## 2 Probabilistic Non-Linear Computation
### Sampling Methods  
The Monte-Carlo (MC) method provides a powerful, adaptable, and accurate method which has been increasingly used. MC replaces an exact or approximate calculation of the probability density of the limit state function by generating a large number of individual evaluations of the function or the input parameters using random realizations \(x_{ik}\) of the underlying distributions for the random variables \(X_{i}\) where the index \"\(k\)\" stands for the \"\(k\)\"-th simulation (\(k=1,2,\ldots,N_{mul}\)), see Figure 1. When combined with a finite element model, the resistance of the analyzed component or the limit state function has to be calculated for a large number of repetitions, yielding an output distribution function. Each set of the \(k\) realizations introduced into the analyzed model leads to a solution. \[G_{k}=G(a_{0},\ x_{1k},\ x_{2k},\ \ldots,\ x_{ik},\ \ldots,\ x_{nk}) \tag{7}\]  
The resulting \(z\) numbers \(g_{k}\) are evaluated statistically according to the basic statistics and lead to the \(p_{f}=z_{0}/z\), where \(z_{0}\) is the number of results violating the design threshold. The correlation between the basic variables, see Figure 1 (right), is set up by a controlled rearrangement of the sample field using simulating annealing. Simulated annealing (simulated cooling/annealing) is a heuristic approximation method. In summary, the MC method provides an approximate solution by performing statistical sampling. It relies on repeated random sampling whereby the error decreases with the square root of \(1/N_{sim}\), where \(N_{sim}\) = amount of simulations; \(p_{i}\) is generated randomly with uniform distribution on the interval \([0-1]\). However, sample points are generated without considering the previously generated sample points. The Latin Hyper Cube Simulation (LHS) method [28] is an extended procedure and is based on the basic idea of the Monte Carlo method. In particular, in the context of statistical sampling, a square grid containing sample positions \"\(Latin\) square\" is used for the LHS method. During sampling there is only one sample in each row and each column accepted, see Figure 2. While \(p_{i}\) is chosen randomly when using the MC method, see Figure 1 left, one \(p_{i}\) is chosen from each of the \(N_{sim}\)-intervals, see Figure 2 left, when using the LHS method. Random samples can be taken one at a time, remembering which samples were taken so far by dividing the cumulative density function (CDF) into \(N_{sim}\) equally probable \(k\) intervals, see Figure 2. It ensures an acceptable accuracy at a low \(N_{sim}\). Figure 1: Monte-Carlo method simulation procedure including and not including correlation between basic variables. The Fractile Based Simulation (FBS) method is processed as the LHS method with a number of \(N_{sim}\)-intervals but taking only a reduced number of \(p_{i}\)s from this sample field for the evaluation procedures into account. The selection of the reduced \(p_{i}\)s is based on predefined fractile values of the CDF, for instance as shown in Figure 3. Hence, the effort of simulations can be reduced while fully including the properties of the LHS sampling field--encompassing the \(N_{sim}\)-intervals. Since several basic variables are generally involved in a problem and the assignment of the \(p_{i}\)s to predefined fractile values is only possible for one basic variable, it is necessary to define a leading basic variable in advance and to formulate the dependencies of the other basic variables on this via the LHS simulation field.