*5.1.1 Continuum methods*

The different continuum methods of design are as under.

input parameters. This method computes separately the essential information in the solution domains from the information at the boundary, which is achieved by the solution of boundary integral equation rather than direct solution of PDEs [26].

Following are the different Hybrid continuum/discontinuum methods of design:

This method of design was developed by Clough et al., (1950). Due to wide application of this method in mining engineering especially tunneling, it get more attention for solving mining problems and popularity in this field [19]. The FEM divide problem into small parts and connect these parts at a point/nodes at the apexes and at the boundaries of meshing/discretization. The FEM has many applications in modeling in rock engineering design due to dealing with nonlinearity,

The unidentified function over each element in FEM estimated through test function having its nodal values of anonymous system (in polynomial form). This practice is the fundamental supposition of FEM. For experimental function, it is mandatory to satisfy the principal of PDFs. In this research the FEM based software Phase2 was used for analysis of stresses and total displacement around tunnel. For experimental

*j*¼1

*j*¼1 *f e*

*Nij* is the shape function or interpolation function; this must be defined into inherent coordinates for use of Gaussian quadratic integration, *M* is the element order. Using shape function the problem original PDFs can be substituted by the

*Nij*u*<sup>e</sup>*

*<sup>i</sup>* (7)

*<sup>i</sup> or Ku* ¼ *F* (8)

function it must be satisfied the principal of PDFs, which is given in Eq. (7).

u*e <sup>i</sup>* <sup>¼</sup> <sup>X</sup> *M*

The different discontinuum methods of design are given below.

*5.1.2 Discontinuum methods*

1.Discrete Element Method (DEM)

*Design Techniques in Rock and Soil Engineering DOI: http://dx.doi.org/10.5772/intechopen.90195*

2.Discrete Fracture Network (DFN)

*5.1.3 Hybrid continuum/Discontinuum*

1.Hybrid FEM/BEM methods

2.Hybrid DEM/DEM methods

3.Hybrid FEM/DEM methods

4.Other hybrid method/models

**6. Finite element method (FEM)**

arithmetical equation as given below.

X *N*

*Ke ij* h i <sup>u</sup>*<sup>e</sup> i* � � <sup>¼</sup> <sup>X</sup> *N*

*j*¼1

Where,

**69**

boundary conditions and heterogeneity problems [26, 27].

**Figure 7.**

*Division of numerical models and methods [24, 25].*


#### *Finite Difference Method (FDM).*

The Finite difference method (FDM) is the direct calculation of PDEs and transmitted the creative PDEs in term of unknown at grid point into a system of algebraic equations by interchange the fractional derivatives with difference at irregular or regular grid forced over problem areas. This system is solved due to establishing the required initial and boundary condition. This method is old but widely applied in the numerical modeling in rock mechanics. This method is based for explicit approach of discreet element method (DEM) [26].

#### *Finite Element Method (FEM).*

The Finite element method (FEM) splits the problem into sub-elements of smaller sizes and shapes with fitting the number of nodes at the vertices and at the side of discretization. FEM is mostly used to estimate the behavior of PDEs at elemental level and for signifying the behavior of elements; it produces the local algebraic equation. After creating the local equation the FEM gathered it according to topographic relation of node and elements and further put it into worldwide system of algebraic equation for receiving the required information after establishing the definite initial and boundary situations.

#### *Boundary Element Method (BEM).*

The Boundary element method is the precise method then FEM and FDM because of its easiness. This method involves the discretization of solution areas at boundary and thus decreases the problem dimension by simplifying the design

input parameters. This method computes separately the essential information in the solution domains from the information at the boundary, which is achieved by the solution of boundary integral equation rather than direct solution of PDEs [26].

*5.1.2 Discontinuum methods*

The different discontinuum methods of design are given below.

