Home > Theory > Hydrodynamics > Coupled Analysis

Coupled analysis

Introduction

With classical uncoupled analysis, the floater is modeled as a boundary condition at the top of the lines imposing their motions. The analysis consists then in solving the system composed by the equations of motions of the different lines. The problem is most of times discretized by the finite element method. The number of unknowns into calculations depend on the mesh refinement but the floater itself adds no degrees of freedom into the model : its motions are an input and are fixed whatever the lines behaviour.

The main objective of a coupled analysis is to capture the potential interactions between the floater and the lines behavior especially the dynamic part of the moorings and the risers response.

To do so, the solution simply consists in adding the floaters six degrees of freedom into the global system. These degrees of freedom must respect the floaters equations of motion.

Several terms of these equations represent the "hydrodynamic characteristics" of the floater. They result from the interactions between the floater's hull and the incident wave since the floater's dimensions are large enough to disturb the incident wave (large body assumption).

The hydrodynamic characteristics are classically computed by a diffraction/radiation software like Diodore. They correspond to transfer functions of efforts or added masses and damping terms which may depend on the wave frequency and the wave incidence. These characteristics shall be imported into DeepLines model through a HDB file.

Note

this topic details the different terms in the equation of motion, which are normally contained in an hydrodynamic database file (.HDB). The structure and content of hydrodynamic database files (.HDB) are described here.

Low and Wave Frequencies

This option is the recommended option to properly address coupled analysis when the dynamic of the risers/mooring lines is accounted for.

With this option, the following equation of the floater motion is added to the global system :

\[M\ddot X (t) + K_{hyd}(t,X)X(t) + F_{rad}^1 (t, \theta_{wave}) = \\ F_{wind}(t,X,\theta_{wind})+ F_{cur}(t,X, \theta_{cur})+ F_{Drag}(t,X)+ F_{WAg}^{(1)}(t,X, \theta_{wave})+ \\F_{Mooring}(t,X,X_{lines},\dot X_{lines},\ddot X_{lines})+ F_{WAg}^{(2)}(t,X, \theta_{wave})\]

where :

$1.~~$ $F_{Wa}^{(1)} (t,X, \theta_{wave})$ The first order wave efforts are deduced from the RAOs of the exciting efforts, they usually gather two terms: the Froude-Krilov and the diffraction forces.

They are withdrawn from a HDB file which provides the RAOs of efforts previously calculated by a diffraction/radiation software like DIODORE; These RAOs depend from the relative wave heading :

\[ F_{Wa}^{(1)} (t,X, \theta_{wave}) = \sum_{j=1}^{N_{h}}{a_jF_{RAO}^{(1)}(\omega_{wave},\theta_{wave})e^{i[\omega_{j}t - k_{j}(xcos\beta+ysin\beta)-\phi ]}} \]

With:

$N_h$ is the number of wave components.
$a_j$ is the wave amplitude of the j-th component.
$\omega_{wave}$ is the wave circular frequency.
$\theta_{wave}$ is the wave heading relative to the floater.
$F_{RAO}^{(1)}(\omega_{wave},\theta_{wave})$ is the RAOs of efforts for this wave circular frequenciy and heading.
$t$ is the time.
$\omega_j$ is the wave component pulsation.
$k_j$ is angular wave vector of the j-th component.
$\beta$ is the wave heading in DeepLines global frame.
$\phi_{RAO}^{(1)}(\omega_{wave},\theta_{wave})$ is the phase angle of the motion RAO for the of j-th wave component.

Note

Specific attention shall be paid when using the option "non-linear Froude-krilov". In that case, only the diffraction terms are read from the HDB file, that's why the option [DETAIL_EXC] shall be set to Yes (Y) in the HDB (see HDB keywords for further details here).

$2.~~$ $F_{rad}^{(1)}(t)$, the wave radiation :

$\hspace{1cm}$ 2.1. For regular wave, this radiation loads give two terms: an added mass and a damping matrix stored in the HDB file which are considered at the specified wave period.

\[ F_{rad}^{(1)}(t) = M_a(\omega_{wave})\ddot X(t)+B_{rad}(\omega_{wave})\dot X(t) \]

$\hspace{1cm}$ 2.2. For irregular wave analysis, the radiation loads are directly computed using the retardation function (so-called memory effect) : $$ F_{rad}^{(1)}(t) = M_a(\infty)\ddot X(t) + \int_0^t R(t-\tau)\dot X(\tau)\mathrm{d}\tau ~, ~~R(t) = \frac 2 \pi \int_0^\infty B_{wa}(\omega)cos(\omega t)\mathrm{d}\omega $$ Experience shows that the memory effect quickly vanishes with time and it is useless calculating all the terms over the whole simulation. In practice, a sliding time window of the order of one hundred seconds is applied to avoid excessive CPUtime consumption.

