## 2 Ship Hydroelasticity Theory in Irregular Waves  
### Potential Flow Theory in Ship Hydroelasticity  
The hydrodynamic part of ship hydroelastic equation is solved by 3D potential flow theory. Sketch of the ship wave interaction problem is shown in Figure 2. In the framework of potential flow theory, the fluid is assumed to be irrotational, inviscid and incompressible. The fluid domain is denoted as \(\Omega_{F}\) and its boundary is composed of free surface boundary \(S_{F}\), body surface boundary \(S_{B}\), bottom boundary \(S_{H}\) and infinity radiation boundary \(S_{\infty}\). The ship is assumed to travel in waves with constant forward speed \(U\). The angle between ship advancing direction and wave incoming direction is defined as wave heading angle \(\beta\); i.e., 0\({}^{\circ}\), 90\({}^{\circ}\), 180\({}^{\circ}\) and 270\({}^{\circ}\) are heading wave, starboard beam wave, following wave and port beam wave, respectively. Three Cartesian coordinate systems are introduced to describe the ship motion in waves. They are the space fixed system \(O\)-\(XYZ\), the translational movement system \(o\)-\(xyz\) and the body fixed system \(G\)-\(x_{b}y_{b}z_{b}\). The space fixed system \(O\)-\(XYZ\) is an earth absolutely coordinate system. The translational  
Figure 1: Structural failure of hull structures. **(a)** MSC NAPOLI; **(b)** MOL COMRORT. movement system _o-xyz_ is moving horizontally with constant speed \(U\) and course angle \(\beta\). The origins of these two coordinate systems are coincident initially. The body fixed system _G-x\({}_{b}\)y\({}_{b}\)z\({}_{b}\)_ is fully fixed on the moving ship with its origin \(G\) coincides with the center of gravity (COG) of ship. The Galilean transformation relationship between the three coordinate systems is:  
\[\left\{ \begin{array}{c}x\\ y\\ z\end{array} \right\}=\left[ \begin{array}{ccc}\cos\beta&\sin\beta&0\\ -\sin\beta&\cos\beta&0\\ 0&0&1\end{array} \right]\left\{ \begin{array}{c}X-Ut\cos\beta\\ Y-Ut\sin\beta\\ Z\end{array} \right\}\approx\left\{ \begin{array}{c}x_{b}\\ y_{b}\\ z_{b}+z_{G}\end{array} \right\} \tag{1}\]  
where \(t\) denotes time, _z\({}_{G}\)_ denotes the vertical height of COG of ship with respect to the undisturbed free surface. The solution of velocity potential is based on the 3D boundary element method (BEM). The fluid disturbance velocity potential under the translational movement system _o-xyz_ can be decomposed into:  
\[\Phi(x,y,z,t)=\text{Re}(\phi(x,y,z)e^{i\omega_{e}t}) \tag{2}\]  
where _phi(x,y,z)_ is the space-dependent part of _\(\Phi\)(x,y,z,t)_, _\(\omega_{e}\)_ is wave encounter frequency which is defined as:  
\[\omega_{e}=\omega\left(1+\frac{U\omega}{g}\cos\beta\right) \tag{3}\]  
where _\(\omega\)_ is the wave natural frequency, \(g\) is acceleration of gravity. The spatial potential _phi(x,y,z)_ can be further decomposed into:  
\[\phi(x,y,z)=\zeta_{a}\phi_{0}(x,y,z)+\zeta_{a}\phi_{d}(x,y,z)+\sum_{r=1}^{m}p_ {ra}\phi_{r}(x,y,z) \tag{4}\]  
where _\(\zeta_{a}\)_ denotes amplitude of incident wave, _\(\phi_{0}\)_ denotes incident wave potential under unit wave amplitude, _\(\phi_{d}\)_ denotes diffraction wave potential under unit wave amplitude, _\(p_{ra}\)_ is the motion amplitude of the _r_th degree of freedom (DOF), _\(\phi_{r}\)_ denotes radiation wave potential under unit motion amplitude of the _r_th order DOF, when \(r\) = 1 - 6 it denotes rigid body 6-DOF motion and when \(r\) = 7 - \(m\) it denotes elastic deformation. The incident wave potential \(\phi_{0}\) can be obtained by the incident wave field of regular waves directly, which is easy to obtain. The boundary value problem for the determination of radiation potential \(\phi_{r}\) and diffraction potential \(\phi_{d}\) in frequency-domain is expressed as follows:  
Figure 2: Description of ship–wave interaction issue. \[\left\{\begin{array}{l}\
abla^{2}\phi_{r}=0(in\;\Omega)\\ -\omega_{e}^{2}\phi_{r}+g_{\frac{\partial}{dz}}\phi_{r}=0(on\;z=0)\\ \frac{\partial\phi_{r}}{\partial n}=\left\{