2D Boundary element method

Short commentary

Here, a complex expression of potential is used because it makes the calculation of sin, cos derivative easy. When potentil of the fluid \( \Phi(x,y,t) \) is defiend as $$\Phi(x,y,t) = \Re {[ \phi(x,y) e^{ i \omega t} ]} $$ where,\( \omega \) is anglar frequency of induced waves. Induced waves potential \( \Phi_0(x,y,t) \) with infinate sea depth with wave height \( \zeta_a \) and wave number \( k \) is written as $$ \Phi_0(x,y,t)= \Re {[\frac{g \zeta_a}{i \omega} \phi_0(x,y) e^{ i \omega t} ]} = \Re {[ \frac{g \zeta_a}{i \omega} e^{ -Ky+iKx} e^{ i \omega t} ]} $$ where \( g \) is the acceleration of gravity.
if linialized boundary condtion on the free surface is imposed, induced wave height \( \zeta_0 \) is calculated at y = 0 as $$ \zeta_0(x,t) = - \frac{1}{g} {(\frac{\partial \Phi_0(x)}{\partial t})}_{y=0} = \Re([-\zeta_a e^{i (Kx+\omega t)}]) = \zeta_a \cos{(Kx+\omega t)}$$ Componet \( \phi(x,y) \) of potentil \( \Phi(x,y,t) \) can be divide into combination of induced wave potential \( \phi_0 \), diffraction wave potential \( \phi_D \), and wave potentials of indeced motion of ship \( \phi_{sway} \), \( \phi_{heave} \), \( \phi_{roll}\), which is called radiation waves, as $$ \phi(x,y) = \frac{g \zeta_a}{i \omega} (\phi_0 + \phi_D) + i\omega(X_{sway} \phi_{sway} + X_{heave} \phi_{heave} + X_{roll} \phi_{roll}) $$ Boundary element method is a method to calculate these unknown potentials; \( \phi_D \), \( \phi_{sway} \), \( \phi_{heave} \), \( \phi_{roll}\) by solving the influence coefficient matrixes of each boundary conditions on ship surfaces, where boundary conditions of free surface are considered in Green funcion.

After calculate these potentials; \( \phi_D \), \( \phi_{sway} \), \( \phi_{heave} \), \( \phi_{roll}\), each forces of potential to ship is evaluated by integration of pressure on all hull surfaces. Based on the phase information, integrated matrixes of faces for radiation condition prodece added mass and damping force in induced wave number \( k \).

These forces; added mass, damping forces, and diffraction forces, is the input of equations of motion for each movement; sway, heave, roll. Solution of these 3 equation of motions make amplitude of each motion; \( X_{sway} \), \( X_{heave} \), and \( X_{roll} \).

Finally, we got all the unknown parameter of potentil of the fluid \( \Phi(x,y,t) \),so now we are able to evaluate the wave pattern \( \zeta(x,t) \) by llinialized equation of $$ \zeta(x,t) = - \frac{1}{g} {(\frac{\partial \Phi(x)}{\partial t})}_{y=0} = \Re{[e^{ i \omega t} \times [- \zeta_a (\phi_0 + \phi_D) + K(X_{sway} \phi_{sway} + X_{heave} \phi_{heave} + X_{roll} \phi_{roll})]]} $$