SUMMARY, CONCLUSIONS AND FUTURE SCOPE
CHAPTER 3 NUMERICAL MODELLING
3.3. GEOMETRY AND MESHING
Griffiths and Fenton (2004) also showed that in case of Normal distribution, local averaging leads to reduction in variance but the mean is not affected. However, local averaging reduces both the mean and the standard deviation in case of a Lognormal distribution. This is attributed to that fact that the mean and variance of a Lognormal distribution are influenced by both the mean and variance of the underlying Normal distribution (Griffiths and Fenton, 2004; Fenton and Griffiths, 2008; Chok, 2009)
dividing a large domain into smaller pieces, working and analyzing the smaller pieces, and then connecting the smaller pieces together to obtain the behavior of the whole domain, similar to the fundamental concept of finite elements. The regions may be simple straight-sided shapes like quadrilaterals or triangles or a free form, multi-sided polygon.
Discretization or meshing is one of the most fundamental aspects of finite element modeling, which is used to classify any given region, surface, or line, into smaller parts to address the global response of the same from localized responses. GeoStudio has its own system and algorithms for meshing, which are designed specifically for the analysis of geotechnical and geo-environmental problems. The default scheme of meshing is fully automatic and there is no need to draw individual
“finite elements” as a default mesh would be generated once the geometry regions are specified.
However, the generated default mesh can be altered globally or locally according to the demand of the problem being analyzed.
One of the main features of a finite element are its constituent nodes. The nodes are used to describe the distribution of the primary unknowns within the element. All finite element equations are formed at the nodes. In a finite element formulation, it is necessary to adopt a model describing the distribution of the primary variable within the element (e.g., total head). The distribution could be linear or curved. For a linear distribution of the primary unknown, the nodes are required only at the corners of the element. With three nodes defined along an edge, a quadratic or any other higher order equation can be utilized for describing the distribution of the primary unknown within the element. The compatibility between different elements are established through their common nodes by considering the distribution of the primary unknown along the common element edge to
be utilized by both the individual elements for their local estimations. The meshing algorithms in GeoStudio ensure element compatibility within regions. A special integer-based algorithm is also included to check the compatibility between regions. This algorithm ensures that common edges between regions have the same number of elements and nodes. Even though the software is very powerful and seeks to ensure mesh compatibility, the user nonetheless needs to be careful about creating adjoining regions. The integer-programming algorithm in GeoStudio seeks to ensure that the same number of element divisions exist between points along an edge of a region. The number of element divisions are automatically adjusted in each region until this condition is satisfied.
Consequently, it is often noticed that the number of divisions along the edge of a region is higher than what was specified as default.
There are different finite element mesh patterns available as default in Geostudio, namely (a) Quads and Triangles (b) Triangles only (c) Rectangular grid of Quads, and (d) Triangular grid of Quads / Triangles. In a finite element formulation, there are many integrals to be determined for achieving local scale equilibrium. For simple element shapes like 3-noded or 4-noded brick (rectangular) elements, it is possible to develop closed-formed solutions to obtain the integrals.
However, for higher-order and more complex element shapes, it is necessary to conduct numerical integration. GeoStudio uses the Gauss quadrature scheme for conducting the numerical integration.
This scheme is involved in sampling the element characteristics at specific points, known as Gauss points, and then adding up the sampled information. Table 3.1 lists some of the element types available in GeoStudio and their available integration points.
Table 3.1 Commonly available element types available in GeoStudio and their corresponding integration points
Element Type Integration Points Comments
4-noded quadrilateral 4 Default
8-noded quadrilateral 4 or 9 4 is the default
3-noded triangle 1 or 3 3 is the default
6-noded triangle 3 Default
GeoStudio presents the results for a Gauss region, but the associated data is actually computed at the exact Gauss integration sampling points. In the present study, the use of mixed ‘quad and triangle’ unstructured mesh with 4-noded quadrilateral and 3-noded triangle element type is utilized. Figure 3.4 shows the typical unstructured mesh of ‘quad and triangle’. The total number of nodes and elements used in the models depends on the number of mesh elements used and the adopted refinements.
Fig. 3.4 Typical ‘quad and triangle’ meshing scheme adopted to represent a slope (adopted from Sigma/W Manual, 2018)
In Rslope2d, once the random field is transformed into the desired lognormal field, it is then mapped onto the finite element mesh, which is established according to the user-defined slope geometry. A typical finite element mesh for a 1:1 slope with a height, H = 10 m, is shown in Fig.
3.5. Each element within the slope geometry is 1 m by 1 m in size and it is assigned a random variable of the particular soil property (i.e. c or φ). The computer model uses 8-noded quadrilateral elements with reduced integration for the generation of gravity loads, stiffness matrix generation and stress redistribution phases of the algorithm (Chok, 2009).
Fig. 3.5 Typical finite element mesh used in Rslope2d RFEM model