2.3.14. Mathematical model for frequency domain problems

2.3.14.1. Boundary element method

In SeaFEM, the Boundary Element Method (BEM) is used to solve problems stated within the framework of the frequency domain analysis. The frequency domain solver of SeaFEM is based on an adaptation of NEMOH, developed by Ecole Centrale de Nantes (for further information about NEMOH, see lheea.ec-nantes.fr/doku.php/emo/nemoh/start). To solve the general equations, the boundary element method is used and applying Green’s function, it allows taking into account the boundary conditions on body, bottom and free surface. BEM solver has the following approach: it decouples the resolution of the linear free surface Boundary Value Problem and the definition of the boundary condition on the body. Assuming an irrotational flow, the velocity can be expressed as the gradient of the velocity potential. The latter assumption along with incompressibility supposition leads to the Laplace equation:

(2.666)\[\begin{aligned} &\nabla ^2 Φ = 0 \end{aligned}\]

The first order dynamic free surface boundary condition is applied:

(2.667)\[\begin{aligned} & Φ_t + g φ = 0 \end{aligned}\]

And the kinematic free surface boundary condition establishes a relation between the free surface elevation φ and Φ:

(2.668)\[\begin{aligned} & φ_t = Φ_z \end{aligned}\]

Introducing (14 1) into (14 2), the equation becomes:

(2.669)\[\begin{aligned} & Φ_{tt} + g Φ_z=0 \end{aligned}\]

Assuming that the solution is of harmonic type, then the last equation becomes:

(2.670)\[\begin{aligned} & Φ = Re \left( Ae^{i\Omega t} \right) \end{aligned}\]

(2.671)\[\begin{aligned} & -\Omega^2 \cdot Φ + g Φ_z = 0 \end{aligned}\]

In the frequency domain, the first order free surface boundary condition becomes:

(2.672)\[\begin{split}\begin{aligned} & Φ_z - K \cdot Φ = 0 \\ & K = \Omega^2/g \end{aligned}\end{split}\]

The velocity potential of the incident wave is defined as:

(2.673)\[\begin{aligned} & φ_0 = \frac{i \cdot g \cdot A}{\Omega} \cdot \frac{cosh⁡[k \cdot (z+h)]}{cosh ⁡k \cdot H} \cdot e^{ -i \cdot k \cdot x \cdot cos⁡ β - i \cdot k \cdot y \cdot sin⁡β } \end{aligned}\]

Where \(k\) fulfils dispersion relation:

(2.674)\[\begin{aligned} & \frac{\Omega^2}{g} = k \cdot tanh ⁡k \cdot H \end{aligned}\]

And \(β\) is the angle between the direction of propagation of the incident wave and the positive x-axis.

As the problem is linearized, the velocity potential can be decomposed into two components: radiation and diffraction.

(2.675)\[\begin{aligned} & φ = φ_R + φ_D \end{aligned}\]

(2.676)\[\begin{aligned} & φ_R = i\cdot \Omega\cdot \sum_{j=1}^6 ξ_j \cdot φ_j \end{aligned}\]

(2.677)\[\begin{aligned} & φ_D = φ_0 + φ_S \end{aligned}\]

Where \(ξ_j\) denotes the amplitude of the body’s movement in its six degrees of freedom and \(φ_j\) the corresponding unit-amplitude radiation potentials.

2.3.14.2. Forward speed corrections

Assuming a forward speed of the bodies as veloctiy \(=U\), direction \(=β\), then a few corrections have to be made to take into account this effect. First, the frequency has to be changed to a frequency of wave encounter:

(2.678)\[\begin{aligned} & \Omega_e = \Omega -k\cdot U\cdot cos⁡β \end{aligned}\]

Here \(k\) is the wave number, \(\Omega\) the frequency without velocity and \(\Omega_e\) the encounter frequency. Other changes to be made are for the added mass and damping matrixes:

(2.679)\[\begin{aligned} & A_{3,5} = -U/\Omega^2 \cdot B_{3,3}^0 \end{aligned}\]

(2.680)\[\begin{aligned} & B_{3,5} = +U \cdot A_{3,3}^0 \end{aligned}\]

(2.681)\[\begin{aligned} & A_{5,3} = +U/\Omega^2 \cdot B_{3,3}^0 \end{aligned}\]

(2.682)\[\begin{aligned} & A_{5,3} = +U/\Omega^2 \cdot B_{3,3}^0 \end{aligned}\]

(2.683)\[\begin{aligned} & B_{5,5} = +U^2/\Omega^2 \cdot A_{3,3}^0 \end{aligned}\]

This formulation has been taken from the one presented in [ ].