# On Two Approaches to Slope Stability Reliability Assessments Using the Random Finite Element Method
## 3 The Random Finite Element Method  
The random finite element method (RFEM) is often used to compute geotechnical structure (e.g., slope) response (e.g., factor of safety and displacements) within a Monte Carlo framework [7]. The procedure is as follows. 1. Generate random property fields, for example, using the local average subdivision (LAS) method [17], based on some soil property statistics, e.g., a distribution type such as normal or lognormal characterised by the mean and standard deviation; spatial correlation structure (type of correlation function; and horizontal and vertical scales of fluctuation, \(\theta_{h}\) and \(\theta_{v}\), respectively). 2. Map random field cell values onto the Gauss points within the finite element mesh modelling the given problem (in this case, a slope stability problem). 3. Carry out a traditional finite element (e.g., slope stability) analysis [15]. 4. Repeat the above steps for multiple realisations in a Monte Carlo analysis, until the output statistics (e.g., the mean and standard deviation of the factor of safety) converge. For a given set of statistics, a probability distribution of the factor of safety can be obtained. Moreover, the potential consequences (e.g., failure volume in the case of slope stability) may also be quantified for each realised factor of safety. An existing LAS program, implemented in [18] and improved in [19], has been used in this paper. The implementation uses the following covariance function,  
\[C\left(\tau_{1},\tau_{2}\right)=\sigma^{2}\exp\left(-\sqrt{\left(\frac{2\tau_ {1}}{\theta_{1}}\right)^{2}+\left(\frac{2\tau_{2}}{\theta_{2}}\right)^{2}}\right) \tag{2}\]  
where \(\tau\) and \(\theta\) are the lag distance and scale of fluctuation, respectively; subscripts 1-2 denote the lateral and vertical coordinate directions, respectively; and \(\sigma^{2}\) is the variance of the relevant soil property. However, because the subdivision algorithm itself is incapable of preserving anisotropy [20], in this implementation, an isotropic random field is initially generated, and this field is then postprocessed to give the target anisotropic random field; that is, by squashing and/or stretching the field in the vertical and/or horizontal directions, respectively. Figures 2-4 show three possible ways of postprocessing an isotropic random field to produce an anisotropic field ready for mapping to a finite element mesh. In each figure, the starting point is an isotropic random field comprising \(512\times 512\) cells (i.e., as has been generated by nine LAS subdivision stages). Figure 2 shows an example of squashing the starting isotropic random field vertically. In this example, the target square domain size is \(D=5.0\) m, the vertical scale of fluctuation is \(\theta_{v}=0.5\) m, and the anisotropy of the heterogeneity is \(\xi=\theta_{h}/\theta_{v}=6\), i.e., the horizontal scale of fluctuation is \(\theta_{h}=3.0\) m. The final cell size is \(d_{c}=0.1\) m, and therefore 50 cells are needed in the horizontal direction and \(50\times 6=300\) cells (before squashing) in the vertical direction, as these are to be squashed and replaced by 50 new square cells. The starting isotropic field has a scale of fluctuation of \(\theta_{iso}=3.0\) m, to ensure that the final target field has the required horizontal and vertical SOFs. Note that, by squashing vertically (and averaging each group of 6 cells to form a single new cell of 0.1 m dimension), the vertical scale of fluctuation is scaled down to the target 0.5 m.  
Figure 3: Postprocessing an isotropic random field to produce an anisotropic field by stretching (\(\xi=6\)) (52 cells after Crop2 to be able to interpolate between the last two columns of cells). Figure 2: Postprocessing an isotropic random field to produce an anisotropic field by squashing (\(\xi=6\)).