In this chapter, the basic formulation of the distinct element method is introduced, which will be followed by more discussions on the blocky and particle formulation of the DEM. This will be followed by some applications of the DEM to several types of problems. As the alternatives to the DEM, DDA and manifold for twodimensional and three-dimensional problems will be introduced with case studies. There are also many new numerical methods that have evolved in the last twenty years. For example, some recent continuity based methods as such as the SEM, SPH and MPM will also be introduced in this chapter, with applications to some slope failure problems in Hong Kong.

Many engineering problems can be formulated as an equivalent optimization problem. In fact, the popular finite element method is a form of optimization problem, where local interpolation function is used to represent the local effect within an element. In limit equilibrium and limit analysis, the use of global optimization search for the critical slip surfaces and the failure mechanism is required for many slope stability, lateral earth pressure and bearing capacity problems. For pile driving back analysis, the whole analysis is just actually an optimization process. In this chapter, some of the classical gradient type, as well as the modern heuristic optimization methods will be introduced.

This chapter will give an overview about the application of reliability analysis in geotechnical engineering problems, and slope stability problem will be chosen for the illustration.