Template:Added mass damping and force matrices definition

We then define the matrices

[math]\displaystyle{ A_{\mu\nu} = \mathrm{Re} \left\{ -\frac{\mathrm{i}}{\omega}\rho\iint_{\partial\Omega_{B}} \phi_{\nu}^{\mathrm{R}} \mathbf{n}_{\mu}\, dS \right\} }[/math]

which is called the added mass matrix and We then define the matrices

[math]\displaystyle{ B_{\mu\nu} = \mathrm{Im} \left\{ \rho\iint_{\partial\Omega_{B}} \phi_{\nu}^{\mathrm{R}} \mathbf{n}_{\mu}\, dS \right\} }[/math]

which is called the damping matrix and the forcing vector is

[math]\displaystyle{ f_{\mu} = \mathrm{i}\omega\rho\iint_{\partial\Omega_{B}} \left(\phi^{\mathrm{I}} + \phi^{\mathrm{D}} \right) \mathbf{n}_{\mu}\, dS }[/math]