# Stochastic Natural Vibration Analyses of Free-Form Shells
## 3 Stochastic and Sensitivity Analysis Method  
The natural vibration analyses are coupled with LHS to investigate the stochastic characteristics of natural frequencies of free-form shells and to explore the sensitivity of the uncertainty factors. LHS [16; 17; 18] is a widely-used sampling method, which allows for the extraction of a large amount of stochastic and sensitivity information with a relatively small sample size. It can ensure that the entire distribution of each input random variable is covered and good at estimating the mean values of output variables [19]. LHS consists of two main steps, sampling and permutation. Sampling is implemented to produce representative samples to describe the distribution of each input random variable, while permutation aims at reducing the correlations between the samples of different random variables. Herein, LHS is used for sampling. \(e_{1},e_{2},\cdots,e_{n_{e}}\) denote the input uncertainty factors. Their respective probability cumulative function is expressed as \(F_{i}=F_{i}(e_{i})\). \(f\) represents the output variable, i.e., the natural frequency in the current study. For a sample size \(n_{s}\), the range of \(F_{i}\) is divided into \(n_{s}\) equal interval zones, which are non-overlapping. One value is randomly selected for each intervals and the sampling. A row of sampled values \(e_{1},e_{2},\cdots,e_{in_{n}}\) is obtained. A \(n_{e}\times n_{s}\) sampling matrix can be generated after all the input random variables are sampled. Subsequently, random permutation is performed, in which the sampled value of every input variable is randomly chosen out of the sample values from the corresponding row in sampling matrix without replacement. To check the sample size, the convergence estimation formulas are given, as follows  
\[check_{1}=\left|\frac{\overline{f}(n_{s})-\overline{f}(n_{s}-ch)}{\overline{f} (n_{s})}\right|\leq\xi_{1};\ check_{2}=\left|\frac{\sigma(n_{s})-\sigma(n_{s}- ch)}{\sigma(n_{s})}\right|\leq\xi_{2} \tag{3}\]  
where \(n_{s}\) is the number of samplings; \(ch\) is added number of samplings; \(\overline{f}(n_{s}-ch)\), \(\overline{f}(n_{s})\) is mean value of output results with \(n_{s}-ch\) samplings and \(n_{s}\) samplings, respectively; \(\sigma(n_{s}-ch)\) and \(\sigma(n_{s})\) is standard deviation of output results with \(n_{s}-ch\) samplings and \(n_{s}\) samplings, respectively; referring the existing work [19; 20; 21], the tolerances \(\xi_{1}\) and \(\xi_{2}\) are determined as values that are less than or equal to 0.01. Based on LHS, the natural frequencies of free-form shells, which involve inherent randomness, is investigated by a partial correlation-based method [17; 18; 19; 22; 23]. The partial correlation coefficient was proposed by [PERSON] and [PERSON] [22], which is used to measure the degree of the linear correlation between the input variable and the model output after making an adjustment to remove the linear effect of all the remaining variables. The derivation of the partial correlation coefficient can be referred to [23]. Herein, the formulation of the partial correlation coefficients between input uncertainty factors \(e_{i}\) and output variables \(f\) (i.e., natural frequencies in the current study) is given, as follows [22; 23]\[S_{i}=-c_{if}/\sqrt{c_{ii}c_{ff}} \tag{4}\]  
where \(c_{if}\), \(c_{ii}\), and \(c_{ff}\) are the elements of matrix \(C\), written as follows  
\[C=\left[ \begin{array}{ccccc}c_{11}&c_{12}&\cdots&c_{1n_{e}}&c_{1f}\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ c_{n_{e}1}&c_{n_{e}2}&\cdots&c_{n_{e}n_{e}}&c_{n_{e}f}\\ c_{f1}&c_{f2}&\cdots&c_{fn_{e}}&c_{ff}\end{array} \right]=\left[ \begin{array}[] {ccccc}r_{11}&r_{12}&\cdots&r_{1n_{e}}&r_{1f}\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ r_{n_{e}1}&r_{n_{e}2}&\cdots&r_{n_{e}n_{e}}&r_{n_{e}f}\\ r_{f1}&r_{f2}&\cdots&r_{fn_{e}}&r_{ff}\end{array}