$3.~~$ $F_{wind}(t,\dot X,\theta_{wind})$ efforts on the floater due to the wind which depend from the relative wind velocity and heading via polar coefficients.

$4.~~$ $F_{cur}(t,\dot X,\theta_{current})$ efforts on the floater due to current which depend from the relative current velocity and heading via polar coefficients.

$5.~~$ $F_{Drag}(t)$ viscous drag efforts on the hull and additional appendices. This term is source of great uncertainties especially for FPSO roll. The drag efforts are often represented by a linear and a quadratic matrix.

$6.~~$ $F_{WA}^{(1)}(t,X,\theta_{wave})$ second order wave efforts could also be added when an irregular wave is defined. There may be of two kinds :

$\hspace{1cm}$ 6.1. The incident wave loads due to the wave kinematics :

Let h be the wave elevation : $\eta = \sum_{i} a_i*cos(k_i x cos\beta + k_i y sin\beta - \omega_i t - \phi_i )$

The second order potential associated to the difference mode:(if $k_i*h >> 1$) comes : $$ \phi^{(2)} = \sum_i \sum_j -\frac {a_i a_j \omega_i \omega_j(\omega_i - \omega_j)}{g(k_i - k_j )th \left[(k_i-k_j)h \right] - (\omega_i -\omega_j)^2} * \frac{ch[(k_i-k_j)(z+h)]}{ch[(k_i-k_j)h]} * sin[(k_i-k_j)(xcos\beta + ysin\beta) - (\omega_i-\omega_j)t - \varphi_i + \varphi_j] $$

For surge and sway, a simplified formulation may be used: $$ F_x = \rho (1+C_{m_{xx}})V \frac {\partial^2{\phi ^{(2)}}} {\partial x \partial t} $$ $$ F_y = \rho (1+C_{m_{yy}})V \frac {\partial^2{\phi ^{(2)}}} {\partial y \partial t} $$

where V is the immersed volume and $C_{m_{xx}}/C_{m_{yy}}$ are the asymptotic added mass for large period.

$\hspace{1cm}$ 6.2. The wave drift loads calculated from the Quadratic transfer Function (QTF) : $$ F_{wave}^{(2)}(t) = F_{(0)} + \sum_{j=1}^{N_h -1} \sum_{k=j+1}^{N_h} a_j a_{j-k} F^{(2)} (\omega_j, \omega_j - \omega_{j-k}) e^{i[(\omega_j - \omega_{j-k})t - (k_j-k_{j-k})(xcos\beta+ysin\beta) - \phi]} $$

where $F^{(0)} = a_ja_jF^{(2)}(\omega_j,0)$ is the constant drift forces. If not provided, the QTF can also be computed from the regular wave drift forces using the Newmans approximation.

$7.~~$ $F_{Mooring}(t,X,X_{lines},\dot X_{lines},\ddot X_{lines})$ restoring forces. This is the coupling term.

$\hspace{1cm}$ 7.1. It is directly provided by the resolution of the global system. Every line defined into the model gives a contribution which depends from its position, speed and acceleration.

$\hspace{1cm}$ 7.2. For simplified analysis, the contribution of selected lines may be calculated using a stiffness matrix which is then added to the floaters hydrostatic stiffness.

Low Frequencies

The floaters motions are assumed to be divided into a low frequencies and a wave frequencies parts, which can be separated from each other. Then, the total motions can be written as such : $$ X(t) = X_{LF}(t)+X_{WF}(t)~~~~~(*) $$

The low frequencies motions $X_{LF}(t)$ is solved in the time domain in parallel with the global system. If only looking at pure low frequency motion, the wave frequncy motion is not taken into account. In that case it is recommended to use the "Implicit scheme" option. If the wave frequency is also requested, the wave frequency motion stored in the hydrodynamic database as first order motion RAOs is computed and added to the low frequency motion.

