Theory: Irregular frequencies

The mapping from the full boundary value problem to integral equations reduces a three-dimensional partial differential equation on the unbounded domain $\fV$ to the two-dimensional problem of finding the unknown $\phi$ on the surface $\SB$. The simplification resulting from this mapping is the key to the success of the boundary integral method: it makes efficient numerical solution tractable. However the mapping also introduces the irregular frequency effect which can cause erroneous numerical results if not accounted for correctly.

A clear explanation of the irregular frequency effect is given by Lee, Newman & Zhu. Below we summarise the key theory and describe how to avoid the numerical errors.

The errors caused by irregular frequencies

The underlying boundary value problem is well-posed and is thought to have a unique solution. Kleinman, for example, proves a uniqueness theorem that is applicable to almost all real-world diffraction problems, since it puts only weak conditions on the geometry of the body. However, it is well known that the integral equations derived by applying Green's theorem do not possess unique solutions: there is a discrete set of wave frequencies for which the determinant of the operators on the left-hand side is zero. This causes the derived matrix equation to be ill-conditioned and causes numerical errors.

It is possible for the matrix equation to have a determinant of zero, causing an exception during the OrcaWave calculation. However, the typical behaviour is that the matrix is ill-conditioned; therefore, no exception will occur, but the results will contain numerical errors. These are usually easy to see by inspecting results graphs, which will show a sharp spike in the neighbourhood of an irregular frequency.

As in this example, irregular frequency effects are usually encountered at the high end of the frequency range. For every geometry there is a first irregular frequency, $\omegahat$, such that all waves with $\omega < \omegahat$ are regular. At frequencies greater than $\omegahat$, more irregular frequencies occur (at increasing density on the real line).

The spectrum of irregular frequencies is determined by the body geometry, $\SB$. To determine exactly which $\omega$ are irregular would require the solution of another, slightly modified, boundary value problem on the domain $\VInt$. In practice this is never done because knowledge of the precise spectrum is not required to remove the effects. If irregular frequency effects are observed, then they can be removed using the technique given below.

Whilst OrcaWave does not have functionality to compute the first irregular frequency $\omegahat$ for you, it can estimate its value using equation (30) of (Chen 2004). The estimate, $\omegahatest$, is the first irregular frequency of a cuboid of length $L$, breadth $B$ and draught $T$, that is large enough to surround the body surface \begin{equation}\label{eqFirstIrregularFrequencyApproximation} \omegahatest = \sqrt{\frac{g k_{11}}{\tanh{k_{11}T}}} \;\;\textrm{where}\;\; k_{11} = \pi\sqrt{L^{-2} + B^{-2}} \end{equation} OrcaWave will issue a warning if (i) your wave environment includes a wave with $\omega > \omegahatest$ and (ii) you have not extended the mesh by including interior surface panels, as described below.

In fact, $\omegahatest$ gives a lower bound on $\omegahat$. For simple (i.e. monohull) bodies, the estimate is usually a good approximation and $\omegahat \simeq \omegahatest$. For complex body shapes, the true value of $\omegahat$ can be significantly larger than $\omegahatest$ and therefore, in some circumstances, it may be safe to ignore the warning. In that case, you should carefully inspect the results for irregular frequency effects.

For the purpose of the validation warning, OrcaWave infers $(L, B, T)$ from the mesh in order to compute $\omegahatest$:

• If you do not perform validation of panel arrangement, OrcaWave uses a simple approach to determine $(L, B, T)$ using the maximum extent of the relevant bodies.
• If you do perform validation of panel arrangement, OrcaWave uses a more sophisticated approach. First, neighbouring panels are identified as those closer than the gap warning level, then this information is used to determine $(L, B, T)$ for each component of the body. This gives a better $\omegahatest$ (i.e. closer to the true $\omegahat$) for bodies such as catamarans or platforms with distinct floating columns, for which each component is much smaller than the overall body. This approach is also more suitable for models with sectional bodies.

Removing irregular frequency effects

OrcaWave can remove irregular frequency effects by solving extended boundary integral equations. OrcaWave follows the methodology described by (Lee, Newman & Zhu 1996) to remove the effects from both the potential and source formulations. Applying this method to the example given above demonstrates how this can successfully remove all effects.

