## 2 Theoretical Background on the ANSYS DesignXplorer(c) Probabilistic Methods
### Monte Carlo Simulation
#### 2.1.1 Direct Sampling (\"Crude\" Monte Carlo)  
The technique sketched above is the simplest Monte Carlo approach for reliability problems but not the most efficient. The basis for its application is outlined below. The probability of limit state violation may be expressed as follows [56]:  
\[p_{f}=J=\int\ldots\int I[G(x)\leq 0]f_{X}(x)dx, \tag{2}\]  
where \(I[]\) is an \"indicator function\" which is equal to one if \([\;]\) is \"true\" and zero if \([\;]\) is \"false\". Thus, the indicator function identifies the integration domain. Equation (1) represents the expected value of \(I[]\). If \(x_{j}\) represents the \(j\)-th vector of random observations from \(f_{x}()\), it follows directly from sample statistics that Equation (3),  
\[p_{f}\approx J_{1}=\frac{1}{N}\sum_{j=1}^{N}I\big{[}G\big{(}\hat{x}_{j}\leq 0 \big{)}\big{]}, \tag{3}\]  
is an unbiased estimator of \(J\). Thus, Equation (3) provides a direct estimate of \(p_{f}\). In exploiting this procedure, three matters are of interest: how to extract most information from the simulation points, how many simulation points are needed for a given accuracy, or, conversely, how to improve the sampling technique to obtain greater accuracy for the same or fewer sample points [55].  
The fundamentals of the Monte Carlo simulation method are well documented in literature [48; 57]. The key functionality of Monte Carlo simulation techniques is the generation of random numbers with a uniform distribution from 0-1 [54]. The DesignXplorer(r) uses the advanced Mersenne Twister (MT) algorithm [58]. The MT algorithm is adopted for generating uniform pseudorandom numbers. In order to accommodate Mersenne-prime period, the MT is a modified variant of a previous generator, the twisted generalized feedback shift register (TGFSR). The MT generator was implemented in portable C-code and passed stringent statistical tests, including diehard. Its merits are due to the efficient algorithms that are unique to polynomial calculations over the two-element field [58]. ANSYS also uses the inverse probability method [48] to generate random numbers with arbitrary distributions. The implementation of the inverse cumulative probability function was verified with the results published by [PERSON] [59] with matching results even for probabilities as low as \(10^{-24}\). It uses the Latin hypercube sampling technique [60; 61] to make the sampling process more efficient, as well as to ensure that the tails of the distribution of the input variables are better represented.  
The interpretation of the results of a Monte Carlo simulation analysis is based on statistical methods. The statistical procedures to calculate, for example, mean values and standard deviations, and to derive histogram plots are available in textbooks on statistics and probabilistic methods (e.g., Reference [62]). For the cumulative distribution function (CDF), the data are sorted in ascending order, and the CDF of the \(i\)-th data point, here denoted with \(F_{i}\), can be derived as follows [54]:  
\[\sum_{k=i}^{N}\frac{N!}{(N-k)!k!}F_{i}^{k}(1-F_{i})^{N-k}=50\%. \tag{4}\]  
In DesignXplorer(r), this equation is solved numerically for \(F_{i}\). From the CDF curve, failure probabilities, as well as the inverse probability, can be derived.  
In ANSYS, sensitivities are evaluated based on correlation coefficients [62]. The stronger a result parameter is correlated with a particular input variable, the more sensitive the result parameter is with respect to the input variable.