Problems in Linear Water-Wave theory in which there is an elastic body.
Finite Element Method (FEM) can be used to analyze any general 3D elastic structure using linear hydroelastic theory.
local FE [math]\displaystyle{ \Rightarrow }[/math] global FE model
Dynamic equation of motion in matrix form can be expressed as:
[math]\displaystyle{
\begin{bmatrix}K\end{bmatrix}\begin{bmatrix}D\end{bmatrix}+
\begin{bmatrix}S\end{bmatrix}\begin{bmatrix}\dot D\end{bmatrix}+
\begin{bmatrix}M\end{bmatrix}\begin{bmatrix}\ddot D\end{bmatrix}=
\begin{bmatrix}F(t)\end{bmatrix} }[/math], where:
[math]\displaystyle{ \begin{bmatrix}K\end{bmatrix} }[/math] is structural stiffness matrix,
[math]\displaystyle{ \begin{bmatrix}S\end{bmatrix} }[/math] is structural damping matrix,
[math]\displaystyle{ \begin{bmatrix}M\end{bmatrix} }[/math] is structural mass matrix,
[math]\displaystyle{ \begin{bmatrix}D\end{bmatrix} }[/math] is generalized nodal displacements vector,
[math]\displaystyle{ \begin{bmatrix}F\end{bmatrix} }[/math] is generalized force vector (fluid forces, gravity forces,...).
Left-hand side of the global FEM matrix equation represents "dry" (in vacuuo) structure, while the right-hand side includes fluid forces (and coupling between the surrounding fluid and the structure).
The eigenvalue problem for the "dry" natural vibrations yields:
[math]\displaystyle{ \begin{bmatrix}K\end{bmatrix}\begin{bmatrix}D\end{bmatrix}+\begin{bmatrix}M\end{bmatrix}\begin{bmatrix}\ddot D\end{bmatrix}=\begin{bmatrix}0\end{bmatrix} }[/math]
If one assumes trial solution as [math]\displaystyle{ \begin{bmatrix}D\end{bmatrix}=\begin{bmatrix}w\end{bmatrix}\,e^{i\omega t} }[/math] then the eigenvalue problem reduces to [math]\displaystyle{ \left( \begin{bmatrix}K\end{bmatrix}-\omega^2 \begin{bmatrix}M\end{bmatrix} \right )\begin{bmatrix}w\end{bmatrix}=\begin{bmatrix}0\end{bmatrix} }[/math]. As a solution of the eigenvalue problem for each natural mode one obtains [math]\displaystyle{ \omega_n }[/math], the n-th dry natural frequency and [math]\displaystyle{ \begin{bmatrix}w_n\end{bmatrix} }[/math], the corresponding dry natural mode.
Generalized nodal displacements vector can be expressed using calculated "dry" structure natural modes:
[math]\displaystyle{ \begin{bmatrix}D\end{bmatrix} }[/math]=[math]\displaystyle{ \begin{bmatrix}W\end{bmatrix}\cdot\begin{bmatrix}\xi\end{bmatrix} }[/math]
[math]\displaystyle{ \begin{bmatrix}W\end{bmatrix} }[/math]=[math]\displaystyle{ \begin{bmatrix}\mathbf{w_1}\,\mathbf{w_2}\,\ldots \end{bmatrix} }[/math] is matrix of dry natural modes, with modes being sorted column-wise,
[math]\displaystyle{ \begin{bmatrix}\xi\end{bmatrix} }[/math]is natural modes coefficients vector (modal amplitudes).
[math]\displaystyle{ \begin{bmatrix}W\end{bmatrix}^T \begin{bmatrix}K\end{bmatrix} \begin{bmatrix}W\end{bmatrix}
\begin{bmatrix}\xi\end{bmatrix}+\begin{bmatrix}W\end{bmatrix}^T \begin{bmatrix}S\end{bmatrix} \begin{bmatrix}W\end{bmatrix}
\begin{bmatrix}\dot\xi\end{bmatrix}
+\begin{bmatrix}W\end{bmatrix}^T \begin{bmatrix}M\end{bmatrix} \begin{bmatrix}W\end{bmatrix}
\begin{bmatrix}\ddot\xi\end{bmatrix} }[/math]=[math]\displaystyle{ \begin{bmatrix}W\end{bmatrix}^T\begin{bmatrix}F(t)\end{bmatrix} }[/math]
[math]\displaystyle{ \begin{bmatrix}k\end{bmatrix} \begin{bmatrix}\xi\end{bmatrix}+\begin{bmatrix}s\end{bmatrix} \begin{bmatrix}\dot\xi\end{bmatrix}+\begin{bmatrix}m\end{bmatrix} \begin{bmatrix}\ddot\xi\end{bmatrix} }[/math]=[math]\displaystyle{ \begin{bmatrix}f(t)\end{bmatrix} }[/math]
[math]\displaystyle{ \begin{bmatrix}k\end{bmatrix}=\begin{bmatrix}W^T\end{bmatrix}\begin{bmatrix}K\end{bmatrix}\begin{bmatrix}W\end{bmatrix} }[/math]is the modal stiffness matrix,
[math]\displaystyle{ \begin{bmatrix}m\end{bmatrix}=\begin{bmatrix}W^T\end{bmatrix}\begin{bmatrix}M\end{bmatrix}\begin{bmatrix}W\end{bmatrix} }[/math]is the modal mass matrix.
Hydroelastic analysis of the general 3D structure is thus preformed using the modal superposition method.
Let us assume time-harmonic motion. Then the following is valid:
[math]\displaystyle{ \begin{bmatrix}\xi(t)\end{bmatrix}=\begin{bmatrix}\tilde{\xi}(\omega)\end{bmatrix}\cdot e^{i \omega t},
\; \begin{bmatrix}f(t)\end{bmatrix}=\begin{bmatrix}\tilde{f}(\omega)\end{bmatrix}\cdot e^{i \omega t} }[/math]
[math]\displaystyle{ \left ( \begin{bmatrix}k\end{bmatrix}+i\omega\begin{bmatrix}s\end{bmatrix}-\omega^2\begin{bmatrix}m\end{bmatrix} \right ) \begin{bmatrix}\tilde{\xi}\end{bmatrix}=\begin{bmatrix}\tilde{f}\end{bmatrix} }[/math]
Modal hydrodynamic forces are calculated by pressure work integration over the wetted surface:
[math]\displaystyle{ \tilde{f}^{hd}_i(t)=-i\omega\rho\iint_{S}\tilde{\phi}\,\mathbf{h_i}\mathbf{n}\,\mbox{d}S }[/math]
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