formulations apply to the circular failure surface only and (2) the critical circular failure surface is a good approximation to the critical solution for some simple problems and is simple to be evaluated. For the circular failure surface, the location of the critical failure surface is usually determined by the method of grids shown in Figure 3.3. There are three control variables in this case: x and y ordinates of the centre of rotation and the radius of the failure surface. Each grid point is used as the centre of rotation while different radii are considered for the circular failure surface, and the minimum factor of safety from different radii is assigned to this grid point. Different factors of safety are hence assigned to different grid points, and the trend of the global minimum can be assessed by drawing the factor of safety contours from the factors of safety associated with the grid points. This method is robust and is simple to operate, but the accuracy will depend on the spacing between the grid points. The specified Figure 3.1 A simple one-dimensional function illustrating the local minima and the
global minimum.
C D
E A B
y
x
Figure 3.2 Region where factors of safety are nearly stationary around the critical failure surface.
y
x Critical failure surface
grid must also be large enough to embrace all the possible local minima and the global minimum to obtain a clear picture about the distribution of the factor of safety. The grid method is simple to implement and is available in most of the commercial slope stability programs.
For the general non-circular failure surface, the number of control variables which is controlled by the number of points for the failure surface is usually much greater than three. To locate the critical failure surface, the geometric method similar to that for the circular failure surface will be very inefficient in application and requires a lot of effort in defining the solution domain for each control variable (though adopted by some commercial programs). Special features of the objective function of the safety factor F for this case include:
1 The objective function of the safety factor F is usually non-smooth, non-convex and discontinuous over the solution domain. Discontinuity of the objective function can be generated by: generation of an unacceptable failure surface;
‘failure to converge’ of the objective function; presence of obstructions in the form of a sheet pile, retaining wall, large boulders, a tension crack or others.
Gradient-type optimization methods are applicable only to the continuous function and will break down if there are discontinuities in the objective function.
2 Chen and Shao (1988) have demonstrated that multiple minima similar to that shown in Figure 3.3 will exist in general. Duncan and Wright (2005) have also shown the existence of multiple local minima even for a simple homogeneous slope which is also illustrated by Cheng et al.
(2007e). The local minimum close to the initial trial will be obtained by
Figure 3.3 Grid method and presence of multiple local minima.
the classical gradient-type optimization methods. If an initial trial close to the global minimum is used, the global minimum can usually be found by classical methods, but a good initial trial is difficult to be established for a general multi-variable problem. The success of a global optimization algorithm to escape from the local minima for an initial solution far from the global minimum is crucial in the slope analysis problem.
3 A good optimization algorithm should be effective and efficient over different topography, soil parameters and loadings. The analysis should also be insensitive to the optimization parameters as well.
Various classical optimization methods for the non-circular failure surface have been proposed and used in the past. Baker and Garber (1978) have proposed the use of the variational principle, but this method is complicated even for a simple slope and is not adopted for practical problems. Moreover, if the gradient of the global minimum is not zero, the variational principle will miss the critical solution. Chen and Shao (1988) and Nguyen (1985) have suggested the use of the simplex method for this problem which is actually suitable only for linear problems. The simplex method has been adopted by the program EMU, developed by Chen, and it works fairly well for simple prob-lems. The authors have however come across many complicated cases in China where manual interaction is required with the simplex method before a good solution can be found. The simplex method also fails to work automatically for cases where the local minimum and global minimum differ by a very small value but differ significantly in the location. Celestino and Duncan (1981) have adopted the alternating variable method while Arai and Tagyo (1985) and Yamagami and Jiang (1997) have adopted the conjugate-gradient method and dynamic programming, respectively. These classical methods are applicable mainly to continuous functions, but they are limited by the presence of the local minimum, as the local minimum close to the initial trial will be obtained in the analysis. There is also a possibility that the global minimum within the solution domain is not given by the condition that the gradient of the objective function ∇f =0, and a good example has been illustrated by Cheng (2003).
The presence of the other local minima or the global minimum will not be obtained by the classical methods unless a good initial trial is adopted, but a good initial trial is difficult to be established for a general problem.
In view of the limitations of the classical optimization methods, the current approach to locate the critical failure surface is the adoption of the heuristic global optimization methods. The term heuristic is used for algorithms which find solutions among all the possible ones, but they do not guarantee that the best will be found; therefore, they may be considered as approximate and not accurate algorithms. These algorithms usually find a solution close to the best one, and they find it fastly and easily. Another important feature is that the requirement of human judgement or interaction should be minimized or even eliminated if possible, and the authors have come across some hydropower projects in China where there are several weak zones (strong local minima) for which nearly all existing methods fail to work well.
Greco (1996) and Malawi et al. (2001) have adopted the Monte Carlo technique for locating the critical slip surface with success for some cases, but there is no precision control on the accuracy of the global minimum.
Zolfaghari et al. (2005) adopted the genetic algorithm while Bolton et al.
(2003) used the leap-frog optimization technique to evaluate the minimum factor of safety. All of the above methods are based on the use of static bounds to the control variables, which means that the solution domain for each control variable is fixed and is pre-determined by engineering experience. Cheng (2003) has developed a procedure which transforms the various constraints and the requirement of a kinematically acceptable failure mechanism to the evaluation of upper and lower bounds of the control variables, and the simulated annealing algorithm is used to determine the critical slip surface. The control variables are defined with dynamic domains which are changing during the solution, and the bounds are controlled by the requirement of a kinematically acceptable failure mechanism. Through such an approach, there is no need to define the pre-determined static solution domain to each control variable based on engineering experience, and a precision control during the search for the critical solution will be possible.
There are two major aspects in the location of the critical failure surface which will be discussed in the following sections, and they are the generation of the trial failure surface and the global optimization algorithms for the search for the critical failure surface.