# Contact Analysis and Friction Prediction of Non-Gaussian Random Surfaces
## 2 Methodology  
### Methodology for Non-Gaussian Rough Surface Generation  
In this section, a generation program based on linear transformation and MA model is developed to produce a non-Gaussian rough surface with expected height distribution and ACF. The ACF describes the general dependence of the values of the dates at one location to their values at another location. It is crucial for generating random rough surfaces because it supplies fundamental information about the wavelength and the amplitude properties of rough surfaces. In this paper, the typically exponential type of ACF is adopted. It can be given below:  
\[R(\tau_{x})=\sigma^{2}\exp\left(-2.3\left(\frac{\tau_{x}}{\tau_{x}}\right)\right) \tag{1}\]  
where \(\sigma\) is standard deviation of the rough surface heights, and \(\tau_{x}\) represents the autocorrelation lengths in the x direction. \(\tau_{x}^{*}\) is the length at which ACF decays to 0.1 of its value in origin position. Based on [PERSON]'s method, the random surface \(z\) can be generated by linear transformation between a random sequence \(\eta\) and a transform matrix \(\alpha\):  
\[z_{i}=\sum\limits_{k=1}^{n}\alpha_{k}\eta_{i+k} \tag{2}\]  
The random sequence is independent and has unit variance; hence, it satisfies the following formula:  
\[E(\eta_{i},\eta_{k})=\left\{ \begin{array}{cc}1&if\ i=k\\ 0&\text{otherwise}\end{array} \right. \tag{3}\]  
According to the property of ACF, the relationship between transform coefficients and the values of ACF can be expressed as:  
\[R_{p}=\sum\limits_{k=1}^{n-p}\alpha_{k}\alpha_{k+p},p=0,1,\ldots,n-1 \tag{4}\]  
Therefore, the transform coefficients can be obtained by solving nonlinear equations represented by Equation (4). Initially, [PERSON] adopted the Newton method to determine the transform. The initial approximation of \(\alpha\) is given as:  
\[\alpha_{i}^{0}=sc_{i} \tag{5}\]  
where \(c_{i}=\frac{R_{i-1}}{n-i+1}\), \(s=\sqrt{\frac{R_{0}}{\sum_{i=1}^{n}c_{i}^{2}}}\). However, with the increased number of nonlinear equations, the convergence problem becomes more severe. Moreover, it requires more computer memory space to store the Jacobian matrix. To cope with the efficiency and the storage space, [PERSON] [10] transformed the nonlinear system of equations to a nonlinear optimization problem and use NCGM to solve. Recently, [PERSON] et al. [12] proposed an improved numerical modeling by transforming the problem described by [PERSON] to the nonlinear squares problem. The transformation can be rewritten as:  
\[\text{min}f(\alpha_{i})=\left\|F(\alpha_{i})\right\|^{2}=\sum\limits_{p=0}^{n -1}\left(R_{p}-\sum\limits_{k=1}^{n-p}\alpha_{k}\alpha_{k+p}\right)^{2} \tag{6}\]  
where \(f(\alpha)\) is the objective function, and its gradient can be given by:  
\[g(\alpha_{i})=\
abla f(\alpha_{i})=\left(\frac{\partial f}{\partial\alpha_{i} },\ldots,\frac{\partial f}{\partial\alpha_{n}}\right)^{T} \tag{7}\]The analytical formula of gradient function with respect to each of the transform coefficient is shown as follows:  
\[ \begin{array}{l}g(a_{i})=\sum\limits_{p=0}^{n-i}2\left(R_{p}-\sum\limits_{k=1}^ {n-p}\alpha_{k}\alpha_{k+p}\right)\left(-\alpha_{i+p}\right)\\ +\sum\limits_{p=0}^{i-1}2\left(R_{p}-\sum\limits_{k=1}^{n-p}\alpha_{k+p} \right)\left(-\alpha_{i-p}\right)\end{array}  \tag{8}\]  
Through this transformation, the NCGM was implemented into Equation (8) for solving the nonlinear system of equations depicted in Equation (4). In addition, the FFT method can be applied to the computation of Equation (4) to improve the calculation efficiency dramatically. Once the transform coefficients are obtained, the Gaussian surface can be generated through convolution between the transform matrix \(\alpha\) and the Gaussian random sequence \(\eta\). Different from Gaussian surfaces, the generation of non-Gaussian surfaces requires that the random sequence \(\eta\) is subject to a non-Gaussian distribution.