Potential formulation

In the potential formulation, the unknown potential $\phi$ on $\SB$ is extended to include values on $\SI$ (the values on $\SI$ should be understood as non-physical). OrcaWave solves the following extended integral equation that overcomes the irregular frequency problem \begin{equation} \begin{aligned} 2\pi\phi(\vec{X}) + \CPVint{\SB}{}\phi(\vec{\xi})\PD{G}{n_{\xi}} \ud S_{\xi} -\nu\int_{\SI}\phi(\vec{\xi}) G \ud S_{\xi} & = \int_{\SB}{}q_B(\vec{\xi}) G \ud S_{\xi} + \int_{\SF} \frac{q_F(\vec{\xi})}{g} G \ud S_{\xi} & & \vec{X}\in \SB \\ -4\pi \phi(\vec{X}) + \int_{\SB} \phi(\vec{\xi}) \PD{G}{n_{\xi}} \ud S_{\xi} - \nu \int_{\SI} \phi(\vec{\xi}) G \ud S_{\xi} & = \int_{\SB} q_B(\vec{\xi}) G \ud S_{\xi} + \int_{\SF} \frac{q_F(\vec{\xi})}{g} G \ud S_{\xi} & & \vec{X}\in \SI \end{aligned} \label{eqBIE-Pot-ClassicalIRR} \end{equation} It can be shown that this extended integral equation has non-zero determinant and a unique solution, even at the irregular frequencies, and further that the solution values of $\phi$ on $\SB$ are the desired solutions of the original problem. OrcaWave solves the matrix equation derived from (\ref{eqBIE-Pot-ClassicalIRR}) and discards the non-physical values of $\phi$ on $\SI$, which removes the numerical errors of any irregular frequencies.

The motivation for equation (\ref{eqBIE-Pot-ClassicalIRR}) is clearly described by (Lee, Newman & Zhu 1996), whilst the mathematical proof that it works can be found in the work of (Kleinman 1982). In order to formulate and solve the extended integral equation (\ref{eqBIE-Pot-ClassicalIRR}), OrcaWave requires mesh panels covering an extended domain: both $\SB$ and $\SI$.

Source formulation

In the source formulation, an extended integral equation can also be derived which has the same ability to remove irregular frequency effects. In this formulation, the extension of the domain implies a different relationship between the source function and the potential \begin{equation} \begin{aligned} \phi(\vec{X}) & = \int_{\SB+\SI} \sigma(\vec{\xi}) G \ud S_{\xi} + \int_{\SF}\frac{q_F(\vec{\xi})}{4\pi g} G \ud S_{\xi} & & \vec{X}\in \fV \end{aligned} \label{eqClassicalSourceDefIRR} \end{equation} and the extended integral equation is \begin{equation} \begin{aligned} 2\pi \sigma(\vec{X}) + \CPVint{\SB + \SI}{} \;\;\;\;\;\;\sigma(\vec{\xi}) \PD{G}{n_{x}} \ud S_{\xi} & = q_B(\vec{X}) - \int_{\SF}\frac{q_F(\vec{\xi})}{4\pi g} \PD{G}{n_{x}} \ud S_{\xi} & & \vec{X} \in \SB \\ -4\pi \sigma(\vec{X}) -\nu \int_{\SB + \SI}\sigma(\vec{\xi}) G \ud S_{\xi} & = -V(\vec{X}) +\nu \int_{\SF}\frac{q_F(\vec{\xi})}{4\pi g} G \ud S_{\xi} & & \vec{X} \in \SI \end{aligned} \label{eqBIE-Sor-ClassicalIRR} \end{equation} where $V(\vec X)$ is a function defined on the interior surface $\SI$. In principle, $V(\vec{X})$ is arbitrary. However, (Lee, Newman & Zhu 1996) show that $V(\vec{X})$ should be continuous with $\PD{\phi}{Z}$ on the waterline, $\CWL$, to avoid introducing a weak singularity in $\sigma(\vec{X})$ which can cause numerical error (see their figure 8). OrcaWave follows the procedure recommended by (Lee, Newman & Zhu 1996) and solves the potential formulation first, which then allows a $V(\vec X)$ to be constructed that meets the continuity condition and avoids introducing a harmful singularity into the source function $\sigma(\vec{X})$.