Mathematical Analyses and Numerical Simulation

It is difficult to study soil because the Problems of soil is complex. It is always inaccurate because of the layered, anisotropy, nonlinear, complex boundary conditions, complex loading conditions, the soil-water interaction, the soil-structure interaction and many simplifying assumptions. With the development of computing technology, it is possible to solve complex issues using numerical methods, so the soil numerical method becomes a new area of rapid development. There are also a number of ways in numerical solution. One of the most vigorous and advantageous one is the Finite Element Method (FEM). Finite Difference Method (FDM), Discrete Element Method (DEM) and adaptive finite element method were also widely used in geotechnical engineering.

Engineering problems that exhibit large scale discontinuous behaviour cannot be solved with a conventional continuum based procedure such as the FEM. The discrete element procedure is used to determine the dynamic contact topology of the bodies. It accounts for complex non-linear interaction phenomena between bodies and numerically solves the equations of motion. Since the DEM is a very computationally intensive procedure, many existing computer codes are limited to modelling either two-dimensional or small three-dimensional problems that employ simple body geometries.

In addition to numerical analysis, there are many mathematical methods have been used for soil analysis recent years. One of the greatest one is the probability and statistics method, which formed a new area--geotechnical stability reliability method. The Monte-Carlo method, for example, is a technique that involves using random numbers and probability to solve problems. The term Monte Carlo Method was coined by S. Ulam and Nicholas Metropolis in reference to games of chance, a popular attraction in Monte Carlo, Monaco (Hoffman, 1998; Metropolis and Ulam, 1949). The Monte Carlo method uses random or pseudo-random numbers to sample from probability distributions and, if sufficiently large numbers of samples are generated and used in a calculation such as that for a factor of safety, a distribution of values for the end product will be generated. The term ‘Monte Carlo’ is believed to have been introduced as a code word to describe this hit-and-miss technique used during secret work on the development of the atomic bomb during World War II (Harr 1987). Today, Monte Carlo techniques can be applied to a wide variety of problems involving random behaviour and a number of algorithms are available for generating random Monte Carlo samples from different types of input probability distributions. But this area is not ripe, there is still great room for development.