## 3 Fourier Series-Based 3D Models with Random \(\omega\)  
On the basis of a superposition of Fourier series expansion of different functions, a 3D model with random localized imperfection (\(\omega\)) is proposed in this section. The Fourier series technique has been widely used for 3D surface modeling [35; 36]. For a function \(\text{f}(x)\) that is periodic on an interval [\(-\)L, L], it can be expressed as the Fourier series, given by  
\[\text{f}(x)=\frac{a_{0}}{2}+\sum_{k=1}^{\infty}[\text{a}_{k}\cos(\frac{k\pi x }{L})+\text{b}_{k}\sin(\frac{k\pi x}{L})], \tag{1}\]  
Figure 6: Identifying probability distribution for \(\omega_{\max}\). (**a**) Goodness of fit test results for 16 different distribution tests; (**b**) probability plot of \(\omega_{\max}\); (**c**) histogram of \(\omega_{\max}\); (**d**) comparison of cumulative probability (CDF) curve against the log-normal distribution. AD, Anderson–Darling.  
where  
\[a_{0}=\frac{1}{L}\int_{-L}^{L}f(x)dx, \tag{2}\]  
\[a_{k}=\frac{1}{L}\int_{-L}^{L}f(x)\cos\biggl{(}\frac{k\pi x}{L}\biggr{)}dx\ (k=\ 0,\ 1,\ 2,\ 3,\ \cdots), \tag{3}\]  
\[b_{k}=\frac{1}{L}\int_{-L}^{L}f(x)\sin\biggl{(}\frac{k\pi x}{L}\biggr{)}dx\ (k=\ 1,\ 2,\ 3,\ \cdots). \tag{4}\]  
Assume a surface consists of \(n\times m\) points in a 3D coordinate system, where X coordinate represents the longitudinal (length) direction, Y represents the transverse (width) direction, and Z represents deviation from the flat surface parallel to the XY plane. For a point (\(x_{i}\), \(y_{j}\), \(z_{ij}\)) (I = 1, 2,..., \(m\); \(j\) = 1, 2,..., \(n\)) on the surface, \(z_{ij}\) governs localized imperfection (\(\omega\)). All \(z_{ij}\) elements comprise an \(n\times m\) matrix [Z], which can be determined by  
\[[\mathbf{Z}]=\ [\mathbf{F}_{1}]+([\mathbf{F}_{2}]-[\mathbf{F}_{1}])[\mathbf{S}], \tag{5}\]  
\[[\mathbf{F}_{1}]=\left[\begin{array}{ccccc}\mathrm{f}_{1}(\mathrm{x}_{1})& \mathrm{f}_{1}(\mathrm{x}_{2})&\cdots&\mathrm{f}_{1}(\mathrm{x}_{i})&\cdots& \mathrm{f}_{1}(\mathrm{x}_{m})\\ \mathrm{f}_{1}(\mathrm{x}_{1})&\mathrm{f}_{1}(\mathrm{x}_{2})&\cdots&\mathrm{ f}_{1}(\mathrm{x}_{i})&\cdots&\mathrm{f}_{1}(\mathrm{x}_{m})\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ \mathrm{f}_{1}(\mathrm{x}_{1})&\mathrm{f}_{1}(\mathrm{x}  
and (\(x_{b}\), \(y_{j}\)) (j = 0, 1,..., m). The generated surface with random \(\omega\) is determined by \([\mathbf{F_{1}}]+([\mathbf{F_{2}}]-[\mathbf{F_{1}}])\), \([\mathbf{S}]\), as shown in Figure 7c.  
The 3D model of a member with random \(\omega\) is finally assembled by four surfaces. It should be mentioned that in order to fit the four faces together, relevant coordinate transformation should be conducted. Coordinate transformation depends on assembling order and the position of the surface in a 3D space. For developing a 3D model of RHS and SHS with round corners, additional curved surfaces representing round corners need to be modelled. The generated 3D model for a typical surface with random \(\omega\) and half-sine edges is shown in Figure 8a, and for a typical member with random \(\omega\) and half-sine edges, it is shown in Figure 8b.  
Figure 7: Development of 3D surface with random \(\omega\). (**a**) Two surfaces determined by \([\mathbf{F_{1}}]\) and \([\mathbf{F_{2}}]\); (**b**) surface determined by \(([\mathbf{F_{2}}]-[\mathbf{F_{1}}])\) [S]; (**c**) surface determined by \([\mathbf{F_{1}}]+([\mathbf{F_{2}}]-[\mathbf{F_{1}}])\) [S].