The floaters equations of motion in low frequencies is as such :

\[ (M+M_a(0))\ddot X_{LF}(t)+B\dot X{LF}(t)+K_{hyd}X_{LF}(t) \\= F_{LF}^{(2)}(t) + ~ F_{Mooring}^{LF}(t,X) + ~ F_{wind}^{LF}(t,\dot X) + ~ F_{cur}^{LF}(t,\dot X) + ~ F_{Drag}^{LF}(t) \]

with :

$F_{wind}(t,dot X,\theta_{wind})$ efforts on the floater due to the wind which depend from the relave wind velocity and heading via polar coefficients.
$F_{cur}^{LF}(t,\dot X,\theta_{current})$ efforts on the floater due to current which depend from the relative current velocity and heading via polar coefficients.
$F_{Drag(t)}$ viscous drag efforts on the hull and additional appendices. This term is source of great uncertainties especially for FPSO roll. The drag efforts are often represented by a linear and a quadratic matrix.
$F_{WA}^{(2)}(t,X,\theta_{wave})$ , the wave drift loads calculated from the Quadratic Transfer Function. If not provided, the QTF can be also be evaluated using the Newmans approximation. $$ F_{wave}^{(2)}(t) = F_{(0)} + \sum_{j=1}^{N_h -1} \sum_{k=j+1}^{N_h} a_j a_{j-k} F^{(2)} (\omega_j, \omega_j - \omega_{j-k}) e^{i[(\omega_j - \omega_{j-k})t - (k_j-k_{j-k})(xcos\beta+ysin\beta) - \phi]} $$ where $F^{(0)} = a_ja_jF^{(2)}(\omega_j,0)$ is the constant drift forces.
Here, the wave radiation loads consist of the added mass and the damping matrix which are considered at their asymptotic values (large periods, w = 0) : $$ F_{rad}^{(1)}(t) = M_a(0)\ddot X(t) + B_{rad}(0)\dot X(t) $$
$F_{Mooring}(t,X,X_{lines},\dot X_{lines},\ddot X_{lines})$ restoring forces. This is the coupling term.

$\hspace{1cm}$ 6.1. It is directly provided by the resolution of the global system. Every line defined into the model gives a contribution which depends from its position, speed and acceleration.

$\hspace{1cm}$ 6.2. For simplified analysis, the contribution of selected lines may be calculated using a stiffness matrix which is then added to the floaters hydrostatic stiffness.

Note

A specific attention shall be paid to the terminology "second order" which may be a bit confusing. Actually, second order loads encompass so-called second order wave loads but also non-linear effects coming from the variation of boundary limits constituted by the floater's hull which is moving.

At first order, an irregular wave can be modelled as a superposition of regular waves, where amplitudes are derived from an energy spectrum. Then, the wave elevation signal at free surface can be written as follows: $$ \eta^{(1)}(x,t) = \sum_i A_i cos[k_i.x-\omega_it+\theta_i] $$

with $A_i^2 = 2S(\omega_i,\beta_i)\delta\omega_i\delta\beta_i$ where $S(\omega)$ is the directional wave spectrum.

At the second order of approximation, other terms appear and in particular some terms of interaction between 2 components of the wave signal, $\omega_i$ and $\omega_j$.

The energy spectrum of the 2nd order components of the wave elevation can be written as follow for the difference mode ( $\omega_i - \omega_j$) in infinite water depth:

\[ S_{\eta_- ^{(2)}} = \frac{1}{2} \int_0^{\infty} S(\omega)S(\omega + \Omega)\left[\frac {w^2 - (w+\Omega)^2}{g}\right] d\omega \]

This energy may excite the floating system out of the wave frequency on its natural periods. It is then an important effect that should be taken into account, in particular for mooring design, since mooring lines tensions are very correlated to floater's motions. This effect is usually taken into account by the QTF in difference mode.

In theory, the 2nd order forces applied on the floater are composed of the following components:

In the non-linear Froude-Krylov approach (FKNL), the hydrostatic forces and first-order forces induced by the incident (undisturbed) waves are computed analytically at each time step, by integration on the instantaneous immersed part of the hull. Therefore that, at the second order, the torsor of hydrostatic forces and the torsor of the incident wave forces contain contributions usually accounted for in $2^{nd}$ order forces : all terms (1), (6) and (7) as well as the explicit contribution of the incident potential (part Fi(1) of the total potential F(1)) in terms (2) (3) and (4).

Term (5) accounts for the second order of the incident potential which is ignored by the nonlinear Froude-Krylov approach.

Therefore, so far, it is not recommended to combine the nonlinear Froude- Krylov approach with QTF since some terms are counted twice.