Wave Radiation Forces

1. Wave Radiation Forces on SIMO Bodies

Wave radiation forces arise from the interaction between one or multiple floating bodies and the waves they generate through their own motion. In the frequency domain, these forces are represented by frequency-dependent added mass and damping coefficients. In the time domain, they are represented by convolution integrals involving retardation functions that capture the fluid memory effects.

The calculation of wave radiation forces depends on the type of SIMO body:

  • Bodies of type 6 DOF - time domain (Body type 1) use added mass at infinite frequency and retardation functions, as further elaborated in Solution by convolution integral.

  • Bodies of type 6 DOF - separated analysis (Body type 2) use added mass at zero frequency, as further elaborated in Separation of motions. It is then assumed that the effect of wave radiation forces on the wave-frequency motion is embedded in the First Order Motion Transfer Function of the body.

  • Bodies of type 3 DOF - time domain (Body type 3) don’t allow wave radiation forces. Instead, hydrodynamic coefficients can be specified as Small Body Hydrodynamic Data (see Small Body Hydrodynamic Data).

  • Bodies of type Prescribed (Body type 4) compute wave radiation forces in the same way as bodies of type 6 DOF - time domain.

The hydrodynamic coefficients for wave radiation forces are typically imported from WADAM or other supported boundary element solver.

Wave radiation forces on a SIMO body due to its own motion are represented by the following model elements:

  • Retardation Function

  • Added Mass at Zero Frequency

  • Added Mass at Infinite Frequency

  • Frequency-Dependent Added Mass

  • Frequency-Dependent Damping

When hydrodynamic interaction is accounted for, typically by importing data from a multi-body WADAM run, the coupling between bodies is represented by the following model elements:

  • Coupled Retardation Function

  • Coupled Added Mass Infinite Frequency

  • Coupled Added Mass Zero Frequency

  • Coupled Frequency Dependent Added Mass

  • Coupled Frequency Dependent Damping

These hydrodynamic coupling elements must be specified for each pair of bodies. Only one coupling element of each type per pair is required, due to the following symmetry properties:

  • Retardation function: \(\mathrm {\boldsymbol{h}(t-\tau)_{i,j}=\boldsymbol{h}(t-\tau)_{j,i}^T}\)

  • Added mass: \(\mathrm {(\boldsymbol{A}(\omega))_{i,j}=(\boldsymbol{A}(\omega))_{j,i}^T}\)

  • Damping: \(\mathrm {(\boldsymbol{B}(\omega))_{i,j}=(\boldsymbol{B}(\omega))_{j,i}^T}\)

where the indices i and j refers to body i and body j and superscript T denoted matrix transpose.

2. Calculation of retardation functions from frequency dependent damping

When importing frequency dependent damping from WADAM or other supported boundary element solver, SIMA will compute retardation functions from the frequency dependent damping. It is also possible to recalculate the retardation functions, for example to change the time step, based on the frequency dependent damping in the SIMA model.

Some coupling terms in the retardation function matrix should be zero, but is slightly non-zero due to numerical noise. Moreover, because the retardation functions tend to zero as time increases, each retardation function can be truncated at a certain time instant after which it is approximately zero. Deciding which coupling terms that can be neglected and the truncation time of those that are kept, is governed by the following input parameters (at import of hydrodynamic data, or recalculation of retardation functions):

  • Cut Factor: Factor used to determine the duration of the significant part of the retardation function (the part remaining after truncating the near-zero part from the right). Large cut factor means large duration (less truncation).

  • Power Of Two: Maximum number of samples in the retardation functions. Must be a power of 2. The actual number of samples will be reduced by truncating the near-zero part from the right side, as determined by Cut Factor.

  • Mass Cut Factor: Factor used together with structural mass to determine which coupling terms in the retardation function matrix that are close to zero at all values of t and can be neglected. Small factor means larger chance of neglecting coupling terms.

In addition, the Time Step used for sampling the retardation functions must be specified. Note that the retardation functions will be resampled using cubic spline interpolation to a new time step equal to the SIMO time increment before each simulation. Moreover, if the SIMO time increment is larger than the simulation time step, the forces from the retardation functions will be computed at each time increment and held constant at intermediate time steps (zero-order hold).

The calculation of retardation functions from frequency dependent damping is documented in detail in here.

3. Linear damping compensating retardation function negative damping

Due to inaccuracies in the potential flow solver and the transformation of frequency dependent damping into retardation functions, the retardation functions may become slightly non-passive, meaning that they may produce non-physical negative damping at some frequencies. During calculation of retardation functions, SIMA will also add a linear damping matrix to compensate for this negative damping. The damping matrix will be diagonal with small positive numbers on the diagonal. Larger values is generally a bad sign, and may indicate that the hydrodynamic data is inaccurate or poorly resolved in the frequency domain. The linear damping matrix is contained in Retardation Function model in SIMA.

4. Optional calculation of added mass at infinite frequency from retardation functions

Infinite frequency added mass should preferably be imported from WADAM or other supported boundary element solver. If the infinite frequency added mass is not present, or if the Use Limiting Frequencies flag is unchecked, the infinite frequency added mass will be estimated from the retardation functions using the method documented here.