Gingivitis

For the classical gradient-type optimization method, once an initial trial is defined the refinement of the critical failure surface will be given by the gradient of the objective function (which can be obtained by a simple finite difference operation). On the other hand, for the heuristic global optimization methods, trial failure surfaces are required to be generated which are controlled by the bounds for each control variable. Different methods in generating the failure surfaces have been proposed by Greco (1996), Malkwai et al. (2001), Cheng (2003), Cheng et al. (2007b,e,f), Bolton et al. (2003), Li et al. (2005) and Zolfaghari et al. (2005). In general, these methods are very similar in the basic operations. The coordinates of the points defining the failure surface are taken as the control variables, and lower and upper bounds are assigned to each control variable. Consider the failure defined by ABCDEF shown in Figure 3.4.

If each control variable is defined over static lower and upper bounds, point D, which is unlikely to be acceptable for a normal problem, can be generated by the random number generator. Since segment CD will be a kink which hinders the development of the failure, D is highly unlikely to be acceptable except for some special cases which will be discussed later.

To generate a convex surface by the method proposed by Cheng (2003), consider a typical failure surface ACDEFB shown in Figure 3.5. The x-ordinates of the two exit ends A and B are taken as the control variables of the objective function and the upper and lower bounds of these two variables

are specified by the engineer (bounds for the first two control variables are fixed). The static bounds for the first two control variables can be defined easily for the present problem with engineering experience. Once the two exit ends A and B of the failure surface are defined, the requirements on the kinematically acceptable mechanism can be implemented as:

1 The x-ordinates of the interior points C, D, E and F of the failure sur-face can be obtained by the uniform division of the horizontal distance between A and B which is Xright–Xleft. The x-ordinates of C, D, E and F are hence not control variables. Alternatively, the division can be made to follow the slope profile and the x-ordinates of the interior points are also not control variables.

2 Points A and B are connected and C1 is determined as a point located vertically above C. The y-ordinate of C1 is the lower value of either: (1) the y-ordinate of the ground profile as determined by the x-ordinate of C;

(2) the y-ordinate of the point lying along the line joining points A and B and determined by the x-ordinate of C. C1 is the upper bound to the y-ordinate of the first inter-slice. The lower bound of the y-ordinate of C (third control variable) is set by Cheng (2003) as C1–AB/4. In fact, such a lower bound can allow for a deep-seated failure surface and is adequate for all the cases that Cheng has encountered. The lower bound of the y-ordinate of C can be set to C1–AB/5 (instead of C1–AB/4 which is a conservative estimation of the lower bound) in most situations without affecting the solution. The y-ordinate of point C is a control variable of the objective function and it is confined within the upper and lower bounds as determined in Step 2.

3 Once a y-ordinate of C is chosen in the simulated annealing analysis, it connects A and C and extrapolates the line to G which is defined by the x-ordinate of point D. The lower bound of the y-ordinate of point D will be point G to maintain a concave failure shape. The upper bound of D Figure 3.4 A failure surface with a kink or non-convex portion.

Source: Reproduced with permission of Taylor & Francis.

A

B

C D

E F

D

which is D1 is determined in the same way as for point C1. If part of the ground profile lies below the line joining B and C and affects the deter-mination of D1 (e.g. point J in Figure 3.5), it connects C and J instead of B and C and determines the upper bound as D2 instead of D1.

4 Perform Step 3 for the remaining points until all upper and lower bounds of the control variables are defined.

5 To allow for a non-concave failure surface which is unlikely to occur in reality, an option where the lower bound of point E will be set to a lower value as determined in Step 3 or the y-ordinate of point D is allowed. The y-ordinate of point E cannot be lower than that of D or else there will be a kink in the failure surface which prevents failure to occur. The lower bound to the y-ordinate is sometimes totally eliminated which is required for problems with a soft band. A non-convex failure surface can hence be generated from the present proposal by removing the lower bound requirement as required in the present method.

