|
Vessel theory: Impulse response and convolution |
|
|
Vessel theory: Impulse response and convolution |
To implement frequency-dependent added mass and damping in the time domain, OrcaFlex uses a method proposed by Cummins (1962), and implemented by Wichers (1979). This method involves calculating the impulse response function (IRF) for the vessel and then applying that IRF at each time step using a convolution integral to account for the past motion of the vessel. In addition, if not specified, the infinite frequency added mass matrix, $\Ainf$, must be calculated and included in the vessel total inertia matrix.
In more detail, at each calculation time $t$ in the simulation, the total added mass and damping load (force and moment) $\vec{l}$ on the vessel (or vessels, in the case of a multibody group) is given by the convolution integral equation \begin{equation} \label{convolution} \vec{l}(t) = -\Ainf\ \vec{x}''(t) -\int_{\tau=0}^\infty \IRF(\tau)\ \vec{x}'(t-\tau)\,\ud\tau \end{equation} where
$\vec{x}''$ and $\vec{x}'$ are the vessel acceleration and velocity, respectively, of the added mass and damping reference origin relative to the global origin, due to primary motion of the vessel(s)
$\tau$ is a time lag integration variable.
This load is applied at the added mass and damping reference origin.
OrcaFlex calculates the IRF using a Fourier integral equation \begin{equation} \label{IRF} \IRF(\tau) = c(\tau)\ \int_{f=0}^\infty 4\ \mat{B}(f)\ \cos(2\pi f \tau)\,\ud\! f \end{equation} where
$\mat{B}(f)$ is the frequency-dependent damping matrix, at frequency $f$
$c(\tau)$ is a cutoff scaling function.
The damping data $\mat{B}(f)$ and the resulting $\IRF(\tau)$ are matrix-valued functions, of frequency and time lag respectively. For a single independent vessel (not in a multibody group) these matrices are $6{\times}6$ matrices; for a multibody group they are $6N{\times}6N$ matrices, where $N$ is the number of vessels in the multibody group.
The integral in equation (\ref{IRF}) is calculated numerically using the following assumptions about the form of the damping:
The convolution integral equation (\ref{convolution}) involves integrating all the way back to the start of the simulation, at every time step. This can cause long simulations to run increasingly slowly as the simulation time progresses. Fortunately, however, impulse response functions decay to zero as the time lag $\tau{\rightarrow}\infty$, so there comes a point beyond which the rest of the integral can safely be neglected. OrcaFlex therefore truncates the impulse response function at the cutoff time, $\Tc$
Truncating the integral improves performance significantly, but it does involve some approximation in the levels of damping and added mass applied. And simply cutting off the integral sharply at $\tau{=}\Tc$ can result in negative damping (energy feed-in) being applied at low frequencies. To avoid this, OrcaFlex instead smoothly scales down the IRF function as $\tau$ approaches $\Tc$, by applying the cutoff scaling function \begin{equation} c(\tau) = \exp\left[ -\left( \frac{3\tau}{\Tc} \right)^2 \right] \end{equation} in equation (\ref{IRF}) above.
The effect of this scaling by $c(\tau)$ is that the impulse response function is smoothly scaled down for large $\tau$, so that the IRF is virtually insignificant beyond the cutoff time, $\tau{=}\Tc$, where the integral is truncated. As a result the danger of energy feed-in is virtually eliminated, but this scaling does introduce approximation errors that can be significant for small values of $\Tc$.
For details of how to manage these errors, see cutoff time.
The infinite frequency added mass matrix $\Ainf$ gives the vessel's instantaneous response to acceleration (as opposed to the IRF, which characterises the response to past motion). If this has not been specified, an estimate $\Aiinf$ for $\Ainf$ can be obtained from the IRF and the added mass matrix $\mat{A}(f_i)$ for the $i$th frequency from the equation \begin{equation} \Aiinf = \mat{A}(f_i) + \frac{1}{2\pi f_i} \int_{s=0}^{\Tc} \IRF(s)\ \sin(2\pi f_i s)\,\ud s \end{equation} where $\{\mat{A}(f_i), i=1,2,\ldots,m\}$ are the added mass matrices given for the $m$ frequencies $f_i$. For a single vessel (not in a multibody group), each of the matrices $\Ainf$ and $\mat{A}(f_i)$ is a $6{\times}6$ matrix; for a multibody group each is a $6N{\times}6N$ matrix, where $N$ is the number of vessels in the multibody group.
In theory, if the data were given for all frequencies and precise integration used to calculate the IRF and these estimates, then each estimate $\Aiinf$ would have the same value. In practice, however, the data are only specified for a finite set of frequencies and the integrations have to be done numerically, giving rise to approximation errors. To reduce these approximation effects, OrcaFlex calculates each of the m separate estimates $\Aiinf$ and takes as $\Ainf$ their mean value.
The added mass and damping data are not independent. In fact, added mass and damping are mathematically related through the Kramers-Kronig relations (see Kotik and Mangulis).
OrcaFlex uses the damping data to determine the IRF, and then uses the IRF in conjunction with the added mass data to estimate the infinite frequency limit of the added mass, $\Ainf$.
If the data are consistent, i.e. obey the Kramers-Kronig relations, then the graphs of added mass against frequency should appear to converge with increasing frequency to the estimated limit $\Ainf$. Convergence towards a limit other than $\Ainf$ may indicate that the data are not consistent. In these circumstances, the results of the calculation may not be reliable.