In Figure 3.5, the control variables are the x-ordinates of A and B the y-ordinates of points C, D, E and F. A control variable vector X is used to store these control variables and the order of the control variables must be in (X_A, X_B, Y_C, Y_D, Y_E, Y_F). For the location of the global minimum of the objective function, engineers need to define only the upper and lower bounds of the first two con-trol variables. An initial trial will be determined in a way similar to the Figure 3.5 Generation of dynamic bounds for the non-circular surface.

Source: Reproduced with permission of Taylor & Francis.

A

C1 D1

D2

D J

E

F B

Xright Xleft

G C

approaches shown above. The upper and lower bounds of the other control vari-ables will then be calculated according to Steps 2 and 3. If the number of slices is n, then the number of control variables will be n + 1. If rock is encountered in the problem, the lower bound determination shown above has to be modified slightly. In Steps 2 and 3, the lower bound will either be the y-ordinate of point G or the y-ordinate of the rock profile as determined by the x-ordinate of D.

For the circular failure surface, there are only three control variables which are the x and y coordinates of the centre of rotation and the radius of the failure sur-face. Cheng (2003) however adopts the x-ordinates of the two exit ends and the radius of the failure surface as the three control variables in analysis as it is eas-ier to define the upper and lower bounds for the two exit ends (see Figure 3.6).

This approach is also used by many commercial programs. The control variable vector X will be (X_A, X_B, r). For the lower and upper bounds of the radius, the lower bound is set to half of the length of line AB which is the minimum possi-ble radius. The upper bound of the radius is set to 50× AB (any value which is not too small will be acceptable). An unacceptable failure surface will not be gen-erated in the analysis and the constraints will control the lower and upper bounds of the radius when the two exit ends are defined. The constraints include:

1 The failure surface cannot cut the ground profile at more than two points within the two exit ends. As seen in Figure 3.6, point C will con-trol the upper bound of the radius.

2 The failure surface cannot cut into the rock stratum which will control the lower bound of the radius.

3 The y-ordinate of the centre of rotation is higher than the y-ordinate of the right exit end. For this case, the last slice cannot be defined. This constraint will also control the lower bound of the radius.

Figure 3.6 Dynamic bounds to the acceptable circular surface.

Source: Reproduced with permission of Taylor & Francis.

A

C

B

Critical circular arc

Unacceptable arc

In the present method, the first two variables which are the x-ordinates for the left and right ends are varied within the user defined lower and upper bounds which are constant during the analysis. Besides these two variables, the bounds for the remaining variables (y-ordinates of the failure surface) are com-puted sequentially according to the guidelines shown above for circular and non-circular failure surfaces. The bounds from the present method are dynamic and are different from the classical simulated annealing methods or other global optimization methods where the bounds remain unchanged dur-ing the analysis. The generation of trial failure surfaces and the search direc-tion will then proceed in accordance with the normal simulated annealing procedure and the global minimum can be located easily with a very high accu-racy under the present proposal. The minimization process in the present for-mulation will depend on the lower and upper bounds of the left and right exit ends shown in Figure 3.7, which can be decided easily with experience and engineering principle. For inexperienced engineers, a wide range can be defined for the lower and upper bounds and the number of trials required for analysis will only increase slightly with the increase in the left and right ranges, which is another major advantage of the approach by Cheng (2003). For example, Cheng found that when the ranges for the left and right exit ends are increased by two times, the number of trials required will remain unchanged in many cases and may increase by less than 15 per cent in some rare cases.

In the present algorithm, the x-ordinates are not considered as the control variables to reduce the number of control variables. This is usually satisfactory as Cheng (2003) found that the y-ordinates are more important than the x-ordinates in the factor of safety. Cheng et al. (2008b) have also proposed that the x-ordinates can be adopted as the control variables. This approach will approximately double the number of control variables, and is considered to be useful only for those problems controlled by a soft band where the factor of safety is highly sensitive to the x-ordinates as well.

Figure 3.7 Domains for the left and right ends decided by engineers to define a search for the global minimum.

Right bound

Left bound